Subgroups — #930

Order 96.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#121	—	3	1	no	maximal, Sylow
7	C₂³×C₃	C₂²	1	1	yes	characteristic
8	C₂²×D₃	C₂²	1	2	yes	—
9	C₂×C₃×C₄	—	2	2	no	—
10	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
12	#67	—	2	4	no	—
13	C₂×Dic₃	C₂²	1	2	yes	—
14	D₁₂	—	2	2	no	—
15	C₂×D₄	—	3	2	no	—
16	C₂×D₄	—	3	1	no	—
17	C₂²×C₄	D₃	1	1	yes	characteristic, complemented
18	C₄⋊C₄	—	3	1	no	—
19	C₂²⋊C₄	—	3	2	no	—
20	C₂²×C₃	—	2	1	no	—
21	C₂²×C₃	—	2	1	no	—
22	C₂²×C₃	D₄	1	2	yes	complemented
23	C₂²×C₃	C₂³	1	1	yes	characteristic
24	C₂×D₃	—	2	2	no	—
25	C₂×D₃	—	2	2	no	—
26	C₂×D₃	—	2	2	no	—
27	C₃×C₄	—	2	1	no	—
28	C₃×C₄	D₄	1	2	yes	complemented
29	Dic₃	—	2	2	no	—
30	C₂³	—	3	2	no	—
31	C₂³	C₂×D₃	1	1	yes	characteristic
32	D₄	—	6	4	no	—
33	D₄	—	6	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	3	2	no	—
36	C₂×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃	—	2	2	no	—
39	C₂×C₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃	#41	1	1	yes	characteristic
42	D₃	—	4	2	no	—
43	C₂²	—	2	1	no	—
44	C₂²	—	2	1	no	—
45	C₂²	—	6	2	no	—
46	C₂²	—	6	2	no	—
47	C₂²	—	6	2	no	—
48	C₂²	D₁₂	1	2	yes	complemented
49	C₂²	C₂²×D₃	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	6	2	no	—
52	C₄	#67	1	2	yes	complemented
53	C₃	#121	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	2	2	no	—
55	C₂	—	12	2	no	—
56	C₂	C₂×D₁₂	1	1	yes	characteristic
57	C₂	#235	1	1	yes	characteristic
58	C₂	C₂×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂³
Fitting	C₂²×C₃×C₄	C₂
hypercenter	C₂²×C₄	D₃
hypocenter	C₃	#121
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄	C₂	semidirect
2. C₂×#67	C₂	semidirect
3. C₂×D₁₂	C₂	semidirect
4. (C₂×D₃)⋊C₄	C₂	semidirect
5. C₄⋊Dic₃	C₂	semidirect
17. C₂²×C₄	D₃	semidirect
22. C₂²×C₃	D₄	semidirect
28. C₃×C₄	D₄	semidirect
48. C₂²	D₁₂	semidirect
52. C₄	#67	semidirect
53. C₃	#121	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄	2
C₂×#67	2
C₂×D₁₂	2
(C₂×D₃)⋊C₄	2
C₄⋊Dic₃	2
#121	3