Subgroups — #86757

Order 384.

1032 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#5817	C₂	1	1	yes	maximal, characteristic, complemented
3	#5816	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#947	C₂	1	1	yes	maximal, characteristic, complemented
6	#5735	C₂	1	2	yes	maximal, complemented
7	#5733	C₂	1	2	yes	maximal, complemented
8	#5732	C₂	1	1	yes	maximal, characteristic, complemented
9	#5724	C₂	1	1	yes	maximal, characteristic, complemented
10	#5647	C₂	1	2	yes	maximal, complemented
11	#5643	C₂	1	1	yes	maximal, characteristic, complemented
12	#5640	C₂	1	2	yes	maximal, complemented
13	#5637	C₂	1	2	yes	maximal, complemented
14	#5634	C₂	1	2	yes	maximal, complemented
15	#5633	C₂	1	1	yes	maximal, characteristic, complemented
16	#5630	C₂	1	1	yes	maximal, characteristic, complemented
17	#5622	C₂	1	1	yes	maximal, characteristic, complemented
18	#5582	C₂	1	1	yes	maximal, characteristic, complemented
19	#5568	C₂	1	1	yes	maximal, characteristic, complemented
20	#5560	C₂	1	2	yes	maximal, complemented
21	#5556	C₂	1	2	yes	maximal, complemented
22	#5550	C₂	1	1	yes	maximal, characteristic, complemented
23	#5538	C₂	1	1	yes	maximal, characteristic, complemented
24	#3561	—	3	1	no	maximal, Sylow
25	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
26	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₂×#239	C₂²	1	1	yes	characteristic
28	C₂×D₃×Q₈	C₂²	1	1	yes	characteristic, complemented
29	C₂×#235	C₂²	1	1	yes	characteristic
30	C₂²×D₁₂	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
33	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
34	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
37	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
39	C₃×C₄×D₄	C₂²	1	2	yes	—
40	#947	C₂²	1	1	yes	characteristic
41	#947	C₂²	1	2	yes	—
42	#947	C₂²	1	1	yes	characteristic
43	#947	C₂²	1	2	yes	—
44	#947	C₂²	1	1	yes	characteristic
45	#947	C₂²	1	1	yes	characteristic
46	#947	C₂²	1	1	yes	characteristic
47	#947	C₂²	1	2	yes	complemented
48	#946	C₂²	1	2	yes	complemented
49	#946	C₂²	1	2	yes	complemented
50	#946	C₂²	1	2	yes	complemented
51	#946	C₂²	1	2	yes	complemented
52	#946	C₂²	1	1	yes	characteristic, complemented
53	Q₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
54	Q₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
55	#944	C₂²	1	2	yes	complemented
56	#944	C₂²	1	1	yes	characteristic, complemented
57	#944	C₂²	1	1	yes	characteristic, complemented
58	#939	C₂²	1	1	yes	characteristic
59	#937	C₂²	1	2	yes	—
60	#935	C₂²	1	1	yes	characteristic
61	#930	C₂²	1	1	yes	characteristic
62	#930	C₂²	1	2	yes	complemented
63	#929	C₂²	1	2	yes	complemented
64	#929	C₂²	1	2	yes	complemented
65	C₄×#67	C₂²	1	2	yes	—
66	C₄×#67	C₂²	1	1	yes	characteristic
67	C₄×#67	C₂²	1	1	yes	characteristic, complemented
68	C₄×#67	C₂²	1	2	yes	complemented
69	C₂×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
70	C₂×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
71	#926	C₂²	1	1	yes	characteristic, complemented
72	#924	C₂²	1	2	yes	—
73	#924	C₂²	1	1	yes	characteristic
74	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
75	#898	C₂²	1	2	yes	complemented
76	#898	C₂²	1	2	yes	complemented
77	#898	C₂²	1	2	yes	complemented
78	#898	C₂²	1	2	yes	complemented
79	#898	C₂²	1	1	yes	characteristic, complemented
80	#897	C₂²	1	2	yes	complemented
81	#897	C₂²	1	2	yes	complemented
82	#896	C₂²	1	2	yes	complemented
83	#896	C₂²	1	2	yes	complemented
84	#896	C₂²	1	2	yes	complemented
85	#896	C₂²	1	2	yes	complemented
86	#896	C₂²	1	2	yes	complemented
87	#895	C₂²	1	1	yes	characteristic
88	#895	C₂²	1	2	yes	—
89	#894	C₂²	1	2	yes	—
90	#894	C₂²	1	2	yes	—
91	#894	C₂²	1	2	yes	complemented
92	#894	C₂²	1	2	yes	complemented
93	#894	C₂²	1	2	yes	complemented
94	#893	C₂²	1	1	yes	characteristic
95	#893	C₂²	1	2	yes	complemented
96	#892	C₂²	1	2	yes	complemented
97	#889	C₂²	1	2	yes	complemented
98	#889	C₂²	1	1	yes	characteristic, complemented
99	#888	C₂²	1	1	yes	characteristic, complemented
100	#887	C₂²	1	1	yes	characteristic, complemented
101	#887	C₂²	1	2	yes	complemented
102	#886	C₂²	1	1	yes	characteristic
103	#885	C₂²	1	2	yes	complemented
104	#885	C₂²	1	2	yes	complemented
105	#885	C₂²	1	1	yes	characteristic, complemented
106	#884	C₂²	1	2	yes	—
107	#884	C₂²	1	2	yes	complemented
108	#883	C₂²	1	2	yes	complemented
109	#882	C₂²	1	2	yes	—
110	D₃×C₂²⋊C₄	C₂²	1	2	yes	—
111	D₃×C₂²⋊C₄	C₂²	1	2	yes	—
112	#879	C₂²	1	1	yes	characteristic, complemented
113	#878	C₂²	1	1	yes	characteristic
114	#878	C₂²	1	2	yes	—
115	#877	C₂²	1	2	yes	—
116	#876	C₂²	1	2	yes	complemented
117	#876	C₂²	1	2	yes	complemented
118	#875	C₂²	1	1	yes	characteristic
119	#875	C₂²	1	2	yes	complemented
120	C₄×D₁₂	C₂²	1	1	yes	characteristic
121	#872	C₂²	1	1	yes	characteristic, complemented
122	#872	C₂²	1	2	yes	complemented
123	C₄×Dic₆	C₂²	1	1	yes	characteristic, complemented
124	#563	—	3	2	no	—
125	#562	—	3	2	no	—
126	#556	—	3	1	no	—
127	#556	—	3	1	no	—
128	#551	—	3	2	no	—
129	#551	—	3	2	no	—
130	D₄×Q₈	—	3	1	no	—
131	D₄×Q₈	D₃	1	1	yes	characteristic, complemented
132	#548	—	3	1	no	—
133	#548	—	3	1	no	—
134	#546	—	3	1	no	—
135	#546	—	3	1	no	—
136	#543	—	3	2	no	—
137	#542	—	3	2	no	—
138	#542	—	3	2	no	—
139	#542	—	3	1	no	—
140	#542	—	3	1	no	—
141	#538	—	3	2	no	—
142	#536	—	3	1	no	—
143	#536	—	3	1	no	—
144	#533	—	3	1	no	—
145	#529	—	3	1	no	—
146	C₂×#124	—	3	1	no	—
147	C₂³×D₃	C₂³	1	2	yes	—
148	C₂×C₃×Q₈	—	4	2	no	—
149	C₂×C₃×Q₈	—	4	2	no	—
150	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
151	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
152	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
153	C₂×C₃×Q₈	C₂³	1	2	yes	—
154	C₂×C₃×Q₈	C₂³	1	2	yes	—
155	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
158	C₂²×C₃×C₄	C₂³	1	2	yes	—
159	C₂²×C₃×C₄	C₂³	1	2	yes	—
160	C₂×#67	C₂³	1	1	yes	characteristic
161	C₂×#67	C₂³	1	2	yes	—
162	C₂×#67	C₂³	1	2	yes	—
163	C₂×#67	C₂³	1	1	yes	characteristic
164	C₂²×Dic₃	C₂³	1	1	yes	characteristic
165	C₂²×Dic₃	C₂³	1	1	yes	characteristic
166	#239	—	4	2	no	—
167	D₃×Q₈	—	4	2	no	—
168	#235	—	4	2	no	—
169	C₂×D₁₂	—	4	2	no	—
170	C₂×D₁₂	C₂³	1	1	yes	characteristic
171	C₂×D₁₂	C₂³	1	1	yes	characteristic
172	C₂×D₁₂	C₂³	1	2	yes	—
173	C₂×D₁₂	C₂³	1	2	yes	—
174	C₂×D₁₂	C₂³	1	2	yes	—
175	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
176	C₂×C₄×D₃	C₂³	1	2	yes	—
177	C₂×C₄×D₃	C₂³	1	2	yes	—
178	C₂×C₄×D₃	C₂³	1	2	yes	—
179	C₂×C₄×D₃	C₂³	1	2	yes	—
180	C₂×C₄×D₃	C₂³	1	2	yes	—
181	C₂×Dic₆	C₂³	1	1	yes	characteristic
182	C₂×Dic₆	C₂³	1	1	yes	characteristic
183	C₂×Dic₆	C₂³	1	2	yes	—
184	C₂×Dic₆	C₂³	1	2	yes	—
185	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
186	C₃×C₄⋊C₄	C₂³	1	2	yes	—
187	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
188	C₃×C₄⋊C₄	C₂³	1	2	yes	—
189	C₃×C₄⋊C₄	C₂³	1	2	yes	—
190	C₃×C₄⋊C₄	C₂³	1	2	yes	—
191	C₃×C₄⋊C₄	C₂³	1	2	yes	—
192	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
193	C₃×C₂²⋊C₄	C₂³	1	1	yes	characteristic
194	C₃×C₂²⋊C₄	C₂³	1	1	yes	characteristic
195	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
196	C₃×C₄²	C₂³	1	1	yes	characteristic
197	C₃×C₄²	C₂³	1	2	yes	—
198	#217	C₂³	1	2	yes	—
199	#217	C₂³	1	1	yes	characteristic
200	#217	C₂³	1	1	yes	characteristic
201	#217	C₂³	1	1	yes	characteristic
202	#217	C₂³	1	1	yes	characteristic
203	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
204	(C₂×D₃)⋊C₄	C₂³	1	1	yes	characteristic
205	(C₂×D₃)⋊C₄	C₂³	1	1	yes	characteristic
206	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
207	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
208	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
209	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
210	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
211	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
212	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
213	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
214	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
215	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
216	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
217	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
218	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
219	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
220	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
221	(C₂×D₃)⋊C₄	C₂³	1	2	yes	—
222	(C₂×D₃)⋊C₄	C₂³	1	1	yes	characteristic
223	(C₂×D₃)⋊C₄	C₂³	1	1	yes	characteristic
224	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
225	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
226	C₄⋊Dic₃	C₂³	1	2	yes	—
227	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
228	C₄⋊Dic₃	C₂³	1	2	yes	—
229	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
230	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
231	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
232	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
233	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
234	Dic₃⋊C₄	C₂³	1	2	yes	—
235	Dic₃⋊C₄	C₂³	1	2	yes	—
236	Dic₃⋊C₄	C₂³	1	2	yes	—
237	Dic₃⋊C₄	C₂³	1	2	yes	—
238	Dic₃⋊C₄	C₂³	1	2	yes	—
239	Dic₃⋊C₄	C₂³	1	2	yes	—
240	Dic₃⋊C₄	C₂³	1	2	yes	—
241	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
242	Dic₃⋊C₄	C₂³	1	1	yes	characteristic
243	Dic₃⋊C₄	C₂³	1	2	yes	—
244	C₄×Dic₃	C₂³	1	1	yes	characteristic
245	C₄×Dic₃	C₂³	1	1	yes	characteristic
246	C₄×Dic₃	C₂³	1	2	yes	—
247	C₄×Dic₃	C₂³	1	1	yes	characteristic
248	C₄×Dic₃	C₂³	1	2	yes	—
249	C₄×Dic₃	C₂³	1	1	yes	characteristic
250	C₄×Dic₃	C₂³	1	1	yes	characteristic
251	C₄×Dic₃	C₂³	1	2	yes	—
252	C₂×#41	—	3	1	no	—
253	C₂×#41	—	3	1	no	—
254	C₂²×Q₈	—	3	1	no	—
255	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
256	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
257	C₂²×D₄	—	3	1	no	—
258	C₄⋊Q₈	—	3	1	no	—
259	C₄⋊Q₈	—	3	1	no	—
260	C₄⋊Q₈	—	3	2	no	—
261	C₄⋊Q₈	—	3	1	no	—
262	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
263	C₄⋊Q₈	C₂×D₃	1	2	yes	complemented
264	#126	—	3	1	no	—
265	#126	—	3	2	no	—
266	#126	—	3	2	no	—
267	#126	—	3	2	no	—
268	#126	—	3	2	no	—
269	#126	—	3	1	no	—
270	#126	—	3	2	no	—
271	#126	—	3	2	no	—
272	#125	—	3	1	no	—
273	#125	—	3	2	no	—
274	#124	—	3	1	no	—
275	#124	—	3	1	no	—
276	#124	—	3	2	no	—
277	#124	—	3	1	no	—
278	#124	—	3	1	no	—
279	#124	—	3	2	no	—
280	#124	—	3	1	no	—
281	#124	—	3	2	no	—
282	#124	—	3	1	no	—
283	#124	—	3	2	no	—
284	#124	—	3	2	no	—
285	#124	—	3	1	no	—
286	#124	—	3	2	no	—
287	#123	—	3	1	no	—
288	#123	—	3	2	no	—
289	#123	—	3	2	no	—
290	#123	—	3	2	no	—
291	#123	—	3	2	no	—
292	#123	—	3	2	no	—
293	#123	—	3	2	no	—
294	#123	—	3	2	no	—
295	#123	—	3	1	no	—
296	#123	—	3	2	no	—
297	C₂²⋊Q₈	—	3	2	no	—
298	C₂²⋊Q₈	—	3	2	no	—
299	C₂²⋊Q₈	—	3	1	no	—
300	C₂²⋊Q₈	—	3	2	no	—
301	C₂²⋊Q₈	—	3	2	no	—
302	C₂²⋊Q₈	—	3	1	no	—
303	C₂²⋊Q₈	—	3	2	no	—
304	C₂²⋊Q₈	—	3	2	no	—
305	C₂²⋊Q₈	—	3	2	no	—
306	C₂²⋊Q₈	—	3	2	no	—
307	C₂²⋊Q₈	—	3	2	no	—
308	C₂²⋊Q₈	—	3	2	no	—
309	C₂²⋊Q₈	—	3	2	no	—
310	C₂²⋊Q₈	—	3	2	no	—
311	C₂²⋊Q₈	—	3	1	no	—
312	C₂²⋊Q₈	—	3	2	no	—
313	C₂²⋊Q₈	C₂×D₃	1	1	yes	characteristic
314	C₂²⋊Q₈	C₂×D₃	1	2	yes	—
315	C₂²⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
316	C₂²⋊Q₈	C₂×D₃	1	2	yes	complemented
317	#121	—	3	1	no	—
318	#121	—	3	2	no	—
319	#121	—	3	1	no	—
320	#121	—	3	1	no	—
321	#121	—	3	2	no	—
322	#121	—	3	2	no	—
323	#121	—	3	2	no	—
324	#120	—	3	2	no	—
325	#120	—	3	2	no	—
326	C₄×Q₈	—	3	1	no	—
327	C₄×Q₈	—	3	1	no	—
328	C₄×Q₈	—	3	2	no	—
329	C₄×Q₈	—	3	1	no	—
330	C₄×Q₈	—	3	1	no	—
331	C₄×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
332	C₄×D₄	—	3	1	no	—
333	C₄×D₄	—	3	1	no	—
334	C₄×D₄	—	3	2	no	—
335	C₄×D₄	—	3	2	no	—
336	C₄×D₄	—	3	1	no	—
337	C₄×D₄	—	3	2	no	—
338	C₄×D₄	—	3	1	no	—
339	C₄×D₄	C₂×D₃	1	1	yes	characteristic
340	C₄×D₄	C₂×D₃	1	2	yes	—
341	#117	—	3	1	no	—
342	#117	—	3	2	no	—
343	#117	—	3	2	no	—
344	#117	—	3	1	no	—
345	#117	—	3	2	no	—
346	C₂×C₂²⋊C₄	—	3	2	no	—
347	C₂×C₂²⋊C₄	—	3	2	no	—
348	C₂×C₂²⋊C₄	—	3	2	no	—
349	C₂×C₂²⋊C₄	—	3	2	no	—
350	C₂×C₄²	—	3	1	no	—
351	C₂³×C₃	C₂⁴	1	1	yes	characteristic
352	C₂³×C₃	C₂⁴	1	1	yes	characteristic
353	C₂²×D₃	—	2	4	no	—
354	C₂²×D₃	—	4	2	no	—
355	C₂²×D₃	—	4	2	no	—
356	C₂²×D₃	C₂⁴	1	2	yes	—
357	C₂²×D₃	C₂⁴	1	2	yes	—
358	C₂²×D₃	C₂⁴	1	1	yes	characteristic
359	C₃×Q₈	—	2	2	no	—
360	C₃×Q₈	—	2	2	no	—
361	C₃×Q₈	—	2	2	no	—
362	C₃×Q₈	—	4	2	no	—
363	C₃×Q₈	—	4	2	no	—
364	C₃×D₄	—	2	2	no	—
365	C₂×C₃×C₄	—	2	2	no	—
366	C₂×C₃×C₄	—	2	2	no	—
367	C₂×C₃×C₄	—	4	2	no	—
368	C₂×C₃×C₄	—	4	2	no	—
369	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
370	C₂×C₃×C₄	C₂⁴	1	2	yes	—
371	C₂×C₃×C₄	C₂⁴	1	2	yes	—
372	C₂×C₃×C₄	C₂⁴	1	2	yes	—
373	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
374	C₂×C₃×C₄	C₂⁴	1	2	yes	—
375	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
376	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
377	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
378	#67	—	2	2	no	—
379	#67	—	4	2	no	—
380	#67	—	4	2	no	—
381	#67	—	4	1	no	—
382	C₂×Dic₃	—	2	2	no	—
383	C₂×Dic₃	—	4	1	no	—
384	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
385	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
386	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
387	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
388	C₂×Dic₃	C₂⁴	1	2	yes	—
389	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
390	C₂×Dic₃	C₂⁴	1	2	yes	—
391	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
392	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
393	D₁₂	—	2	2	no	—
394	D₁₂	—	2	2	no	—
395	D₁₂	—	4	2	no	—
396	D₁₂	—	4	2	no	—
397	D₁₂	—	4	2	no	—
398	C₄×D₃	—	2	2	no	—
399	C₄×D₃	—	4	2	no	—
400	C₄×D₃	—	4	2	no	—
401	C₄×D₃	—	4	2	no	—
402	C₄×D₃	—	4	2	no	—
403	C₄×D₃	—	4	2	no	—
404	Dic₆	—	2	2	no	—
405	Dic₆	—	4	2	no	—
406	Dic₆	—	4	1	no	—
407	Dic₆	—	4	2	no	—
408	C₂⁴	—	3	2	no	—
409	#41	—	12	2	no	—
410	#41	—	12	2	no	—
411	C₂×Q₈	—	3	1	no	—
412	C₂×Q₈	—	3	1	no	—
413	C₂×Q₈	—	3	2	no	—
414	C₂×Q₈	—	3	2	no	—
415	C₂×Q₈	—	4	2	no	—
416	C₂×Q₈	—	4	2	no	—
417	C₂×Q₈	—	12	2	no	—
418	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
419	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
420	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
421	C₂×Q₈	C₂²×D₃	1	2	yes	—
422	C₂×Q₈	C₂²×D₃	1	2	yes	—
423	C₂×D₄	—	3	1	no	—
424	C₂×D₄	—	3	2	no	—
425	C₂×D₄	—	3	1	no	—
426	C₂×D₄	—	3	2	no	—
427	C₂×D₄	—	3	1	no	—
428	C₂×D₄	—	3	1	no	—
429	C₂×D₄	—	3	2	no	—
430	C₂×D₄	—	3	2	no	—
431	C₂×D₄	—	3	2	no	—
432	C₂×D₄	—	12	2	no	—
433	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
434	C₂²×C₄	—	3	1	no	—
435	C₂²×C₄	—	3	1	no	—
436	C₂²×C₄	—	3	1	no	—
437	C₂²×C₄	—	3	2	no	—
438	C₂²×C₄	—	3	2	no	—
439	C₂²×C₄	—	3	2	no	—
440	C₂²×C₄	—	3	2	no	—
441	C₂²×C₄	—	3	2	no	—
442	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
443	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
444	C₂²×C₄	C₂²×D₃	1	2	yes	—
445	C₂²×C₄	C₂²×D₃	1	2	yes	—
446	C₄⋊C₄	—	3	1	no	—
447	C₄⋊C₄	—	3	1	no	—
448	C₄⋊C₄	—	3	1	no	—
449	C₄⋊C₄	—	3	1	no	—
450	C₄⋊C₄	—	3	2	no	—
451	C₄⋊C₄	—	3	2	no	—
452	C₄⋊C₄	—	3	1	no	—
453	C₄⋊C₄	—	3	1	no	—
454	C₄⋊C₄	—	3	2	no	—
455	C₄⋊C₄	—	3	1	no	—
456	C₄⋊C₄	—	3	1	no	—
457	C₄⋊C₄	—	3	2	no	—
458	C₄⋊C₄	—	3	2	no	—
459	C₄⋊C₄	—	3	1	no	—
460	C₄⋊C₄	—	3	2	no	—
461	C₄⋊C₄	—	3	2	no	—
462	C₄⋊C₄	—	3	2	no	—
463	C₄⋊C₄	—	3	1	no	—
464	C₄⋊C₄	—	3	2	no	—
465	C₄⋊C₄	—	3	2	no	—
466	C₄⋊C₄	C₂²×D₃	1	2	yes	—
467	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
468	C₄⋊C₄	C₂²×D₃	1	2	yes	—
469	C₄⋊C₄	C₂²×D₃	1	2	yes	—
470	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂²×D₃	1	2	yes	—
472	C₄⋊C₄	C₂²×D₃	1	2	yes	—
473	C₂²⋊C₄	—	3	1	no	—
474	C₂²⋊C₄	—	3	2	no	—
475	C₂²⋊C₄	—	3	1	no	—
476	C₂²⋊C₄	—	3	1	no	—
477	C₂²⋊C₄	—	3	1	no	—
478	C₂²⋊C₄	—	3	2	no	—
479	C₂²⋊C₄	—	3	2	no	—
480	C₂²⋊C₄	—	3	2	no	—
481	C₂²⋊C₄	—	3	2	no	—
482	C₂²⋊C₄	—	3	2	no	—
483	C₂²⋊C₄	—	3	2	no	—
484	C₂²⋊C₄	—	3	2	no	—
485	C₂²⋊C₄	—	3	2	no	—
486	C₂²⋊C₄	—	3	2	no	—
487	C₂²⋊C₄	—	3	2	no	—
488	C₂²⋊C₄	—	3	2	no	—
489	C₂²⋊C₄	—	3	2	no	—
490	C₂²⋊C₄	—	3	2	no	—
491	C₂²⋊C₄	—	3	2	no	—
492	C₂²⋊C₄	—	3	1	no	—
493	C₂²⋊C₄	—	3	1	no	—
494	C₂²⋊C₄	—	3	1	no	—
495	C₂²⋊C₄	—	3	2	no	—
496	C₂²⋊C₄	—	3	2	no	—
497	C₂²⋊C₄	—	3	1	no	—
498	C₂²⋊C₄	—	3	2	no	—
499	C₂²⋊C₄	C₂²×D₃	1	1	yes	characteristic
500	C₂²⋊C₄	C₂²×D₃	1	2	yes	—
501	C₂²⋊C₄	C₂²×D₃	1	1	yes	characteristic
502	C₂²⋊C₄	C₂²×D₃	1	2	yes	—
503	C₄²	—	3	1	no	—
504	C₄²	—	3	1	no	—
505	C₄²	—	3	1	no	—
506	C₄²	—	3	1	no	—
507	C₄²	—	3	2	no	—
508	C₄²	—	3	2	no	—
509	C₄²	—	3	1	no	—
510	C₄²	—	3	2	no	—
511	C₄²	C₂²×D₃	1	1	yes	characteristic
512	C₄²	C₂²×D₃	1	2	yes	—
513	C₂²×C₃	—	2	1	no	—
514	C₂²×C₃	—	2	1	no	—
515	C₂²×C₃	—	2	1	no	—
516	C₂²×C₃	—	2	1	no	—
517	C₂²×C₃	—	2	1	no	—
518	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
519	C₂²×C₃	C₂⁵	1	1	yes	characteristic
520	C₂×D₃	—	2	1	no	—
521	C₂×D₃	—	2	2	no	—
522	C₂×D₃	—	2	2	no	—
523	C₂×D₃	—	2	1	no	—
524	C₂×D₃	—	2	2	no	—
525	C₂×D₃	—	2	2	no	—
526	C₂×D₃	—	2	2	no	—
527	C₂×D₃	—	2	2	no	—
528	C₂×D₃	—	2	1	no	—
529	C₂×D₃	—	8	4	no	—
530	C₃×C₄	—	2	2	no	—
531	C₃×C₄	—	2	1	no	—
532	C₃×C₄	—	2	2	no	—
533	C₃×C₄	—	2	1	no	—
534	C₃×C₄	—	2	2	no	—
535	C₃×C₄	—	2	1	no	—
536	C₃×C₄	—	2	1	no	—
537	C₃×C₄	—	2	2	no	—
538	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
539	Dic₃	—	2	2	no	—
540	Dic₃	—	2	1	no	—
541	Dic₃	—	2	1	no	—
542	Dic₃	—	2	1	no	—
543	Dic₃	—	2	1	no	—
544	Dic₃	—	2	1	no	—
545	Dic₃	—	2	1	no	—
546	Dic₃	—	2	2	no	—
547	Dic₃	—	2	1	no	—
548	C₂³	—	3	2	no	—
549	C₂³	—	3	2	no	—
550	C₂³	—	3	1	no	—
551	C₂³	—	6	4	no	—
552	C₂³	—	12	2	no	—
553	C₂³	—	12	2	no	—
554	C₂³	C₂³×D₃	1	1	yes	characteristic
555	C₂³	C₂³×D₃	1	1	yes	characteristic
556	Q₈	—	2	2	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	2	2	no	—
559	Q₈	—	4	2	no	—
560	Q₈	—	4	2	no	—
561	Q₈	—	6	2	no	—
562	Q₈	—	12	2	no	—
563	Q₈	—	12	1	no	—
564	Q₈	—	12	2	no	—
565	D₄	—	2	2	no	—
566	D₄	—	6	2	no	—
567	D₄	—	6	2	no	—
568	D₄	—	6	2	no	—
569	D₄	—	12	2	no	—
570	D₄	—	12	2	no	—
571	D₄	—	12	2	no	—
572	D₄	—	12	2	no	—
573	D₄	—	12	2	no	—
574	D₄	—	12	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	3	1	no	—
578	C₂×C₄	—	3	1	no	—
579	C₂×C₄	—	3	1	no	—
580	C₂×C₄	—	3	1	no	—
581	C₂×C₄	—	3	1	no	—
582	C₂×C₄	—	3	2	no	—
583	C₂×C₄	—	3	1	no	—
584	C₂×C₄	—	3	2	no	—
585	C₂×C₄	—	3	1	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	6	2	no	—
589	C₂×C₄	—	6	2	no	—
590	C₂×C₄	—	12	2	no	—
591	C₂×C₄	—	12	2	no	—
592	C₂×C₄	—	12	2	no	—
593	C₂×C₄	—	12	2	no	—
594	C₂×C₄	—	12	2	no	—
595	C₂×C₄	—	12	1	no	—
596	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
597	C₂×C₄	C₂³×D₃	1	2	yes	—
598	C₂×C₄	C₂³×D₃	1	2	yes	—
599	C₂×C₄	C₂³×D₃	1	2	yes	—
600	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
601	C₂×C₄	C₂³×D₃	1	2	yes	—
602	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
603	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
604	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
605	C₂×C₃	—	2	2	no	—
606	C₂×C₃	—	4	1	no	—
607	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
608	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
609	C₂×C₃	#585	1	1	yes	characteristic
610	D₃	—	4	1	no	—
611	D₃	—	4	2	no	—
612	D₃	—	4	2	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	6	1	no	—
619	C₂²	—	6	2	no	—
620	C₂²	—	6	2	no	—
621	C₂²	—	6	1	no	—
622	C₂²	—	6	2	no	—
623	C₂²	—	6	2	no	—
624	C₂²	—	6	2	no	—
625	C₂²	—	6	2	no	—
626	C₂²	—	6	1	no	—
627	C₂²	—	24	4	no	—
628	C₂²	#1007	1	2	yes	—
629	C₂²	C₂⁴×D₃	1	1	yes	characteristic
630	C₄	—	2	1	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	2	no	—
638	C₄	—	6	1	no	—
639	C₄	—	6	2	no	—
640	C₄	—	6	1	no	—
641	C₄	—	6	1	no	—
642	C₄	—	6	1	no	—
643	C₄	—	6	1	no	—
644	C₄	—	6	2	no	—
645	C₄	—	6	1	no	—
646	C₄	—	6	1	no	—
647	C₄	#1004	1	2	yes	—
648	C₃	#3561	1	1	yes	characteristic, complemented, Sylow
649	C₂	—	2	2	no	—
650	C₂	—	4	1	no	—
651	C₂	—	12	2	no	—
652	C₂	—	12	2	no	—
653	C₂	—	12	1	no	—
654	C₂	C₂×#1004	1	1	yes	characteristic
655	C₂	C₂×#1007	1	1	yes	characteristic
656	C₂	#5968	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×D₄×Q₈	C₂
hypercenter	D₄×Q₈	D₃
hypocenter	C₃	#3561
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×D₄×Q₈	C₂	semidirect
2. #5817	C₂	semidirect
3. #5816	C₂	semidirect
4. Q₈×#67	C₂	semidirect
5. C₂×#947	C₂	semidirect
6. #5735	C₂	semidirect
7. #5733	C₂	semidirect
8. #5732	C₂	semidirect
9. #5724	C₂	semidirect
10. #5647	C₂	semidirect
11. #5643	C₂	semidirect
12. #5640	C₂	semidirect
13. #5637	C₂	semidirect
14. #5634	C₂	semidirect
15. #5633	C₂	semidirect
16. #5630	C₂	semidirect
17. #5622	C₂	semidirect
18. #5582	C₂	semidirect
19. #5568	C₂	semidirect
20. #5560	C₂	semidirect
21. #5556	C₂	semidirect
22. #5550	C₂	semidirect
23. #5538	C₂	semidirect
26. C₂²×C₃×Q₈	C₂²	semidirect
28. C₂×D₃×Q₈	C₂²	semidirect
31. C₃×C₄⋊Q₈	C₂²	semidirect
32. C₃×C₄⋊Q₈	C₂²	semidirect
35. C₃×C₂²⋊Q₈	C₂²	semidirect
36. C₃×C₂²⋊Q₈	C₂²	semidirect
37. C₃×C₄×Q₈	C₂²	semidirect
47. #947	C₂²	semidirect
48. #946	C₂²	semidirect
49. #946	C₂²	semidirect
50. #946	C₂²	semidirect
51. #946	C₂²	semidirect
52. #946	C₂²	semidirect
53. Q₈×Dic₃	C₂²	semidirect
54. Q₈×Dic₃	C₂²	semidirect
55. #944	C₂²	semidirect
56. #944	C₂²	semidirect
57. #944	C₂²	semidirect
62. #930	C₂²	semidirect
63. #929	C₂²	semidirect
64. #929	C₂²	semidirect
67. C₄×#67	C₂²	semidirect
68. C₄×#67	C₂²	semidirect
71. #926	C₂²	semidirect
75. #898	C₂²	semidirect
76. #898	C₂²	semidirect
77. #898	C₂²	semidirect
78. #898	C₂²	semidirect
79. #898	C₂²	semidirect
80. #897	C₂²	semidirect
81. #897	C₂²	semidirect
82. #896	C₂²	semidirect
83. #896	C₂²	semidirect
84. #896	C₂²	semidirect
85. #896	C₂²	semidirect
86. #896	C₂²	semidirect
91. #894	C₂²	semidirect
92. #894	C₂²	semidirect
93. #894	C₂²	semidirect
95. #893	C₂²	semidirect
96. #892	C₂²	semidirect
97. #889	C₂²	semidirect
98. #889	C₂²	semidirect
99. #888	C₂²	semidirect
100. #887	C₂²	semidirect
101. #887	C₂²	semidirect
103. #885	C₂²	semidirect
104. #885	C₂²	semidirect
105. #885	C₂²	semidirect
107. #884	C₂²	semidirect
108. #883	C₂²	semidirect
112. #879	C₂²	semidirect
116. #876	C₂²	semidirect
117. #876	C₂²	semidirect
119. #875	C₂²	semidirect
121. #872	C₂²	semidirect
122. #872	C₂²	semidirect
123. C₄×Dic₆	C₂²	semidirect
131. D₄×Q₈	D₃	semidirect
256. C₂²×Q₈	C₂×D₃	semidirect
262. C₄⋊Q₈	C₂×D₃	semidirect
263. C₄⋊Q₈	C₂×D₃	semidirect
315. C₂²⋊Q₈	C₂×D₃	semidirect
316. C₂²⋊Q₈	C₂×D₃	semidirect
331. C₄×Q₈	C₂×D₃	semidirect
648. C₃	#3561	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×D₄×Q₈	2
#5817	2
#5816	2
Q₈×#67	2
C₂×#947	2
#5735	2
#5733	2
#5732	2
#5724	2
#5647	2
#5643	2
#5640	2
#5637	2
#5634	2
#5633	2
#5630	2
#5622	2
#5582	2
#5568	2
#5560	2
#5556	2
#5550	2
#5538	2
#3561	3