Subgroups — C₂×C₃×#124 #5856

Order 192.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#124	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁴×C₃	C₂²	1	2	yes	—
8	C₂×C₃×Q₈	—	2	4	no	—
9	C₂×C₃×Q₈	C₂²	1	4	yes	complemented
10	C₂×C₃×D₄	—	2	4	no	—
11	C₂×C₃×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄	C₂²	1	2	yes	—
13	C₂²×C₃×C₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊C₄	C₂²	1	16	yes	complemented
16	C₃×C₄²	C₂²	1	4	yes	complemented
17	C₂²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×D₄	C₂×C₃	1	1	yes	characteristic
19	#124	C₂×C₃	1	8	yes	direct product factor
20	C₂×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
21	C₂×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂³×C₃	—	2	4	no	—
23	C₂³×C₃	—	2	8	no	—
24	C₂³×C₃	C₂³	1	4	yes	—
25	C₂³×C₃	C₂³	1	1	yes	characteristic
26	C₃×Q₈	—	2	8	no	—
27	C₃×D₄	—	2	8	no	—
28	C₂×C₃×C₄	—	2	4	no	—
29	C₂×C₃×C₄	—	2	4	no	—
30	C₂×C₃×C₄	D₄	1	4	yes	—
31	C₂×C₃×C₄	C₂³	1	4	yes	—
32	C₂×C₃×C₄	C₂³	1	4	yes	—
33	C₂×C₃×C₄	C₂³	1	2	yes	—
34	C₂⁴	C₂²×C₃	1	2	yes	—
35	C₂×Q₈	—	2	4	no	—
36	C₂×Q₈	C₂²×C₃	1	4	yes	complemented
37	C₂×D₄	—	2	4	no	—
38	C₂×D₄	C₂²×C₃	1	4	yes	—
39	C₂²×C₄	C₂²×C₃	1	2	yes	—
40	C₂²×C₄	C₂²×C₃	1	2	yes	—
41	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²⋊C₄	C₂²×C₃	1	16	yes	complemented
43	C₄²	C₂²×C₃	1	4	yes	complemented
44	C₂²×C₃	—	2	4	no	—
45	C₂²×C₃	—	2	8	no	—
46	C₂²×C₃	—	4	8	no	—
47	C₂²×C₃	C₂×D₄	1	2	yes	—
48	C₂²×C₃	#41	1	4	yes	—
49	C₂²×C₃	C₂⁴	1	1	yes	characteristic
50	C₃×C₄	—	2	4	no	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	C₂×D₄	1	4	yes	—
53	C₂³	—	2	4	no	—
54	C₂³	—	2	8	no	—
55	C₂³	C₂³×C₃	1	4	yes	—
56	C₂³	C₂³×C₃	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	D₄	—	2	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	C₃×D₄	1	4	yes	—
62	C₂×C₄	C₂³×C₃	1	4	yes	—
63	C₂×C₄	C₂³×C₃	1	4	yes	—
64	C₂×C₄	C₂³×C₃	1	2	yes	—
65	C₂×C₃	—	4	4	no	—
66	C₂×C₃	#124	1	4	yes	direct product factor
67	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₃	C₂×#41	1	2	yes	—
69	C₂²	—	2	4	no	—
70	C₂²	—	2	8	no	—
71	C₂²	—	4	8	no	—
72	C₂²	C₂×C₃×D₄	1	2	yes	—
73	C₂²	C₃×#41	1	4	yes	—
74	C₂²	C₂⁴×C₃	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	2	4	no	—
77	C₄	C₂×C₃×D₄	1	4	yes	—
78	C₃	C₂×#124	1	1	yes	characteristic, direct product factor, Sylow
79	C₂	—	4	4	no	—
80	C₂	C₃×#124	1	4	yes	direct product factor
81	C₂	C₂²×C₃×D₄	1	1	yes	characteristic
82	C₂	C₂×C₃×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
3. C₃×#124	C₂	direct (N × H)
4. C₂×C₃×C₂²⋊C₄	C₂	semidirect
5. C₂×C₃×C₄²	C₂	semidirect
6. C₂×#124	C₃	direct (N × H)
9. C₂×C₃×Q₈	C₂²	semidirect
15. C₃×C₂²⋊C₄	C₂²	semidirect
16. C₃×C₄²	C₂²	semidirect
17. C₂²×Q₈	C₂×C₃	semidirect
19. #124	C₂×C₃	direct (N × H)
20. C₂×C₂²⋊C₄	C₂×C₃	semidirect
21. C₂×C₄²	C₂×C₃	semidirect
36. C₂×Q₈	C₂²×C₃	semidirect
42. C₂²⋊C₄	C₂²×C₃	semidirect
43. C₄²	C₂²×C₃	semidirect
66. C₂×C₃	#124	direct (N × H)
78. C₃	C₂×#124	direct (N × H)
80. C₂	C₃×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂²×C₃×D₄	2
C₃×#124	2
C₂×C₃×C₂²⋊C₄	2
C₂×C₃×C₄²	2
C₂×#124	3

Subgroups — C2×C3×#124 #5856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#124 #5856