Subgroups — #33726

Order 256.

245 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160	C₂	1	2	yes	maximal
2	#2254	C₂	1	2	yes	maximal, complemented
3	#2253	C₂	1	4	yes	maximal, complemented
4	#2252	C₂	1	2	yes	maximal, complemented
5	#2158	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
7	#533	—	2	2	no	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#500	C₂²	1	4	yes	—
10	#499	—	2	2	no	—
11	#499	C₂²	1	4	yes	—
12	#498	—	2	2	no	—
13	#427	—	2	2	no	—
14	#427	C₂²	1	4	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#367	C₂²	1	8	yes	—
17	#366	C₂²	1	8	yes	—
18	C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₁₆	C₂²	1	2	yes	—
20	C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	—	2	4	no	—
24	#125	—	2	4	no	—
25	C₂²⋊Q₈	—	4	4	no	—
26	C₄×Q₈	—	4	4	no	—
27	#117	—	2	4	no	—
28	C₂×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₁₆	Q₈	1	8	yes	complemented
30	C₂×C₁₆	C₂³	1	2	yes	—
31	C₂×C₁₆	C₂³	1	2	yes	—
32	#107	—	2	4	no	—
33	#107	C₂³	1	8	yes	—
34	#106	—	2	4	no	—
35	#106	—	2	4	no	—
36	C₄×C₈	D₄	1	1	yes	characteristic
37	C₄×C₈	D₄	1	1	yes	characteristic
38	C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₈	C₂³	1	1	yes	characteristic
40	C₂×Q₈	—	4	4	no	—
41	C₂²×C₄	D₈	1	2	yes	—
42	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈	Q₁₆	1	4	yes	—
44	C₂×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈	C₂×Q₈	1	4	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄	—	2	8	no	—
50	C₄⋊C₄	—	4	8	no	—
51	C₄⋊C₄	—	4	4	no	—
52	C₂²⋊C₄	—	4	4	no	—
53	C₄²	—	4	4	no	—
54	C₄²	D₈	1	2	yes	—
55	C₄²	C₂×D₄	1	1	yes	characteristic
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₁₆	C₂×Q₈	1	4	yes	—
58	C₁₆	C₂×Q₈	1	4	yes	—
59	C₂³	C₂×D₈	1	1	yes	characteristic
60	Q₈	—	8	8	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	4	8	no	—
63	C₂×C₄	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄	C₂×D₈	1	2	yes	—
65	C₂×C₄	C₂×D₈	1	2	yes	—
66	C₂×C₄	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
69	C₈	C₄⋊Q₈	1	2	yes	—
70	C₈	C₂×Q₁₆	1	2	yes	—
71	C₈	C₂×Q₁₆	1	2	yes	—
72	C₈	C₂²×Q₈	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	#500	1	2	yes	—
75	C₂²	C₂²×D₈	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	8	8	no	—
78	C₄	#500	1	2	yes	—
79	C₄	#508	1	4	yes	—
80	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	C₂×#500	1	1	yes	characteristic
84	C₂	C₂×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂²×D₈
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2254	C₂	semidirect
3. #2253	C₂	semidirect
4. #2252	C₂	semidirect
29. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
#2254	2
#2253	2
#2252	2
#2158	2
C₂×C₄×C₁₆	2