Subgroups — #5734

Order 192.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#947	C₂	1	4	yes	maximal, complemented
3	#895	C₂	1	4	yes	maximal, complemented
4	#892	C₂	1	4	yes	maximal, complemented
5	C₄×D₁₂	C₂	1	2	yes	maximal, complemented
6	#555	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×D₁₂	C₂²	1	2	yes	—
9	C₂×D₁₂	C₂²	1	4	yes	—
10	C₂×C₄×D₃	C₂²	1	4	yes	—
11	C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
15	C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₄×Dic₃	C₂²	1	4	yes	—
17	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
18	#124	—	3	4	no	—
19	#121	—	3	4	no	—
20	C₄×D₄	—	3	2	no	—
21	#117	—	3	4	no	—
22	C₂²×D₃	C₂³	1	4	yes	—
23	C₃×Q₈	—	4	2	no	—
24	C₂×C₃×C₄	C₂³	1	4	yes	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	2	yes	—
27	C₂×Dic₃	C₂³	1	4	yes	—
28	D₁₂	—	2	4	no	—
29	D₁₂	—	4	4	no	—
30	C₄×D₃	—	2	8	no	—
31	C₂×Q₈	C₂×D₃	1	2	yes	—
32	C₂×D₄	—	3	2	no	—
33	C₂×D₄	—	3	4	no	—
34	C₂²×C₄	—	3	4	no	—
35	C₄⋊C₄	—	3	2	no	—
36	C₄⋊C₄	C₂×D₃	1	4	yes	—
37	C₂²⋊C₄	—	3	8	no	—
38	C₂²⋊C₄	—	3	4	no	—
39	C₄²	—	3	4	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₃	—	2	4	no	—
43	C₂×D₃	—	2	4	no	—
44	C₂×D₃	—	2	4	no	—
45	C₃×C₄	—	2	4	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	#41	1	4	yes	—
48	Dic₃	—	2	4	no	—
49	C₂³	—	3	4	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	6	4	no	—
52	D₄	—	12	4	no	—
53	C₂×C₄	—	3	4	no	—
54	C₂×C₄	—	6	8	no	—
55	C₂×C₄	C₂²×D₃	1	4	yes	—
56	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃	1	2	yes	—
58	C₂×C₃	C₂×#41	1	2	yes	—
59	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
60	D₃	—	4	4	no	—
61	C₂²	—	6	4	no	—
62	C₂²	—	6	4	no	—
63	C₂²	—	6	4	no	—
64	C₂²	C₂³×D₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	2	1	no	—
67	C₄	—	6	4	no	—
68	C₄	#239	1	4	yes	—
69	C₃	#555	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	12	4	no	—
71	C₂	#1004	1	1	yes	characteristic
72	C₂	C₂×#239	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄⋊Q₈	C₂	semidirect
2. #947	C₂	semidirect
3. #895	C₂	semidirect
4. #892	C₂	semidirect
5. C₄×D₁₂	C₂	semidirect
17. C₄⋊Q₈	D₃	semidirect
69. C₃	#555	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊Q₈	2
#947	2
#895	2
#892	2
C₄×D₁₂	2
#555	3