Subgroups — #10814892

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic
2	#82515	C₂	1	2	yes	maximal, complemented
3	#72085	C₂	1	2	yes	maximal
4	#71646	C₂	1	2	yes	maximal, complemented
5	#70660	C₂	1	2	yes	maximal, complemented
6	#70291	C₂	1	4	yes	maximal, complemented
7	#69792	C₂	1	1	yes	maximal, characteristic, complemented
8	#69757	C₂	1	1	yes	maximal, characteristic, complemented
9	#22803	—	3	1	no	maximal, Sylow
10	C₂×C₃×#135	C₂²	1	1	yes	characteristic
11	C₃×C₄×#41	C₂²	1	2	yes	—
12	C₃×C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
13	C₃×C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×SD₁₆	C₄	1	2	yes	complemented
15	C₃×C₄×SD₁₆	C₄	1	2	yes	complemented
16	C₃×C₄×D₈	C₄	1	1	yes	characteristic, complemented
17	C₃×C₄×D₈	C₄	1	1	yes	characteristic, complemented
18	C₃×#427	C₂²	1	1	yes	characteristic
19	C₃×#416	C₂²	1	2	yes	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×#837	C₂²	1	4	yes	—
22	#5234	C₂²	1	4	yes	—
23	C₂×#819	C₂²	1	2	yes	—
24	C₂×#819	C₂²	1	2	yes	—
25	C₂×#815	C₂²	1	2	yes	—
26	#5000	—	2	4	no	—
27	#4997	C₂²	1	4	yes	—
28	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	#4926	—	2	2	no	—
30	#4925	—	2	2	no	—
31	#4924	C₂²	1	2	yes	—
32	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
33	#4550	C₂²	1	2	yes	—
34	#4532	C₂²	1	4	yes	—
35	C₄×#135	D₃	1	1	yes	characteristic
36	#2925	—	3	2	no	—
37	#1975	—	3	2	no	—
38	#1929	—	3	4	no	—
39	#1848	—	3	1	no	—
40	#1806	—	3	2	no	—
41	#1773	—	3	2	no	—
42	#1770	—	3	1	no	—
43	C₂×C₃×#41	C₂³	1	2	yes	—
44	C₃×#135	—	2	2	no	—
45	C₃×#135	—	2	2	no	—
46	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	—
48	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
52	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
53	C₃×C₄×D₄	—	2	2	no	—
54	C₃×C₄×D₄	—	2	2	no	—
55	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
56	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
57	C₃×#117	—	2	2	no	—
58	C₃×#117	C₂³	1	2	yes	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	2	1	no	—
62	C₂×C₄×Dic₃	C₂³	1	2	yes	—
63	C₂×#208	C₂³	1	4	yes	—
64	C₂²×C₃⋊C₈	C₂³	1	2	yes	—
65	C₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₃×#106	C₂×C₄	1	2	yes	—
68	C₃×#103	C₂×C₄	1	2	yes	—
69	C₃×#103	C₂×C₄	1	2	yes	—
70	C₃×#102	C₂×C₄	1	2	yes	—
71	C₃×#102	C₂×C₄	1	2	yes	—
72	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	#837	—	4	4	no	—
77	C₃⋊#95	—	2	1	no	—
78	#830	—	2	4	no	—
79	#830	—	2	4	no	—
80	#819	—	2	4	no	—
81	#819	—	4	2	no	—
82	#815	—	2	4	no	—
83	C₄⋊(C₃⋊C₈)	—	2	2	no	—
84	C₄⋊(C₃⋊C₈)	—	2	2	no	—
85	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
86	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
87	#803	—	2	4	no	—
88	C₄×C₃⋊C₈	—	2	4	no	—
89	C₂×#135	C₂×D₃	1	1	yes	characteristic
90	C₄×#41	C₂×D₃	1	2	yes	—
91	C₄×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
92	C₄×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
93	C₄×SD₁₆	Dic₃	1	2	yes	complemented
94	C₄×SD₁₆	Dic₃	1	2	yes	complemented
95	C₄×D₈	Dic₃	1	1	yes	characteristic, complemented
96	C₄×D₈	Dic₃	1	1	yes	characteristic, complemented
97	#433	—	6	4	no	—
98	#432	—	6	2	no	—
99	#431	—	6	2	no	—
100	C₂×#108	—	3	2	no	—
101	C₂×#108	—	3	2	no	—
102	#427	C₂×D₃	1	1	yes	characteristic
103	#423	—	3	2	no	—
104	C₂×C₄⋊C₈	—	3	1	no	—
105	C₂×#104	—	3	4	no	—
106	#416	C₂×D₃	1	2	yes	—
107	#408	—	3	4	no	—
108	#405	—	3	4	no	—
109	C₂×#97	—	3	2	no	—
110	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	3	1	no	—
112	#339	—	3	4	no	—
113	#336	—	3	2	no	—
114	C₃×#41	—	2	2	no	—
115	C₃×#41	—	2	2	no	—
116	C₃×#41	—	2	2	no	—
117	C₃×#41	—	2	2	no	—
118	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	C₂²×C₄	1	2	yes	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	#41	1	1	yes	characteristic
125	C₂²×C₃×C₄	#41	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
127	C₂²×Dic₃	—	2	2	no	—
128	C₃×Q₁₆	—	2	2	no	—
129	C₃×SD₁₆	—	2	4	no	—
130	C₃×D₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×C₈	#41	1	2	yes	—
138	C₂×C₃×C₈	#41	1	2	yes	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
142	C₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
143	C₃×C₂²⋊C₄	—	2	2	no	—
144	C₃×C₂²⋊C₄	—	2	2	no	—
145	C₃×C₂²⋊C₄	—	2	2	no	—
146	C₃×C₄²	—	2	2	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	—	2	2	no	—
149	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
150	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
151	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
153	C₄⋊Dic₃	—	4	2	no	—
154	C₄×Dic₃	—	2	2	no	—
155	C₄×Dic₃	—	2	2	no	—
156	#208	—	2	4	no	—
157	#208	—	2	4	no	—
158	C₂×C₃⋊C₈	—	2	2	no	—
159	C₂×C₃⋊C₈	—	2	2	no	—
160	C₂×C₃⋊C₈	—	2	4	no	—
161	C₂×C₃⋊C₈	C₂×D₄	1	4	yes	—
162	C₂×#41	C₂²×D₃	1	2	yes	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	C₂×Q₁₆	C₂×Dic₃	1	1	yes	characteristic
166	C₂×SD₁₆	C₂×Dic₃	1	2	yes	—
167	C₂×D₈	C₂×Dic₃	1	1	yes	characteristic
168	C₂×M₁₆	—	3	4	no	—
169	C₂²×C₈	—	3	2	no	—
170	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
172	C₄×Q₈	C₂×Dic₃	1	2	yes	—
173	C₄×Q₈	C₂×Dic₃	1	2	yes	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	C₂×Dic₃	1	2	yes	—
177	C₄×D₄	C₂×Dic₃	1	2	yes	—
178	#117	—	2	2	no	—
179	#117	C₂²×D₃	1	2	yes	—
180	C₂×C₄⋊C₄	—	6	1	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	3	2	no	—
183	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
184	#108	—	6	4	no	—
185	#108	—	12	2	no	—
186	#107	C₂×Dic₃	1	1	yes	characteristic
187	#107	C₂×Dic₃	1	1	yes	characteristic
188	#106	C₂×Dic₃	1	2	yes	—
189	C₄⋊C₈	—	3	2	no	—
190	C₄⋊C₈	—	3	2	no	—
191	C₄⋊C₈	—	6	2	no	—
192	C₄⋊C₈	—	6	2	no	—
193	#104	—	12	4	no	—
194	#103	C₂×Dic₃	1	2	yes	—
195	#103	C₂×Dic₃	1	2	yes	—
196	#102	C₂×Dic₃	1	2	yes	—
197	#102	C₂×Dic₃	1	2	yes	—
198	C₂²⋊C₈	—	6	4	no	—
199	C₂²⋊C₈	—	6	4	no	—
200	#97	—	6	4	no	—
201	#97	—	6	4	no	—
202	C₄×C₈	—	6	4	no	—
203	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
204	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
207	#95	—	6	1	no	—
208	C₂³×C₃	—	2	2	no	—
209	C₂³×C₃	C₂×#41	1	1	yes	characteristic
210	C₃×Q₈	—	2	2	no	—
211	C₃×Q₈	—	2	2	no	—
212	C₃×D₄	—	2	2	no	—
213	C₃×D₄	—	2	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	—	4	4	no	—
224	C₂×C₃×C₄	—	8	2	no	—
225	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
233	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
235	C₂×Dic₃	—	4	2	no	—
236	C₂×Dic₃	—	4	2	no	—
237	C₂×Dic₃	—	4	2	no	—
238	C₃×C₈	—	2	1	no	—
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	#117	1	2	yes	—
241	C₃×C₈	#117	1	2	yes	—
242	C₃⋊C₈	—	2	4	no	—
243	C₃⋊C₈	—	4	4	no	—
244	#41	—	2	2	no	—
245	#41	—	2	2	no	—
246	#41	—	2	2	no	—
247	#41	—	2	2	no	—
248	C₂×Q₈	C₂²×Dic₃	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	C₂²×Dic₃	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	6	2	no	—
255	C₂²×C₄	#237	1	1	yes	characteristic
256	C₂²×C₄	#237	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
258	Q₁₆	—	2	2	no	—
259	SD₁₆	—	2	4	no	—
260	D₈	—	2	2	no	—
261	M₁₆	—	6	4	no	—
262	M₁₆	—	6	4	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	3	4	no	—
266	C₂×C₈	—	6	2	no	—
267	C₂×C₈	—	6	2	no	—
268	C₂×C₈	—	6	4	no	—
269	C₂×C₈	#235	1	2	yes	—
270	C₂×C₈	#235	1	2	yes	—
271	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	12	2	no	—
278	C₄⋊C₄	C₂²×Dic₃	1	2	yes	—
279	C₄⋊C₄	C₂²×Dic₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	6	2	no	—
287	C₄²	—	6	2	no	—
288	C₄²	D₃×D₄	1	1	yes	characteristic
289	C₄²	D₃×D₄	1	1	yes	characteristic
290	C₄²	C₂²×Dic₃	1	1	yes	characteristic
291	C₄²	C₂²×Dic₃	1	1	yes	characteristic
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	4	2	no	—
295	C₂²×C₃	—	4	2	no	—
296	C₂²×C₃	—	4	2	no	—
297	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
298	C₂²×C₃	#555	1	1	yes	characteristic
299	C₂²×C₃	#556	1	1	yes	characteristic
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	4	2	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	4	2	no	—
306	C₃×C₄	#444	1	2	yes	—
307	C₃×C₄	C₂×#117	1	1	yes	characteristic
308	C₃×C₄	#520	1	1	yes	characteristic
309	C₃×C₄	#547	1	1	yes	characteristic
310	C₃×C₄	#550	1	1	yes	characteristic
311	Dic₃	—	8	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	C₂×#237	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	12	2	no	—
330	C₂×C₄	—	12	2	no	—
331	C₂×C₄	—	12	2	no	—
332	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
333	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
334	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
335	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#235	1	1	yes	characteristic
337	C₂×C₄	C₂×#235	1	1	yes	characteristic
338	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
339	C₂×C₄	#1009	1	1	yes	characteristic
340	C₂×C₄	#1010	1	1	yes	characteristic
341	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	6	4	no	—
345	C₈	—	12	4	no	—
346	C₈	#926	1	2	yes	—
347	C₈	#926	1	2	yes	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	2	1	no	—
350	C₂×C₃	—	8	2	no	—
351	C₂×C₃	#2354	1	1	yes	characteristic
352	C₂×C₃	C₂×#444	1	1	yes	characteristic
353	C₂×C₃	#2957	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	#5542	1	1	yes	characteristic
360	C₂²	#5563	1	1	yes	characteristic
361	C₂²	C₂×D₄×Dic₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	24	2	no	—
369	C₄	#5197	1	2	yes	—
370	C₄	#5551	1	1	yes	characteristic
371	C₄	#5553	1	1	yes	characteristic
372	C₄	C₂×#926	1	1	yes	characteristic
373	C₄	#5827	1	1	yes	characteristic
374	C₃	#22803	1	1	yes	characteristic, complemented, Sylow
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	8	2	no	—
378	C₂	#82337	1	1	yes	characteristic
379	C₂	C₂×#5197	1	1	yes	characteristic
380	C₂	#83910	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×C₄×#135	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂×C₃×#135	C₂²
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₄×#135	D₃
hypocenter	C₃	#22803
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #82515	C₂	semidirect
4. #71646	C₂	semidirect
5. #70660	C₂	semidirect
6. #70291	C₂	semidirect
7. #69792	C₂	semidirect
8. #69757	C₂	semidirect
12. C₃×C₄×Q₁₆	C₄	semidirect
13. C₃×C₄×Q₁₆	C₄	semidirect
14. C₃×C₄×SD₁₆	C₄	semidirect
15. C₃×C₄×SD₁₆	C₄	semidirect
16. C₃×C₄×D₈	C₄	semidirect
17. C₃×C₄×D₈	C₄	semidirect
91. C₄×Q₁₆	Dic₃	semidirect
92. C₄×Q₁₆	Dic₃	semidirect
93. C₄×SD₁₆	Dic₃	semidirect
94. C₄×SD₁₆	Dic₃	semidirect
95. C₄×D₈	Dic₃	semidirect
96. C₄×D₈	Dic₃	semidirect
374. C₃	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#135	2
#82515	2
#72085	2
#71646	2
#70660	2
#70291	2
#69792	2
#69757	2
#22803	3