Subgroups — #893

Order 96.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃	C₂	1	2	yes	maximal, complemented
3	C₃×C₄⋊C₄	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	C₄×D₄	—	3	1	no	maximal, Sylow
7	C₂²×D₃	C₂²	1	2	yes	—
8	C₂×C₃×C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃	C₂²	1	1	yes	characteristic
12	D₁₂	C₄	1	4	yes	complemented
13	C₄×D₃	—	2	4	no	—
14	C₂×D₄	—	3	1	no	—
15	C₂²×C₄	—	3	2	no	—
16	C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
17	C₂²⋊C₄	—	3	2	no	—
18	C₄²	—	3	1	no	—
19	C₂²×C₃	C₂³	1	1	yes	characteristic
20	C₂×D₃	—	2	2	no	—
21	C₂×D₃	—	2	2	no	—
22	C₂×D₃	C₂×C₄	1	4	yes	complemented
23	C₃×C₄	—	2	2	no	—
24	C₃×C₄	C₂×C₄	1	2	yes	complemented
25	Dic₃	—	2	1	no	—
26	Dic₃	D₄	1	2	yes	complemented
27	C₂³	—	3	2	no	—
28	D₄	—	3	4	no	—
29	C₂×C₄	—	3	1	no	—
30	C₂×C₄	—	3	1	no	—
31	C₂×C₄	—	6	4	no	—
32	C₂×C₄	C₂×D₃	1	2	yes	—
33	C₂×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃	#41	1	1	yes	characteristic
37	D₃	—	2	4	no	—
38	C₂²	—	3	4	no	—
39	C₂²	—	6	2	no	—
40	C₂²	—	6	2	no	—
41	C₂²	C₂²×D₃	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	3	2	no	—
44	C₄	—	6	1	no	—
45	C₄	C₄×D₃	1	2	yes	complemented
46	C₃	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	6	4	no	—
48	C₂	C₂×C₄×D₃	1	1	yes	characteristic
49	C₂	D₃×D₄	1	1	yes	characteristic
50	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	C₂×D₁₂	C₂
hypercenter	C₄⋊C₄	D₃
hypocenter	C₃	C₄×D₄
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. C₂×C₄×D₃	C₂	semidirect
3. C₃×C₄⋊C₄	C₂	semidirect
4. (C₂×D₃)⋊C₄	C₂	semidirect
5. C₄×Dic₃	C₂	semidirect
12. D₁₂	C₄	semidirect
16. C₄⋊C₄	D₃	semidirect
22. C₂×D₃	C₂×C₄	semidirect
24. C₃×C₄	C₂×C₄	semidirect
26. Dic₃	D₄	semidirect
45. C₄	C₄×D₃	semidirect
46. C₃	C₄×D₄	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
C₄×Dic₃	2
C₄×D₄	3