Subgroups — #916

Order 96.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#239	C₂	1	1	yes	maximal, characteristic, complemented
2	#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#216	C₂	1	1	yes	maximal, characteristic, complemented
5	#213	C₂	1	1	yes	maximal, characteristic, complemented
6	#204	C₂	1	1	yes	maximal, characteristic, complemented
7	D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#135	—	3	1	no	maximal, Sylow
9	C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₃×D₄	C₂²	1	1	yes	characteristic
11	#67	—	2	1	no	—
12	C₂×Dic₃	—	2	1	no	—
13	D₁₂	—	2	1	no	—
14	D₁₂	C₂²	1	1	yes	characteristic
15	C₄×D₃	—	2	1	no	—
16	C₄×D₃	C₂²	1	1	yes	characteristic
17	Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₈	C₂²	1	1	yes	characteristic
19	C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#41	—	3	1	no	—
21	#41	—	3	1	no	—
22	Q₁₆	—	3	1	no	—
23	SD₁₆	—	3	1	no	—
24	SD₁₆	D₃	1	1	yes	characteristic, complemented
25	D₈	—	3	1	no	—
26	C₂×C₈	—	3	1	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂×D₃	—	2	1	no	—
29	C₂×D₃	D₄	1	1	yes	characteristic
30	C₃×C₄	—	2	1	no	—
31	C₃×C₄	C₂³	1	1	yes	characteristic
32	Dic₃	—	2	1	no	—
33	Dic₃	D₄	1	1	yes	characteristic
34	Q₈	—	3	1	no	—
35	Q₈	C₂×D₃	1	1	yes	characteristic
36	D₄	—	3	1	no	—
37	D₄	—	6	1	no	—
38	D₄	—	6	1	no	—
39	D₄	C₂×D₃	1	1	yes	characteristic
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	—	6	1	no	—
42	C₂×C₄	—	6	1	no	—
43	C₈	—	3	1	no	—
44	C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₃	—	4	1	no	—
46	C₂×C₃	C₂×D₄	1	1	yes	characteristic
47	D₃	—	2	1	no	—
48	D₃	—	4	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	—	3	1	no	—
51	C₂²	—	6	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	3	1	no	—
54	C₄	—	6	1	no	—
55	C₄	C₂²×D₃	1	1	yes	characteristic
56	C₃	#135	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	4	1	no	—
58	C₂	—	6	1	no	—
59	C₂	—	12	1	no	—
60	C₂	D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₄
commutator, cosocle	C₃×C₄	C₂³
Frattini	C₄	C₂²×D₃
Fitting	C₃×SD₁₆	C₂
socle	C₂×C₃	C₂×D₄
hypercenter	SD₁₆	D₃
hypocenter	C₃	#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #239	C₂	semidirect
2. #237	C₂	semidirect
3. C₃×SD₁₆	C₂	semidirect
4. #216	C₂	semidirect
5. #213	C₂	semidirect
6. #204	C₂	semidirect
7. D₃×C₈	C₂	semidirect
24. SD₁₆	D₃	semidirect
56. C₃	#135	semidirect

Maximal subgroups

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

subgroup	index
#239	2
#237	2
C₃×SD₁₆	2
#216	2
#213	2
#204	2
D₃×C₈	2
#135	3