Subgroups — #5225

Order 192.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	1	yes	maximal, characteristic
3	C₂×#208	C₂	1	1	yes	maximal, characteristic
4	#835	C₂	1	4	yes	maximal, complemented
5	#419	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₈	—	2	2	no	—
7	C₂×C₃×Q₈	—	2	1	no	—
8	C₂×C₃×Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	—	2	1	no	—
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	#217	—	2	1	no	—
13	C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₄×Dic₃	—	2	1	no	—
15	#208	—	2	2	no	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂²×Q₈	D₃	1	1	yes	characteristic
18	C₂×M₁₆	—	3	1	no	—
19	#117	—	3	1	no	—
20	#103	—	3	4	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₃×Q₈	—	2	4	no	—
23	C₃×Q₈	—	2	2	no	—
24	C₃×Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄	—	2	4	no	—
26	C₂×C₃×C₄	—	2	2	no	—
27	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄	D₄	1	2	yes	—
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂×Dic₃	—	2	2	no	—
32	C₃⋊C₈	—	2	2	no	—
33	C₂×Q₈	—	2	2	no	—
34	C₂×Q₈	—	2	1	no	—
35	C₂×Q₈	Dic₃	1	4	yes	complemented
36	C₂×Q₈	C₂×D₃	1	2	yes	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
39	M₁₆	—	6	2	no	—
40	C₂×C₈	—	3	2	no	—
41	C₄⋊C₄	—	3	2	no	—
42	C₂²⋊C₄	—	6	1	no	—
43	C₄²	—	6	1	no	—
44	C₂²×C₃	—	2	2	no	—
45	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄	—	2	4	no	—
48	C₃×C₄	C₂²⋊C₄	1	2	yes	—
49	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	Dic₃	—	4	2	no	—
52	C₂³	#67	1	1	yes	characteristic
53	Q₈	—	2	4	no	—
54	Q₈	—	2	2	no	—
55	Q₈	C₂×Dic₃	1	4	yes	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	C₂×Dic₃	1	2	yes	—
60	C₂×C₄	#67	1	1	yes	characteristic
61	C₂×C₄	#67	1	2	yes	—
62	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₈	—	6	2	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃	#137	1	2	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	#217	1	2	yes	—
69	C₂²	C₂×#67	1	1	yes	characteristic
70	C₄	—	2	4	no	—
71	C₄	—	12	2	no	—
72	C₄	#217	1	2	yes	—
73	C₄	C₂²×Dic₃	1	1	yes	characteristic
74	C₄	C₂×#67	1	1	yes	characteristic
75	C₃	#419	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	2	no	—
77	C₂	#942	1	2	yes	—
78	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	#419
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
4. #835	C₂	semidirect
8. C₂×C₃×Q₈	C₄	semidirect
35. C₂×Q₈	Dic₃	semidirect
75. C₃	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
#926	2
C₂×#208	2
#835	2
#419	3