Subgroups — #895

Order 96.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	#121	—	3	1	no	maximal, Sylow
7	C₂²×D₃	C₂²	1	2	yes	—
8	C₂²×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×Dic₃	C₂²	1	1	yes	characteristic
12	D₁₂	—	2	4	no	—
13	D₁₂	—	2	2	no	—
14	C₄×D₃	—	2	2	no	—
15	C₂×D₄	—	3	2	no	—
16	C₂×D₄	—	3	1	no	—
17	C₂²×C₄	—	3	1	no	—
18	C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
19	C₂²⋊C₄	—	3	2	no	—
20	C₂²×C₃	C₂³	1	1	yes	characteristic
21	C₂×D₃	—	2	1	no	—
22	C₂×D₃	—	2	1	no	—
23	C₂×D₃	—	2	2	no	—
24	C₂×D₃	—	2	2	no	—
25	C₂×D₃	—	2	2	no	—
26	C₂×D₃	D₄	1	2	yes	complemented
27	C₃×C₄	—	2	2	no	—
28	C₃×C₄	D₄	1	2	yes	complemented
29	Dic₃	—	2	1	no	—
30	C₂³	—	3	1	no	—
31	C₂³	—	3	2	no	—
32	D₄	—	6	4	no	—
33	D₄	—	6	2	no	—
34	C₂×C₄	—	3	1	no	—
35	C₂×C₄	—	6	2	no	—
36	C₂×C₄	C₂×D₃	1	2	yes	—
37	C₂×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃	#41	1	1	yes	characteristic
41	D₃	—	2	2	no	—
42	D₃	—	4	2	no	—
43	C₂²	—	3	2	no	—
44	C₂²	—	6	1	no	—
45	C₂²	—	6	1	no	—
46	C₂²	—	6	2	no	—
47	C₂²	—	6	2	no	—
48	C₂²	—	6	2	no	—
49	C₂²	C₂²×D₃	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	6	1	no	—
52	C₄	D₁₂	1	2	yes	complemented
53	C₃	#121	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	6	2	no	—
55	C₂	—	12	2	no	—
56	C₂	C₂×D₁₂	1	1	yes	characteristic
57	C₂	D₃×D₄	1	1	yes	characteristic
58	C₂	#239	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₂	C₂	semidirect
2. C₂×D₁₂	C₂	semidirect
3. C₂×C₄×D₃	C₂	semidirect
4. C₃×C₄⋊C₄	C₂	semidirect
5. (C₂×D₃)⋊C₄	C₂	semidirect
18. C₄⋊C₄	D₃	semidirect
26. C₂×D₃	D₄	semidirect
28. C₃×C₄	D₄	semidirect
52. C₄	D₁₂	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
C₂×D₁₂	2
C₂×C₄×D₃	2
C₃×C₄⋊C₄	2
(C₂×D₃)⋊C₄	2
#121	3