Subgroups — #5577

Order 192.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#239	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#895	C₂	1	6	yes	maximal, complemented
4	#874	C₂	1	3	yes	maximal, complemented
5	C₄×D₁₂	C₂	1	3	yes	maximal, complemented
6	#550	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	#239	—	2	8	no	—
9	C₂×D₁₂	C₂²	1	3	yes	—
10	C₂×D₁₂	C₂²	1	6	yes	—
11	C₂×D₁₂	C₂²	1	6	yes	—
12	C₂×C₄×D₃	C₂²	1	6	yes	—
13	C₃×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²	C₂²	1	3	yes	—
15	(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
16	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×#41	—	3	2	no	—
18	#127	—	3	3	no	—
19	#121	—	3	6	no	—
20	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
21	C₄×D₄	—	3	3	no	—
22	C₂²×D₃	C₂³	1	6	yes	—
23	C₃×Q₈	D₄	1	4	yes	complemented
24	C₂×C₃×C₄	C₂³	1	3	yes	—
25	C₂×C₃×C₄	C₂³	1	3	yes	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	C₂³	1	2	yes	—
28	D₁₂	—	2	12	no	—
29	D₁₂	—	2	6	no	—
30	D₁₂	—	4	6	no	—
31	C₄×D₃	—	2	12	no	—
32	#41	—	6	8	no	—
33	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
34	C₂×D₄	—	3	3	no	—
35	C₂×D₄	—	3	6	no	—
36	C₂×D₄	—	3	6	no	—
37	C₂²×C₄	—	3	6	no	—
38	C₄⋊C₄	—	3	1	no	—
39	C₄⋊C₄	C₂×D₃	1	3	yes	—
40	C₂²⋊C₄	—	3	6	no	—
41	C₄²	C₂×D₃	1	3	yes	—
42	C₂²×C₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₃	—	2	6	no	—
44	C₂×D₃	—	2	6	no	—
45	C₂×D₃	—	2	6	no	—
46	C₃×C₄	—	2	3	no	—
47	C₃×C₄	C₂×D₄	1	6	yes	—
48	C₃×C₄	#41	1	2	yes	complemented
49	Dic₃	—	2	2	no	—
50	C₂³	—	3	6	no	—
51	Q₈	D₁₂	1	4	yes	complemented
52	D₄	—	6	12	no	—
53	D₄	—	6	6	no	—
54	D₄	—	12	6	no	—
55	C₂×C₄	—	3	2	no	—
56	C₂×C₄	—	6	12	no	—
57	C₂×C₄	C₂²×D₃	1	3	yes	—
58	C₂×C₄	C₂²×D₃	1	3	yes	—
59	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₃	C₂×#41	1	1	yes	characteristic
62	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
63	D₃	—	4	6	no	—
64	C₂²	—	6	6	no	—
65	C₂²	—	6	6	no	—
66	C₂²	—	6	6	no	—
67	C₂²	C₂³×D₃	1	1	yes	characteristic
68	C₄	—	2	3	no	—
69	C₄	—	6	2	no	—
70	C₄	C₂×D₁₂	1	6	yes	—
71	C₄	#239	1	2	yes	complemented
72	C₃	#550	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	12	6	no	—
74	C₂	C₂²×D₁₂	1	1	yes	characteristic
75	C₂	C₂×#239	1	1	yes	characteristic
76	C₂	#1009	1	1	yes	characteristic

	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	#550
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. C₂×#239	C₂	semidirect
2. C₃×C₄×Q₈	C₂	semidirect
3. #895	C₂	semidirect
4. #874	C₂	semidirect
5. C₄×D₁₂	C₂	semidirect
20. C₄×Q₈	D₃	semidirect
23. C₃×Q₈	D₄	semidirect
48. C₃×C₄	#41	semidirect
51. Q₈	D₁₂	semidirect
71. C₄	#239	semidirect
72. C₃	#550	semidirect

Maximal subgroups

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

subgroup	index
C₂×#239	2
C₃×C₄×Q₈	2
#895	2
#874	2
C₄×D₁₂	2
#550	3