Subgroups — #33930

Order 256.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3395	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#2241	C₂	1	3	yes	maximal
4	#2227	C₂	1	3	yes	maximal
5	#2209	C₂	1	3	yes	maximal
6	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₃₂	C₂	1	3	yes	maximal
8	Q₈²	—	2	1	no	—
9	#557	—	2	1	no	—
10	C₂×Q₃₂	C₂²	1	3	yes	—
11	#500	C₂²	1	3	yes	—
12	#494	C₂²	1	3	yes	—
13	#477	—	2	3	no	—
14	#462	—	2	3	no	—
15	C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆	C₂²	1	3	yes	—
17	C₄×Q₁₆	C₂²	1	3	yes	—
18	#366	C₂²	1	3	yes	—
19	C₄⋊C₁₆	C₂²	1	3	yes	—
20	#358	C₂²	1	1	yes	characteristic
21	#358	C₂²	1	3	yes	—
22	#358	C₂²	1	6	yes	—
23	C₄×C₁₆	C₂²	1	3	yes	—
24	C₂×Q₁₆	—	2	3	no	—
25	C₂×Q₁₆	C₂³	1	3	yes	—
26	C₂×Q₁₆	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈	—	2	3	no	—
28	C₄⋊Q₈	—	2	3	no	—
29	C₄⋊Q₈	—	4	3	no	—
30	#125	—	4	3	no	—
31	C₄×Q₈	—	2	3	no	—
32	C₄×Q₈	—	2	3	no	—
33	C₄×Q₈	—	4	1	no	—
34	C₄×Q₈	—	4	1	no	—
35	C₄×Q₈	D₄	1	1	yes	characteristic
36	Q₃₂	—	2	6	no	—
37	C₂×C₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₁₆	C₂³	1	3	yes	—
39	#107	—	2	3	no	—
40	#107	C₂³	1	3	yes	—
41	#107	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	D₄	1	3	yes	—
43	#103	—	2	3	no	—
44	#103	—	2	3	no	—
45	C₄×C₈	C₂³	1	3	yes	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	—	2	3	no	—
48	C₂×Q₈	—	4	3	no	—
49	C₂×Q₈	D₈	1	1	yes	characteristic
50	Q₁₆	—	2	3	no	—
51	Q₁₆	—	2	1	no	—
52	Q₁₆	—	2	3	no	—
53	Q₁₆	—	8	3	no	—
54	Q₁₆	#41	1	2	yes	—
55	C₂×C₈	C₂×D₄	1	3	yes	—
56	C₂×C₈	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	2	3	no	—
59	C₄⋊C₄	—	4	3	no	—
60	C₄⋊C₄	—	4	3	no	—
61	C₄⋊C₄	—	4	3	no	—
62	C₄⋊C₄	—	4	3	no	—
63	C₄⋊C₄	—	4	3	no	—
64	C₄⋊C₄	D₈	1	3	yes	—
65	C₄²	—	4	3	no	—
66	C₄²	—	4	3	no	—
67	C₄²	C₂×D₄	1	3	yes	—
68	C₁₆	—	2	3	no	—
69	C₁₆	—	2	1	no	—
70	Q₈	—	2	2	no	—
71	Q₈	—	4	1	no	—
72	Q₈	—	4	3	no	—
73	Q₈	—	4	3	no	—
74	Q₈	—	16	3	no	—
75	Q₈	Q₃₂	1	4	yes	complemented
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	3	no	—
79	C₂×C₄	—	4	3	no	—
80	C₂×C₄	C₂×D₈	1	3	yes	—
81	C₂×C₄	C₂×D₈	1	3	yes	—
82	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
83	C₈	—	2	3	no	—
84	C₈	C₂×#41	1	1	yes	characteristic
85	C₈	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂²	C₂²×D₈	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	—	4	2	no	—
89	C₄	—	8	3	no	—
90	C₄	—	8	1	no	—
91	C₄	—	8	3	no	—
92	C₄	C₂×Q₃₂	1	6	yes	—
93	C₄	#548	1	1	yes	characteristic
94	C₄	#577	1	1	yes	characteristic
95	C₂	#3358	1	1	yes	characteristic
96	C₂	C₂²×Q₃₂	1	1	yes	characteristic
97	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
75. Q₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#3395	2
Q₈×Q₁₆	2
#2241	2
#2227	2
#2209	2
Q₈×C₁₆	2
C₄×Q₃₂	2