Subgroups — #33827

Order 256.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3177	C₂	1	2	yes	maximal
2	#2229	C₂	1	4	yes	maximal, complemented
3	#2219	C₂	1	2	yes	maximal
4	#2216	C₂	1	2	yes	maximal
5	#2209	C₂	1	4	yes	maximal, complemented
6	#2151	C₂	1	1	yes	maximal, characteristic
7	#541	—	2	2	no	—
8	C₂×Q₃₂	C₂²	1	2	yes	—
9	C₂×Q₃₂	C₂²	1	2	yes	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#499	C₂²	1	2	yes	—
13	#494	C₂²	1	2	yes	—
14	#491	—	2	2	no	—
15	#470	—	2	2	no	—
16	#467	C₂²	1	4	yes	—
17	#441	—	2	2	no	—
18	C₄×Q₁₆	C₂²	1	4	yes	—
19	#405	C₂²	1	1	yes	characteristic
20	#367	C₂²	1	2	yes	—
21	#366	C₂²	1	2	yes	—
22	C₄⋊C₁₆	C₂²	1	4	yes	—
23	#358	C₂²	1	4	yes	—
24	#358	C₂²	1	4	yes	—
25	C₂×Q₁₆	—	2	2	no	—
26	C₂×Q₁₆	C₂³	1	4	yes	—
27	C₂×M₁₆	D₄	1	1	yes	characteristic
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈	—	2	2	no	—
30	#126	—	4	4	no	—
31	#125	—	2	2	no	—
32	#125	—	4	4	no	—
33	C₂²⋊Q₈	—	2	4	no	—
34	C₄×Q₈	—	2	4	no	—
35	#117	D₄	1	1	yes	characteristic
36	Q₃₂	—	4	2	no	—
37	Q₃₂	—	4	2	no	—
38	M₃₂	—	4	1	no	—
39	C₂×C₁₆	—	4	1	no	—
40	C₂×C₁₆	C₂³	1	2	yes	—
41	C₂×C₁₆	C₂³	1	2	yes	—
42	#107	C₂³	1	4	yes	—
43	#106	—	2	2	no	—
44	#103	—	2	4	no	—
45	#103	—	2	4	no	—
46	#97	D₄	1	2	yes	—
47	C₄×C₈	C₂³	1	2	yes	—
48	C₂×Q₈	—	2	4	no	—
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	Q₁₆	—	2	4	no	—
51	Q₁₆	—	2	4	no	—
52	Q₁₆	—	4	4	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	C₂×D₄	1	2	yes	—
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄	—	2	4	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	D₈	1	2	yes	—
65	C₂²⋊C₄	—	4	4	no	—
66	C₂²⋊C₄	D₈	1	2	yes	—
67	C₄²	—	4	4	no	—
68	C₄²	C₂×D₄	1	2	yes	—
69	C₁₆	—	2	2	no	—
70	C₁₆	—	2	2	no	—
71	C₂³	C₂×D₈	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂×D₈	1	4	yes	—
79	C₂×C₄	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	2	no	—
83	C₈	H(5,𝔽₂)	1	1	yes	characteristic
84	C₈	2₋¹⁺⁴	1	1	yes	characteristic
85	C₂²	—	2	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	C₂²×D₈	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	1	no	—
90	C₄	—	8	4	no	—
91	C₄	—	8	4	no	—
92	C₄	#537	1	1	yes	characteristic
93	C₄	#575	1	1	yes	characteristic
94	C₂	—	4	1	no	—
95	C₂	#3245	1	1	yes	characteristic
96	C₂	#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ 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. #2229	C₂	semidirect
5. #2209	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3177	2
#2229	2
#2219	2
#2216	2
#2209	2
#2151	2