Subgroups — #4470

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#930	C₂	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
4	#327	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂×#67	—	2	1	no	—
7	C₂²×Dic₃	—	2	1	no	—
8	C₂×D₁₂	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×C₈	—	2	1	no	—
10	(C₂×D₃)⋊C₄	—	2	1	no	—
11	C₄⋊Dic₃	C₄	1	1	yes	characteristic, complemented
12	#121	—	3	1	no	—
13	C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
14	#95	—	3	1	no	—
15	C₂³×C₃	D₄	1	1	yes	characteristic
16	C₂²×D₃	—	2	1	no	—
17	C₂×C₃×C₄	—	4	1	no	—
18	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄	D₄	1	1	yes	characteristic
20	#67	—	4	1	no	—
21	#67	—	4	1	no	—
22	C₂×Dic₃	—	2	2	no	—
23	C₂×Dic₃	—	2	1	no	—
24	C₂×Dic₃	—	4	1	no	—
25	C₂×Dic₃	—	4	1	no	—
26	D₁₂	—	4	1	no	—
27	C₃×C₈	—	4	1	no	—
28	C₂×D₄	—	3	1	no	—
29	C₂×D₄	—	6	1	no	—
30	C₂²×C₄	—	6	1	no	—
31	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₄⋊C₄	—	3	1	no	—
34	C₂²⋊C₄	—	6	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	—	2	1	no	—
37	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₃	D₈	1	1	yes	characteristic
39	C₂²×C₃	SD₁₆	1	1	yes	characteristic
40	C₂×D₃	—	4	1	no	—
41	C₂×D₃	—	4	1	no	—
42	C₂×D₃	—	4	1	no	—
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	2	1	no	—
45	Dic₃	—	4	2	no	—
46	Dic₃	—	4	1	no	—
47	C₂³	—	6	1	no	—
48	C₂³	D₁₂	1	1	yes	characteristic
49	D₄	—	12	1	no	—
50	D₄	—	12	1	no	—
51	D₄	—	12	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	—	12	1	no	—
56	C₂×C₄	—	12	1	no	—
57	C₂×C₄	C₄×D₃	1	1	yes	characteristic
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	#99	1	1	yes	characteristic
63	C₂×C₃	#102	1	1	yes	characteristic
64	C₂×C₃	#104	1	1	yes	characteristic
65	D₃	—	8	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	12	1	no	—
69	C₂²	—	12	1	no	—
70	C₂²	—	12	1	no	—
71	C₂²	#204	1	1	yes	characteristic
72	C₂²	D₂₄	1	1	yes	characteristic
73	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	12	2	no	—
77	C₄	—	12	1	no	—
78	C₃	#327	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	24	1	no	—
82	C₂	#806	1	1	yes	characteristic
83	C₂	#821	1	1	yes	characteristic
84	C₂	#825	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₃×C₂²⋊C₈	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	#930	C₂
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#327
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₂²⋊C₈	C₂	semidirect
3. C₃⋊#95	C₂	semidirect
8. C₂×D₁₂	C₄	semidirect
11. C₄⋊Dic₃	C₄	semidirect
13. C₂²⋊C₈	D₃	semidirect
78. C₃	#327	semidirect

Maximal subgroups

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

subgroup	index
#930	2
C₃×C₂²⋊C₈	2
C₃⋊#95	2
#327	3