Subgroups — #7153

Order 256.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂²×C₁₆	—	2	1	no	—
6	#431	C₄	1	1	yes	characteristic
7	#431	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆	—	2	1	no	—
10	C₄⋊C₁₆	—	2	1	no	—
11	C₄⋊C₁₆	C₄	1	2	yes	—
12	C₄⋊C₁₆	C₄	1	2	yes	—
13	#322	—	2	1	no	—
14	C₈²	—	2	1	no	—
15	C₂²×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₈	Q₈	1	1	yes	characteristic
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	M₃₂	—	4	1	no	—
19	M₃₂	—	4	1	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	—	4	1	no	—
23	C₂×C₁₆	—	4	1	no	—
24	C₄⋊C₈	—	2	1	no	—
25	C₄⋊C₈	C₈	1	2	yes	—
26	C₂²⋊C₈	—	2	1	no	—
27	C₂²⋊C₈	C₈	1	2	yes	—
28	C₄×C₈	—	2	1	no	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄	M₁₆	1	1	yes	characteristic
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈	D₈	1	1	yes	characteristic
46	C₂×C₈	SD₁₆	1	1	yes	characteristic
47	C₂×C₈	SD₁₆	1	1	yes	characteristic
48	C₂×C₈	Q₁₆	1	1	yes	characteristic
49	C₄²	C₄²	1	1	yes	characteristic
50	C₄²	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²	C₂×C₈	1	1	yes	characteristic
52	C₄²	M₁₆	1	1	yes	characteristic
53	C₁₆	—	4	2	no	—
54	C₁₆	—	4	2	no	—
55	C₂³	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	#95	1	1	yes	characteristic
59	C₂×C₄	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄	#97	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄	#106	1	1	yes	characteristic
65	C₂×C₄	#107	1	1	yes	characteristic
66	C₂×C₄	#108	1	1	yes	characteristic
67	C₂×C₄	#108	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	4	2	no	—
71	C₈	—	4	2	no	—
72	C₈	#102	1	1	yes	characteristic
73	C₈	#102	1	1	yes	characteristic
74	C₈	#103	1	1	yes	characteristic
75	C₈	#103	1	1	yes	characteristic
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	#334	1	1	yes	characteristic
79	C₂²	#335	1	1	yes	characteristic
80	C₂²	#336	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	#334	1	1	yes	characteristic
85	C₄	#335	1	1	yes	characteristic
86	C₄	#340	1	1	yes	characteristic
87	C₄	#341	1	1	yes	characteristic
88	C₄	#350	1	2	yes	—
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	#1277	1	1	yes	characteristic
92	C₂	#1378	1	1	yes	characteristic
93	C₂	#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2150	2
C₂×C₄⋊C₁₆	2
#1450	2