Subgroups — C₃×#518 #5848

Order 192.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×D₄	C₂	1	8	yes	maximal, complemented
3	C₃×#117	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	#518	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁴×C₃	C₂²	1	2	yes	—
7	C₂×C₃×D₄	C₄	1	8	yes	complemented
8	C₂×C₃×D₄	C₂²	1	4	yes	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄	C₂²	1	8	yes	complemented
11	C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
12	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
14	C₃×C₄²	C₂²	1	4	yes	complemented
15	C₂²×D₄	C₂×C₃	1	1	yes	characteristic
16	C₄×D₄	C₂×C₃	1	8	yes	complemented
17	#117	C₂×C₃	1	2	yes	complemented
18	C₂×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
19	C₂³×C₃	—	4	4	no	—
20	C₂³×C₃	C₂×C₄	1	8	yes	complemented
21	C₂³×C₃	C₂³	1	1	yes	characteristic
22	C₂³×C₃	C₂³	1	4	yes	—
23	C₃×D₄	C₂×C₄	1	16	yes	complemented
24	C₂×C₃×C₄	—	4	8	no	—
25	C₂×C₃×C₄	C₂×C₄	1	4	yes	complemented
26	C₂×C₃×C₄	C₂³	1	8	yes	—
27	C₂×C₃×C₄	C₂³	1	2	yes	—
28	C₂⁴	C₂²×C₃	1	2	yes	—
29	C₂×D₄	C₃×C₄	1	8	yes	complemented
30	C₂×D₄	C₂²×C₃	1	4	yes	—
31	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
32	C₂²×C₄	C₂²×C₃	1	8	yes	complemented
33	C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
34	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
35	C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
36	C₄²	C₂²×C₃	1	4	yes	complemented
37	C₂²×C₃	—	2	8	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	4	8	no	—
40	C₂²×C₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₃	C₂²×C₄	1	8	yes	—
42	C₂²×C₃	C₂⁴	1	1	yes	characteristic
43	C₃×C₄	—	2	8	no	—
44	C₃×C₄	C₂²×C₄	1	4	yes	—
45	C₂³	—	4	4	no	—
46	C₂³	C₂×C₃×C₄	1	8	yes	complemented
47	C₂³	C₂³×C₃	1	1	yes	characteristic
48	C₂³	C₂³×C₃	1	4	yes	—
49	D₄	C₂×C₃×C₄	1	16	yes	complemented
50	C₂×C₄	—	4	8	no	—
51	C₂×C₄	C₂×C₃×C₄	1	4	yes	complemented
52	C₂×C₄	C₂³×C₃	1	8	yes	—
53	C₂×C₄	C₂³×C₃	1	2	yes	—
54	C₂×C₃	—	2	2	no	—
55	C₂×C₃	—	2	8	no	—
56	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
57	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
58	C₂²	—	2	8	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	8	no	—
61	C₂²	C₂²×C₃×C₄	1	2	yes	—
62	C₂²	C₂²×C₃×C₄	1	8	yes	—
63	C₂²	C₂⁴×C₃	1	1	yes	characteristic
64	C₄	—	2	8	no	—
65	C₄	C₂²×C₃×C₄	1	4	yes	—
66	C₃	#518	1	1	yes	characteristic, direct product factor, Sylow
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	C₂³×C₃×C₄	1	1	yes	characteristic
70	C₂	C₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×D₄	C₂	semidirect
3. C₃×#117	C₂	semidirect
4. C₂×C₃×C₂²⋊C₄	C₂	semidirect
5. #518	C₃	direct (N × H)
7. C₂×C₃×D₄	C₄	semidirect
10. C₂²×C₃×C₄	C₂²	semidirect
11. C₃×C₄⋊C₄	C₂²	semidirect
13. C₃×C₂²⋊C₄	C₂²	semidirect
14. C₃×C₄²	C₂²	semidirect
16. C₄×D₄	C₂×C₃	semidirect
17. #117	C₂×C₃	semidirect
18. C₂×C₂²⋊C₄	C₂×C₃	semidirect
20. C₂³×C₃	C₂×C₄	semidirect
23. C₃×D₄	C₂×C₄	semidirect
25. C₂×C₃×C₄	C₂×C₄	semidirect
29. C₂×D₄	C₃×C₄	semidirect
32. C₂²×C₄	C₂²×C₃	semidirect
33. C₄⋊C₄	C₂²×C₃	semidirect
35. C₂²⋊C₄	C₂²×C₃	semidirect
36. C₄²	C₂²×C₃	semidirect
46. C₂³	C₂×C₃×C₄	semidirect
49. D₄	C₂×C₃×C₄	semidirect
51. C₂×C₄	C₂×C₃×C₄	semidirect
66. C₃	#518	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×C₄×D₄	2
C₃×#117	2
C₂×C₃×C₂²⋊C₄	2
#518	3

Subgroups — C3×#518 #5848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#518 #5848