Subgroups — #78628

Order 384.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#67	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
5	#5126	C₂	1	1	yes	maximal, characteristic, complemented
6	#5118	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#4730	C₂	1	2	yes	maximal, complemented
9	#4729	C₂	1	2	yes	maximal, complemented
10	#4728	C₂	1	2	yes	maximal, complemented
11	#67⋊C₈	C₂	1	2	yes	maximal, complemented
12	D₄×M₁₆	—	3	1	no	maximal, Sylow
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂²×#67	C₄	1	1	yes	characteristic
15	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#217	C₄	1	1	yes	characteristic
19	C₄×#67	C₄	1	1	yes	characteristic
20	C₄×#67	C₄	1	1	yes	characteristic
21	C₄×#67	C₄	1	1	yes	characteristic
22	C₄×#67	C₄	1	1	yes	characteristic
23	C₄×#67	C₂²	1	1	yes	characteristic
24	C₄×#67	C₂²	1	1	yes	characteristic
25	C₄×#67	C₂²	1	1	yes	characteristic
26	C₄×#67	C₂²	1	1	yes	characteristic
27	C₂×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
28	C₂×Dic₃⋊C₄	C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
30	C₂×#208	—	2	4	no	—
31	C₂×#208	C₂²	1	1	yes	characteristic
32	C₂×#208	C₂²	1	1	yes	characteristic
33	C₂×#208	C₂²	1	2	yes	complemented
34	C₂²×C₃⋊C₈	C₂²	1	2	yes	complemented
35	D₃×M₁₆	—	2	4	no	—
36	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
37	C₂×D₃×C₈	C₂²	1	2	yes	complemented
38	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
39	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
41	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
42	#815	C₂²	1	2	yes	complemented
43	Dic₃⋊C₈	C₂²	1	2	yes	complemented
44	Dic₃⋊C₈	C₂²	1	2	yes	complemented
45	C₈×Dic₃	C₂²	1	2	yes	complemented
46	C₂²×M₁₆	—	3	1	no	—
47	C₂²×M₁₆	—	3	1	no	—
48	C₂×C₄×D₄	—	3	1	no	—
49	#436	—	3	2	no	—
50	#435	—	3	2	no	—
51	#435	—	3	2	no	—
52	C₈×D₄	—	3	2	no	—
53	#423	—	3	1	no	—
54	#407	—	3	1	no	—
55	#407	D₃	1	1	yes	characteristic, complemented
56	C₄×M₁₆	—	3	1	no	—
57	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂×#67	—	2	4	no	—
68	C₂×#67	C₂×C₄	1	1	yes	characteristic
69	C₂×#67	C₂×C₄	1	1	yes	characteristic
70	C₂×#67	C₂×C₄	1	1	yes	characteristic
71	C₂×#67	C₂×C₄	1	1	yes	characteristic
72	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
73	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₃	—	2	2	no	—
76	C₂×C₄×D₃	—	2	2	no	—
77	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
81	C₃×M₁₆	—	2	2	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₂×C₃×C₈	C₂³	1	2	yes	—
84	C₂×C₃×C₈	C₂³	1	2	yes	—
85	#217	C₂×C₄	1	1	yes	characteristic
86	#217	C₂×C₄	1	1	yes	characteristic
87	#217	C₂×C₄	1	1	yes	characteristic
88	#217	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
90	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
91	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
92	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
93	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
94	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
95	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
96	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
99	C₄×Dic₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₃	C₂³	1	1	yes	characteristic
101	#208	—	2	2	no	—
102	#208	—	2	4	no	—
103	#208	D₄	1	4	yes	complemented
104	C₂×C₃⋊C₈	—	2	4	no	—
105	C₂×C₃⋊C₈	C₂³	1	2	yes	—
106	C₂×C₃⋊C₈	C₂³	1	2	yes	—
107	ΓL(1,𝔽₂₅)	—	2	4	no	—
108	D₃×C₈	—	2	4	no	—
109	C₂²×D₄	—	3	1	no	—
110	C₂³×C₄	—	3	1	no	—
111	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
112	C₂×M₁₆	—	3	1	no	—
113	C₂×M₁₆	—	3	1	no	—
114	C₂×M₁₆	—	3	2	no	—
115	C₂×M₁₆	—	3	2	no	—
116	C₂×M₁₆	—	6	4	no	—
117	C₂×M₁₆	—	6	4	no	—
118	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
119	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
120	C₂²×C₈	—	3	2	no	—
121	C₂²×C₈	—	3	2	no	—
122	C₄×D₄	—	3	1	no	—
123	C₄×D₄	—	3	1	no	—
124	C₄×D₄	—	3	1	no	—
125	C₄×D₄	—	3	1	no	—
126	C₄×D₄	—	3	1	no	—
127	C₄×D₄	—	3	1	no	—
128	C₄×D₄	—	3	1	no	—
129	C₄×D₄	—	3	1	no	—
130	C₂×C₄⋊C₄	—	3	1	no	—
131	C₂×C₂²⋊C₄	—	3	1	no	—
132	C₂×C₂²⋊C₄	—	3	1	no	—
133	C₂×C₄²	—	3	1	no	—
134	C₄⋊C₈	—	3	2	no	—
135	C₄⋊C₈	—	3	2	no	—
136	C₂²⋊C₈	—	3	2	no	—
137	C₂²⋊C₈	—	3	2	no	—
138	C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
139	C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
140	#97	—	3	2	no	—
141	C₄×C₈	—	3	2	no	—
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	—	4	1	no	—
145	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
146	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
147	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×D₃	—	2	2	no	—
149	C₂²×D₃	—	2	2	no	—
150	C₂²×D₃	—	4	1	no	—
151	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
164	C₂×C₃×C₄	#41	1	2	yes	—
165	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
166	#67	—	2	2	no	—
167	#67	—	2	2	no	—
168	#67	—	2	2	no	—
169	#67	M₁₆	1	4	yes	complemented
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	2	no	—
172	C₂×Dic₃	—	4	1	no	—
173	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
174	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
176	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
177	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
178	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
179	C₄×D₃	—	2	2	no	—
180	C₄×D₃	—	2	2	no	—
181	C₄×D₃	—	2	1	no	—
182	C₄×D₃	—	2	1	no	—
183	C₄×D₃	—	2	1	no	—
184	C₄×D₃	—	2	1	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃⋊C₈	—	2	2	no	—
188	C₃⋊C₈	C₂×D₄	1	4	yes	complemented
189	C₂⁴	—	3	1	no	—
190	C₂⁴	C₄×D₃	1	1	yes	characteristic
191	C₂×D₄	—	3	1	no	—
192	C₂×D₄	—	3	1	no	—
193	C₂×D₄	—	3	1	no	—
194	C₂×D₄	—	3	1	no	—
195	C₂×D₄	—	6	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	3	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂²×C₄	—	6	2	no	—
206	C₂²×C₄	—	6	2	no	—
207	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
208	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
209	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	3	4	no	—
216	M₁₆	—	6	4	no	—
217	M₁₆	—	6	4	no	—
218	M₁₆	—	6	2	no	—
219	C₂×C₈	—	3	2	no	—
220	C₂×C₈	—	3	2	no	—
221	C₂×C₈	—	6	4	no	—
222	C₂×C₈	—	6	4	no	—
223	C₂×C₈	C₂²×D₃	1	2	yes	—
224	C₂×C₈	C₂²×D₃	1	2	yes	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	3	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₂²⋊C₄	—	3	1	no	—
230	C₂²⋊C₄	—	3	1	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	—	3	1	no	—
234	C₂²⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	3	1	no	—
239	C₄²	—	3	1	no	—
240	C₄²	—	3	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	C₄×D₄	1	2	yes	—
250	C₂²×C₃	C₂×M₁₆	1	2	yes	complemented
251	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
252	C₂×D₃	—	2	1	no	—
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	4	4	no	—
258	C₂×D₃	C₂×M₁₆	1	2	yes	complemented
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	C₄×D₄	1	2	yes	—
262	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
263	C₃×C₄	C₂×#41	1	1	yes	characteristic
264	Dic₃	—	2	1	no	—
265	Dic₃	—	2	1	no	—
266	Dic₃	—	2	1	no	—
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	1	no	—
269	Dic₃	C₂×M₁₆	1	2	yes	complemented
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	3	1	no	—
273	C₂³	—	3	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	6	2	no	—
276	C₂³	—	6	2	no	—
277	C₂³	—	12	1	no	—
278	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
279	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
280	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
281	D₄	—	3	4	no	—
282	D₄	—	6	2	no	—
283	D₄	—	6	2	no	—
284	D₄	—	6	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	—	6	2	no	—
301	C₂×C₄	—	6	2	no	—
302	C₂×C₄	—	6	1	no	—
303	C₂×C₄	—	6	1	no	—
304	C₂×C₄	—	6	1	no	—
305	C₂×C₄	—	6	1	no	—
306	C₂×C₄	—	12	1	no	—
307	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
310	C₂×C₄	D₃×D₄	1	2	yes	—
311	C₂×C₄	#237	1	2	yes	—
312	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	3	4	no	—
316	C₈	—	6	2	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	4	1	no	—
320	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
321	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
322	C₂×C₃	#568	1	1	yes	characteristic
323	D₃	—	2	2	no	—
324	D₃	—	4	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	3	2	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	6	1	no	—
335	C₂²	—	6	1	no	—
336	C₂²	—	6	1	no	—
337	C₂²	—	6	1	no	—
338	C₂²	—	6	1	no	—
339	C₂²	—	12	4	no	—
340	C₂²	#67⋊C₄	1	2	yes	—
341	C₂²	D₃×M₁₆	1	2	yes	complemented
342	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	3	2	no	—
346	C₄	—	6	1	no	—
347	C₄	—	6	1	no	—
348	C₄	—	6	1	no	—
349	C₄	—	6	1	no	—
350	C₄	—	6	1	no	—
351	C₄	#67⋊C₄	1	2	yes	—
352	C₄	C₂×D₃×D₄	1	1	yes	characteristic
353	C₄	C₂×#237	1	1	yes	characteristic
354	C₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	6	2	no	—
359	C₂	—	12	1	no	—
360	C₂	C₂×#67⋊C₄	1	1	yes	characteristic
361	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
362	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#407	C₂
prime radical	C₂²×#67	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#407	D₃
hypocenter	C₃	D₄×M₁₆
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₂²×#208	C₂	semidirect
3. C₂×D₃×M₁₆	C₂	semidirect
4. C₃×#407	C₂	semidirect
5. #5126	C₂	semidirect
6. #5118	C₂	semidirect
7. Dic₃×M₁₆	C₂	semidirect
8. #4730	C₂	semidirect
9. #4729	C₂	semidirect
10. #4728	C₂	semidirect
11. #67⋊C₈	C₂	semidirect
33. C₂×#208	C₂²	semidirect
34. C₂²×C₃⋊C₈	C₂²	semidirect
36. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
37. C₂×D₃×C₈	C₂²	semidirect
38. C₃×C₂²⋊C₈	C₂²	semidirect
39. C₃×C₂²⋊C₈	C₂²	semidirect
40. (C₂×D₃)⋊C₈	C₂²	semidirect
41. (C₂×D₃)⋊C₈	C₂²	semidirect
42. #815	C₂²	semidirect
43. Dic₃⋊C₈	C₂²	semidirect
44. Dic₃⋊C₈	C₂²	semidirect
45. C₈×Dic₃	C₂²	semidirect
55. #407	D₃	semidirect
103. #208	D₄	semidirect
138. C₂²⋊C₈	C₂×D₃	semidirect
139. C₂²⋊C₈	C₂×D₃	semidirect
169. #67	M₁₆	semidirect
188. C₃⋊C₈	C₂×D₄	semidirect
250. C₂²×C₃	C₂×M₁₆	semidirect
258. C₂×D₃	C₂×M₁₆	semidirect
269. Dic₃	C₂×M₁₆	semidirect
341. C₂²	D₃×M₁₆	semidirect
354. C₃	D₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#67	2
C₂²×#208	2
C₂×D₃×M₁₆	2
C₃×#407	2
#5126	2
#5118	2
Dic₃×M₁₆	2
#4730	2
#4729	2
#4728	2
#67⋊C₈	2
D₄×M₁₆	3