Subgroups — #882

Order 96.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₂	C₂	1	2	yes	maximal, complemented
4	C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	#120	—	3	1	no	maximal, Sylow
7	C₂³×C₃	C₂²	1	1	yes	characteristic
8	C₂²×D₃	—	2	2	no	—
9	C₂²×D₃	—	2	4	no	—
10	C₂²×D₃	C₂²	1	2	yes	—
11	C₂²×D₃	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₄	C₂²	1	2	yes	complemented
13	#67	—	2	2	no	—
14	C₂×Dic₃	C₂²	1	1	yes	characteristic, complemented
15	D₁₂	—	2	4	no	—
16	C₂⁴	—	3	1	no	—
17	C₂×D₄	—	3	1	no	—
18	C₂×D₄	—	3	2	no	—
19	C₂²⋊C₄	—	3	2	no	—
20	C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
21	C₂²×C₃	—	2	2	no	—
22	C₂²×C₃	D₄	1	2	yes	complemented
23	C₂²×C₃	C₂³	1	1	yes	characteristic
24	C₂×D₃	—	2	8	no	—
25	C₂×D₃	—	2	2	no	—
26	C₂×D₃	—	2	2	no	—
27	C₂×D₃	—	2	2	no	—
28	C₂×D₃	—	2	1	no	—
29	C₂×D₃	D₄	1	4	yes	complemented
30	C₃×C₄	—	2	2	no	—
31	Dic₃	—	2	1	no	—
32	C₂³	—	3	2	no	—
33	C₂³	—	3	1	no	—
34	C₂³	—	6	4	no	—
35	C₂³	—	6	2	no	—
36	C₂³	C₂×D₃	1	1	yes	characteristic
37	D₄	—	6	4	no	—
38	D₄	—	6	2	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	C₂×D₃	1	2	yes	complemented
41	C₂×C₃	—	2	2	no	—
42	C₂×C₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃	C₂×D₄	1	2	yes	—
44	D₃	—	2	4	no	—
45	D₃	—	4	1	no	—
46	C₂²	—	2	2	no	—
47	C₂²	—	3	4	no	—
48	C₂²	—	6	8	no	—
49	C₂²	—	6	2	no	—
50	C₂²	—	6	2	no	—
51	C₂²	—	6	1	no	—
52	C₂²	—	6	2	no	—
53	C₂²	D₁₂	1	2	yes	complemented
54	C₂²	C₂²×D₃	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	6	1	no	—
57	C₃	#120	1	1	yes	characteristic, complemented, Sylow
58	C₂	—	2	2	no	—
59	C₂	—	6	4	no	—
60	C₂	—	12	1	no	—
61	C₂	C₂×D₁₂	1	1	yes	characteristic
62	C₂	D₃×D₄	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂³
Fitting	C₃×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	D₃
hypocenter	C₃	#120
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. C₂³×D₃	C₂	semidirect
2. C₂×#67	C₂	semidirect
3. C₂×D₁₂	C₂	semidirect
4. C₃×C₂²⋊C₄	C₂	semidirect
5. (C₂×D₃)⋊C₄	C₂	semidirect
11. C₂²×D₃	C₂²	semidirect
12. C₂×C₃×C₄	C₂²	semidirect
14. C₂×Dic₃	C₂²	semidirect
20. C₂²⋊C₄	D₃	semidirect
22. C₂²×C₃	D₄	semidirect
29. C₂×D₃	D₄	semidirect
40. C₂×C₄	C₂×D₃	semidirect
53. C₂²	D₁₂	semidirect
57. C₃	#120	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₃	2
C₂×#67	2
C₂×D₁₂	2
C₃×C₂²⋊C₄	2
(C₂×D₃)⋊C₄	2
#120	3