Subgroups — #153392

Order 512.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1424	C₂	1	1	yes	maximal, characteristic
2	#13628	C₂	1	1	yes	maximal, characteristic
3	#13171	C₂	1	2	yes	maximal
4	#7478	C₂	1	2	yes	maximal
5	#7472	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	C₂×#366	C₂²	1	2	yes	—
10	#2108	C₂²	1	1	yes	characteristic
11	#1952	—	2	2	no	—
12	#1424	—	2	2	no	—
13	C₄⋊C₃₂	—	2	2	no	—
14	#1387	—	2	2	no	—
15	C₂×M₃₂	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	C₂×#108	—	2	2	no	—
18	#428	—	4	2	no	—
19	C₂×#107	—	2	2	no	—
20	#405	D₄	1	1	yes	characteristic
21	M₆₄	—	2	2	no	—
22	C₂×C₃₂	—	2	2	no	—
23	C₂×C₃₂	—	2	1	no	—
24	C₂×C₃₂	D₄	1	2	yes	—
25	#368	—	2	2	no	—
26	#368	Q₈	1	4	yes	—
27	#366	—	2	2	no	—
28	#366	—	2	2	no	—
29	#346	—	2	1	no	—
30	C₄×C₁₆	—	2	1	no	—
31	#343	—	4	2	no	—
32	C₂×M₁₆	—	4	2	no	—
33	C₂×M₁₆	D₈	1	1	yes	characteristic
34	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
35	#117	D₈	1	1	yes	characteristic
36	C₂×C₄⋊C₄	—	4	2	no	—
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂×Q₈	1	2	yes	—
41	C₂×C₁₆	#41	1	2	yes	—
42	C₂×C₁₆	#41	1	1	yes	characteristic
43	#108	—	2	4	no	—
44	#108	—	4	2	no	—
45	#107	—	4	2	no	—
46	#107	—	4	2	no	—
47	#107	—	8	2	no	—
48	#106	—	8	2	no	—
49	#97	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₃₂	—	2	2	no	—
52	C₃₂	—	4	1	no	—
53	C₂²×C₄	—	8	2	no	—
54	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	4	4	no	—
57	M₁₆	—	8	2	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	8	2	no	—
60	C₂×C₈	#121	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₈	#135	1	1	yes	characteristic
63	C₂×C₈	#137	1	2	yes	—
64	C₄⋊C₄	—	8	2	no	—
65	C₄⋊C₄	—	8	2	no	—
66	C₄⋊C₄	D₁₆	1	2	yes	—
67	C₂²⋊C₄	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₁₆	—	4	1	no	—
70	C₁₆	#123	1	1	yes	characteristic
71	C₁₆	#125	1	2	yes	—
72	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
73	C₂³	#508	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	16	2	no	—
76	C₂×C₄	—	32	2	no	—
77	C₂×C₄	#466	1	1	yes	characteristic
78	C₂×C₄	#474	1	2	yes	—
79	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
80	C₂×C₄	#510	1	2	yes	—
81	C₈	—	4	1	no	—
82	C₈	—	8	4	no	—
83	C₈	#398	1	1	yes	characteristic
84	C₈	#470	1	1	yes	characteristic
85	C₈	#479	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	#2215	1	1	yes	characteristic
88	C₂²	#2229	1	2	yes	—
89	C₄	—	4	2	no	—
90	C₄	—	32	2	no	—
91	C₄	#2086	1	1	yes	characteristic
92	C₄	#2219	1	1	yes	characteristic
93	C₄	#2224	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#13514	1	1	yes	characteristic
96	C₂	#13702	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ 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×#1424	2
#13628	2
#13171	2
#7478	2
#7472	2