Subgroups — #892

Order 96.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×Dic₃	C₂	1	2	yes	maximal, complemented
6	#117	—	3	1	no	maximal, Sylow
7	C₂²×D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₃	C₂²	1	2	yes	—
11	C₂×Dic₃	C₂²	1	1	yes	characteristic
12	C₄×D₃	C₄	1	4	yes	complemented
13	C₂²×C₄	—	3	1	no	—
14	C₄⋊C₄	—	3	1	no	—
15	C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
16	C₂²⋊C₄	—	3	2	no	—
17	C₄²	—	3	2	no	—
18	C₂²×C₃	C₂³	1	1	yes	characteristic
19	C₂×D₃	—	2	1	no	—
20	C₂×D₃	—	2	1	no	—
21	C₂×D₃	C₂×C₄	1	2	yes	—
22	C₃×C₄	—	2	2	no	—
23	C₃×C₄	C₂×C₄	1	2	yes	—
24	Dic₃	—	2	2	no	—
25	Dic₃	C₂×C₄	1	2	yes	—
26	C₂³	—	3	1	no	—
27	C₂×C₄	—	3	1	no	—
28	C₂×C₄	—	3	4	no	—
29	C₂×C₄	—	3	2	no	—
30	C₂×C₄	C₂×D₃	1	2	yes	—
31	C₂×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃	#41	1	1	yes	characteristic
34	C₂×C₃	#41	1	1	yes	characteristic
35	D₃	—	2	2	no	—
36	C₂²	—	3	2	no	—
37	C₂²	—	6	1	no	—
38	C₂²	—	6	1	no	—
39	C₂²	C₂²×D₃	1	1	yes	characteristic
40	C₄	—	2	2	no	—
41	C₄	—	3	2	no	—
42	C₄	—	6	2	no	—
43	C₄	C₄×D₃	1	2	yes	—
44	C₃	#117	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	6	2	no	—
46	C₂	C₂×C₄×D₃	1	1	yes	characteristic
47	C₂	#237	1	1	yes	characteristic
48	C₂	#239	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄
Fitting	C₃×C₄⋊C₄	C₂
socle, cosocle	C₂²×C₃	C₂³
prime radical	C₂²×D₃	C₂²
hypercenter	C₄⋊C₄	D₃
hypocenter	C₃	#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊C₄	C₂	semidirect
4. C₄⋊Dic₃	C₂	semidirect
5. C₄×Dic₃	C₂	semidirect
12. C₄×D₃	C₄	semidirect
15. C₄⋊C₄	D₃	semidirect
44. C₃	#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃	2
C₃×C₄⋊C₄	2
(C₂×D₃)⋊C₄	2
C₄⋊Dic₃	2
C₄×Dic₃	2
#117	3