Subgroups — #11786192

Order 1152.

768 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	6	yes	maximal
3	#69439	—	3	4	no	maximal
4	C₂×C₃²×C₄²	C₂²	1	3	yes	—
5	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	—
6	C₃²×#95	C₄	1	4	yes	—
7	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	12	no	—
8	C₃×#376	D₃	1	4	yes	—
9	C₂×C₄⋊(C₃⋊C₈)	—	3	24	no	—
10	C₂²×C₃²×C₄	C₂×C₄	1	6	yes	—
11	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
12	C₃²×C₄²	D₄	1	6	yes	—
13	C₃²×C₄²	Q₈	1	6	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	—	2	12	no	—
15	#1843	—	9	1	no	Sylow
16	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
17	C₂²×C₃⋊C₈	—	3	16	no	—
18	C₃×#95	Dic₃	1	16	yes	—
19	C₄⋊(C₃⋊C₈)	—	6	48	no	—
20	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₄	—	2	12	no	—
22	C₂×C₃²×C₄	C₄⋊C₄	1	12	yes	—
23	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
24	C₂×C₃²×C₄	C₂×Q₈	1	3	yes	—
25	C₃⋊(C₃⋊C₈)	—	4	4	no	—
26	C₂×C₄⋊C₈	—	9	6	no	—
27	#376	C₃⋊D₃	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂×Dic₃	1	24	yes	—
29	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
30	C₃×C₄²	Dic₆	1	24	yes	—
31	C₃×C₄²	D₁₂	1	24	yes	—
32	C₂×C₃⋊C₈	—	6	48	no	—
33	C₂²×C₃²	C₂×C₄⋊C₄	1	3	yes	—
34	C₂²×C₃²	#131	1	4	yes	—
35	C₃²×C₄	—	2	12	no	—
36	C₃²×C₄	#127	1	1	yes	characteristic
37	C₃²×C₄	C₄⋊Q₈	1	3	yes	—
38	C₂²×C₈	—	9	4	no	—
39	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
40	C₄⋊C₈	—	18	12	no	—
41	#95	C₃⋊Dic₃	1	4	yes	—
42	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
43	C₂×C₃×C₄	—	2	48	no	—
44	C₂×C₃×C₄	C₄⋊Dic₃	1	48	yes	—
45	C₂×C₃×C₄	C₂×Dic₆	1	12	yes	—
46	C₂×C₃×C₄	C₂×D₁₂	1	12	yes	—
47	C₃⋊C₈	—	12	16	no	—
48	C₂×C₃²	#384	1	1	yes	characteristic
49	C₂×C₃²	#424	1	6	yes	—
50	C₂²×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
51	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
52	C₂×C₈	—	18	12	no	—
53	C₄²	#634	1	6	yes	—
54	C₄²	C₄⋊(C₃⋊D₃)	1	6	yes	—
55	C₂²×C₃	C₂×C₄⋊Dic₃	1	12	yes	—
56	C₂²×C₃	#948	1	16	yes	—
57	C₃×C₄	—	2	48	no	—
58	C₃×C₄	#869	1	12	yes	—
59	C₃×C₄	#874	1	4	yes	—
60	C₃²	#1843	1	1	yes	characteristic, complemented, Sylow
61	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	12	yes	—
64	C₂×C₄	C₂×#634	1	3	yes	—
65	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
66	C₈	—	36	4	no	—
67	C₂×C₃	#4935	1	4	yes	—
68	C₂×C₃	#4999	1	24	yes	—
69	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	3	yes	—
70	C₂²	#64157	1	4	yes	—
71	C₄	—	2	12	no	—
72	C₄	#64078	1	3	yes	—
73	C₄	#64083	1	1	yes	characteristic
74	C₃	#69439	1	4	yes	complemented
75	C₂	#10592634	1	1	yes	characteristic
76	C₂	#10592698	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₃²×#376	C₂
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#376	C₃⋊D₃
hypocenter	C₃²	#1843
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
60. C₃²	#1843	semidirect
74. C₃	#69439	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#376	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#69439	3