Subgroups — #4975

Order 192.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
5	#384	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	2	yes	—
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	C₂²	1	2	yes	—
9	C₂²×Dic₃	C₂²	1	2	yes	—
10	C₃×C₄⋊C₄	—	2	2	no	—
11	C₄⋊Dic₃	—	2	2	no	—
12	Dic₃⋊C₄	—	2	8	no	—
13	C₄×Dic₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₄	—	3	4	no	—
15	C₂×C₄⋊C₄	—	3	1	no	—
16	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×C₄²	—	3	1	no	—
18	C₂³×C₃	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₄	—	2	4	no	—
20	C₂×C₃×C₄	—	2	2	no	—
21	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄	D₄	1	2	yes	—
23	C₂×C₃×C₄	Q₈	1	2	yes	—
24	C₂×Dic₃	—	2	4	no	—
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	C₂×C₄	1	4	yes	—
27	C₂×Dic₃	D₄	1	4	yes	—
28	C₂×Dic₃	Q₈	1	4	yes	—
29	C₂²×C₄	—	3	2	no	—
30	C₂²×C₄	—	3	2	no	—
31	C₂²×C₄	C₂×D₃	1	2	yes	—
32	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	6	8	no	—
35	C₄⋊C₄	—	6	2	no	—
36	C₄²	—	3	4	no	—
37	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₃	C₂×D₄	1	2	yes	—
39	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₃	C₂×Q₈	1	2	yes	—
42	C₃×C₄	—	4	2	no	—
43	C₃×C₄	C₄⋊C₄	1	4	yes	complemented
44	Dic₃	—	4	2	no	—
45	Dic₃	C₄⋊C₄	1	8	yes	complemented
46	C₂³	C₂²×D₃	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	3	4	no	—
50	C₂×C₄	—	3	4	no	—
51	C₂×C₄	—	3	4	no	—
52	C₂×C₄	—	6	4	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	Dic₆	1	2	yes	—
55	C₂×C₄	C₄×D₃	1	2	yes	—
56	C₂×C₄	#67	1	2	yes	—
57	C₂×C₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃	#127	1	1	yes	characteristic
60	C₂×C₃	C₄⋊Q₈	1	2	yes	—
61	C₂×C₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²	C₂×Dic₆	1	1	yes	characteristic
63	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
64	C₂²	D₃×D₄	1	2	yes	—
65	C₂²	D₃×Q₈	1	2	yes	—
66	C₂²	C₂×#67	1	1	yes	characteristic
67	C₄	—	3	8	no	—
68	C₄	—	4	2	no	—
69	C₄	—	12	2	no	—
70	C₄	Dic₃⋊C₄	1	4	yes	complemented
71	C₃	#384	1	1	yes	characteristic, complemented, Sylow
72	C₂	#888	1	2	yes	—
73	C₂	D₃×C₄⋊C₄	1	2	yes	—
74	C₂	C₂×Dic₃⋊C₄	1	1	yes	characteristic
75	C₂	#940	1	1	yes	characteristic
76	C₂	#944	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#384
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₃	C₄	semidirect
43. C₃×C₄	C₄⋊C₄	semidirect
45. Dic₃	C₄⋊C₄	semidirect
70. C₄	Dic₃⋊C₄	semidirect
71. C₃	#384	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
C₂×C₄⋊Dic₃	2
C₂×Dic₃⋊C₄	2
C₂×C₄×Dic₃	2
#384	3