Subgroups — #12654

Order 256.

191 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1726	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	2	yes	maximal
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₂×#97	C₄	1	1	yes	characteristic
9	C₂×#97	C₄	1	1	yes	characteristic
10	C₂×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#376	C₄	1	1	yes	characteristic
15	C₄⋊C₁₆	C₄	1	8	yes	complemented
16	#346	—	2	2	no	—
17	C₄×C₁₆	—	2	2	no	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₆	C₂×C₄	1	4	yes	—
31	#97	—	2	2	no	—
32	#97	—	2	2	no	—
33	#97	C₈	1	4	yes	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₂×C₄	1	4	yes	—
36	C₄×C₈	D₄	1	2	yes	—
37	C₄×C₈	Q₈	1	2	yes	—
38	#95	C₈	1	4	yes	—
39	C₂²×C₄	C₂×C₈	1	2	yes	—
40	C₂²×C₄	C₂×C₈	1	2	yes	—
41	C₂²×C₄	C₂×C₈	1	2	yes	—
42	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	C₄²	1	2	yes	—
48	C₂×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₈	C₂×C₈	1	2	yes	—
50	C₂×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₈	M₁₆	1	2	yes	—
52	C₂×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈	#41	1	2	yes	—
56	C₄²	—	2	1	no	—
57	C₄²	—	2	1	no	—
58	C₄²	C₄²	1	2	yes	—
59	C₄²	C₄⋊C₄	1	2	yes	—
60	C₄²	C₂×C₈	1	2	yes	—
61	C₄²	M₁₆	1	2	yes	—
62	C₁₆	—	2	4	no	—
63	C₁₆	—	2	4	no	—
64	C₂³	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	C₄×C₈	1	4	yes	—
70	C₂×C₄	C₄⋊C₈	1	4	yes	—
71	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄	#117	1	1	yes	characteristic
74	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	2	2	no	—
79	C₈	C₄⋊C₈	1	4	yes	—
80	C₈	C₄×D₄	1	2	yes	—
81	C₈	C₄×Q₈	1	2	yes	—
82	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²	#431	1	1	yes	characteristic
85	C₂²	#504	1	4	yes	—
86	C₄	—	2	4	no	—
87	C₄	—	2	4	no	—
88	C₄	—	4	2	no	—
89	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄	C₄×M₁₆	1	1	yes	characteristic
91	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄	#431	1	1	yes	characteristic
93	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
94	C₂	#2108	1	2	yes	—
95	C₂	#2151	1	2	yes	—
96	C₂	#2164	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. C₄⋊C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆	2
C₂×#346	2
C₂×C₄×C₁₆	2
#1726	2
#1374	2