Subgroups — #13515

Order 256.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1942	C₂	1	2	yes	maximal
4	#1380	C₂	1	2	yes	maximal
5	#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#107	C₂²	1	2	yes	—
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#389	—	2	2	no	—
12	#366	—	2	2	no	—
13	C₄⋊C₁₆	—	2	2	no	—
14	#340	—	2	2	no	—
15	C₂²×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊C₄	—	4	2	no	—
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	D₄	1	2	yes	—
25	#107	—	2	2	no	—
26	#107	—	2	2	no	—
27	#107	—	2	2	no	—
28	#107	Q₈	1	4	yes	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	—	2	1	no	—
31	#95	—	4	2	no	—
32	C₂²×C₄	—	4	2	no	—
33	C₂²×C₄	—	4	2	no	—
34	C₂²×C₄	D₈	1	1	yes	characteristic
35	C₂²×C₄	D₈	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈	C₂×Q₈	1	2	yes	—
41	C₂×C₈	#41	1	2	yes	—
42	C₂×C₈	#41	1	1	yes	characteristic
43	C₄⋊C₄	—	2	4	no	—
44	C₄⋊C₄	—	4	2	no	—
45	C₄⋊C₄	—	4	2	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄⋊C₄	—	8	4	no	—
48	C₄²	—	2	1	no	—
49	C₄²	—	2	1	no	—
50	C₁₆	—	2	2	no	—
51	C₁₆	—	4	1	no	—
52	C₂³	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	4	4	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	Q₃₂	1	2	yes	—
64	C₂×C₄	#121	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄	#135	1	1	yes	characteristic
67	C₂×C₄	#137	1	2	yes	—
68	C₈	—	4	1	no	—
69	C₈	#123	1	1	yes	characteristic
70	C₈	#125	1	2	yes	—
71	C₈	C₄⋊Q₈	1	1	yes	characteristic
72	C₂²	#466	1	1	yes	characteristic
73	C₂²	#474	1	2	yes	—
74	C₂²	C₂×Q₃₂	1	1	yes	characteristic
75	C₂²	#508	1	1	yes	characteristic
76	C₂²	#509	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	4	1	no	—
79	C₄	—	8	4	no	—
80	C₄	—	16	2	no	—
81	C₄	#398	1	1	yes	characteristic
82	C₄	#470	1	1	yes	characteristic
83	C₄	#479	1	2	yes	—
84	C₂	#2086	1	1	yes	characteristic
85	C₂	#2216	1	1	yes	characteristic
86	C₂	#2218	1	1	yes	characteristic
87	C₂	#2227	1	2	yes	—
88	C₂	#2228	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#366	2
C₂×C₄⋊C₁₆	2
#1942	2
#1380	2