Subgroups — #7015

Order 256.

207 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈	C₄	1	4	yes	—
4	C₂×C₄×C₈	—	2	2	no	—
5	C₂×C₄×C₈	—	2	2	no	—
6	C₂×C₄×C₈	C₄	1	2	yes	—
7	C₄³	C₂²	1	1	yes	characteristic
8	#336	—	2	2	no	—
9	C₂²×C₈	—	2	2	no	—
10	C₂²×C₈	—	2	4	no	—
11	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²	C₂×C₄	1	2	yes	—
13	C₂×C₄²	D₄	1	2	yes	—
14	C₂×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₄²	Q₈	1	1	yes	characteristic
16	C₄⋊C₈	—	2	4	no	—
17	C₄⋊C₈	C₈	1	8	yes	complemented
18	C₄×C₈	—	2	8	no	—
19	C₄×C₈	—	2	2	no	—
20	C₄×C₈	—	4	2	no	—
21	C₄×C₈	—	4	2	no	—
22	C₄×C₈	C₈	1	4	yes	complemented
23	C₂²×C₄	C₄²	1	1	yes	characteristic
24	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈	—	2	8	no	—
29	C₂×C₈	—	2	4	no	—
30	C₂×C₈	—	2	8	no	—
31	C₂×C₈	—	4	4	no	—
32	C₂×C₈	—	4	4	no	—
33	C₄²	—	2	4	no	—
34	C₄²	—	2	2	no	—
35	C₄²	C₂×C₈	1	2	yes	—
36	C₄²	C₂×C₈	1	4	yes	—
37	C₄²	M₁₆	1	2	yes	—
38	C₄²	M₁₆	1	4	yes	—
39	C₄²	D₈	1	1	yes	characteristic
40	C₄²	SD₁₆	1	2	yes	—
41	C₄²	Q₁₆	1	1	yes	characteristic
42	C₂³	#95	1	1	yes	characteristic
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄	C₄×C₈	1	2	yes	—
48	C₂×C₄	#97	1	1	yes	characteristic
49	C₂×C₄	#97	1	2	yes	—
50	C₂×C₄	C₂²⋊C₈	1	2	yes	—
51	C₂×C₄	C₂²⋊C₈	1	4	yes	—
52	C₂×C₄	#102	1	2	yes	—
53	C₂×C₄	#103	1	2	yes	—
54	C₂×C₄	#104	1	4	yes	—
55	C₂×C₄	C₄⋊C₈	1	4	yes	—
56	C₂×C₄	C₄⋊C₈	1	2	yes	—
57	C₂×C₄	#106	1	1	yes	characteristic
58	C₂×C₄	#107	1	1	yes	characteristic
59	C₂×C₄	#108	1	2	yes	—
60	C₈	—	2	8	no	—
61	C₈	—	4	8	no	—
62	C₂²	#336	1	1	yes	characteristic
63	C₂²	#336	1	2	yes	—
64	C₂²	#339	1	2	yes	—
65	C₂²	#340	1	1	yes	characteristic
66	C₂²	#341	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	4	no	—
69	C₄	C₈²	1	1	yes	characteristic
70	C₄	#322	1	1	yes	characteristic
71	C₄	#322	1	2	yes	—
72	C₄	D₄⋊C₈	1	4	yes	—
73	C₄	Q₈⋊C₈	1	4	yes	—
74	C₄	#334	1	2	yes	—
75	C₄	#335	1	2	yes	—
76	C₂	#1273	1	1	yes	characteristic
77	C₂	#1276	1	1	yes	characteristic
78	C₂	#1277	1	1	yes	characteristic
79	C₂	#1278	1	2	yes	—
80	C₂	#1279	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₄
commutator	C₄	C₈²
Frattini, cosocle	C₄³	C₂²
socle, prime radical	C₂³	#95
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
17. C₄⋊C₈	C₈	semidirect
22. C₄×C₈	C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₄⋊C₈	2
C₄²×C₈	2