Subgroups — #5212

Order 192.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#217	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
4	C₃⋊#95	C₂	1	2	yes	maximal, complemented
5	#380	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄	—	2	4	no	—
8	C₂²×C₃×C₄	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃	C₂²	1	4	yes	—
12	#217	—	2	4	no	—
13	C₄⋊Dic₃	—	2	4	no	—
14	C₂³×C₄	D₃	1	1	yes	characteristic
15	C₂×C₄⋊C₄	—	3	2	no	—
16	C₂×C₂²⋊C₄	—	3	2	no	—
17	#95	—	3	2	no	—
18	C₂³×C₃	—	2	2	no	—
19	C₂³×C₃	—	2	2	no	—
20	C₂³×C₃	C₂×C₄	1	2	yes	—
21	C₂³×C₃	D₄	1	2	yes	—
22	C₂³×C₃	Q₈	1	2	yes	—
23	C₂³×C₃	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	—	2	8	no	—
26	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄	D₄	1	4	yes	—
29	C₂×Dic₃	—	2	4	no	—
30	C₂×Dic₃	—	2	4	no	—
31	C₂×Dic₃	—	2	4	no	—
32	C₂⁴	C₂×D₃	1	1	yes	characteristic
33	C₂²×C₄	—	2	4	no	—
34	C₂²×C₄	—	3	4	no	—
35	C₂²×C₄	Dic₃	1	4	yes	complemented
36	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₄⋊C₄	—	6	4	no	—
39	C₂²⋊C₄	—	6	4	no	—
40	C₂²×C₃	—	2	2	no	—
41	C₂²×C₃	—	2	2	no	—
42	C₂²×C₃	—	2	4	no	—
43	C₂²×C₃	—	2	4	no	—
44	C₂²×C₃	C₄⋊C₄	1	4	yes	complemented
45	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₃	C₂×D₄	1	2	yes	—
48	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₃	#41	1	2	yes	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	C₂²⋊C₄	1	4	yes	complemented
52	Dic₃	—	4	4	no	—
53	C₂³	—	2	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	Dic₆	1	2	yes	—
56	C₂³	D₁₂	1	2	yes	—
57	C₂³	C₂×Dic₃	1	2	yes	—
58	C₂³	C₂²×D₃	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	8	no	—
61	C₂×C₄	—	6	4	no	—
62	C₂×C₄	—	6	4	no	—
63	C₂×C₄	—	6	4	no	—
64	C₂×C₄	C₂×Dic₃	1	2	yes	—
65	C₂×C₄	C₂×Dic₃	1	2	yes	—
66	C₂×C₄	#67	1	4	yes	—
67	C₂×C₃	—	2	4	no	—
68	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃	#117	1	1	yes	characteristic
71	C₂×C₃	#121	1	2	yes	—
72	C₂×C₃	C₂²⋊Q₈	1	2	yes	—
73	C₂²	—	2	4	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	2	2	no	—
77	C₂²	C₄⋊Dic₃	1	4	yes	complemented
78	C₂²	C₂×Dic₆	1	1	yes	characteristic
79	C₂²	C₂×D₁₂	1	1	yes	characteristic
80	C₂²	#235	1	2	yes	—
81	C₂²	C₂²×Dic₃	1	1	yes	characteristic
82	C₂²	C₂×#67	1	2	yes	—
83	C₄	—	2	2	no	—
84	C₄	—	12	4	no	—
85	C₄	#217	1	4	yes	complemented
86	C₃	#380	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	2	4	no	—
88	C₂	#924	1	2	yes	—
89	C₂	C₂×C₄⋊Dic₃	1	1	yes	characteristic
90	C₂	#926	1	1	yes	characteristic
91	C₂	#930	1	2	yes	—
92	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄
Fitting	C₂³×C₃×C₄	C₂
socle, cosocle	C₂³×C₃	C₂³
prime radical	C₂⁴×C₃	C₂²
hypercenter	C₂³×C₄	D₃
hypocenter	C₃	#380
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊Dic₃	C₂	semidirect
4. C₃⋊#95	C₂	semidirect
8. C₂²×C₃×C₄	C₄	semidirect
35. C₂²×C₄	Dic₃	semidirect
44. C₂²×C₃	C₄⋊C₄	semidirect
51. C₃×C₄	C₂²⋊C₄	semidirect
77. C₂²	C₄⋊Dic₃	semidirect
85. C₄	#217	semidirect
86. C₃	#380	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂×#217	2
C₂×C₄⋊Dic₃	2
C₃⋊#95	2
#380	3