Finite Group Databasebeta

Subgroups — #513290

Order 512.

298 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33947	C₂	1	2	yes	maximal, complemented
2	#13734	C₂	1	2	yes	maximal, complemented
3	#13732	C₂	1	1	yes	maximal, characteristic, complemented
4	#13707	C₂	1	2	yes	maximal, complemented
5	#13707	C₂	1	2	yes	maximal, complemented
6	#13703	C₂	1	2	yes	maximal, complemented
7	#13660	C₂	1	1	yes	maximal, characteristic, complemented
8	#13654	C₂	1	2	yes	maximal
9	C₄×D₃₂	C₂	1	1	yes	maximal, characteristic
10	#3380	—	2	2	no	—
11	C₂×D₃₂	C₂²	1	1	yes	characteristic
12	#2253	C₂²	1	2	yes	—
13	#2253	C₂²	1	2	yes	—
14	#2252	C₂²	1	2	yes	—
15	#2228	—	2	2	no	—
16	#2228	—	2	2	no	—
17	#2226	—	2	2	no	—
18	Q₈×C₁₆	C₂²	1	1	yes	characteristic
19	C₄×D₁₆	C₂²	1	2	yes	—
20	C₄×D₁₆	C₂²	1	2	yes	—
21	C₄×D₁₆	C₂²	1	2	yes	—
22	#1424	C₂²	1	2	yes	—
23	#1424	C₂²	1	2	yes	—
24	#1423	C₂²	1	1	yes	characteristic
25	#1423	C₂²	1	1	yes	characteristic
26	#1423	C₂²	1	2	yes	—
27	#1423	C₂²	1	1	yes	characteristic
28	C₄⋊C₃₂	C₂²	1	2	yes	—
29	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
30	#1415	C₂²	1	2	yes	—
31	#1415	C₂²	1	2	yes	—
32	#1415	C₂²	1	2	yes	—
33	#1397	C₂²	1	2	yes	—
34	C₄×C₃₂	C₂²	1	1	yes	characteristic
35	#554	—	4	2	no	—
36	C₂×D₁₆	C₂³	1	2	yes	—
37	#500	—	2	2	no	—
38	#499	—	2	2	no	—
39	#499	—	2	2	no	—
40	#478	—	4	2	no	—
41	#478	—	4	2	no	—
42	#476	—	4	2	no	—
43	C₈×Q₈	D₄	1	1	yes	characteristic
44	C₄×D₈	—	2	2	no	—
45	C₄×D₈	—	2	2	no	—
46	C₄×D₈	—	2	2	no	—
47	D₃₂	Q₈	1	4	yes	complemented
48	C₂×C₃₂	C₂³	1	1	yes	characteristic
49	C₂×C₃₂	C₂³	1	2	yes	—
50	C₂×C₃₂	C₂³	1	1	yes	characteristic
51	#367	—	2	2	no	—
52	#367	—	2	2	no	—
53	#366	—	2	2	no	—
54	#366	C₂³	1	2	yes	—
55	#366	C₂³	1	2	yes	—
56	#366	C₂³	1	2	yes	—
57	C₄⋊C₁₆	D₄	1	2	yes	—
58	C₄⋊C₁₆	D₄	1	1	yes	characteristic
59	#357	—	2	2	no	—
60	#357	—	2	2	no	—
61	#357	—	2	2	no	—
62	C₄×C₁₆	C₂³	1	1	yes	characteristic
63	C₄×C₁₆	C₂³	1	2	yes	—
64	C₂×D₈	—	2	2	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	#125	—	4	2	no	—
67	#125	—	4	2	no	—
68	C₂²⋊Q₈	—	8	2	no	—
69	C₂²⋊Q₈	—	8	2	no	—
70	C₂²⋊Q₈	—	8	2	no	—
71	C₄×Q₈	D₈	1	1	yes	characteristic
72	C₄×D₄	—	4	2	no	—
73	C₄×D₄	—	4	2	no	—
74	C₄×D₄	—	4	2	no	—
75	C₂×C₄⋊C₄	—	8	2	no	—
76	D₁₆	—	2	2	no	—
77	D₁₆	C₂×Q₈	1	4	yes	—
78	C₂×C₁₆	C₂×D₄	1	2	yes	—
79	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
80	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
81	#107	—	2	2	no	—
82	#107	—	2	2	no	—
83	#107	—	2	2	no	—
84	#107	—	4	2	no	—
85	#106	—	4	2	no	—
86	#106	—	4	2	no	—
87	C₄⋊C₈	D₈	1	2	yes	—
88	C₄⋊C₈	D₈	1	1	yes	characteristic
89	#102	—	4	2	no	—
90	#102	—	4	2	no	—
91	#102	—	4	2	no	—
92	C₄×C₈	C₂×D₄	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	2	yes	—
94	C₃₂	—	2	1	no	—
95	C₃₂	—	2	2	no	—
96	C₃₂	C₂×Q₈	1	2	yes	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	D₁₆	1	1	yes	characteristic
99	C₂×D₄	—	4	2	no	—
100	C₂²×C₄	—	8	2	no	—
101	C₂²×C₄	—	8	2	no	—
102	C₂²×C₄	—	8	2	no	—
103	D₈	—	2	4	no	—
104	D₈	—	4	2	no	—
105	C₂×C₈	C₂×D₈	1	1	yes	characteristic
106	C₂×C₈	C₂×D₈	1	2	yes	—
107	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄⋊C₄	—	8	2	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	D₁₆	1	2	yes	—
117	C₄⋊C₄	D₁₆	1	1	yes	characteristic
118	C₂²⋊C₄	—	8	2	no	—
119	C₂²⋊C₄	—	8	2	no	—
120	C₂²⋊C₄	—	8	2	no	—
121	C₄²	C₂×D₈	1	1	yes	characteristic
122	C₄²	C₂×D₈	1	2	yes	—
123	C₁₆	—	2	2	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	C₂²×Q₈	1	1	yes	characteristic
126	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂³	—	8	2	no	—
128	Q₈	—	4	1	no	—
129	Q₈	—	32	2	no	—
130	D₄	—	4	4	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	16	4	no	—
136	C₂×C₄	—	32	2	no	—
137	C₂×C₄	—	32	2	no	—
138	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
139	C₂×C₄	C₂×D₁₆	1	2	yes	—
140	C₂×C₄	C₂×D₁₆	1	2	yes	—
141	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
142	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	D₄×Q₈	1	1	yes	characteristic
146	C₈	#578	1	1	yes	characteristic
147	C₂²	—	8	4	no	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	C₂²×D₁₆	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	16	2	no	—
156	C₄	—	16	2	no	—
157	C₄	—	16	2	no	—
158	C₄	Q₈×D₈	1	1	yes	characteristic
159	C₄	#3417	1	1	yes	characteristic
160	C₂	—	16	4	no	—
161	C₂	Q₈×D₁₆	1	1	yes	characteristic
162	C₂	#33981	1	1	yes	characteristic
163	C₂	#33982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×D₃₂	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33947	C₂	semidirect
2. #13734	C₂	semidirect
3. #13732	C₂	semidirect
4. #13707	C₂	semidirect
5. #13707	C₂	semidirect
6. #13703	C₂	semidirect
7. #13660	C₂	semidirect
47. D₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33947	2
#13734	2
#13732	2
#13707	2
#13703	2
#13660	2
#13654	2
C₄×D₃₂	2