Finite Group Databasebeta

Subgroups — #146076

Order 512.

212 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 72 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	#13268	C₂	1	1	yes	maximal, characteristic
2	#12886	C₂	1	1	yes	maximal, characteristic
3	#8663	C₂	1	1	yes	maximal, characteristic
4	#8651	C₂	1	1	yes	maximal, characteristic
5	#7281	C₂	1	1	yes	maximal, characteristic
6	#7278	C₂	1	1	yes	maximal, characteristic
7	#7159	C₂	1	1	yes	maximal, characteristic
8	C₂×#446	C₂²	1	1	yes	characteristic
9	#2150	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	1	yes	characteristic
12	#1844	C₂²	1	1	yes	characteristic
13	#1803	—	2	1	no	—
14	#1564	C₂²	1	1	yes	characteristic
15	C₂×#335	C₂²	1	1	yes	characteristic
16	#1487	—	2	1	no	—
17	#1383	—	2	1	no	—
18	#1383	—	2	1	no	—
19	#1377	—	2	1	no	—
20	#1377	—	2	1	no	—
21	#1301	—	2	1	no	—
22	#1296	—	2	1	no	—
23	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
24	C₂×M₃₂	—	2	1	no	—
25	C₂×M₃₂	D₄	1	1	yes	characteristic
26	C₂×M₃₂	D₄	1	1	yes	characteristic
27	#446	—	2	1	no	—
28	#446	—	2	1	no	—
29	#446	—	2	1	no	—
30	#446	—	2	1	no	—
31	#431	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#106	—	2	1	no	—
35	C₂×#106	—	2	1	no	—
36	#423	—	2	1	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
40	C₂×#103	—	4	1	no	—
41	C₂×#97	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#384	—	2	1	no	—
44	C₄⋊C₁₆	—	2	1	no	—
45	C₄⋊C₁₆	—	2	1	no	—
46	#346	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	#340	—	4	1	no	—
49	#335	—	2	1	no	—
50	#335	—	2	1	no	—
51	#335	—	2	1	no	—
52	#334	—	2	1	no	—
53	Q₈⋊C₈	—	4	2	no	—
54	C₂²×Q₈	SD₁₆	1	1	yes	characteristic
55	C₂×M₁₆	—	4	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	4	1	no	—
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	#41	1	1	yes	characteristic
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₄×Q₈	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	4	1	no	—
66	C₂×C₄⋊C₄	—	4	1	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	M₃₂	—	2	1	no	—
72	M₃₂	—	2	1	no	—
73	M₃₂	—	2	1	no	—
74	M₃₂	—	2	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	—	4	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	#108	—	4	1	no	—
81	#108	—	4	1	no	—
82	#108	—	8	1	no	—
83	#106	—	4	1	no	—
84	#106	—	4	1	no	—
85	#106	—	8	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄⋊C₈	—	4	1	no	—
95	#103	—	8	2	no	—
96	C₂²⋊C₈	—	4	1	no	—
97	#97	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	8	1	no	—
111	C₂²×C₄	#121	1	1	yes	characteristic
112	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₄	#135	1	1	yes	characteristic
114	M₁₆	—	8	1	no	—
115	M₁₆	—	8	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	8	1	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	—	8	1	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	#123	1	1	yes	characteristic
127	C₂×C₈	#124	1	1	yes	characteristic
128	C₂×C₈	#126	1	1	yes	characteristic
129	C₂×C₈	#126	1	1	yes	characteristic
130	C₂×C₈	#135	1	1	yes	characteristic
131	C₂×C₈	#135	1	1	yes	characteristic
132	C₂×C₈	#137	1	1	yes	characteristic
133	C₂×C₈	#137	1	1	yes	characteristic
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄⋊C₄	—	8	1	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄⋊C₄	—	16	1	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	1	no	—
145	C₄²	#123	1	1	yes	characteristic
146	C₄²	#123	1	1	yes	characteristic
147	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄²	#137	1	1	yes	characteristic
149	C₁₆	—	4	1	no	—
150	C₁₆	—	4	1	no	—
151	C₁₆	—	8	1	no	—
152	C₂³	#465	1	1	yes	characteristic
153	Q₈	—	4	2	no	—
154	Q₈	—	4	1	no	—
155	Q₈	—	4	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	16	1	no	—
164	C₂×C₄	—	16	1	no	—
165	C₂×C₄	—	16	1	no	—
166	C₂×C₄	#397	1	1	yes	characteristic
167	C₂×C₄	#464	1	1	yes	characteristic
168	C₂×C₄	#464	1	1	yes	characteristic
169	C₂×C₄	#468	1	1	yes	characteristic
170	C₂×C₄	#472	1	1	yes	characteristic
171	C₂×C₄	#473	1	1	yes	characteristic
172	C₂×C₄	#481	1	1	yes	characteristic
173	C₂×C₄	#482	1	1	yes	characteristic
174	C₂×C₄	#483	1	1	yes	characteristic
175	C₂×C₄	#485	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	8	1	no	—
179	C₈	—	16	1	no	—
180	C₈	—	16	1	no	—
181	C₈	—	16	1	no	—
182	C₈	#485	1	1	yes	characteristic
183	C₈	#485	1	1	yes	characteristic
184	C₈	#488	1	1	yes	characteristic
185	C₈	#491	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#1678	1	1	yes	characteristic
189	C₂²	#1680	1	1	yes	characteristic
190	C₂²	#2077	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	8	1	no	—
195	C₄	—	8	1	no	—
196	C₄	—	32	1	no	—
197	C₄	#1698	1	1	yes	characteristic
198	C₄	#1700	1	1	yes	characteristic
199	C₄	#2074	1	1	yes	characteristic
200	C₄	#2078	1	1	yes	characteristic
201	C₄	#2238	1	1	yes	characteristic
202	C₄	#2239	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#10626	1	1	yes	characteristic
206	C₂	#13501	1	1	yes	characteristic
207	C₂	#13502	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ 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
#13268	2
#12886	2
#8663	2
#8651	2
#7281	2
#7278	2
#7159	2