Subgroups — C₃×D₄² #5875

Order 192.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	4	yes	maximal, complemented
2	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#121	C₂	1	4	yes	maximal, complemented
4	C₃×#120	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
6	D₄²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁴×C₃	C₂²	1	4	yes	complemented
8	C₂×C₃×D₄	—	2	16	no	—
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂×C₃×D₄	C₂²	1	8	yes	complemented
11	C₂×C₃×D₄	C₂²	1	4	yes	complemented
12	C₂×C₃×D₄	C₂²	1	2	yes	complemented
13	C₂²×C₃×C₄	C₂²	1	4	yes	complemented
14	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
15	C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
16	C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄	C₂×C₃	1	4	yes	complemented
19	#127	C₂×C₃	1	1	yes	characteristic, complemented
20	#121	C₂×C₃	1	4	yes	complemented
21	#120	C₂×C₃	1	4	yes	complemented
22	C₄×D₄	C₂×C₃	1	2	yes	complemented
23	C₂³×C₃	—	2	16	no	—
24	C₂³×C₃	—	4	4	no	—
25	C₂³×C₃	C₂³	1	4	yes	—
26	C₂³×C₃	C₂³	1	4	yes	—
27	C₃×D₄	—	2	16	no	—
28	C₃×D₄	—	2	8	no	—
29	C₃×D₄	—	4	2	no	—
30	C₃×D₄	D₄	1	8	yes	direct product factor
31	C₂×C₃×C₄	—	2	8	no	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂³	1	4	yes	—
34	C₂×C₃×C₄	C₂³	1	2	yes	—
35	C₂⁴	C₂²×C₃	1	4	yes	complemented
36	C₂×D₄	—	2	16	no	—
37	C₂×D₄	C₂²×C₃	1	2	yes	—
38	C₂×D₄	C₂²×C₃	1	8	yes	complemented
39	C₂×D₄	C₂²×C₃	1	4	yes	complemented
40	C₂×D₄	C₂²×C₃	1	2	yes	complemented
41	C₂²×C₄	C₂²×C₃	1	4	yes	complemented
42	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
43	C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
44	C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
45	C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
46	C₂²×C₃	—	2	4	no	—
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	—	2	8	no	—
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	4	16	no	—
51	C₂²×C₃	C₂×D₄	1	8	yes	complemented
52	C₂²×C₃	C₂⁴	1	1	yes	characteristic
53	C₃×C₄	—	2	4	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	C₂×D₄	1	4	yes	complemented
56	C₂³	—	2	16	no	—
57	C₂³	—	4	4	no	—
58	C₂³	C₂³×C₃	1	4	yes	—
59	C₂³	C₂³×C₃	1	4	yes	—
60	D₄	—	2	16	no	—
61	D₄	—	2	8	no	—
62	D₄	—	4	2	no	—
63	D₄	C₃×D₄	1	8	yes	direct product factor
64	C₂×C₄	—	2	8	no	—
65	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
66	C₂×C₄	C₂³×C₃	1	4	yes	—
67	C₂×C₄	C₂³×C₃	1	2	yes	—
68	C₂×C₃	—	2	8	no	—
69	C₂×C₃	—	4	4	no	—
70	C₂×C₃	C₂²×D₄	1	2	yes	—
71	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂²	—	2	4	no	—
73	C₂²	—	2	4	no	—
74	C₂²	—	2	8	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	4	16	no	—
77	C₂²	C₂×C₃×D₄	1	8	yes	complemented
78	C₂²	C₂⁴×C₃	1	1	yes	characteristic
79	C₄	—	2	4	no	—
80	C₄	—	2	1	no	—
81	C₄	C₂×C₃×D₄	1	4	yes	complemented
82	C₃	D₄²	1	1	yes	characteristic, direct product factor, Sylow
83	C₂	—	2	8	no	—
84	C₂	—	4	4	no	—
85	C₂	C₂²×C₃×D₄	1	2	yes	—
86	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
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. C₂²×C₃×D₄	C₂	semidirect
2. C₃×#127	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. C₃×#120	C₂	semidirect
5. C₃×C₄×D₄	C₂	semidirect
6. D₄²	C₃	direct (N × H)
7. C₂⁴×C₃	C₂²	semidirect
10. C₂×C₃×D₄	C₂²	semidirect
11. C₂×C₃×D₄	C₂²	semidirect
12. C₂×C₃×D₄	C₂²	semidirect
13. C₂²×C₃×C₄	C₂²	semidirect
14. C₃×C₄⋊C₄	C₂²	semidirect
15. C₃×C₂²⋊C₄	C₂²	semidirect
16. C₃×C₂²⋊C₄	C₂²	semidirect
17. C₃×C₄²	C₂²	semidirect
18. C₂²×D₄	C₂×C₃	semidirect
19. #127	C₂×C₃	semidirect
20. #121	C₂×C₃	semidirect
21. #120	C₂×C₃	semidirect
22. C₄×D₄	C₂×C₃	semidirect
30. C₃×D₄	D₄	direct (N × H)
35. C₂⁴	C₂²×C₃	semidirect
38. C₂×D₄	C₂²×C₃	semidirect
39. C₂×D₄	C₂²×C₃	semidirect
40. C₂×D₄	C₂²×C₃	semidirect
41. C₂²×C₄	C₂²×C₃	semidirect
42. C₄⋊C₄	C₂²×C₃	semidirect
43. C₂²⋊C₄	C₂²×C₃	semidirect
44. C₂²⋊C₄	C₂²×C₃	semidirect
45. C₄²	C₂²×C₃	semidirect
51. C₂²×C₃	C₂×D₄	semidirect
55. C₃×C₄	C₂×D₄	semidirect
63. D₄	C₃×D₄	direct (N × H)
77. C₂²	C₂×C₃×D₄	semidirect
81. C₄	C₂×C₃×D₄	semidirect
82. C₃	D₄²	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×D₄	2
C₃×#127	2
C₃×#121	2
C₃×#120	2
C₃×C₄×D₄	2
D₄²	3

Subgroups — C3×D42 #5875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄² #5875