Subgroups — #10711297

Order 768.

400 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4895	C₂	1	1	yes	maximal, characteristic
2	#71442	C₂	1	1	yes	maximal, characteristic
3	C₂×#4511	C₂	1	1	yes	maximal, characteristic, complemented
4	#70693	C₂	1	1	yes	maximal, characteristic, complemented
5	#70687	C₂	1	1	yes	maximal, characteristic
6	C₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	#67731	C₂	1	1	yes	maximal, characteristic, complemented
8	#12825	—	3	1	no	maximal, Sylow
9	C₂×#907	—	2	1	no	—
10	C₂×#904	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₃×#108	C₂²	1	1	yes	characteristic
14	C₂×#825	—	2	1	no	—
15	C₂×#822	—	2	1	no	—
16	#5118	—	2	1	no	—
17	C₂×#819	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×#812	C₂²	1	1	yes	characteristic
20	#4895	—	2	1	no	—
21	#4895	—	2	1	no	—
22	#4895	C₄	1	2	yes	—
23	#4895	C₄	1	2	yes	—
24	#4517	—	2	1	no	—
25	#4516	—	2	1	no	—
26	#4511	—	2	1	no	—
27	#4511	—	2	1	no	—
28	#4496	—	2	1	no	—
29	#4495	—	2	1	no	—
30	C₂×#443	—	3	1	no	—
31	C₂×#364	—	3	1	no	—
32	#2148	—	3	1	no	—
33	#2148	—	3	1	no	—
34	#1848	—	3	1	no	—
35	#1385	—	3	1	no	—
36	#1385	D₃	1	1	yes	characteristic, complemented
37	C₂×#235	—	2	1	no	—
38	C₂×C₃×M₁₆	—	2	1	no	—
39	C₂×C₃×M₁₆	—	2	1	no	—
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₃	D₄	1	1	yes	characteristic
42	C₂×#208	—	2	1	no	—
43	C₂²×C₃⋊C₈	D₄	1	1	yes	characteristic
44	#907	—	2	2	no	—
45	#907	—	2	2	no	—
46	#907	—	4	1	no	—
47	#907	—	4	1	no	—
48	C₂×Dic₁₂	D₄	1	1	yes	characteristic
49	#904	—	2	1	no	—
50	#904	—	2	1	no	—
51	#904	C₂×C₄	1	2	yes	—
52	#904	Q₈	1	2	yes	—
53	C₂×D₂₄	D₄	1	1	yes	characteristic
54	C₂×#204	—	2	1	no	—
55	C₂×ΓL(1,𝔽₂₅)	—	4	1	no	—
56	C₂×D₃×C₈	—	4	1	no	—
57	C₃×M₃₂	—	2	1	no	—
58	C₃×M₃₂	—	2	1	no	—
59	C₂×C₃×C₁₆	—	2	1	no	—
60	C₃×#108	—	2	1	no	—
61	C₃×#108	—	2	1	no	—
62	C₃×#108	—	2	1	no	—
63	C₃×#108	C₂×C₄	1	2	yes	—
64	#825	—	2	2	no	—
65	#825	—	2	2	no	—
66	#822	—	4	1	no	—
67	#822	—	4	1	no	—
68	#819	—	2	1	no	—
69	#819	—	2	1	no	—
70	Dic₃⋊C₈	—	4	2	no	—
71	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₃	D₄	1	1	yes	characteristic
74	C₈×Dic₃	D₄	1	1	yes	characteristic
75	#812	—	2	1	no	—
76	#812	—	2	1	no	—
77	C₂×C₃⋊C₁₆	—	2	1	no	—
78	C₂×#135	—	3	1	no	—
79	C₂×#131	—	6	1	no	—
80	C₂×M₃₂	—	3	1	no	—
81	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
82	#443	—	3	2	no	—
83	#443	—	3	2	no	—
84	#443	—	6	1	no	—
85	#443	—	6	1	no	—
86	C₂×#108	—	3	1	no	—
87	C₂×#108	—	6	1	no	—
88	C₂×#108	C₂×D₃	1	1	yes	characteristic
89	C₂×#108	C₂×D₃	1	1	yes	characteristic
90	#423	—	6	1	no	—
91	C₂×#104	—	6	1	no	—
92	C₂×C₄×C₈	—	3	1	no	—
93	#364	—	6	1	no	—
94	#364	—	6	1	no	—
95	#362	—	6	1	no	—
96	#362	—	6	1	no	—
97	#361	—	6	1	no	—
98	#361	—	6	1	no	—
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×#67	—	4	1	no	—
101	C₂²×Dic₃	—	2	1	no	—
102	#235	—	2	2	no	—
103	#235	—	2	2	no	—
104	#235	—	4	1	no	—
105	#235	—	4	1	no	—
106	C₂×D₁₂	—	2	1	no	—
107	C₂×C₄×D₃	—	4	1	no	—
108	C₂×Dic₆	—	2	1	no	—
109	C₃×M₁₆	—	2	2	no	—
110	C₃×M₁₆	—	4	1	no	—
111	C₃×M₁₆	—	4	1	no	—
112	C₃×M₁₆	—	4	1	no	—
113	C₂×C₃×C₈	—	4	1	no	—
114	C₂×C₃×C₈	—	4	1	no	—
115	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₃×C₈	#41	1	1	yes	characteristic
120	C₂×C₃×C₈	#41	1	1	yes	characteristic
121	C₄×Dic₃	—	2	1	no	—
122	C₄×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
124	#208	—	4	1	no	—
125	#208	—	4	1	no	—
126	C₂×C₃⋊C₈	—	4	1	no	—
127	C₂×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₃⋊C₈	D₈	1	1	yes	characteristic
130	C₂×C₃⋊C₈	D₈	1	1	yes	characteristic
131	C₂×C₃⋊C₈	SD₁₆	1	1	yes	characteristic
132	C₂×C₃⋊C₈	SD₁₆	1	1	yes	characteristic
133	Dic₁₂	—	2	1	no	—
134	Dic₁₂	C₄⋊C₄	1	2	yes	—
135	D₂₄	—	2	1	no	—
136	D₂₄	C₄⋊C₄	1	2	yes	—
137	#204	—	2	2	no	—
138	#204	—	2	2	no	—
139	ΓL(1,𝔽₂₅)	—	4	2	no	—
140	ΓL(1,𝔽₂₅)	—	4	2	no	—
141	D₃×C₈	—	4	2	no	—
142	D₃×C₈	—	4	2	no	—
143	C₃×C₁₆	—	4	1	no	—
144	C₃⋊C₁₆	—	4	1	no	—
145	C₂×#41	—	6	1	no	—
146	#135	—	3	2	no	—
147	#135	—	3	2	no	—
148	#135	—	6	1	no	—
149	#135	—	6	1	no	—
150	C₂×Q₁₆	—	3	1	no	—
151	C₂×SD₁₆	—	6	1	no	—
152	C₂×D₈	—	3	1	no	—
153	#131	—	6	2	no	—
154	#131	—	6	2	no	—
155	#131	—	12	1	no	—
156	#131	—	12	1	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	6	1	no	—
160	C₂×M₁₆	—	12	1	no	—
161	C₂²×C₈	—	3	1	no	—
162	C₂²×C₈	—	12	1	no	—
163	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₄²	—	3	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	6	1	no	—
168	M₃₂	—	6	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	6	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	—	6	1	no	—
175	#108	—	6	1	no	—
176	#108	—	12	1	no	—
177	#108	—	12	1	no	—
178	#108	C₄×D₃	1	2	yes	—
179	C₄⋊C₈	—	12	2	no	—
180	#104	—	6	2	no	—
181	#104	—	6	2	no	—
182	C₄×C₈	—	3	1	no	—
183	C₄×C₈	—	3	1	no	—
184	C₄×C₈	—	3	1	no	—
185	C₄×C₈	—	3	1	no	—
186	C₂³×C₃	#137	1	1	yes	characteristic
187	C₂²×D₃	—	4	1	no	—
188	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	#121	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
193	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
194	#67	—	4	2	no	—
195	#67	—	4	2	no	—
196	C₂×Dic₃	—	2	1	no	—
197	C₂×Dic₃	—	2	1	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	1	no	—
200	D₁₂	—	2	2	no	—
201	D₁₂	—	4	1	no	—
202	C₄×D₃	—	4	2	no	—
203	C₄×D₃	—	4	2	no	—
204	Dic₆	—	2	2	no	—
205	Dic₆	—	4	1	no	—
206	C₃×C₈	—	4	2	no	—
207	C₃×C₈	—	8	1	no	—
208	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₃×C₈	#117	1	1	yes	characteristic
210	C₃×C₈	#121	1	1	yes	characteristic
211	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
212	C₃⋊C₈	—	8	1	no	—
213	C₃⋊C₈	#102	1	1	yes	characteristic
214	C₃⋊C₈	#102	1	1	yes	characteristic
215	C₃⋊C₈	#102	1	1	yes	characteristic
216	C₃⋊C₈	#102	1	1	yes	characteristic
217	#41	—	6	2	no	—
218	#41	—	6	2	no	—
219	#41	—	12	1	no	—
220	#41	—	12	1	no	—
221	C₂×Q₈	—	6	1	no	—
222	C₂×D₄	—	6	1	no	—
223	C₂×D₄	—	12	1	no	—
224	C₂²×C₄	—	6	1	no	—
225	C₂²×C₄	—	12	1	no	—
226	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
227	Q₁₆	—	3	2	no	—
228	Q₁₆	—	6	1	no	—
229	SD₁₆	—	6	2	no	—
230	SD₁₆	—	6	2	no	—
231	D₈	—	3	2	no	—
232	D₈	—	6	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	M₁₆	—	12	2	no	—
238	M₁₆	—	12	2	no	—
239	M₁₆	—	12	1	no	—
240	M₁₆	—	12	1	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	3	1	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	12	2	no	—
250	C₂×C₈	—	12	2	no	—
251	C₂×C₈	—	12	1	no	—
252	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
253	C₂×C₈	#235	1	1	yes	characteristic
254	C₂×C₈	#235	1	1	yes	characteristic
255	C₂×C₈	D₃×D₄	1	1	yes	characteristic
256	C₂×C₈	D₃×D₄	1	1	yes	characteristic
257	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
258	C₄²	—	3	1	no	—
259	C₄²	—	3	1	no	—
260	C₄²	—	6	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₆	—	12	1	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	#417	1	1	yes	characteristic
265	C₂²×C₃	#461	1	1	yes	characteristic
266	C₂²×C₃	#474	1	1	yes	characteristic
267	C₂×D₃	—	4	2	no	—
268	C₂×D₃	—	8	2	no	—
269	C₃×C₄	#380	1	1	yes	characteristic
270	C₃×C₄	C₂×#102	1	1	yes	characteristic
271	C₃×C₄	#459	1	1	yes	characteristic
272	C₃×C₄	#476	1	1	yes	characteristic
273	Dic₃	—	2	2	no	—
274	Dic₃	—	4	1	no	—
275	Dic₃	—	4	2	no	—
276	C₂³	—	12	1	no	—
277	C₂³	#918	1	1	yes	characteristic
278	Q₈	—	6	2	no	—
279	Q₈	—	12	1	no	—
280	D₄	—	6	2	no	—
281	D₄	—	12	2	no	—
282	D₄	—	12	2	no	—
283	D₄	—	12	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	6	1	no	—
286	C₂×C₄	—	12	2	no	—
287	C₂×C₄	—	12	1	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	—	12	2	no	—
290	C₂×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
291	C₂×C₄	#884	1	1	yes	characteristic
292	C₂×C₄	#896	1	1	yes	characteristic
293	C₂×C₄	D₃×D₈	1	1	yes	characteristic
294	C₂×C₄	D₃×SD₁₆	1	1	yes	characteristic
295	C₂×C₄	#914	1	1	yes	characteristic
296	C₈	—	3	1	no	—
297	C₈	—	3	1	no	—
298	C₈	—	3	1	no	—
299	C₈	—	3	1	no	—
300	C₈	—	4	2	no	—
301	C₈	—	8	1	no	—
302	C₈	—	24	1	no	—
303	C₈	#872	1	1	yes	characteristic
304	C₈	#884	1	1	yes	characteristic
305	C₈	D₃×C₄⋊C₄	1	1	yes	characteristic
306	C₈	#896	1	1	yes	characteristic
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	#1802	1	1	yes	characteristic
311	D₃	—	8	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	12	2	no	—
314	C₂²	—	24	2	no	—
315	C₂²	#4803	1	1	yes	characteristic
316	C₂²	#4818	1	1	yes	characteristic
317	C₂²	#4890	1	1	yes	characteristic
318	C₄	—	6	2	no	—
319	C₄	—	12	1	no	—
320	C₄	—	12	2	no	—
321	C₄	#4667	1	1	yes	characteristic
322	C₄	D₃×#102	1	1	yes	characteristic
323	C₄	#4786	1	1	yes	characteristic
324	C₄	#4870	1	1	yes	characteristic
325	C₃	#12825	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	24	2	no	—
330	C₂	#68892	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4870
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1385	C₂
socle	C₂×C₃	#1802
prime radical	C₂×#904	C₂²
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1385	D₃
hypocenter	C₃	#12825
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

Complements

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

N	H ≅ G/N	type
3. C₂×#4511	C₂	semidirect
4. #70693	C₂	semidirect
6. C₃×#1385	C₂	semidirect
7. #67731	C₂	semidirect
36. #1385	D₃	semidirect
325. C₃	#12825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4895	2
#71442	2
C₂×#4511	2
#70693	2
#70687	2
C₃×#1385	2
#67731	2
#12825	3