Subgroups — #10693958

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×#4484	C₂	1	4	yes	maximal
4	C₄²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#9637	—	3	1	no	maximal, Sylow
6	C₂×#869	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4935	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₃	—	2	2	no	—
13	#4929	C₄	1	2	yes	—
14	#4928	—	2	2	no	—
15	C₂×C₄×C₃⋊C₈	—	2	2	no	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	#4484	—	2	8	no	—
18	C₄×C₄⋊Q₈	—	3	1	no	—
19	#1842	D₃	1	1	yes	characteristic
20	C₄²×C₈	—	3	1	no	—
21	C₂×Q₈⋊C₈	—	3	4	no	—
22	C₂²×Dic₆	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	2	yes	—
27	C₂×C₃×C₄²	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₃⋊C₄	—	2	2	no	—
33	C₂×C₄×Dic₃	—	2	2	no	—
34	C₂²×C₃⋊C₈	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	C₈	1	4	yes	complemented
37	C₄×Dic₆	—	2	4	no	—
38	C₄×Dic₆	—	2	4	no	—
39	C₃×C₄⋊C₈	—	2	4	no	—
40	C₃×C₄⋊C₈	—	4	4	no	—
41	C₃⋊#95	—	2	4	no	—
42	C₄×C₃⋊C₈	—	2	8	no	—
43	C₄×C₃⋊C₈	—	2	2	no	—
44	C₄×C₃⋊C₈	D₄	1	4	yes	—
45	C₂×C₄⋊Q₈	—	3	1	no	—
46	C₂×C₄×Q₈	—	3	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
49	C₂×C₄×C₈	—	3	2	no	—
50	C₂×C₄×C₈	—	6	2	no	—
51	#391	—	3	2	no	—
52	#389	—	6	2	no	—
53	#384	—	3	1	no	—
54	C₄×C₄⋊C₄	—	6	2	no	—
55	C₄³	C₂×D₃	1	1	yes	characteristic
56	Q₈⋊C₈	—	6	8	no	—
57	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₃	—	2	2	no	—
63	C₂²×Dic₃	—	2	2	no	—
64	C₂×Dic₆	—	2	4	no	—
65	C₂×Dic₆	—	2	4	no	—
66	C₂×Dic₆	—	2	2	no	—
67	C₂×Dic₆	C₂×C₈	1	4	yes	—
68	C₂×C₃×C₈	—	4	2	no	—
69	C₂×C₃×C₈	—	4	4	no	—
70	C₃×C₄²	—	2	4	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	C₂×C₈	1	2	yes	—
73	C₃×C₄²	M₁₆	1	2	yes	—
74	C₃×C₄²	SD₁₆	1	4	yes	—
75	C₃×C₄²	Q₁₆	1	4	yes	—
76	C₃×C₄²	C₂×D₄	1	2	yes	—
77	C₃×C₄²	#41	1	2	yes	—
78	C₄⋊Dic₃	—	2	8	no	—
79	C₄⋊Dic₃	—	2	4	no	—
80	C₄⋊Dic₃	—	4	4	no	—
81	Dic₃⋊C₄	—	4	4	no	—
82	C₄×Dic₃	—	4	2	no	—
83	C₄×Dic₃	—	4	2	no	—
84	C₂×C₃⋊C₈	—	2	8	no	—
85	C₂×C₃⋊C₈	—	2	4	no	—
86	C₂²×Q₈	—	3	2	no	—
87	C₂²×C₈	—	2	2	no	—
88	C₂²×C₈	—	6	2	no	—
89	C₄⋊Q₈	—	3	4	no	—
90	C₄⋊Q₈	—	6	2	no	—
91	C₄×Q₈	—	6	4	no	—
92	C₄×Q₈	—	6	4	no	—
93	C₂×C₄⋊C₄	—	3	2	no	—
94	C₂×C₄⋊C₄	—	6	2	no	—
95	C₂×C₄⋊C₄	—	6	2	no	—
96	C₂×C₄⋊C₄	—	6	2	no	—
97	C₂×C₄²	—	6	2	no	—
98	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
99	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄²	D₁₂	1	2	yes	—
101	C₂×C₄²	#67	1	2	yes	—
102	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	4	4	no	—
105	C₄×C₈	—	3	4	no	—
106	C₄×C₈	—	6	8	no	—
107	C₄×C₈	—	6	2	no	—
108	#95	—	6	4	no	—
109	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	C₂²⋊C₈	1	4	yes	—
114	C₂×C₃×C₄	#103	1	4	yes	—
115	C₂×C₃×C₄	#103	1	4	yes	—
116	C₂×C₃×C₄	#104	1	4	yes	—
117	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
118	C₂×C₃×C₄	#121	1	2	yes	—
119	C₂×C₃×C₄	#124	1	1	yes	characteristic
120	C₂×C₃×C₄	#127	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₃×C₄	#131	1	2	yes	—
124	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
125	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
126	C₂×Dic₃	—	2	4	no	—
127	C₂×Dic₃	—	4	2	no	—
128	C₂×Dic₃	—	4	2	no	—
129	C₂×Dic₃	—	4	2	no	—
130	C₂×Dic₃	—	4	4	no	—
131	Dic₆	—	2	8	no	—
132	Dic₆	—	4	4	no	—
133	C₃×C₈	—	8	2	no	—
134	C₃⋊C₈	—	2	8	no	—
135	C₂×Q₈	—	3	4	no	—
136	C₂×Q₈	—	6	2	no	—
137	C₂×Q₈	—	6	4	no	—
138	C₂×Q₈	—	6	4	no	—
139	C₂²×C₄	—	6	2	no	—
140	C₂²×C₄	—	6	2	no	—
141	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
145	C₂²×C₄	C₂×#67	1	1	yes	characteristic
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	6	8	no	—
149	C₂×C₈	—	6	4	no	—
150	C₄⋊C₄	—	6	8	no	—
151	C₄⋊C₄	—	6	4	no	—
152	C₄⋊C₄	—	12	4	no	—
153	C₄⋊C₄	—	12	4	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	12	2	no	—
157	C₄²	—	12	2	no	—
158	C₄²	D₃×C₈	1	2	yes	—
159	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
160	C₄²	#214	1	4	yes	—
161	C₄²	#216	1	4	yes	—
162	C₄²	D₃×D₄	1	2	yes	—
163	C₄²	#239	1	2	yes	—
164	C₂²×C₃	#390	1	1	yes	characteristic
165	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₂²×C₃	#408	1	1	yes	characteristic
167	C₂²×C₃	C₂×#103	1	2	yes	—
168	C₂²×C₃	C₂×#104	1	2	yes	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	Q₈⋊C₈	1	8	yes	complemented
172	C₃×C₄	C₈×D₄	1	2	yes	—
173	C₃×C₄	#436	1	2	yes	—
174	C₃×C₄	#487	1	2	yes	—
175	C₃×C₄	#492	1	1	yes	characteristic
176	C₃×C₄	#494	1	1	yes	characteristic
177	Dic₃	—	4	4	no	—
178	Dic₃	—	8	2	no	—
179	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	Q₈	—	6	8	no	—
181	Q₈	—	12	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	6	4	no	—
186	C₂×C₄	—	12	2	no	—
187	C₂×C₄	—	12	2	no	—
188	C₂×C₄	—	12	4	no	—
189	C₂×C₄	—	12	2	no	—
190	C₂×C₄	#810	1	4	yes	—
191	C₂×C₄	#810	1	4	yes	—
192	C₂×C₄	(C₂×D₃)⋊C₈	1	4	yes	—
193	C₂×C₄	#825	1	4	yes	—
194	C₂×C₄	#893	1	2	yes	—
195	C₂×C₄	#895	1	2	yes	—
196	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
198	C₂×C₄	#907	1	2	yes	—
199	C₂×C₄	C₂×#214	1	2	yes	—
200	C₂×C₄	#940	1	1	yes	characteristic
201	C₂×C₄	C₂×#216	1	2	yes	—
202	C₂×C₄	#947	1	1	yes	characteristic
203	C₈	—	6	8	no	—
204	C₈	—	8	2	no	—
205	C₂×C₃	C₂×Q₈⋊C₈	1	2	yes	—
206	C₂×C₃	#1493	1	2	yes	—
207	C₂×C₃	#1954	1	1	yes	characteristic
208	C₂×C₃	#1956	1	1	yes	characteristic
209	C₂×C₃	#1962	1	1	yes	characteristic
210	C₂²	C₂×#810	1	2	yes	—
211	C₂²	#4988	1	1	yes	characteristic
212	C₂²	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
213	C₂²	#5129	1	1	yes	characteristic
214	C₂²	C₂×#825	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	2	no	—
217	C₄	—	12	4	no	—
218	C₄	—	24	2	no	—
219	C₄	#4484	1	8	yes	complemented
220	C₄	#4834	1	2	yes	—
221	C₄	#4838	1	2	yes	—
222	C₄	#5076	1	1	yes	characteristic
223	C₄	#5092	1	1	yes	characteristic
224	C₄	#5093	1	2	yes	—
225	C₃	#9637	1	1	yes	characteristic, complemented, Sylow
226	C₂	#69579	1	1	yes	characteristic
227	C₂	C₂×#4484	1	2	yes	—
228	C₂	#70470	1	1	yes	characteristic
229	C₂	#70501	1	2	yes	—
230	C₂	#70512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1842	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1842	D₃
hypocenter	C₃	#9637
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
36. #869	C₈	semidirect
171. C₃×C₄	Q₈⋊C₈	semidirect
219. C₄	#4484	semidirect
225. C₃	#9637	semidirect

Maximal subgroups

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

subgroup	index
C₄×#869	2
C₃×#1842	2
C₂×#4484	2
C₄²×C₃⋊C₈	2
#9637	3