Subgroups — #10950899

Order 768.

516 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78469	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#71447	C₂	1	1	yes	maximal, characteristic, complemented
4	#71444	C₂	1	2	yes	maximal, complemented
5	#71443	C₂	1	1	yes	maximal, characteristic
6	#69082	C₂	1	1	yes	maximal, characteristic, complemented
7	#69078	C₂	1	1	yes	maximal, characteristic, complemented
8	#69077	C₂	1	2	yes	maximal, complemented
9	#69062	C₂	1	1	yes	maximal, characteristic, complemented
10	#69059	C₂	1	2	yes	maximal, complemented
11	#69045	C₂	1	1	yes	maximal, characteristic, complemented
12	#33750	—	3	1	no	maximal, Sylow
13	#5479	—	2	2	no	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	#5116	C₂²	1	2	yes	—
18	#5115	C₂²	1	2	yes	—
19	#5114	—	2	2	no	—
20	C₂×Dic₂₄	C₂²	1	1	yes	characteristic
21	C₂×#800	C₂²	1	2	yes	—
22	C₂×D₄₈	C₂²	1	1	yes	characteristic
23	C₄×Dic₁₂	C₂²	1	2	yes	—
24	#4697	—	2	2	no	—
25	#4696	C₂²	1	2	yes	—
26	C₄×D₂₄	C₂²	1	2	yes	—
27	C₄×#204	—	2	2	no	—
28	#4683	C₂²	1	2	yes	—
29	C₃×#346	C₂²	1	1	yes	characteristic
30	C₃×#346	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#4509	C₂²	1	2	yes	—
34	#4509	C₂²	1	2	yes	—
35	#4505	C₂²	1	1	yes	characteristic
36	#4505	C₂²	1	1	yes	characteristic
37	#4504	C₂²	1	2	yes	—
38	#4503	C₂²	1	2	yes	—
39	#4503	C₂²	1	2	yes	—
40	#3168	—	3	2	no	—
41	#2253	—	3	1	no	—
42	#2249	—	3	1	no	—
43	#2242	—	3	2	no	—
44	#2220	—	3	2	no	—
45	#2219	—	3	1	no	—
46	#2218	—	3	1	no	—
47	#2177	—	3	1	no	—
48	#2176	—	3	1	no	—
49	#2175	—	3	2	no	—
50	C₄×M₃₂	D₃	1	1	yes	characteristic, complemented
51	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
54	#930	—	2	2	no	—
55	#929	—	4	2	no	—
56	C₄×#67	—	4	2	no	—
57	#924	—	2	2	no	—
58	C₂×Dic₁₂	C₂³	1	2	yes	—
59	C₂×D₂₄	C₂³	1	2	yes	—
60	C₂×#204	—	2	2	no	—
61	#876	—	4	2	no	—
62	#875	—	2	2	no	—
63	C₄×D₁₂	—	2	2	no	—
64	#872	—	4	2	no	—
65	#870	—	2	2	no	—
66	C₄×Dic₆	—	2	2	no	—
67	C₃×M₃₂	—	2	2	no	—
68	C₃×M₃₂	—	2	2	no	—
69	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₃×C₄×C₈	D₄	1	1	yes	characteristic
74	C₃×C₄×C₈	D₄	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
77	#821	—	2	2	no	—
78	#821	—	2	2	no	—
79	#818	C₂³	1	2	yes	—
80	#818	C₂³	1	2	yes	—
81	#817	—	2	2	no	—
82	#816	—	2	2	no	—
83	#816	—	2	2	no	—
84	Dic₂₄	—	2	2	no	—
85	#800	—	2	4	no	—
86	D₄₈	—	2	2	no	—
87	#529	—	6	2	no	—
88	C₂×Q₃₂	—	3	1	no	—
89	C₂×SD₃₂	—	3	2	no	—
90	C₂×D₁₆	—	3	1	no	—
91	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
92	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
93	#499	—	3	2	no	—
94	#495	—	3	2	no	—
95	#488	—	6	2	no	—
96	#467	—	3	2	no	—
97	#466	—	3	2	no	—
98	#465	—	6	2	no	—
99	C₄×Q₁₆	—	3	2	no	—
100	C₄×SD₁₆	—	6	2	no	—
101	C₄×D₈	—	3	2	no	—
102	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
103	#367	—	3	1	no	—
104	#367	—	3	1	no	—
105	#366	—	3	2	no	—
106	#358	—	3	2	no	—
107	#358	—	3	2	no	—
108	#357	—	3	2	no	—
109	#357	—	3	2	no	—
110	#346	C₂×D₃	1	1	yes	characteristic
111	#346	C₂×D₃	1	1	yes	characteristic
112	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
113	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	D₈	1	2	yes	—
115	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂×#67	—	4	2	no	—
117	C₂×D₁₂	—	2	2	no	—
118	C₂×C₄×D₃	—	4	2	no	—
119	C₂×Dic₆	—	2	2	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
127	C₃×C₄²	D₈	1	2	yes	—
128	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
129	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
130	#217	—	4	2	no	—
131	(C₂×D₃)⋊C₄	—	4	2	no	—
132	(C₂×D₃)⋊C₄	—	4	2	no	—
133	(C₂×D₃)⋊C₄	—	4	2	no	—
134	(C₂×D₃)⋊C₄	—	4	2	no	—
135	C₄⋊Dic₃	—	2	2	no	—
136	C₄⋊Dic₃	—	2	2	no	—
137	Dic₃⋊C₄	—	4	2	no	—
138	Dic₃⋊C₄	—	4	2	no	—
139	Dic₃⋊C₄	—	4	2	no	—
140	Dic₃⋊C₄	—	4	2	no	—
141	C₄×Dic₃	—	4	2	no	—
142	Dic₁₂	—	2	2	no	—
143	Dic₁₂	—	2	2	no	—
144	D₂₄	—	2	2	no	—
145	D₂₄	—	2	2	no	—
146	#204	—	4	4	no	—
147	C₃×C₁₆	—	2	1	no	—
148	C₃×C₁₆	—	2	1	no	—
149	C₃×C₁₆	#41	1	2	yes	—
150	C₃×C₁₆	#41	1	2	yes	—
151	C₂×Q₁₆	—	3	2	no	—
152	C₂×SD₁₆	—	6	2	no	—
153	C₂×D₈	—	3	2	no	—
154	C₂²×C₈	D₁₂	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
156	#126	—	12	2	no	—
157	#125	—	6	2	no	—
158	#124	—	6	2	no	—
159	#123	—	12	2	no	—
160	C₂²⋊Q₈	—	6	2	no	—
161	#121	—	6	2	no	—
162	C₄×Q₈	—	6	2	no	—
163	C₄×D₄	—	6	2	no	—
164	C₄×D₄	—	12	2	no	—
165	#117	—	12	2	no	—
166	C₂×C₄²	D₁₂	1	1	yes	characteristic
167	Q₃₂	—	6	2	no	—
168	SD₃₂	—	6	4	no	—
169	D₁₆	—	6	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
176	#107	—	3	2	no	—
177	#107	—	3	2	no	—
178	#106	—	6	2	no	—
179	#103	—	6	2	no	—
180	#103	—	6	2	no	—
181	#102	—	6	2	no	—
182	#102	—	6	2	no	—
183	C₄×C₈	D₁₂	1	1	yes	characteristic
184	C₄×C₈	D₁₂	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
187	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
188	C₂²×D₃	—	4	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
193	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
195	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
196	#67	—	8	4	no	—
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	2	no	—
200	D₁₂	—	4	2	no	—
201	D₁₂	—	4	2	no	—
202	C₄×D₃	—	8	4	no	—
203	Dic₆	—	4	2	no	—
204	Dic₆	—	4	2	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	#135	1	2	yes	—
208	C₃×C₈	C₂×#41	1	1	yes	characteristic
209	C₃×C₈	C₂×#41	1	1	yes	characteristic
210	C₂×Q₈	—	6	2	no	—
211	C₂×D₄	—	6	2	no	—
212	C₂×D₄	—	12	2	no	—
213	C₂²×C₄	—	12	2	no	—
214	C₂²×C₄	D₂₄	1	2	yes	—
215	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
216	Q₁₆	—	6	2	no	—
217	Q₁₆	—	6	2	no	—
218	SD₁₆	—	12	4	no	—
219	D₈	—	6	2	no	—
220	D₈	—	6	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
225	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
226	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
228	C₄⋊C₄	—	6	2	no	—
229	C₄⋊C₄	—	6	2	no	—
230	C₄⋊C₄	—	12	2	no	—
231	C₄⋊C₄	—	12	2	no	—
232	C₄⋊C₄	—	12	2	no	—
233	C₄⋊C₄	—	12	2	no	—
234	C₂²⋊C₄	—	12	2	no	—
235	C₂²⋊C₄	—	12	2	no	—
236	C₂²⋊C₄	—	12	2	no	—
237	C₂²⋊C₄	—	12	2	no	—
238	C₂²⋊C₄	—	12	2	no	—
239	C₄²	—	12	2	no	—
240	C₄²	D₂₄	1	2	yes	—
241	C₄²	C₂×D₁₂	1	1	yes	characteristic
242	C₄²	C₂×D₁₂	1	1	yes	characteristic
243	C₁₆	—	2	1	no	—
244	C₁₆	—	2	1	no	—
245	C₁₆	#235	1	2	yes	—
246	C₁₆	#235	1	2	yes	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
250	C₂×D₃	—	8	2	no	—
251	C₂×D₃	—	8	2	no	—
252	C₂×D₃	—	8	2	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	#532	1	1	yes	characteristic
257	C₃×C₄	C₂×#135	1	1	yes	characteristic
258	Dic₃	—	8	2	no	—
259	Dic₃	—	8	2	no	—
260	Dic₃	—	8	2	no	—
261	C₂³	—	12	2	no	—
262	C₂³	C₂×D₂₄	1	1	yes	characteristic
263	Q₈	—	12	2	no	—
264	Q₈	—	12	2	no	—
265	D₄	—	12	2	no	—
266	D₄	—	12	2	no	—
267	D₄	—	24	4	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	—	24	4	no	—
275	C₂×C₄	C₂×D₂₄	1	2	yes	—
276	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
277	C₂×C₄	C₂×D₂₄	1	2	yes	—
278	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	#904	1	2	yes	—
282	C₈	C₂×#235	1	1	yes	characteristic
283	C₈	C₂×#235	1	1	yes	characteristic
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	#3169	1	1	yes	characteristic
286	C₂×C₃	#3414	1	2	yes	—
287	D₃	—	16	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	24	2	no	—
291	C₂²	—	24	2	no	—
292	C₂²	—	24	2	no	—
293	C₂²	C₂²×D₂₄	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	24	2	no	—
298	C₄	—	24	2	no	—
299	C₄	—	24	2	no	—
300	C₄	#5477	1	1	yes	characteristic
301	C₄	C₂×#904	1	1	yes	characteristic
302	C₃	#33750	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	4	1	no	—
304	C₂	—	48	2	no	—
305	C₂	#78470	1	1	yes	characteristic
306	C₂	#82009	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×C₄×M₃₂	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	#71443	C₂
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#33750
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. #78469	C₂	semidirect
2. C₃×C₄×M₃₂	C₂	semidirect
3. #71447	C₂	semidirect
4. #71444	C₂	semidirect
6. #69082	C₂	semidirect
7. #69078	C₂	semidirect
8. #69077	C₂	semidirect
9. #69062	C₂	semidirect
10. #69059	C₂	semidirect
11. #69045	C₂	semidirect
50. C₄×M₃₂	D₃	semidirect
302. C₃	#33750	semidirect

Maximal subgroups

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

subgroup	index
#78469	2
C₃×C₄×M₃₂	2
#71447	2
#71444	2
#71443	2
#69082	2
#69078	2
#69077	2
#69062	2
#69059	2
#69045	2
#33750	3