Finite Group Databasebeta

Subgroups — #11918517

Order 1152.

2234 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 2234 conjugacy classes of subgroups — too many to draw. Only its 750 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10596758	C₂	1	1	yes	maximal, characteristic
2	#10596648	C₂	1	6	yes	maximal, complemented
3	#10596647	C₂	1	6	yes	maximal, complemented
4	#10596640	C₂	1	9	yes	maximal, complemented
5	C₂×#64287	C₂	1	9	yes	maximal, complemented
6	C₃²×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	#64367	C₄	1	6	yes	—
8	#41×C₃⋊D₃	C₂²	1	6	yes	—
9	#64361	C₄	1	9	yes	—
10	C₂×#3830	C₂²	1	9	yes	—
11	C₂×#3792	C₂²	1	18	yes	complemented
12	C₂²×#622	C₂²	1	6	yes	complemented
13	#64289	C₂²	1	2	yes	complemented
14	#64289	C₂²	1	6	yes	complemented
15	#64287	C₂²	1	18	yes	complemented
16	#64287	C₂²	1	36	yes	complemented
17	#64287	C₂²	1	18	yes	complemented
18	#64283	C₂²	1	18	yes	complemented
19	C₂×#3788	C₂²	1	9	yes	complemented
20	C₂×#3787	C₂²	1	9	yes	complemented
21	#5966	—	6	2	no	—
22	C₃²×#41	C₂×C₄	1	6	yes	—
23	C₂×C₃²×D₄	C₂×C₄	1	9	yes	—
24	C₂×#638	C₂×C₄	1	18	yes	—
25	C₂²×C₃⋊Dic₃	C₂³	1	6	yes	—
26	#3832	C₂³	1	2	yes	—
27	Q₈×C₃⋊D₃	C₂×C₄	1	6	yes	—
28	#3830	C₂×C₄	1	36	yes	—
29	#3830	C₂³	1	18	yes	—
30	D₄×C₃⋊D₃	C₂×C₄	1	18	yes	—
31	#3828	C₂×C₄	1	18	yes	—
32	C₂×C₄×C₃⋊D₃	C₂³	1	9	yes	—
33	C₂×#634	C₂×C₄	1	9	yes	—
34	#3792	C₂³	1	36	yes	—
35	C₂×#622	C₂³	1	9	yes	—
36	C₂×#622	C₂³	1	9	yes	—
37	C₂×#622	C₂³	1	18	yes	—
38	#3788	C₂³	1	18	yes	—
39	#3788	C₂³	1	6	yes	—
40	#3787	C₂³	1	6	yes	—
41	#3787	C₂³	1	18	yes	—
42	#3573	—	9	1	no	maximal, Sylow
43	C₃×H(5,𝔽₂)	—	2	2	no	—
44	#1010	—	6	12	no	—
45	D₃×#41	—	6	12	no	—
46	#1004	—	6	18	no	—
47	C₂×#237	—	6	18	no	—
48	C₂³×C₃²	C₂²×C₄	1	6	yes	—
49	C₂²×C₃⋊D₃	C₂²×C₄	1	9	yes	—
50	C₃²×Q₈	C₂²×C₄	1	2	yes	—
51	C₃²×D₄	C₂²×C₄	1	18	yes	—
52	C₂×C₃²×C₄	C₂²×C₄	1	9	yes	—
53	#638	C₂²×C₄	1	36	yes	—
54	C₂×C₃⋊Dic₃	C₂²×C₄	1	18	yes	—
55	C₂×C₃⋊Dic₃	C₂⁴	1	9	yes	—
56	C₄⋊(C₃⋊D₃)	C₂²×C₄	1	6	yes	—
57	C₄×C₃⋊D₃	C₂²×C₄	1	18	yes	—
58	C₄×C₃⋊D₃	C₂⁴	1	6	yes	—
59	#634	C₂²×C₄	1	18	yes	—
60	#622	C₂⁴	1	1	yes	characteristic
61	#622	C₂⁴	1	6	yes	—
62	#622	C₂⁴	1	9	yes	—
63	#585	—	9	1	no	—
64	#568	—	9	9	no	—
65	#568	—	9	6	no	—
66	C₂×#131	—	9	9	no	—
67	C₂×#131	—	9	6	no	—
68	C₃×#41	—	2	12	no	—
69	C₂×C₃×D₄	—	2	18	no	—
70	C₂×#67	—	6	36	no	—
71	C₂²×Dic₃	—	6	12	no	—
72	#239	—	6	4	no	—
73	D₃×Q₈	—	6	12	no	—
74	#237	—	6	36	no	—
75	#237	—	6	72	no	—
76	D₃×D₄	—	6	36	no	—
77	#235	—	6	36	no	—
78	C₂×C₄×D₃	—	6	18	no	—
79	C₂×Dic₆	—	6	18	no	—
80	C₂²×C₃²	—	4	6	no	—
81	C₂²×C₃²	C₂³×C₄	1	9	yes	—
82	C₂×C₃⋊D₃	—	4	9	no	—
83	C₂×C₃⋊D₃	C₂³×C₄	1	6	yes	—
84	C₃²×C₄	C₂³×C₄	1	6	yes	—
85	C₃⋊Dic₃	C₂³×C₄	1	9	yes	—
86	C₃⋊Dic₃	C₂⁵	1	1	yes	characteristic
87	2₋¹⁺⁴	—	9	6	no	—
88	H(5,𝔽₂)	—	9	9	no	—
89	H(5,𝔽₂)	#157	1	1	yes	characteristic
90	C₂×#41	—	9	6	no	—
91	C₂×#41	—	9	9	no	—
92	#131	—	9	18	no	—
93	#131	—	9	36	no	—
94	#131	—	9	18	no	—
95	#131	—	9	2	no	—
96	#131	—	9	6	no	—
97	C₂×M₁₆	—	9	9	no	—
98	C₂×M₁₆	—	9	18	no	—
99	C₂×M₁₆	—	9	18	no	—
100	C₂²×C₈	—	9	9	no	—
101	C₂²×C₈	—	9	6	no	—
102	C₂³×C₃	—	2	12	no	—
103	C₂²×D₃	—	6	18	no	—
104	C₃×Q₈	—	2	4	no	—
105	C₃×D₄	—	2	36	no	—
106	C₂×C₃×C₄	—	2	18	no	—
107	#67	—	6	72	no	—
108	C₂×Dic₃	—	6	36	no	—
109	C₂×Dic₃	—	6	18	no	—
110	D₁₂	—	6	12	no	—
111	C₄×D₃	—	6	36	no	—
112	C₄×D₃	—	6	12	no	—
113	Dic₆	—	6	36	no	—
114	C₂×C₃²	—	2	9	no	—
115	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
116	C₃⋊D₃	—	2	6	no	—
117	#41	—	9	2	no	—
118	#41	—	9	18	no	—
119	#41	—	9	36	no	—
120	#41	—	9	18	no	—
121	#41	C₂×#157	1	6	yes	—
122	C₂×Q₈	—	9	6	no	—
123	C₂×Q₈	—	9	9	no	—
124	C₂×D₄	—	9	18	no	—
125	C₂×D₄	—	9	18	no	—
126	C₂×D₄	C₂×#157	1	9	yes	—
127	C₂²×C₄	—	9	6	no	—
128	C₂²×C₄	—	9	9	no	—
129	M₁₆	—	9	18	no	—
130	M₁₆	—	9	6	no	—
131	M₁₆	—	9	36	no	—
132	C₂×C₈	—	9	6	no	—
133	C₂×C₈	—	9	9	no	—
134	C₂×C₈	—	9	18	no	—
135	C₂×C₈	—	9	9	no	—
136	C₂×C₈	—	9	18	no	—
137	C₂²×C₃	—	2	18	no	—
138	C₂²×C₃	—	8	12	no	—
139	C₂×D₃	—	6	12	no	—
140	C₂×D₃	—	24	18	no	—
141	C₃×C₄	—	2	12	no	—
142	Dic₃	—	6	18	no	—
143	Dic₃	—	6	2	no	—
144	C₃²	#3573	1	1	yes	characteristic, complemented, Sylow
145	C₂³	—	9	9	no	—
146	C₂³	C₂²×#157	1	6	yes	—
147	Q₈	—	9	18	no	—
148	Q₈	C₂²×#157	1	2	yes	—
149	D₄	—	9	36	no	—
150	D₄	—	9	6	no	—
151	D₄	C₂²×#157	1	18	yes	—
152	C₂×C₄	—	9	9	no	—
153	C₂×C₄	—	9	18	no	—
154	C₂×C₄	—	9	6	no	—
155	C₂×C₄	—	9	18	no	—
156	C₂×C₄	C₂²×#157	1	9	yes	—
157	C₈	—	9	1	no	—
158	C₈	—	9	6	no	—
159	C₈	—	9	9	no	—
160	C₂×C₃	—	2	2	no	—
161	C₂×C₃	—	4	18	no	—
162	D₃	—	12	12	no	—
163	C₂²	—	4	6	no	—
164	C₂²	—	9	6	no	—
165	C₂²	—	36	9	no	—
166	C₂²	C₂³×#157	1	9	yes	—
167	C₄	—	9	1	no	—
168	C₄	—	9	9	no	—
169	C₄	C₂³×#157	1	6	yes	—
170	C₃	—	2	2	no	—
171	C₂	—	2	9	no	—
172	C₂	—	18	6	no	—
173	C₂	C₂⁴×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#157
commutator, socle	C₂×C₃²	C₂⁴×C₄
Fitting	C₃²×H(5,𝔽₂)	C₄
prime radical	#10596758	C₂
cosocle	C₃⋊Dic₃	C₂⁵
hypercenter	H(5,𝔽₂)	#157
hypocenter	C₃²	#3573
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #10596648	C₂	semidirect
3. #10596647	C₂	semidirect
4. #10596640	C₂	semidirect
5. C₂×#64287	C₂	semidirect
11. C₂×#3792	C₂²	semidirect
12. C₂²×#622	C₂²	semidirect
13. #64289	C₂²	semidirect
14. #64289	C₂²	semidirect
15. #64287	C₂²	semidirect
16. #64287	C₂²	semidirect
17. #64287	C₂²	semidirect
18. #64283	C₂²	semidirect
19. C₂×#3788	C₂²	semidirect
20. C₂×#3787	C₂²	semidirect
144. C₃²	#3573	semidirect

Maximal subgroups

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

subgroup	index
#10596758	2
#10596648	2
#10596647	2
#10596640	2
C₂×#64287	2
#3573	9