Subgroups — #11704205

Order 768.

7130 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#518	C₂	1	1	yes	maximal, characteristic
2	C₂×#5798	C₂	1	2	yes	maximal
3	#85860	C₂	1	32	yes	maximal, complemented
4	#85766	C₂	1	16	yes	maximal, complemented
5	#85725	C₂	1	4	yes	maximal, complemented
6	C₂×#5483	C₂	1	8	yes	maximal, complemented
7	#62843	—	3	1	no	maximal, Sylow
8	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₃×#518	C₂²	1	16	yes	—
10	C₂×C₃×C₄×D₄	C₂²	1	8	yes	—
11	C₂×C₃×#117	C₂²	1	2	yes	—
12	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂²×#217	C₂²	1	8	yes	—
14	#5798	C₂²	1	16	yes	—
15	#5798	C₂²	1	16	yes	—
16	C₂×D₄×Dic₃	C₂²	1	16	yes	—
17	C₂×#926	C₂²	1	4	yes	—
18	C₂×#924	C₂²	1	32	yes	complemented
19	#5543	C₂²	1	64	yes	complemented
20	#5539	C₂²	1	128	yes	complemented
21	D₄×Dic₆	C₂²	1	64	yes	complemented
22	#5519	C₂²	1	32	yes	complemented
23	#5517	C₂²	1	32	yes	complemented
24	#5500	C₂²	1	16	yes	complemented
25	#5483	C₂²	1	16	yes	—
26	#5483	C₂²	1	64	yes	complemented
27	#5483	C₂²	1	32	yes	complemented
28	C₂×#878	C₂²	1	64	yes	complemented
29	#5470	C₂²	1	16	yes	complemented
30	#3501	—	3	32	no	—
31	#3501	—	3	16	no	—
32	#3458	—	3	4	no	—
33	C₂×#543	—	3	8	no	—
34	C₂×#518	—	3	2	no	—
35	C₂×#518	D₃	1	1	yes	characteristic
36	C₂⁵×C₃	C₂³	1	2	yes	—
37	C₂²×C₃×D₄	Q₈	1	16	yes	complemented
38	C₂²×C₃×D₄	C₂³	1	8	yes	—
39	C₂²×C₃×D₄	C₂³	1	4	yes	—
40	C₂³×C₃×C₄	C₂³	1	8	yes	—
41	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂³×Dic₃	C₂³	1	16	yes	—
43	C₂²×Dic₆	C₂³	1	32	yes	complemented
44	C₃×C₄×D₄	C₂³	1	64	yes	—
45	C₃×#117	C₂³	1	16	yes	—
46	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₃×C₂²⋊C₄	C₂³	1	32	yes	—
48	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
49	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
50	C₂×C₃×C₄²	C₂³	1	4	yes	—
51	C₂×#217	C₂³	1	32	yes	—
52	C₂×#217	C₂³	1	8	yes	—
53	C₂×#217	C₂³	1	16	yes	—
54	C₂×#217	C₂³	1	32	yes	—
55	D₄×Dic₃	C₂³	1	128	yes	—
56	#926	C₂³	1	16	yes	—
57	#926	C₂³	1	16	yes	—
58	C₂×C₄⋊Dic₃	C₂³	1	8	yes	—
59	C₂×C₄⋊Dic₃	C₂³	1	32	yes	complemented
60	#924	C₂³	1	64	yes	—
61	#924	C₂³	1	128	yes	complemented
62	C₂×Dic₃⋊C₄	C₂³	1	64	yes	complemented
63	C₂×C₄×Dic₃	C₂³	1	8	yes	—
64	#890	C₂³	1	64	yes	complemented
65	#888	C₂³	1	64	yes	complemented
66	#878	C₂³	1	128	yes	—
67	#878	C₂³	1	256	yes	complemented
68	#870	C₂³	1	32	yes	complemented
69	#869	C₂³	1	32	yes	complemented
70	C₄×Dic₆	C₂³	1	64	yes	complemented
71	C₂³×D₄	C₂×D₃	1	1	yes	characteristic
72	#554	—	3	128	no	—
73	#554	—	3	64	no	—
74	D₄×Q₈	—	3	64	no	—
75	#544	—	3	32	no	—
76	#544	—	3	16	no	—
77	#543	—	3	16	no	—
78	#543	—	3	64	no	—
79	#543	—	3	32	no	—
80	#533	—	3	32	no	—
81	#533	—	3	16	no	—
82	C₂×C₂²⋊Q₈	—	3	32	no	—
83	C₂×C₂²⋊Q₈	—	3	64	no	—
84	#518	—	3	16	no	—
85	#518	—	3	16	no	—
86	#518	C₂×D₃	1	16	yes	—
87	C₂×C₄×D₄	—	3	16	no	—
88	C₂×C₄×D₄	C₂×D₃	1	8	yes	—
89	C₂×#117	—	3	4	no	—
90	C₂×#117	C₂×D₃	1	2	yes	—
91	C₂²×C₂²⋊C₄	—	3	8	no	—
92	C₂²×C₂²⋊C₄	C₂×D₃	1	4	yes	—
93	C₂⁴×C₃	—	8	4	no	—
94	C₂⁴×C₃	C₂×Q₈	1	16	yes	complemented
95	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
96	C₂⁴×C₃	C₂⁴	1	4	yes	—
97	C₂⁴×C₃	C₂⁴	1	8	yes	—
98	C₂×C₃×D₄	C₂×Q₈	1	64	yes	complemented
99	C₂×C₃×D₄	C₂×Q₈	1	32	yes	complemented
100	C₂×C₃×D₄	C₂⁴	1	16	yes	—
101	C₂²×C₃×C₄	—	8	8	no	—
102	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	complemented
103	C₂²×C₃×C₄	C₂⁴	1	32	yes	—
104	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
105	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
106	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
107	C₂²×Dic₃	—	8	16	no	—
108	C₂²×Dic₃	C₂⁴	1	16	yes	—
109	C₂²×Dic₃	C₂⁴	1	64	yes	—
110	C₂×Dic₆	—	8	32	no	—
111	C₂×Dic₆	C₂⁴	1	64	yes	—
112	C₃×C₄⋊C₄	C₂⁴	1	16	yes	—
113	C₃×C₂²⋊C₄	C₂⁴	1	16	yes	—
114	C₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
115	C₃×C₄²	C₂⁴	1	16	yes	—
116	#217	C₂⁴	1	16	yes	—
117	#217	C₂⁴	1	64	yes	—
118	#217	C₂⁴	1	16	yes	—
119	C₄⋊Dic₃	C₂⁴	1	64	yes	—
120	C₄⋊Dic₃	C₂⁴	1	32	yes	—
121	Dic₃⋊C₄	C₂⁴	1	128	yes	—
122	C₄×Dic₃	C₂⁴	1	32	yes	—
123	C₂⁵	C₂²×D₃	1	2	yes	—
124	C₂²×Q₈	—	3	32	no	—
125	C₂²×D₄	Dic₆	1	16	yes	complemented
126	C₂²×D₄	C₂²×D₃	1	8	yes	—
127	C₂²×D₄	C₂²×D₃	1	4	yes	—
128	C₂³×C₄	—	3	16	no	—
129	C₂³×C₄	C₂²×D₃	1	8	yes	—
130	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	3	64	no	—
132	C₄⋊Q₈	—	3	32	no	—
133	#125	—	3	64	no	—
134	#125	—	3	32	no	—
135	C₂²⋊Q₈	—	3	64	no	—
136	C₂²⋊Q₈	—	3	128	no	—
137	C₂²⋊Q₈	—	3	128	no	—
138	C₂²⋊Q₈	—	3	256	no	—
139	C₄×Q₈	—	3	64	no	—
140	C₄×D₄	—	3	128	no	—
141	C₄×D₄	C₂²×D₃	1	64	yes	—
142	#117	—	3	16	no	—
143	#117	—	3	16	no	—
144	#117	C₂²×D₃	1	16	yes	—
145	C₂×C₄⋊C₄	—	3	8	no	—
146	C₂×C₄⋊C₄	—	3	64	no	—
147	C₂×C₄⋊C₄	—	3	32	no	—
148	C₂×C₄⋊C₄	C₂²×D₃	1	4	yes	—
149	C₂×C₂²⋊C₄	—	3	8	no	—
150	C₂×C₂²⋊C₄	—	3	16	no	—
151	C₂×C₂²⋊C₄	—	3	32	no	—
152	C₂×C₂²⋊C₄	—	3	32	no	—
153	C₂×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
155	C₂×C₂²⋊C₄	C₂²×D₃	1	32	yes	—
156	C₂×C₄²	—	3	8	no	—
157	C₂×C₄²	C₂²×D₃	1	4	yes	—
158	C₂³×C₃	—	4	16	no	—
159	C₂³×C₃	—	4	8	no	—
160	C₂³×C₃	—	4	2	no	—
161	C₂³×C₃	—	8	16	no	—
162	C₂³×C₃	C₂²×Q₈	1	4	yes	—
163	C₂³×C₃	C₂²×Q₈	1	16	yes	—
164	C₂³×C₃	C₂²×Q₈	1	32	yes	—
165	C₂³×C₃	C₂⁵	1	1	yes	characteristic
166	C₂³×C₃	C₂⁵	1	2	yes	—
167	C₂³×C₃	C₂⁵	1	8	yes	—
168	C₃×D₄	C₂²×Q₈	1	64	yes	—
169	C₂×C₃×C₄	—	4	32	no	—
170	C₂×C₃×C₄	—	4	8	no	—
171	C₂×C₃×C₄	C₂²×Q₈	1	16	yes	—
172	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
173	C₂×C₃×C₄	C₂⁵	1	16	yes	—
174	C₂×C₃×C₄	C₂⁵	1	4	yes	—
175	C₂×Dic₃	—	4	64	no	—
176	C₂×Dic₃	—	4	16	no	—
177	C₂×Dic₃	C₂⁵	1	32	yes	—
178	Dic₆	—	4	64	no	—
179	C₂⁴	—	8	4	no	—
180	C₂⁴	C₂×Dic₆	1	16	yes	complemented
181	C₂⁴	C₂³×D₃	1	1	yes	characteristic
182	C₂⁴	C₂³×D₃	1	4	yes	—
183	C₂⁴	C₂³×D₃	1	8	yes	—
184	C₂×Q₈	—	3	64	no	—
185	C₂×Q₈	—	24	32	no	—
186	C₂×D₄	C₂×Dic₆	1	64	yes	complemented
187	C₂×D₄	C₂×Dic₆	1	32	yes	complemented
188	C₂×D₄	C₂³×D₃	1	16	yes	—
189	C₂²×C₄	—	3	16	no	—
190	C₂²×C₄	—	3	64	no	—
191	C₂²×C₄	—	8	8	no	—
192	C₂²×C₄	—	24	16	no	—
193	C₂²×C₄	C₂×Dic₆	1	8	yes	complemented
194	C₂²×C₄	C₂³×D₃	1	8	yes	—
195	C₂²×C₄	C₂³×D₃	1	32	yes	—
196	C₂²×C₄	C₂³×D₃	1	4	yes	—
197	C₂²×C₄	C₂³×D₃	1	2	yes	—
198	C₄⋊C₄	—	3	128	no	—
199	C₄⋊C₄	—	3	64	no	—
200	C₄⋊C₄	—	3	32	no	—
201	C₄⋊C₄	C₂³×D₃	1	16	yes	—
202	C₂²⋊C₄	—	3	16	no	—
203	C₂²⋊C₄	—	3	16	no	—
204	C₂²⋊C₄	—	3	64	no	—
205	C₂²⋊C₄	C₂³×D₃	1	16	yes	—
206	C₂²⋊C₄	C₂³×D₃	1	32	yes	—
207	C₄²	—	3	32	no	—
208	C₄²	C₂³×D₃	1	16	yes	—
209	C₂²×C₃	—	2	16	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	4	4	no	—
213	C₂²×C₃	—	4	32	no	—
214	C₂²×C₃	—	4	16	no	—
215	C₂²×C₃	C₂³×Q₈	1	2	yes	—
216	C₂²×C₃	C₂³×Q₈	1	4	yes	—
217	C₂²×C₃	C₂³×Q₈	1	16	yes	—
218	C₂²×C₃	C₂⁶	1	1	yes	characteristic
219	C₃×C₄	—	2	16	no	—
220	C₃×C₄	C₂³×Q₈	1	8	yes	—
221	Dic₃	—	2	32	no	—
222	C₂³	—	4	8	no	—
223	C₂³	—	4	16	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	16	no	—
226	C₂³	C₂²×Dic₆	1	4	yes	—
227	C₂³	C₂²×Dic₆	1	16	yes	—
228	C₂³	C₂²×Dic₆	1	32	yes	—
229	C₂³	C₂⁴×D₃	1	1	yes	characteristic
230	C₂³	C₂⁴×D₃	1	2	yes	—
231	C₂³	C₂⁴×D₃	1	8	yes	—
232	Q₈	—	12	64	no	—
233	D₄	C₂²×Dic₆	1	64	yes	—
234	C₂×C₄	—	3	32	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	32	no	—
237	C₂×C₄	—	12	16	no	—
238	C₂×C₄	—	12	64	no	—
239	C₂×C₄	C₂²×Dic₆	1	16	yes	—
240	C₂×C₄	C₂²×Dic₆	1	8	yes	—
241	C₂×C₄	C₂⁴×D₃	1	16	yes	—
242	C₂×C₄	C₂⁴×D₃	1	4	yes	—
243	C₂×C₃	—	2	4	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	16	no	—
246	C₂×C₃	C₂⁴×Q₈	1	1	yes	characteristic
247	C₂×C₃	H(7,𝔽₂)	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	16	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	16	no	—
253	C₂²	—	4	32	no	—
254	C₂²	C₂³×Dic₆	1	2	yes	—
255	C₂²	C₂³×Dic₆	1	4	yes	—
256	C₂²	C₂³×Dic₆	1	16	yes	—
257	C₂²	C₂⁵×D₃	1	1	yes	characteristic
258	C₄	—	2	16	no	—
259	C₄	—	6	32	no	—
260	C₄	C₂³×Dic₆	1	8	yes	—
261	C₃	#62843	1	1	yes	characteristic, complemented, Sylow
262	C₂	—	2	2	no	—
263	C₂	—	2	4	no	—
264	C₂	—	2	16	no	—
265	C₂	C₂⁴×Dic₆	1	1	yes	characteristic
266	C₂	#87568	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁶
Fitting	C₂×C₃×#518	C₂
prime radical	C₂³×C₃×D₄	C₂²
hypercenter	C₂×#518	D₃
hypocenter	C₃	#62843
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
3. #85860	C₂	semidirect
4. #85766	C₂	semidirect
5. #85725	C₂	semidirect
6. C₂×#5483	C₂	semidirect
18. C₂×#924	C₂²	semidirect
19. #5543	C₂²	semidirect
20. #5539	C₂²	semidirect
21. D₄×Dic₆	C₂²	semidirect
22. #5519	C₂²	semidirect
23. #5517	C₂²	semidirect
24. #5500	C₂²	semidirect
26. #5483	C₂²	semidirect
27. #5483	C₂²	semidirect
28. C₂×#878	C₂²	semidirect
29. #5470	C₂²	semidirect
37. C₂²×C₃×D₄	Q₈	semidirect
43. C₂²×Dic₆	C₂³	semidirect
59. C₂×C₄⋊Dic₃	C₂³	semidirect
61. #924	C₂³	semidirect
62. C₂×Dic₃⋊C₄	C₂³	semidirect
64. #890	C₂³	semidirect
65. #888	C₂³	semidirect
67. #878	C₂³	semidirect
68. #870	C₂³	semidirect
69. #869	C₂³	semidirect
70. C₄×Dic₆	C₂³	semidirect
94. C₂⁴×C₃	C₂×Q₈	semidirect
98. C₂×C₃×D₄	C₂×Q₈	semidirect
99. C₂×C₃×D₄	C₂×Q₈	semidirect
102. C₂²×C₃×C₄	C₂×Q₈	semidirect
125. C₂²×D₄	Dic₆	semidirect
180. C₂⁴	C₂×Dic₆	semidirect
186. C₂×D₄	C₂×Dic₆	semidirect
187. C₂×D₄	C₂×Dic₆	semidirect
193. C₂²×C₄	C₂×Dic₆	semidirect
261. C₃	#62843	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#518	2
C₂×#5798	2
#85860	2
#85766	2
#85725	2
C₂×#5483	2
#62843	3