Finite Group Databasebeta

Subgroups — #145577

Order 512.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#33732	C₂	1	1	yes	maximal, characteristic
2	#13273	C₂	1	1	yes	maximal, characteristic
3	#13271	C₂	1	1	yes	maximal, characteristic
4	#13164	C₂	1	1	yes	maximal, characteristic
5	#11956	C₂	1	1	yes	maximal, characteristic
6	C₂×#1366	C₂	1	1	yes	maximal, characteristic
7	#7162	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×#499	C₂²	1	1	yes	characteristic
10	#2255	—	2	2	no	—
11	#2253	—	2	1	no	—
12	#2253	—	2	1	no	—
13	#2159	—	2	1	no	—
14	#2159	—	2	1	no	—
15	#2153	—	2	1	no	—
16	C₂×#364	C₂²	1	1	yes	characteristic
17	C₂×#364	C₂²	1	1	yes	characteristic
18	C₂×#364	C₂²	1	1	yes	characteristic
19	C₄×M₃₂	C₂²	1	1	yes	characteristic
20	#1772	C₂²	1	1	yes	characteristic
21	#1389	—	2	1	no	—
22	#1386	—	2	1	no	—
23	#1384	—	2	1	no	—
24	#1383	—	2	1	no	—
25	#1378	—	2	1	no	—
26	#1366	—	2	2	no	—
27	#1366	—	2	1	no	—
28	#1366	—	2	1	no	—
29	#533	—	2	1	no	—
30	C₂×#125	—	2	1	no	—
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	D₄	1	1	yes	characteristic
35	C₂×M₃₂	D₄	1	1	yes	characteristic
36	#500	—	2	1	no	—
37	#499	—	2	1	no	—
38	#499	—	2	1	no	—
39	#499	—	2	1	no	—
40	#499	—	2	1	no	—
41	#499	D₄	1	2	yes	—
42	#499	D₄	1	1	yes	characteristic
43	#499	D₄	1	1	yes	characteristic
44	#498	—	2	1	no	—
45	C₂×#108	D₄	1	1	yes	characteristic
46	#427	—	2	1	no	—
47	#427	—	2	1	no	—
48	#427	D₄	1	1	yes	characteristic
49	C₂×#107	—	2	1	no	—
50	C₂×#106	—	2	1	no	—
51	#405	—	2	1	no	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#367	—	2	2	no	—
54	#367	—	2	2	no	—
55	#366	—	2	2	no	—
56	#366	—	2	2	no	—
57	#364	—	2	1	no	—
58	#364	—	2	1	no	—
59	#364	—	2	1	no	—
60	#364	—	2	1	no	—
61	#364	—	2	1	no	—
62	#364	—	2	1	no	—
63	#364	—	4	2	no	—
64	#346	—	2	1	no	—
65	C₄×C₁₆	—	2	1	no	—
66	#339	—	2	1	no	—
67	C₂×M₁₆	—	2	1	no	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₄⋊Q₈	—	4	1	no	—
71	#125	—	2	2	no	—
72	#125	—	2	1	no	—
73	#125	—	2	1	no	—
74	#125	—	4	1	no	—
75	#125	—	8	1	no	—
76	C₂²⋊Q₈	—	8	1	no	—
77	C₄×Q₈	—	8	1	no	—
78	#117	—	2	1	no	—
79	#117	—	4	1	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄⋊C₄	—	8	1	no	—
82	C₂×C₄²	#41	1	1	yes	characteristic
83	M₃₂	—	2	1	no	—
84	M₃₂	—	2	1	no	—
85	M₃₂	—	4	1	no	—
86	M₃₂	—	4	1	no	—
87	M₃₂	—	4	1	no	—
88	M₃₂	—	4	1	no	—
89	M₃₂	—	4	1	no	—
90	M₃₂	—	4	1	no	—
91	M₃₂	SD₁₆	1	2	yes	—
92	M₃₂	Q₁₆	1	2	yes	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	—	4	1	no	—
97	C₂×C₁₆	—	4	1	no	—
98	#108	—	2	1	no	—
99	#108	Q₁₆	1	2	yes	—
100	#107	—	2	2	no	—
101	#107	—	2	1	no	—
102	#107	—	2	1	no	—
103	#107	—	2	1	no	—
104	#107	—	4	1	no	—
105	#107	—	4	1	no	—
106	#106	—	2	2	no	—
107	#106	—	2	1	no	—
108	#106	—	2	1	no	—
109	#106	—	2	1	no	—
110	#106	—	4	1	no	—
111	#106	—	4	1	no	—
112	#97	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	#41	1	1	yes	characteristic
117	C₄×C₈	#41	1	1	yes	characteristic
118	C₂×Q₈	—	8	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	8	1	no	—
121	C₂²×C₄	#121	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	#120	1	1	yes	characteristic
126	C₂×C₈	#121	1	1	yes	characteristic
127	C₂×C₈	#121	1	1	yes	characteristic
128	C₂×C₈	#124	1	1	yes	characteristic
129	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₈	#135	1	1	yes	characteristic
133	C₂×C₈	#137	1	1	yes	characteristic
134	C₂×C₈	#137	1	1	yes	characteristic
135	C₂×C₈	#137	1	1	yes	characteristic
136	C₂×C₈	#137	1	1	yes	characteristic
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	8	2	no	—
142	C₄⋊C₄	—	8	1	no	—
143	C₄⋊C₄	—	8	1	no	—
144	C₄⋊C₄	—	8	2	no	—
145	C₄⋊C₄	—	8	1	no	—
146	C₄⋊C₄	—	8	1	no	—
147	C₂²⋊C₄	—	4	1	no	—
148	C₂²⋊C₄	—	8	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	—	8	1	no	—
152	C₄²	#123	1	1	yes	characteristic
153	C₄²	#123	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	8	1	no	—
157	C₁₆	—	8	1	no	—
158	C₁₆	—	8	1	no	—
159	C₂³	#472	1	1	yes	characteristic
160	Q₈	—	16	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	16	2	no	—
169	C₂×C₄	#394	1	1	yes	characteristic
170	C₂×C₄	#461	1	1	yes	characteristic
171	C₂×C₄	#462	1	1	yes	characteristic
172	C₂×C₄	#467	1	1	yes	characteristic
173	C₂×C₄	#470	1	1	yes	characteristic
174	C₂×C₄	#472	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	#448	1	1	yes	characteristic
177	C₈	#451	1	1	yes	characteristic
178	C₈	#462	1	1	yes	characteristic
179	C₈	#462	1	1	yes	characteristic
180	C₈	#464	1	1	yes	characteristic
181	C₈	#464	1	1	yes	characteristic
182	C₈	#487	1	1	yes	characteristic
183	C₈	#491	1	1	yes	characteristic
184	C₂²	—	4	1	no	—
185	C₂²	#1676	1	1	yes	characteristic
186	C₂²	#1695	1	1	yes	characteristic
187	C₂²	#2045	1	1	yes	characteristic
188	C₄	—	4	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	16	1	no	—
191	C₄	—	16	2	no	—
192	C₄	—	16	1	no	—
193	C₄	—	16	1	no	—
194	C₄	#1675	1	1	yes	characteristic
195	C₄	#1696	1	1	yes	characteristic
196	C₄	#2043	1	1	yes	characteristic
197	C₄	#2056	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#10543	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#10543
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#33732	2
#13273	2
#13271	2
#13164	2
#11956	2
C₂×#1366	2
#7162	2