Subgroups — #33474

Order 256.

253 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	4	yes	maximal, complemented
3	Q₈×C₁₆	C₂	1	4	yes	maximal, complemented
4	#2151	C₂	1	2	yes	maximal
5	#2150	C₂	1	1	yes	maximal, characteristic
6	#2108	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#533	C₄	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#446	C₄	1	4	yes	—
13	C₈×Q₈	C₂²	1	4	yes	—
14	#424	C₄	1	2	yes	—
15	#423	C₄	1	1	yes	characteristic
16	#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆	C₂²	1	4	yes	—
19	C₄⋊C₁₆	C₂²	1	8	yes	—
20	#346	C₂²	1	4	yes	—
21	C₄×C₁₆	C₂²	1	2	yes	—
22	C₄×C₁₆	C₂²	1	2	yes	—
23	C₄×C₁₆	C₂²	1	4	yes	—
24	C₂×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈	C₈	1	2	yes	—
28	#125	C₈	1	2	yes	—
29	C₂²⋊Q₈	C₈	1	4	yes	—
30	C₄×Q₈	C₂×C₄	1	4	yes	—
31	#117	C₂×C₄	1	2	yes	—
32	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
33	M₃₂	—	2	4	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	Q₈	1	4	yes	complemented
36	C₂×C₁₆	C₂³	1	2	yes	—
37	C₂×C₁₆	C₂³	1	4	yes	—
38	C₂×C₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₆	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	C₂×C₄	1	4	yes	—
41	C₄⋊C₈	C₂×C₄	1	8	yes	—
42	#97	C₂×C₄	1	4	yes	—
43	C₄×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₈	C₂³	1	2	yes	—
45	C₄×C₈	C₂³	1	4	yes	—
46	C₂×Q₈	C₂×C₈	1	4	yes	—
47	C₂²×C₄	C₂×C₈	1	2	yes	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	M₁₆	—	2	4	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₂²×C₄	1	2	yes	—
52	C₂×C₈	C₂²×C₄	1	4	yes	—
53	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈	C₂×Q₈	1	2	yes	—
55	C₂×C₈	#41	1	2	yes	—
56	C₂×C₈	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄	C₂×C₈	1	4	yes	—
58	C₄⋊C₄	C₂×C₈	1	8	yes	—
59	C₄⋊C₄	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄	C₂×C₈	1	4	yes	—
61	C₄²	C₂×C₈	1	1	yes	characteristic
62	C₄²	C₂×C₈	1	1	yes	characteristic
63	C₄²	C₂²×C₄	1	2	yes	—
64	C₄²	C₂²×C₄	1	4	yes	—
65	C₁₆	—	2	2	no	—
66	C₁₆	—	2	4	no	—
67	C₁₆	C₂×Q₈	1	2	yes	—
68	C₁₆	C₂×Q₈	1	2	yes	—
69	C₂³	C₂²×C₈	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	C₄×Q₈	1	2	yes	—
73	C₂×C₄	C₂²×C₈	1	2	yes	—
74	C₂×C₄	C₂²×C₈	1	8	yes	—
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×C₈	1	2	yes	—
77	C₂×C₄	#131	1	2	yes	—
78	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	2	4	no	—
81	C₈	C₄×Q₈	1	2	yes	—
82	C₈	C₂²×Q₈	1	1	yes	characteristic
83	C₈	C₂×#41	1	1	yes	characteristic
84	C₂²	—	2	2	no	—
85	C₂²	C₈×Q₈	1	2	yes	—
86	C₂²	C₂³×C₈	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	2	8	no	—
89	C₄	C₈×Q₈	1	2	yes	—
90	C₄	#504	1	4	yes	—
91	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄	C₂×#131	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
95	C₂	C₂×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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
3. Q₈×C₁₆	C₂	semidirect
35. 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
Q₈×C₁₆	2
#2151	2
#2150	2
#2108	2
C₂×C₄×C₁₆	2