Subgroups — #10950798

Order 768.

484 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79728	C₂	1	2	yes	maximal
2	C₃×C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#71504	C₂	1	1	yes	maximal, characteristic
4	#71499	C₂	1	1	yes	maximal, characteristic, complemented
5	#70909	C₂	1	2	yes	maximal, complemented
6	#69328	C₂	1	1	yes	maximal, characteristic
7	#69321	C₂	1	1	yes	maximal, characteristic, complemented
8	#69229	C₂	1	2	yes	maximal
9	#69218	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#69051	C₂	1	1	yes	maximal, characteristic
12	#33741	—	3	1	no	maximal, Sylow
13	#5583	—	2	2	no	—
14	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
16	#5188	C₂²	1	2	yes	—
17	#5183	C₂²	1	2	yes	—
18	C₂×#829	C₂²	1	2	yes	—
19	#5030	—	2	2	no	—
20	C₂×Dic₂₄	C₂²	1	1	yes	characteristic
21	C₂×#798	C₂²	1	1	yes	characteristic
22	C₂×D₃×C₁₆	C₂²	1	1	yes	characteristic
23	#4887	C₂²	1	2	yes	—
24	#4878	C₂²	1	2	yes	—
25	#4816	—	2	2	no	—
26	#4797	—	2	2	no	—
27	C₄×Dic₁₂	C₂²	1	2	yes	—
28	#4689	C₂²	1	1	yes	characteristic
29	C₃×#366	C₂²	1	1	yes	characteristic
30	C₃×#358	C₂²	1	2	yes	—
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#4564	C₂²	1	2	yes	—
33	#4504	C₂²	1	1	yes	characteristic
34	#4503	C₂²	1	2	yes	—
35	#4502	C₂²	1	1	yes	characteristic
36	Dic₃×C₁₆	C₂²	1	1	yes	characteristic
37	#4492	C₂²	1	2	yes	—
38	#4490	C₂²	1	2	yes	—
39	#4461	C₂²	1	1	yes	characteristic
40	#3177	—	3	2	no	—
41	#2253	—	3	1	no	—
42	#2247	—	3	1	no	—
43	#2245	—	3	2	no	—
44	#2219	—	3	2	no	—
45	#2216	—	3	1	no	—
46	#2216	—	3	1	no	—
47	#2176	—	3	2	no	—
48	C₄×Q₃₂	—	3	1	no	—
49	C₄×Q₃₂	D₃	1	1	yes	characteristic, complemented
50	#2108	—	3	1	no	—
51	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
52	C₃×C₄×Q₈	—	2	2	no	—
53	#946	—	2	2	no	—
54	#944	—	2	2	no	—
55	C₂×#216	—	2	2	no	—
56	C₂×Dic₁₂	C₂³	1	2	yes	—
57	C₂×ΓL(1,𝔽₂₅)	D₄	1	1	yes	characteristic
58	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
59	#898	—	4	2	no	—
60	#897	—	2	2	no	—
61	#890	—	2	2	no	—
62	#889	—	4	2	no	—
63	#876	—	4	2	no	—
64	#872	D₄	1	1	yes	characteristic
65	#870	—	4	2	no	—
66	C₄×Dic₆	—	2	2	no	—
67	C₃×Q₃₂	—	2	2	no	—
68	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₃×#107	C₂³	1	2	yes	—
71	C₃×#103	—	2	2	no	—
72	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
73	#835	—	2	2	no	—
74	#829	—	2	4	no	—
75	#818	C₂³	1	2	yes	—
76	#816	—	2	2	no	—
77	#815	D₄	1	1	yes	characteristic
78	C₈×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
81	#810	—	2	2	no	—
82	#808	—	2	2	no	—
83	#803	D₄	1	1	yes	characteristic
84	Dic₂₄	—	2	2	no	—
85	#798	—	2	2	no	—
86	D₃×C₁₆	—	2	2	no	—
87	#541	—	6	2	no	—
88	C₂×Q₃₂	—	3	2	no	—
89	C₂×Q₃₂	—	3	1	no	—
90	C₂×Q₃₂	C₂×D₃	1	1	yes	characteristic
91	C₂×M₃₂	—	3	1	no	—
92	C₂²×C₁₆	—	3	1	no	—
93	#499	—	3	2	no	—
94	#494	—	3	2	no	—
95	#491	—	6	2	no	—
96	#470	—	6	2	no	—
97	#467	—	3	2	no	—
98	#467	—	3	2	no	—
99	#441	—	6	2	no	—
100	C₄×Q₁₆	—	3	2	no	—
101	C₄×Q₁₆	C₂×D₃	1	2	yes	—
102	#405	—	3	1	no	—
103	#367	—	3	2	no	—
104	#366	—	3	1	no	—
105	#366	C₂×D₃	1	1	yes	characteristic
106	#358	—	3	2	no	—
107	#358	—	3	2	no	—
108	#358	—	3	2	no	—
109	#358	C₂×D₃	1	2	yes	—
110	#346	—	3	1	no	—
111	#346	—	3	1	no	—
112	C₄×C₁₆	—	3	1	no	—
113	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₄×D₃	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₆	—	2	2	no	—
117	C₃×Q₁₆	—	2	2	no	—
118	C₃×Q₁₆	—	2	2	no	—
119	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	4	2	no	—
123	C₃×C₄²	—	4	2	no	—
124	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
125	#216	—	4	4	no	—
126	(C₂×D₃)⋊C₄	—	4	2	no	—
127	(C₂×D₃)⋊C₄	—	4	2	no	—
128	(C₂×D₃)⋊C₄	D₈	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	D₈	1	1	yes	characteristic
130	C₄⋊Dic₃	—	2	2	no	—
131	C₄⋊Dic₃	—	4	2	no	—
132	C₄⋊Dic₃	—	4	2	no	—
133	Dic₃⋊C₄	—	4	2	no	—
134	Dic₃⋊C₄	—	4	2	no	—
135	Dic₃⋊C₄	—	4	2	no	—
136	Dic₃⋊C₄	—	4	2	no	—
137	Dic₃⋊C₄	—	4	2	no	—
138	Dic₃⋊C₄	D₈	1	1	yes	characteristic
139	Dic₃⋊C₄	D₈	1	1	yes	characteristic
140	C₄×Dic₃	—	4	2	no	—
141	C₄×Dic₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
144	Dic₁₂	—	2	2	no	—
145	Dic₁₂	—	2	2	no	—
146	ΓL(1,𝔽₂₅)	—	2	2	no	—
147	D₃×C₈	—	2	1	no	—
148	D₃×C₈	—	2	1	no	—
149	C₃×C₁₆	—	2	1	no	—
150	C₃×C₁₆	#41	1	2	yes	—
151	C₃⋊C₁₆	—	2	1	no	—
152	C₃⋊C₁₆	#41	1	2	yes	—
153	C₂×Q₁₆	—	3	2	no	—
154	C₂×Q₁₆	—	6	2	no	—
155	C₂×Q₁₆	C₂²×D₃	1	2	yes	—
156	C₂×M₁₆	—	3	1	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₄⋊Q₈	—	6	2	no	—
159	#126	—	12	2	no	—
160	#126	—	12	2	no	—
161	#125	—	6	2	no	—
162	#125	—	12	2	no	—
163	#125	—	12	2	no	—
164	C₂²⋊Q₈	—	6	2	no	—
165	C₂²⋊Q₈	—	6	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	6	2	no	—
168	#117	—	3	1	no	—
169	Q₃₂	—	2	2	no	—
170	Q₃₂	—	6	4	no	—
171	Q₃₂	—	6	2	no	—
172	M₃₂	—	6	2	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	—	3	1	no	—
175	C₂×C₁₆	—	6	2	no	—
176	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
178	#107	—	3	2	no	—
179	#107	C₂²×D₃	1	2	yes	—
180	#106	—	6	2	no	—
181	#103	—	2	2	no	—
182	#103	—	6	2	no	—
183	#103	—	6	2	no	—
184	#103	—	6	2	no	—
185	#97	—	3	1	no	—
186	#97	—	3	1	no	—
187	C₄×C₈	—	3	1	no	—
188	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
189	C₂²×D₃	C₂×D₈	1	1	yes	characteristic
190	C₃×Q₈	—	4	2	no	—
191	C₃×Q₈	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	C₂×D₈	1	1	yes	characteristic
200	C₂×Dic₃	C₂×D₈	1	1	yes	characteristic
201	C₂×Dic₃	C₂×D₈	1	1	yes	characteristic
202	C₄×D₃	—	2	1	no	—
203	C₄×D₃	—	2	1	no	—
204	Dic₆	—	4	2	no	—
205	Dic₆	—	4	2	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	C₂×#41	1	1	yes	characteristic
208	C₃×C₈	C₂×#41	1	1	yes	characteristic
209	C₃⋊C₈	—	2	1	no	—
210	C₃⋊C₈	#135	1	2	yes	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	6	2	no	—
213	C₂²×C₄	—	3	1	no	—
214	Q₁₆	—	2	2	no	—
215	Q₁₆	—	2	2	no	—
216	Q₁₆	—	6	2	no	—
217	Q₁₆	—	6	2	no	—
218	Q₁₆	—	12	4	no	—
219	M₁₆	—	6	2	no	—
220	C₂×C₈	—	3	1	no	—
221	C₂×C₈	—	3	1	no	—
222	C₂×C₈	—	6	1	no	—
223	C₂×C₈	—	6	1	no	—
224	C₂×C₈	D₃×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	6	2	no	—
231	C₄⋊C₄	—	12	2	no	—
232	C₄⋊C₄	—	12	2	no	—
233	C₄⋊C₄	—	12	2	no	—
234	C₄⋊C₄	—	12	2	no	—
235	C₄⋊C₄	—	12	2	no	—
236	C₄⋊C₄	—	12	2	no	—
237	C₄⋊C₄	—	12	2	no	—
238	C₂²⋊C₄	—	3	1	no	—
239	C₂²⋊C₄	—	3	1	no	—
240	C₂²⋊C₄	—	12	2	no	—
241	C₂²⋊C₄	—	12	2	no	—
242	C₄²	—	3	1	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	12	2	no	—
245	C₄²	D₃×D₄	1	1	yes	characteristic
246	C₁₆	—	2	1	no	—
247	C₁₆	—	3	2	no	—
248	C₁₆	—	6	1	no	—
249	C₁₆	#235	1	2	yes	—
250	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
251	C₂×D₃	—	2	1	no	—
252	C₂×D₃	—	2	1	no	—
253	C₂×D₃	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	8	2	no	—
257	C₃×C₄	—	8	2	no	—
258	C₃×C₄	#532	1	1	yes	characteristic
259	C₃×C₄	C₂×#135	1	1	yes	characteristic
260	Dic₃	—	2	1	no	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	8	2	no	—
264	Dic₃	—	8	2	no	—
265	C₂³	—	3	1	no	—
266	Q₈	—	4	2	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	12	2	no	—
269	Q₈	—	12	2	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	2	no	—
278	C₂×C₄	—	12	2	no	—
279	C₂×C₄	D₃×D₈	1	1	yes	characteristic
280	C₂×C₄	D₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	3	2	no	—
284	C₈	—	6	1	no	—
285	C₈	C₂×#235	1	1	yes	characteristic
286	C₈	D₃×#41	1	1	yes	characteristic
287	C₂×C₃	#3169	1	1	yes	characteristic
288	C₂×C₃	#3417	1	1	yes	characteristic
289	C₂×C₃	#3417	1	1	yes	characteristic
290	D₃	—	4	1	no	—
291	C₂²	—	6	1	no	—
292	C₂²	—	6	1	no	—
293	C₂²	—	6	1	no	—
294	C₂²	C₂×D₃×D₈	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	6	1	no	—
298	C₄	—	6	1	no	—
299	C₄	—	6	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	24	2	no	—
303	C₄	—	24	2	no	—
304	C₄	#5548	1	1	yes	characteristic
305	C₄	D₃×#135	1	1	yes	characteristic
306	C₃	#33741	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	12	1	no	—
308	C₂	#79598	1	1	yes	characteristic
309	C₂	#82028	1	1	yes	characteristic
310	C₂	#82033	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×C₄×Q₃₂	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	C₂²×D₃	C₂×D₈
hypercenter	C₄×Q₃₂	D₃
hypocenter	C₃	#33741
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₃₂	C₂	semidirect
4. #71499	C₂	semidirect
5. #70909	C₂	semidirect
7. #69321	C₂	semidirect
9. #69218	C₂	semidirect
10. C₄×Dic₂₄	C₂	semidirect
49. C₄×Q₃₂	D₃	semidirect
306. C₃	#33741	semidirect

Maximal subgroups

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

subgroup	index
#79728	2
C₃×C₄×Q₃₂	2
#71504	2
#71499	2
#70909	2
#69328	2
#69321	2
#69229	2
#69218	2
C₄×Dic₂₄	2
#69051	2
#33741	3