Subgroups — C₂×C₃×C₄×Q₈ #5846

Order 192.

298 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
2	C₃×C₄×Q₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂×C₃×C₄²	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×Q₈	C₄	1	8	yes	direct product factor
7	C₂×C₃×Q₈	C₂²	1	4	yes	—
8	C₂²×C₃×C₄	C₂²	1	3	yes	—
9	C₂²×C₃×C₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊C₄	C₂²	1	12	yes	—
12	C₃×C₄²	C₂²	1	12	yes	—
13	C₂²×Q₈	C₂×C₃	1	1	yes	characteristic
14	C₄×Q₈	C₂×C₃	1	8	yes	direct product factor
15	C₂×C₄⋊C₄	C₂×C₃	1	3	yes	—
16	C₂×C₄²	C₂×C₃	1	3	yes	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₃×Q₈	C₂×C₄	1	16	yes	direct product factor
19	C₂×C₃×C₄	—	2	6	no	—
20	C₂×C₃×C₄	C₂×C₄	1	12	yes	—
21	C₂×C₃×C₄	Q₈	1	4	yes	direct product factor
22	C₂×C₃×C₄	C₂³	1	6	yes	—
23	C₂×C₃×C₄	C₂³	1	6	yes	—
24	C₂×C₃×C₄	C₂³	1	2	yes	—
25	C₂×Q₈	C₃×C₄	1	8	yes	direct product factor
26	C₂×Q₈	C₂²×C₃	1	4	yes	—
27	C₂²×C₄	C₂²×C₃	1	3	yes	—
28	C₂²×C₄	C₂²×C₃	1	3	yes	—
29	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₄⋊C₄	C₂²×C₃	1	12	yes	—
31	C₄²	C₂²×C₃	1	12	yes	—
32	C₂²×C₃	C₂²×C₄	1	2	yes	—
33	C₂²×C₃	C₂×Q₈	1	2	yes	—
34	C₂²×C₃	#41	1	2	yes	—
35	C₂²×C₃	C₂⁴	1	1	yes	characteristic
36	C₃×C₄	—	2	6	no	—
37	C₃×C₄	C₂²×C₄	1	12	yes	—
38	C₃×C₄	C₂×Q₈	1	4	yes	direct product factor
39	C₂³	C₂³×C₃	1	1	yes	characteristic
40	Q₈	C₂×C₃×C₄	1	16	yes	direct product factor
41	C₂×C₄	—	2	6	no	—
42	C₂×C₄	C₂×C₃×C₄	1	12	yes	—
43	C₂×C₄	C₃×Q₈	1	4	yes	direct product factor
44	C₂×C₄	C₂³×C₃	1	6	yes	—
45	C₂×C₄	C₂³×C₃	1	6	yes	—
46	C₂×C₄	C₂³×C₃	1	2	yes	—
47	C₂×C₃	C₄×Q₈	1	4	yes	direct product factor
48	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₃	C₂×#41	1	1	yes	characteristic
51	C₂²	C₂²×C₃×C₄	1	2	yes	—
52	C₂²	C₂×C₃×Q₈	1	2	yes	—
53	C₂²	C₃×#41	1	2	yes	—
54	C₂²	C₂⁴×C₃	1	1	yes	characteristic
55	C₄	—	2	6	no	—
56	C₄	C₂²×C₃×C₄	1	12	yes	—
57	C₄	C₂×C₃×Q₈	1	4	yes	direct product factor
58	C₃	C₂×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
59	C₂	C₃×C₄×Q₈	1	4	yes	direct product factor
60	C₂	C₂³×C₃×C₄	1	1	yes	characteristic
61	C₂	C₂²×C₃×Q₈	1	1	yes	characteristic
62	C₂	C₂×C₃×#41	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₈	C₂	direct (N × H)
5. C₂×C₄×Q₈	C₃	direct (N × H)
6. C₂×C₃×Q₈	C₄	direct (N × H)
14. C₄×Q₈	C₂×C₃	direct (N × H)
18. C₃×Q₈	C₂×C₄	direct (N × H)
21. C₂×C₃×C₄	Q₈	direct (N × H)
25. C₂×Q₈	C₃×C₄	direct (N × H)
38. C₃×C₄	C₂×Q₈	direct (N × H)
40. Q₈	C₂×C₃×C₄	direct (N × H)
43. C₂×C₄	C₃×Q₈	direct (N × H)
47. C₂×C₃	C₄×Q₈	direct (N × H)
57. C₄	C₂×C₃×Q₈	direct (N × H)
58. C₃	C₂×C₄×Q₈	direct (N × H)
59. C₂	C₃×C₄×Q₈	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
C₃×C₄×Q₈	2
C₂×C₃×C₄⋊C₄	2
C₂×C₃×C₄²	2
C₂×C₄×Q₈	3

Subgroups — C2×C3×C4×Q8 #5846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×Q₈ #5846