Subgroups — #5688

Order 192.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#896	C₂	1	4	yes	maximal
3	#892	C₂	1	2	yes	maximal
4	#888	C₂	1	2	yes	maximal, complemented
5	#887	C₂	1	4	yes	maximal, complemented
6	C₄²×D₃	C₂	1	1	yes	maximal, characteristic
7	#870	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	—	3	1	no	maximal, Sylow
9	C₂×C₄×D₃	C₂²	1	2	yes	—
10	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₆	C₂²	1	4	yes	—
12	C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₄²	C₂²	1	1	yes	characteristic
15	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
16	C₄⋊Dic₃	C₂²	1	2	yes	—
17	Dic₃⋊C₄	C₂²	1	4	yes	—
18	Dic₃⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₃	C₂²	1	2	yes	—
20	C₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₃	C₂²	1	4	yes	—
22	C₄⋊Q₈	—	3	2	no	—
23	#125	—	3	1	no	—
24	#125	D₃	1	1	yes	characteristic, complemented
25	C₂²⋊Q₈	—	3	4	no	—
26	C₄×Q₈	—	3	4	no	—
27	#117	—	3	2	no	—
28	C₂×C₄²	—	3	1	no	—
29	C₂²×D₃	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	2	yes	—
31	C₂×C₃×C₄	C₂³	1	4	yes	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₃	C₂³	1	4	yes	—
34	C₂×Dic₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₃	C₂³	1	2	yes	—
36	C₄×D₃	—	2	4	no	—
37	C₄×D₃	Q₈	1	4	yes	complemented
38	Dic₆	—	2	8	no	—
39	C₂×Q₈	—	3	4	no	—
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	3	2	no	—
42	C₄⋊C₄	—	3	2	no	—
43	C₄⋊C₄	—	3	4	no	—
44	C₄⋊C₄	—	3	4	no	—
45	C₄⋊C₄	C₂×D₃	1	2	yes	—
46	C₄⋊C₄	C₂×D₃	1	4	yes	—
47	C₂²⋊C₄	—	3	4	no	—
48	C₄²	—	3	2	no	—
49	C₄²	—	3	1	no	—
50	C₄²	—	3	4	no	—
51	C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₃	—	2	2	no	—
54	C₂×D₃	C₂×Q₈	1	2	yes	—
55	C₃×C₄	—	2	4	no	—
56	C₃×C₄	—	2	2	no	—
57	C₃×C₄	C₂×Q₈	1	2	yes	—
58	Dic₃	—	2	4	no	—
59	Dic₃	C₂×Q₈	1	2	yes	—
60	Dic₃	#41	1	4	yes	—
61	C₂³	—	3	1	no	—
62	Q₈	—	6	8	no	—
63	C₂×C₄	—	3	1	no	—
64	C₂×C₄	—	3	2	no	—
65	C₂×C₄	—	3	4	no	—
66	C₂×C₄	—	3	4	no	—
67	C₂×C₄	—	6	4	no	—
68	C₂×C₄	C₂²×D₃	1	2	yes	—
69	C₂×C₄	C₂²×D₃	1	4	yes	—
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₃	C₂×#41	1	2	yes	—
73	D₃	—	2	2	no	—
74	C₂²	—	3	2	no	—
75	C₂²	—	6	2	no	—
76	C₂²	C₂³×D₃	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	3	2	no	—
80	C₄	—	3	4	no	—
81	C₄	—	6	4	no	—
82	C₄	D₃×Q₈	1	2	yes	—
83	C₃	#533	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	6	2	no	—
85	C₂	C₂×D₃×Q₈	1	1	yes	characteristic
86	C₂	D₃×#41	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#125	C₂
prime radical	C₂²×D₃	C₂³
hypercenter	#125	D₃
hypocenter	C₃	#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	semidirect
4. #888	C₂	semidirect
5. #887	C₂	semidirect
7. #870	C₂	semidirect
24. #125	D₃	semidirect
37. C₄×D₃	Q₈	semidirect
83. C₃	#533	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
#896	2
#892	2
#888	2
#887	2
C₄²×D₃	2
#870	2
#533	3