Subgroups — #13540

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1946	C₂	1	1	yes	maximal, characteristic
2	#1849	C₂	1	1	yes	maximal, characteristic
3	#1389	C₂	1	2	yes	maximal
4	#1388	C₂	1	2	yes	maximal
5	#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	—	2	1	no	—
8	C₂×#108	C₂²	1	2	yes	—
9	C₂×#108	C₂²	1	1	yes	characteristic
10	#427	C₂²	1	1	yes	characteristic
11	#424	—	2	1	no	—
12	#423	—	2	1	no	—
13	C₂×#97	C₂²	1	1	yes	characteristic
14	#339	—	2	1	no	—
15	C₂×M₁₆	—	2	1	no	—
16	C₂×M₁₆	—	2	2	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	#117	—	2	1	no	—
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	M₃₂	—	2	2	no	—
22	M₃₂	—	2	2	no	—
23	C₂×C₁₆	—	2	2	no	—
24	#108	—	2	2	no	—
25	#108	—	2	4	no	—
26	#108	—	4	2	no	—
27	#107	—	2	1	no	—
28	#106	Q₈	1	2	yes	—
29	C₄⋊C₈	—	4	2	no	—
30	C₄⋊C₈	—	4	2	no	—
31	#97	—	2	2	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
34	M₁₆	—	4	1	no	—
35	M₁₆	—	4	2	no	—
36	M₁₆	—	4	2	no	—
37	M₁₆	—	4	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	D₈	1	2	yes	—
43	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈	#41	1	2	yes	—
45	C₂×C₈	#41	1	2	yes	—
46	C₂×C₈	#41	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₂²⋊C₄	—	4	1	no	—
49	C₄²	—	2	1	no	—
50	C₄²	—	4	1	no	—
51	C₄²	SD₁₆	1	2	yes	—
52	C₁₆	—	4	2	no	—
53	C₂³	#137	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄	#123	1	1	yes	characteristic
59	C₂×C₄	#124	1	1	yes	characteristic
60	C₂×C₄	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄	#137	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	8	1	no	—
65	C₈	—	8	2	no	—
66	C₈	#125	1	2	yes	—
67	C₈	#126	1	2	yes	—
68	C₂²	—	4	1	no	—
69	C₂²	#474	1	1	yes	characteristic
70	C₂²	#483	1	1	yes	characteristic
71	C₂²	#491	1	1	yes	characteristic
72	C₄	—	4	2	no	—
73	C₄	—	8	2	no	—
74	C₄	#400	1	1	yes	characteristic
75	C₄	#475	1	1	yes	characteristic
76	C₄	#480	1	1	yes	characteristic
77	C₄	#486	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	#2099	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2099
prime radical	C₂³	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ 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
#1946	2
#1849	2
#1389	2
#1388	2
#1375	2