Subgroups — #4485

Order 192.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#874	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#331	—	3	1	no	maximal, Sylow
5	C₂×D₁₂	—	2	1	no	—
6	C₂×D₁₂	—	2	1	no	—
7	C₂×D₁₂	C₄	1	2	yes	—
8	C₂×C₃×C₈	—	2	1	no	—
9	C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊C₈	—	2	1	no	—
11	#127	—	3	1	no	—
12	C₄⋊C₈	—	3	1	no	—
13	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
14	C₂²×D₃	—	2	2	no	—
15	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₄	D₄	1	1	yes	characteristic
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	D₁₂	—	4	2	no	—
19	D₁₂	—	4	1	no	—
20	D₁₂	—	4	1	no	—
21	D₁₂	—	4	1	no	—
22	D₁₂	—	4	1	no	—
23	C₃×C₈	—	4	1	no	—
24	C₃⋊C₈	—	4	1	no	—
25	C₂×D₄	—	3	2	no	—
26	C₂×D₄	—	6	1	no	—
27	C₂×D₄	—	6	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	6	1	no	—
30	C₄²	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₃	—	4	2	no	—
33	C₂×D₃	—	4	2	no	—
34	C₂×D₃	—	4	2	no	—
35	C₃×C₄	—	2	1	no	—
36	C₃×C₄	D₈	1	1	yes	characteristic
37	C₃×C₄	D₈	1	1	yes	characteristic
38	C₃×C₄	SD₁₆	1	1	yes	characteristic
39	C₃×C₄	SD₁₆	1	1	yes	characteristic
40	C₂³	—	6	2	no	—
41	D₄	—	12	2	no	—
42	D₄	—	12	1	no	—
43	D₄	—	12	1	no	—
44	D₄	—	12	1	no	—
45	D₄	—	12	1	no	—
46	C₂×C₄	C₄×D₃	1	1	yes	characteristic
47	C₂×C₄	D₁₂	1	1	yes	characteristic
48	C₂×C₄	#67	1	1	yes	characteristic
49	C₈	—	4	1	no	—
50	C₈	—	12	1	no	—
51	C₂×C₃	#100	1	1	yes	characteristic
52	C₂×C₃	#102	1	1	yes	characteristic
53	C₂×C₃	#102	1	1	yes	characteristic
54	D₃	—	8	2	no	—
55	C₂²	—	12	2	no	—
56	C₂²	—	12	2	no	—
57	C₂²	—	12	2	no	—
58	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	#204	1	1	yes	characteristic
61	C₄	D₂₄	1	1	yes	characteristic
62	C₄	#213	1	1	yes	characteristic
63	C₄	#215	1	1	yes	characteristic
64	C₃	#331	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	24	2	no	—
66	C₂	#809	1	1	yes	characteristic
67	C₂	#821	1	1	yes	characteristic
68	C₂	#823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₄²	C₂×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	#874	C₂
cosocle	C₃×C₄²	C₂²
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#331
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊C₈	C₂	semidirect
3. C₄⋊(C₃⋊C₈)	C₂	semidirect
13. C₄⋊C₈	D₃	semidirect
64. C₃	#331	semidirect

Maximal subgroups

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

subgroup	index
#874	2
C₃×C₄⋊C₈	2
C₄⋊(C₃⋊C₈)	2
#331	3