Subgroups — #33475

Order 256.

245 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	8	yes	maximal, complemented
3	#2151	C₂	1	2	yes	maximal
4	#2150	C₂	1	1	yes	maximal, characteristic
5	#2108	C₂	1	2	yes	maximal
6	C₂×#346	C₂	1	1	yes	maximal, characteristic
7	#533	C₄	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	#446	C₄	1	4	yes	—
12	C₈×Q₈	C₂²	1	4	yes	—
13	#424	C₄	1	2	yes	—
14	#423	C₄	1	1	yes	characteristic
15	#405	C₂²	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	C₂²	1	4	yes	—
18	C₄⋊C₁₆	C₂²	1	8	yes	—
19	#346	C₂²	1	4	yes	—
20	#346	C₂²	1	4	yes	—
21	C₄×C₁₆	C₂²	1	4	yes	—
22	C₂×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈	C₈	1	2	yes	—
26	#125	C₈	1	2	yes	—
27	C₂²⋊Q₈	C₈	1	4	yes	—
28	C₄×Q₈	C₂×C₄	1	4	yes	—
29	#117	C₂×C₄	1	2	yes	—
30	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
31	M₃₂	—	2	4	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	Q₈	1	4	yes	complemented
34	C₂×C₁₆	C₂³	1	4	yes	—
35	C₂×C₁₆	C₂³	1	2	yes	—
36	C₂×C₁₆	C₂³	1	2	yes	—
37	C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₄⋊C₈	C₂×C₄	1	8	yes	—
39	#97	C₂×C₄	1	4	yes	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂³	1	4	yes	—
42	C₄×C₈	C₂³	1	2	yes	—
43	C₂×Q₈	C₂×C₈	1	4	yes	—
44	C₂²×C₄	C₂×C₈	1	2	yes	—
45	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
46	M₁₆	—	2	4	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	C₂²×C₄	1	2	yes	—
49	C₂×C₈	C₂²×C₄	1	4	yes	—
50	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	2	yes	—
52	C₂×C₈	#41	1	2	yes	—
53	C₂×C₈	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄	C₂×C₈	1	4	yes	—
55	C₄⋊C₄	C₂×C₈	1	8	yes	—
56	C₄⋊C₄	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄	C₂×C₈	1	4	yes	—
58	C₄²	C₂×C₈	1	1	yes	characteristic
59	C₄²	C₂×C₈	1	1	yes	characteristic
60	C₄²	C₂²×C₄	1	2	yes	—
61	C₄²	C₂²×C₄	1	4	yes	—
62	C₁₆	—	2	2	no	—
63	C₁₆	—	2	4	no	—
64	C₁₆	C₂×Q₈	1	4	yes	—
65	C₂³	C₂²×C₈	1	1	yes	characteristic
66	Q₈	—	4	4	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	C₄×Q₈	1	2	yes	—
69	C₂×C₄	C₂²×C₈	1	2	yes	—
70	C₂×C₄	C₂²×C₈	1	2	yes	—
71	C₂×C₄	C₂²×C₈	1	8	yes	—
72	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄	#131	1	2	yes	—
74	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	C₄×Q₈	1	2	yes	—
78	C₈	C₂²×Q₈	1	1	yes	characteristic
79	C₈	C₂×#41	1	1	yes	characteristic
80	C₂²	—	2	2	no	—
81	C₂²	C₈×Q₈	1	2	yes	—
82	C₂²	C₂³×C₈	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	2	2	no	—
85	C₄	—	2	8	no	—
86	C₄	C₈×Q₈	1	2	yes	—
87	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄	C₂×#131	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
91	C₂	#3407	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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. #2183	C₂	semidirect
33. 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
#2960	2
#2183	2
#2151	2
#2150	2
#2108	2
C₂×#346	2