Subgroups — #4564

Order 192.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#358	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₈	—	2	1	no	—
6	C₃×Q₁₆	—	2	1	no	—
7	C₃×Q₁₆	C₄	1	2	yes	complemented
8	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃	—	2	1	no	—
10	C₃⋊C₁₆	—	2	1	no	—
11	C₂×Q₁₆	D₃	1	1	yes	characteristic
12	C₂×C₁₆	—	3	1	no	—
13	#107	—	3	1	no	—
14	C₃×Q₈	—	2	2	no	—
15	C₃×Q₈	—	4	1	no	—
16	C₂×C₃×C₄	—	4	1	no	—
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	C₂×Dic₃	—	4	1	no	—
19	C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×Q₈	—	2	1	no	—
22	Q₁₆	—	2	1	no	—
23	Q₁₆	Dic₃	1	2	yes	complemented
24	C₂×C₈	C₂×D₃	1	1	yes	characteristic
25	C₄⋊C₄	—	6	1	no	—
26	C₁₆	—	6	1	no	—
27	C₂²×C₃	D₈	1	1	yes	characteristic
28	C₃×C₄	—	4	2	no	—
29	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₃×C₄	SD₁₆	1	1	yes	characteristic
31	Dic₃	—	8	1	no	—
32	Q₈	—	2	2	no	—
33	Q₈	—	4	1	no	—
34	C₂×C₄	—	4	1	no	—
35	C₂×C₄	—	12	1	no	—
36	C₂×C₄	#67	1	1	yes	characteristic
37	C₈	C₂×Dic₃	1	1	yes	characteristic
38	C₈	#67	1	1	yes	characteristic
39	C₂×C₃	#102	1	1	yes	characteristic
40	C₂×C₃	SD₃₂	1	1	yes	characteristic
41	C₂×C₃	Q₃₂	1	1	yes	characteristic
42	C₂²	#213	1	1	yes	characteristic
43	C₄	—	4	2	no	—
44	C₄	—	24	1	no	—
45	C₄	#214	1	1	yes	characteristic
46	C₄	#217	1	1	yes	characteristic
47	C₃	#358	1	1	yes	characteristic, complemented, Sylow
48	C₂	#828	1	1	yes	characteristic
49	C₂	#829	1	1	yes	characteristic
50	C₂	#832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#213
commutator	C₃×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₃
Fitting	C₂×C₃×Q₁₆	C₂
socle, prime radical	C₂²×C₃	D₈
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂×Q₁₆	D₃
hypocenter	C₃	#358
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
7. C₃×Q₁₆	C₄	semidirect
23. Q₁₆	Dic₃	semidirect
47. C₃	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₁₆	2
#818	2
C₂×C₃⋊C₁₆	2
#358	3