Subgroups — C₃×#543 #5873

Order 192.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
2	C₂×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal
3	#543	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁴×C₃	C₂²	1	1	yes	characteristic
5	C₂×C₃×Q₈	C₂²	1	4	yes	complemented
6	C₂²×C₃×C₄	C₂²	1	6	yes	—
7	C₃×C₄⋊C₄	C₂²	1	12	yes	complemented
8	C₃×C₂²⋊C₄	C₂²	1	12	yes	—
9	C₂²⋊Q₈	C₂×C₃	1	12	yes	complemented
10	C₂×C₂²⋊C₄	C₂×C₃	1	3	yes	—
11	C₂³×C₃	—	4	2	no	—
12	C₂³×C₃	Q₈	1	4	yes	complemented
13	C₂³×C₃	C₂³	1	3	yes	—
14	C₃×Q₈	—	4	4	no	—
15	C₂×C₃×C₄	—	4	6	no	—
16	C₂×C₃×C₄	C₂³	1	12	yes	—
17	C₂⁴	C₂²×C₃	1	1	yes	characteristic
18	C₂×Q₈	C₂²×C₃	1	4	yes	complemented
19	C₂²×C₄	C₂²×C₃	1	6	yes	—
20	C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
21	C₂²⋊C₄	C₂²×C₃	1	12	yes	—
22	C₂²×C₃	—	2	6	no	—
23	C₂²×C₃	—	4	4	no	—
24	C₂²×C₃	C₂×Q₈	1	6	yes	—
25	C₂²×C₃	C₂⁴	1	1	yes	characteristic
26	C₃×C₄	—	2	12	no	—
27	C₂³	—	4	2	no	—
28	C₂³	C₃×Q₈	1	4	yes	complemented
29	C₂³	C₂³×C₃	1	3	yes	—
30	Q₈	—	4	4	no	—
31	C₂×C₄	—	4	6	no	—
32	C₂×C₄	C₂³×C₃	1	12	yes	—
33	C₂×C₃	—	2	6	no	—
34	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
36	C₂²	—	2	6	no	—
37	C₂²	—	4	4	no	—
38	C₂²	C₂×C₃×Q₈	1	6	yes	—
39	C₂²	C₂⁴×C₃	1	1	yes	characteristic
40	C₄	—	2	12	no	—
41	C₃	#543	1	1	yes	characteristic, direct product factor, Sylow
42	C₂	—	2	6	no	—
43	C₂	C₂²×C₃×Q₈	1	1	yes	characteristic
44	C₂	C₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₂²⋊Q₈	C₂	semidirect
3. #543	C₃	direct (N × H)
5. C₂×C₃×Q₈	C₂²	semidirect
7. C₃×C₄⋊C₄	C₂²	semidirect
9. C₂²⋊Q₈	C₂×C₃	semidirect
12. C₂³×C₃	Q₈	semidirect
18. C₂×Q₈	C₂²×C₃	semidirect
20. C₄⋊C₄	C₂²×C₃	semidirect
28. C₂³	C₃×Q₈	semidirect
41. C₃	#543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₂²⋊Q₈	2
C₂×C₃×C₂²⋊C₄	2
#543	3

Subgroups — C3×#543 #5873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#543 #5873