Subgroups — C₃×C₄²×Q₈ #84136

Order 384.

684 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₈	C₂	1	3	yes	maximal
2	C₃×C₄×C₄⋊C₄	C₂	1	9	yes	maximal
3	C₃×C₄³	C₂	1	3	yes	maximal
4	C₄²×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
6	C₃×C₄×Q₈	C₄	1	24	yes	direct product factor
7	C₂×C₃×C₄⋊C₄	C₂²	1	9	yes	—
8	C₂×C₃×C₄²	C₂²	1	9	yes	—
9	C₂×C₃×C₄²	C₂²	1	9	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₃×#95	C₂²	1	6	yes	—
12	C₂×C₄×Q₈	C₂×C₃	1	3	yes	—
13	C₄×C₄⋊C₄	C₂×C₃	1	9	yes	—
14	C₄³	C₂×C₃	1	3	yes	—
15	C₂×C₃×Q₈	C₂×C₄	1	12	yes	—
16	C₂²×C₃×C₄	C₂³	1	9	yes	—
17	C₂²×C₃×C₄	C₂³	1	3	yes	—
18	C₂²×C₃×C₄	C₂³	1	3	yes	—
19	C₃×C₄⋊C₄	C₂×C₄	1	36	yes	—
20	C₃×C₄²	—	2	18	no	—
21	C₃×C₄²	C₂×C₄	1	36	yes	—
22	C₃×C₄²	Q₈	1	4	yes	direct product factor
23	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
24	C₄×Q₈	C₃×C₄	1	24	yes	direct product factor
25	C₂×C₄⋊C₄	C₂²×C₃	1	9	yes	—
26	C₂×C₄²	C₂²×C₃	1	9	yes	—
27	C₂×C₄²	C₂²×C₃	1	9	yes	—
28	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
29	#95	C₂²×C₃	1	6	yes	—
30	C₂³×C₃	C₂⁴	1	1	yes	characteristic
31	C₃×Q₈	C₄²	1	16	yes	direct product factor
32	C₂×C₃×C₄	—	2	18	no	—
33	C₂×C₃×C₄	C₂²×C₄	1	18	yes	—
34	C₂×C₃×C₄	C₂²×C₄	1	18	yes	—
35	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
36	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
37	C₂×C₃×C₄	#41	1	6	yes	—
38	C₂×Q₈	C₂×C₃×C₄	1	12	yes	—
39	C₂²×C₄	C₂³×C₃	1	9	yes	—
40	C₂²×C₄	C₂³×C₃	1	3	yes	—
41	C₂²×C₄	C₂³×C₃	1	3	yes	—
42	C₄⋊C₄	C₂×C₃×C₄	1	36	yes	—
43	C₄²	—	2	18	no	—
44	C₄²	C₂×C₃×C₄	1	36	yes	—
45	C₄²	C₃×Q₈	1	4	yes	direct product factor
46	C₂²×C₃	C₂³×C₄	1	3	yes	—
47	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
48	C₂²×C₃	C₂×#41	1	3	yes	—
49	C₃×C₄	—	2	18	no	—
50	C₃×C₄	C₂×C₄²	1	12	yes	—
51	C₃×C₄	C₄×Q₈	1	12	yes	direct product factor
52	C₂³	C₂⁴×C₃	1	1	yes	characteristic
53	Q₈	C₃×C₄²	1	16	yes	direct product factor
54	C₂×C₄	—	2	18	no	—
55	C₂×C₄	C₂²×C₃×C₄	1	18	yes	—
56	C₂×C₄	C₂²×C₃×C₄	1	18	yes	—
57	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
58	C₂×C₄	C₂×C₃×Q₈	1	6	yes	—
59	C₂×C₄	C₃×#41	1	6	yes	—
60	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
61	C₂×C₃	C₂×C₄×Q₈	1	3	yes	—
62	C₂×C₃	C₄×#41	1	3	yes	—
63	C₂²	C₂³×C₃×C₄	1	3	yes	—
64	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
65	C₂²	C₂×C₃×#41	1	3	yes	—
66	C₄	—	2	18	no	—
67	C₄	C₂×C₃×C₄²	1	12	yes	—
68	C₄	C₃×C₄×Q₈	1	12	yes	direct product factor
69	C₃	C₄²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
71	C₂	C₂×C₃×C₄×Q₈	1	3	yes	—
72	C₂	C₃×C₄×#41	1	3	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	C₂²
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄²×Q₈	C₃	direct (N × H)
6. C₃×C₄×Q₈	C₄	direct (N × H)
22. C₃×C₄²	Q₈	direct (N × H)
24. C₄×Q₈	C₃×C₄	direct (N × H)
31. C₃×Q₈	C₄²	direct (N × H)
45. C₄²	C₃×Q₈	direct (N × H)
51. C₃×C₄	C₄×Q₈	direct (N × H)
53. Q₈	C₃×C₄²	direct (N × H)
68. C₄	C₃×C₄×Q₈	direct (N × H)
69. C₃	C₄²×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
C₂×C₃×C₄×Q₈	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄³	2
C₄²×Q₈	3

Subgroups — C3×C42×Q8 #84136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×Q₈ #84136