Subgroups — #885

Order 96.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#217	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#124	—	3	1	no	maximal, Sylow
9	C₂³×C₃	C₂²	1	1	yes	characteristic
10	C₂²×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
13	#67	—	2	2	no	—
14	C₂×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃	C₂²	1	1	yes	characteristic
17	Dic₆	—	2	2	no	—
18	C₂×Q₈	—	3	1	no	—
19	C₂×D₄	—	3	1	no	—
20	C₂²⋊C₄	—	3	1	no	—
21	C₂²⋊C₄	—	3	1	no	—
22	C₂²⋊C₄	—	3	1	no	—
23	C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
24	C₄²	—	3	1	no	—
25	C₂²×C₃	—	2	1	no	—
26	C₂²×C₃	—	2	1	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂²×C₃	C₂³	1	1	yes	characteristic
29	C₂×D₃	—	2	1	no	—
30	C₂×D₃	—	2	1	no	—
31	C₂×D₃	—	2	1	no	—
32	C₃×C₄	—	2	1	no	—
33	C₃×C₄	—	2	1	no	—
34	Dic₃	—	2	1	no	—
35	Dic₃	—	2	1	no	—
36	Dic₃	D₄	1	2	yes	—
37	C₂³	—	3	1	no	—
38	C₂³	C₂×D₃	1	1	yes	characteristic
39	Q₈	—	6	2	no	—
40	D₄	—	6	2	no	—
41	C₂×C₄	—	3	1	no	—
42	C₂×C₄	—	3	1	no	—
43	C₂×C₄	—	3	1	no	—
44	C₂×C₄	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃	—	4	1	no	—
47	C₂×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃	#41	1	1	yes	characteristic
49	C₂×C₃	#41	1	1	yes	characteristic
50	D₃	—	4	1	no	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	6	1	no	—
55	C₂²	—	6	1	no	—
56	C₂²	—	6	1	no	—
57	C₂²	C₂²×D₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	3	2	no	—
61	C₄	—	6	1	no	—
62	C₄	—	6	1	no	—
63	C₃	#124	1	1	yes	characteristic, complemented, Sylow
64	C₂	—	4	1	no	—
65	C₂	—	12	1	no	—
66	C₂	#235	1	1	yes	characteristic
67	C₂	D₃×D₄	1	1	yes	characteristic
68	C₂	#237	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂³
Fitting	C₃×C₂²⋊C₄	C₂
prime radical	C₂×#67	C₂
hypercenter	C₂²⋊C₄	D₃
hypocenter	C₃	#124
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
2. C₂×Dic₆	C₂	semidirect
3. C₃×C₂²⋊C₄	C₂	semidirect
4. #217	C₂	semidirect
5. (C₂×D₃)⋊C₄	C₂	semidirect
6. (C₂×D₃)⋊C₄	C₂	semidirect
7. C₄×Dic₃	C₂	semidirect
23. C₂²⋊C₄	D₃	semidirect
63. C₃	#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67	2
C₂×Dic₆	2
C₃×C₂²⋊C₄	2
#217	2
(C₂×D₃)⋊C₄	2
C₄×Dic₃	2
#124	3