Subgroups — #5470

Order 192.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	2	yes	maximal
3	#924	C₂	1	4	yes	maximal
4	#870	C₂	1	2	yes	maximal, complemented
5	#869	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₆	C₂	1	4	yes	maximal, complemented
7	#533	—	3	1	no	maximal, Sylow
8	C₂²×C₃×C₄	C₂²	1	2	yes	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₆	C₂²	1	4	yes	—
11	C₃×C₄²	C₂²	1	2	yes	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₃×C₄²	C₂²	1	1	yes	characteristic
14	#217	C₂²	1	4	yes	—
15	C₄⋊Dic₃	C₂²	1	4	yes	—
16	C₄⋊Dic₃	C₂²	1	4	yes	—
17	Dic₃⋊C₄	C₂²	1	8	yes	—
18	C₄×Dic₃	C₂²	1	4	yes	—
19	C₄⋊Q₈	—	3	2	no	—
20	#125	—	3	2	no	—
21	C₂²⋊Q₈	—	3	4	no	—
22	C₄×Q₈	—	3	4	no	—
23	#117	—	3	2	no	—
24	C₂×C₄²	D₃	1	1	yes	characteristic
25	C₂³×C₃	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄	—	2	4	no	—
27	C₂×C₃×C₄	Q₈	1	4	yes	complemented
28	C₂×C₃×C₄	C₂³	1	2	yes	—
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	2	yes	—
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	C₂³	1	8	yes	—
33	Dic₆	—	2	8	no	—
34	C₂×Q₈	—	3	4	no	—
35	C₂²×C₄	C₂×D₃	1	2	yes	—
36	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₄⋊C₄	—	3	4	no	—
38	C₄⋊C₄	—	3	8	no	—
39	C₄⋊C₄	—	3	4	no	—
40	C₂²⋊C₄	—	3	4	no	—
41	C₄²	—	3	4	no	—
42	C₄²	C₂×D₃	1	1	yes	characteristic
43	C₄²	C₂×D₃	1	1	yes	characteristic
44	C₄²	C₂×D₃	1	2	yes	—
45	C₂²×C₃	—	2	2	no	—
46	C₂²×C₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₃	C₂⁴	1	1	yes	characteristic
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	C₂×Q₈	1	2	yes	—
50	C₃×C₄	C₂×Q₈	1	2	yes	—
51	C₃×C₄	#41	1	4	yes	—
52	Dic₃	—	2	8	no	—
53	C₂³	C₂²×D₃	1	1	yes	characteristic
54	Q₈	—	6	8	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	3	8	no	—
57	C₂×C₄	Dic₆	1	4	yes	complemented
58	C₂×C₄	C₂²×D₃	1	2	yes	—
59	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃	1	2	yes	—
61	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
62	C₂×C₃	—	2	2	no	—
63	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₃	C₂×#41	1	2	yes	—
65	C₂²	—	2	2	no	—
66	C₂²	C₂×Dic₆	1	2	yes	—
67	C₂²	C₂³×D₃	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	6	8	no	—
70	C₄	C₂×Dic₆	1	2	yes	—
71	C₄	C₂×Dic₆	1	2	yes	—
72	C₄	#235	1	4	yes	—
73	C₃	#533	1	1	yes	characteristic, complemented, Sylow
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₆	1	1	yes	characteristic
76	C₂	C₂×#235	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂×C₃×C₄²	C₂
prime radical	C₂³×C₃	C₂³
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #870	C₂	semidirect
5. #869	C₂	semidirect
6. C₄×Dic₆	C₂	semidirect
27. C₂×C₃×C₄	Q₈	semidirect
57. C₂×C₄	Dic₆	semidirect
73. C₃	#533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²	2
#926	2
#924	2
#870	2
#869	2
C₄×Dic₆	2
#533	3