Subgroups — C₄×Q₈² #34170

Order 256.

1162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²	C₂	1	1	yes	maximal, characteristic
2	#2369	C₂	1	9	yes	maximal
3	Q₈×C₄⋊C₄	C₂	1	6	yes	maximal
4	C₄×C₄⋊Q₈	C₂	1	9	yes	maximal
5	C₄²×Q₈	C₂	1	6	yes	maximal
6	Q₈²	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊Q₈	C₂²	1	9	yes	—
8	C₂×C₄×Q₈	C₂²	1	6	yes	—
9	C₂×C₄×Q₈	C₂²	1	6	yes	—
10	C₂×C₄×Q₈	C₂²	1	6	yes	—
11	C₂×C₄×Q₈	C₂²	1	2	yes	—
12	#391	C₂²	1	18	yes	—
13	#389	C₂²	1	36	yes	—
14	#384	C₂²	1	9	yes	—
15	C₄×C₄⋊C₄	C₂²	1	18	yes	—
16	C₄×C₄⋊C₄	C₂²	1	18	yes	—
17	C₄×C₄⋊C₄	C₂²	1	18	yes	—
18	C₄³	C₂²	1	9	yes	—
19	C₂²×Q₈	C₂³	1	6	yes	—
20	C₂²×Q₈	C₂³	1	2	yes	—
21	C₄⋊Q₈	C₂×C₄	1	72	yes	—
22	C₄×Q₈	—	2	24	no	—
23	C₄×Q₈	—	4	12	no	—
24	C₄×Q₈	C₂×C₄	1	48	yes	—
25	C₄×Q₈	Q₈	1	16	yes	direct product factor
26	C₂×C₄⋊C₄	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄	C₂³	1	18	yes	—
28	C₂×C₄⋊C₄	C₂³	1	18	yes	—
29	C₂×C₄⋊C₄	C₂³	1	6	yes	—
30	C₂×C₄²	C₂³	1	18	yes	—
31	C₂×C₄²	C₂³	1	9	yes	—
32	C₂×C₄²	C₂³	1	9	yes	—
33	C₂×C₄²	C₂³	1	9	yes	—
34	C₂×C₄²	C₂³	1	6	yes	—
35	#95	C₂³	1	36	yes	—
36	C₂×Q₈	—	4	6	no	—
37	C₂×Q₈	—	4	6	no	—
38	C₂×Q₈	C₂²×C₄	1	24	yes	—
39	C₂×Q₈	C₂²×C₄	1	8	yes	—
40	C₂×Q₈	C₂×Q₈	1	8	yes	—
41	C₂×Q₈	#41	1	8	yes	—
42	C₂²×C₄	C₂⁴	1	9	yes	—
43	C₂²×C₄	C₂⁴	1	9	yes	—
44	C₂²×C₄	C₂⁴	1	6	yes	—
45	C₂²×C₄	C₂⁴	1	6	yes	—
46	C₂²×C₄	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄	—	2	36	no	—
48	C₄⋊C₄	—	4	18	no	—
49	C₄⋊C₄	C₂²×C₄	1	72	yes	—
50	C₄⋊C₄	C₂×Q₈	1	24	yes	—
51	C₄²	—	2	36	no	—
52	C₄²	—	2	18	no	—
53	C₄²	—	4	9	no	—
54	C₄²	C₂²×C₄	1	36	yes	—
55	C₄²	C₂×Q₈	1	24	yes	—
56	C₂³	C₂⁵	1	1	yes	characteristic
57	Q₈	—	4	24	no	—
58	Q₈	C₄×Q₈	1	32	yes	direct product factor
59	C₂×C₄	—	2	6	no	—
60	C₂×C₄	—	2	9	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	—	2	9	no	—
63	C₂×C₄	—	2	18	no	—
64	C₂×C₄	—	2	9	no	—
65	C₂×C₄	C₂³×C₄	1	18	yes	—
66	C₂×C₄	C₂³×C₄	1	12	yes	—
67	C₂×C₄	C₂²×Q₈	1	12	yes	—
68	C₂×C₄	C₂²×Q₈	1	12	yes	—
69	C₂×C₄	C₂²×Q₈	1	4	yes	—
70	C₂×C₄	C₂×#41	1	12	yes	—
71	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
72	C₂²	C₂⁴×C₄	1	1	yes	characteristic
73	C₂²	C₂³×Q₈	1	2	yes	—
74	C₂²	C₂²×#41	1	2	yes	—
75	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²	#585	1	1	yes	characteristic
77	C₄	—	2	18	no	—
78	C₄	—	2	12	no	—
79	C₄	—	4	9	no	—
80	C₄	C₂×C₄×Q₈	1	24	yes	—
81	C₄	Q₈²	1	4	yes	direct product factor
82	C₂	C₂²×C₄×Q₈	1	2	yes	—
83	C₂	C₄×H(5,𝔽₂)	1	1	yes	characteristic
84	C₂	C₂×Q₈²	1	1	yes	characteristic
85	C₂	Q₈×#41	1	2	yes	—
86	C₂	#3483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₄
Frattini, socle, prime radical, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
6. Q₈²	C₄	direct (N × H)
25. C₄×Q₈	Q₈	direct (N × H)
58. Q₈	C₄×Q₈	direct (N × H)
81. C₄	Q₈²	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×Q₈²	2
#2369	2
Q₈×C₄⋊C₄	2
C₄×C₄⋊Q₈	2
C₄²×Q₈	2

Subgroups — C4×Q82 #34170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈² #34170