Finite Group Databasebeta

← #511717

Subgroups — #511717

Order 512.

414 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33258C211yesmaximal, characteristic
2#33258C211yesmaximal, characteristic
3#13538C212yesmaximal
4#13537C212yesmaximal
5#13516C214yesmaximal
6#13506C212yesmaximal
7#13506C212yesmaximal
8#12879C211yesmaximal, characteristic
9#2859—21no—
10#2859—21no—
11C2×#367C2214yes—
12C2×C4⋊C16C2212yes—
13#1942—21no—
14#1942—21no—
15#1942—21no—
16#1942—21no—
17#1942C2212yes—
18#1942C2212yes—
19#1942C2212yes—
20#1942C2212yes—
21#1845C2211yescharacteristic
22#1845C2211yescharacteristic
23#1845C2211yescharacteristic
24#1845C2211yescharacteristic
25#1834—21no—
26#1834—21no—
27C42×C8C2211yescharacteristic
28#1380C2214yes—
29#1380C2214yes—
30#1380C2214yes—
31#1374C2212yes—
32#1374C2212yes—
33C2×#125—41no—
34C2×#125—41no—
35C22×C16C2314yes—
36C2×#107—21no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#107—21no—
40C2×#107C2312yes—
41C2×#107C2312yes—
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48C2×C4×C8C2311yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#389—22no—
51#389—22no—
52#389—22no—
53#389—22no—
54#389—41no—
55#389—41no—
56#389—41no—
57#389—41no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#382—41no—
63#382—41no—
64C43D411yescharacteristic
65#367—44no—
66C4⋊C16—42no—
67#340—22no—
68#340—22no—
69#340—22no—
70#340—22no—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2411yescharacteristic
75#125—84no—
76#125—84no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C42D811yescharacteristic
94C2×C42D811yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—24no—
101C2×C16—24no—
102C2×C16—24no—
103#107—22no—
104#107—22no—
105#107—24no—
106#107—24no—
107#107—22no—
108#107—22no—
109#107—42no—
110#107—42no—
111#107—42no—
112#107—42no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—21no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—21no—
119C4×C8—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D811yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4C2×D811yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4C2×D811yescharacteristic
133C22×C4C2×D811yescharacteristic
134C22×C4C22×D411yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8#13712yes—
143C2×C8#13712yes—
144C2×C8H(5,𝔽2)11yescharacteristic
145C2×C82−1+412yes—
146C2×C82−1+412yes—
147C2×C82−1+411yescharacteristic
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—84no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C4⋊C4—82no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—22no—
175C42—22no—
176C42—41no—
177C42SD3214yes—
178C16—44no—
179C23C22×D811yescharacteristic
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4C2×SD3212yes—
206C2×C4C2×SD3212yes—
207C2×C4C2×SD3212yes—
208C2×C4#50912yes—
209C2×C4#51012yes—
210C2×C4#53611yescharacteristic
211C2×C4#53611yescharacteristic
212C2×C4#53711yescharacteristic
213C2×C4C2×#13711yescharacteristic
214C2×C4C2×#13711yescharacteristic
215C2×C4#57511yescharacteristic
216C8—22no—
217C8—22no—
218C8—41no—
219C8#56312yes—
220C8#56412yes—
221C22#322611yescharacteristic
222C22#322611yescharacteristic
223C22#324511yescharacteristic
224C22C22×SD3211yescharacteristic
225C22C2×#50911yescharacteristic
226C22C2×#51011yescharacteristic
227C22#341411yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—41no—
234C4—162no—
235C4—162no—
236C4—162no—
237C4—162no—
238C4#282211yescharacteristic
239C4#323611yescharacteristic
240C4#323611yescharacteristic
241C4#325511yescharacteristic
242C2#3284811yescharacteristic
243C2#3379311yescharacteristic
244C2#3379511yescharacteristic
245C2#3380011yescharacteristic
246C2#3380211yescharacteristic
247C2#3381211yescharacteristic
248C2#3382011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332582
#135382
#135372
#135162
#135062
#128792