Finite Group Databasebeta

Subgroups — #513288

Order 512.

287 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 122 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	#33950	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#13734	C₂	1	2	yes	maximal
4	#13733	C₂	1	1	yes	maximal, characteristic
5	#13708	C₂	1	2	yes	maximal
6	#13706	C₂	1	1	yes	maximal, characteristic
7	#13704	C₂	1	1	yes	maximal, characteristic
8	#13704	C₂	1	2	yes	maximal
9	#13660	C₂	1	1	yes	maximal, characteristic
10	#13653	C₂	1	2	yes	maximal
11	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic
12	#3383	—	2	1	no	—
13	Q₈×Q₁₆	—	2	1	no	—
14	C₂×Q₆₄	C₂²	1	1	yes	characteristic
15	#2253	C₂²	1	2	yes	—
16	#2252	C₂²	1	2	yes	—
17	#2252	C₂²	1	1	yes	characteristic
18	#2252	C₂²	1	1	yes	characteristic
19	#2229	—	2	2	no	—
20	#2227	—	2	1	no	—
21	#2227	—	2	2	no	—
22	#2225	—	2	1	no	—
23	Q₈×C₁₆	C₂²	1	1	yes	characteristic
24	C₄×Q₃₂	C₂²	1	1	yes	characteristic
25	C₄×Q₃₂	C₂²	1	1	yes	characteristic
26	C₄×Q₃₂	C₂²	1	2	yes	—
27	C₄×Q₃₂	C₂²	1	2	yes	—
28	#1424	C₂²	1	2	yes	—
29	#1424	C₂²	1	2	yes	—
30	#1424	C₂²	1	2	yes	—
31	#1423	C₂²	1	1	yes	characteristic
32	#1423	C₂²	1	2	yes	—
33	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
34	C₄⋊C₃₂	C₂²	1	2	yes	—
35	#1416	C₂²	1	1	yes	characteristic
36	#1416	C₂²	1	1	yes	characteristic
37	#1416	C₂²	1	2	yes	—
38	#1416	C₂²	1	2	yes	—
39	#1397	C₂²	1	2	yes	—
40	C₄×C₃₂	C₂²	1	1	yes	characteristic
41	Q₈²	—	4	1	no	—
42	#557	—	4	1	no	—
43	C₂×Q₃₂	C₂³	1	1	yes	characteristic
44	C₂×Q₃₂	C₂³	1	1	yes	characteristic
45	#500	—	2	2	no	—
46	#500	—	2	1	no	—
47	#500	—	2	1	no	—
48	#499	—	2	2	no	—
49	#479	—	4	2	no	—
50	#477	—	4	1	no	—
51	#477	—	4	2	no	—
52	#475	—	4	1	no	—
53	C₈×Q₈	D₄	1	1	yes	characteristic
54	C₄×Q₁₆	—	2	2	no	—
55	C₄×Q₁₆	—	2	1	no	—
56	C₄×Q₁₆	—	2	1	no	—
57	C₄×Q₁₆	—	2	2	no	—
58	Q₆₄	Q₈	1	4	yes	complemented
59	C₂×C₃₂	C₂³	1	1	yes	characteristic
60	C₂×C₃₂	C₂³	1	2	yes	—
61	C₂×C₃₂	C₂³	1	1	yes	characteristic
62	#367	—	2	2	no	—
63	#366	—	2	1	no	—
64	#366	—	2	2	no	—
65	#366	—	2	1	no	—
66	#366	C₂³	1	1	yes	characteristic
67	#366	C₂³	1	1	yes	characteristic
68	#366	C₂³	1	2	yes	—
69	#366	C₂³	1	2	yes	—
70	C₄⋊C₁₆	D₄	1	1	yes	characteristic
71	C₄⋊C₁₆	D₄	1	2	yes	—
72	#358	—	2	2	no	—
73	#358	—	2	1	no	—
74	#358	—	2	2	no	—
75	#358	—	2	1	no	—
76	C₄×C₁₆	C₂³	1	1	yes	characteristic
77	C₄×C₁₆	C₂³	1	2	yes	—
78	C₂×Q₁₆	—	2	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	4	1	no	—
81	C₄⋊Q₈	—	4	2	no	—
82	C₄⋊Q₈	—	4	1	no	—
83	C₄⋊Q₈	—	8	1	no	—
84	C₄⋊Q₈	—	8	1	no	—
85	C₄⋊Q₈	—	8	2	no	—
86	#125	—	4	2	no	—
87	#125	—	8	2	no	—
88	C₄×Q₈	—	4	1	no	—
89	C₄×Q₈	—	4	2	no	—
90	C₄×Q₈	—	4	2	no	—
91	C₄×Q₈	—	4	1	no	—
92	C₄×Q₈	—	8	1	no	—
93	C₄×Q₈	—	8	1	no	—
94	C₄×Q₈	D₈	1	1	yes	characteristic
95	Q₃₂	—	2	1	no	—
96	Q₃₂	—	2	1	no	—
97	Q₃₂	C₂×Q₈	1	2	yes	—
98	Q₃₂	C₂×Q₈	1	2	yes	—
99	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
100	C₂×C₁₆	C₂×D₄	1	2	yes	—
101	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	2	2	no	—
104	#107	—	2	2	no	—
105	#107	—	2	1	no	—
106	#107	—	4	2	no	—
107	#107	—	4	1	no	—
108	#107	—	4	1	no	—
109	#106	—	4	2	no	—
110	C₄⋊C₈	D₈	1	1	yes	characteristic
111	C₄⋊C₈	D₈	1	2	yes	—
112	#103	—	4	2	no	—
113	#103	—	4	2	no	—
114	#103	—	4	1	no	—
115	#103	—	4	1	no	—
116	C₄×C₈	C₂×D₄	1	2	yes	—
117	C₄×C₈	C₂×D₄	1	1	yes	characteristic
118	C₃₂	—	2	1	no	—
119	C₃₂	—	2	2	no	—
120	C₃₂	C₂×Q₈	1	2	yes	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	4	1	no	—
123	C₂×Q₈	—	8	1	no	—
124	C₂×Q₈	—	8	2	no	—
125	C₂×Q₈	—	8	1	no	—
126	C₂×Q₈	D₁₆	1	1	yes	characteristic
127	Q₁₆	—	2	2	no	—
128	Q₁₆	—	2	2	no	—
129	Q₁₆	—	4	1	no	—
130	Q₁₆	—	4	1	no	—
131	C₂×C₈	C₂×D₈	1	2	yes	—
132	C₂×C₈	C₂×D₈	1	1	yes	characteristic
133	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₄⋊C₄	—	8	2	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄⋊C₄	—	8	2	no	—
143	C₄⋊C₄	—	8	2	no	—
144	C₄⋊C₄	—	8	1	no	—
145	C₄⋊C₄	—	8	2	no	—
146	C₄⋊C₄	—	8	1	no	—
147	C₄⋊C₄	D₁₆	1	1	yes	characteristic
148	C₄⋊C₄	D₁₆	1	2	yes	—
149	C₄²	—	8	2	no	—
150	C₄²	—	8	1	no	—
151	C₄²	—	8	2	no	—
152	C₄²	—	8	1	no	—
153	C₄²	C₂×D₈	1	2	yes	—
154	C₄²	C₂×D₈	1	1	yes	characteristic
155	C₁₆	—	2	1	no	—
156	C₁₆	—	2	2	no	—
157	C₁₆	C₂²×Q₈	1	1	yes	characteristic
158	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
159	Q₈	—	2	2	no	—
160	Q₈	—	4	2	no	—
161	Q₈	—	4	2	no	—
162	Q₈	—	8	1	no	—
163	Q₈	—	8	1	no	—
164	Q₈	—	16	2	no	—
165	Q₈	—	32	2	no	—
166	Q₈	—	32	1	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	C₂×D₁₆	1	2	yes	—
174	C₂×C₄	C₂×D₁₆	1	2	yes	—
175	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	D₄×Q₈	1	1	yes	characteristic
181	C₈	#578	1	1	yes	characteristic
182	C₂²	C₂²×D₁₆	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	16	2	no	—
189	C₄	—	16	1	no	—
190	C₄	—	16	2	no	—
191	C₄	—	16	1	no	—
192	C₄	#2264	1	2	yes	—
193	C₄	Q₈×D₈	1	1	yes	characteristic
194	C₄	#3417	1	1	yes	characteristic
195	C₂	Q₈×D₁₆	1	1	yes	characteristic
196	C₂	C₂×#2264	1	1	yes	characteristic
197	C₂	#33983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
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
58. Q₆₄	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33950	2
Q₈×Q₃₂	2
#13734	2
#13733	2
#13708	2
#13706	2
#13704	2
#13660	2
#13653	2
C₄×Q₆₄	2