Finite Group Databasebeta

← #511400

Subgroups — #511400

Order 512.

417 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 128 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#28012C212yesmaximal
2#13529C212yesmaximal
3#13475C212yesmaximal
4#13450C212yesmaximal
5#13240C211yesmaximal, characteristic
6#13178C212yesmaximal
7#13167C211yesmaximal, characteristic
8#13160C211yesmaximal, characteristic
9#13133C212yesmaximal
10C22×Q32C2211yescharacteristic
11#2514—22no—
12C2×#367C2211yescharacteristic
13C2×#367C2211yescharacteristic
14C2×#366C2211yescharacteristic
15C2×#366C2212yes—
16C2×C4⋊C16C2212yes—
17C2×C4⋊C16C2211yescharacteristic
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#346C2211yescharacteristic
22#2074—22no—
23#2066—22no—
24#1969C2212yes—
25#1969C2212yes—
26#1969C2212yes—
27#1942C2212yes—
28#1927—22no—
29#1846C2212yes—
30C4×#107C2212yes—
31C8×C4⋊C4C2211yescharacteristic
32#1380C2212yes—
33#1380C2212yes—
34#1374C2212yes—
35C22×Q16C2312yes—
36C2×Q32—22no—
37C2×Q32—41no—
38C2×Q32—41no—
39C22×C16C2312yes—
40C22×C16C2311yescharacteristic
41C22×C16C2311yescharacteristic
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#106—22no—
46C2×C4⋊C8D411yescharacteristic
47C2×#103—22no—
48C2×#103—22no—
49C2×#103—22no—
50C2×C4×C8C2312yes—
51C2×C4×C8C2311yescharacteristic
52#401—42no—
53#400—42no—
54#391—22no—
55#391—22no—
56#391—22no—
57#389—22no—
58#387—42no—
59#383—42no—
60#382—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4D411yescharacteristic
63#367—22no—
64#367—41no—
65#366—41no—
66#366—42no—
67C4⋊C16—24no—
68C4⋊C16—41no—
69#358—24no—
70#358—42no—
71#358—42no—
72#346—41no—
73#340—22no—
74#340—22no—
75#336D412yes—
76C22×Q8—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8C2411yescharacteristic
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4D812yes—
91C2×C42—42no—
92C2×C42C2×D412yes—
93C2×C42C2×D411yescharacteristic
94Q32—44no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—22no—
100C2×C16—22no—
101C2×C16—22no—
102C2×C16#4112yes—
103C2×C16#4112yes—
104#107—22no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—22no—
109#107—22no—
110#106—82no—
111C4⋊C8—41no—
112#103—44no—
113#103—82no—
114#103—82no—
115C4×C8—22no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—22no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95D811yescharacteristic
130#95D811yescharacteristic
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C2×D811yescharacteristic
142C22×C4C2×D812yes—
143C22×C4C2×D812yes—
144C22×C4C2×D811yescharacteristic
145C22×C4C22×D411yescharacteristic
146Q16—42no—
147Q16—42no—
148Q16—42no—
149Q16—42no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—21no—
156C2×C8C2×#4111yescharacteristic
157C2×C8C2×#4111yescharacteristic
158C2×C8H(5,𝔽2)11yescharacteristic
159C2×C8H(5,𝔽2)11yescharacteristic
160C2×C82−1+411yescharacteristic
161C2×C82−1+411yescharacteristic
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—84no—
170C42—21no—
171C42—22no—
172C42—21no—
173C42—22no—
174C42—84no—
175C16—22no—
176C16—22no—
177C16—42no—
178C23C22×D811yescharacteristic
179Q8—82no—
180Q8—82no—
181Q8—82no—
182Q8—82no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
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×C4#53711yescharacteristic
206C2×C4#54711yescharacteristic
207C2×C4#55011yescharacteristic
208C2×C4#57511yescharacteristic
209C2×C4#57611yescharacteristic
210C2×C4#57811yescharacteristic
211C8—42no—
212C8—41no—
213C8#53811yescharacteristic
214C8#53911yescharacteristic
215C8#55611yescharacteristic
216C8#56311yescharacteristic
217C22#324511yescharacteristic
218C22#333411yescharacteristic
219C22#339111yescharacteristic
220C22#341412yes—
221C22#341712yes—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—42no—
226C4—162no—
227C4—162no—
228C4—162no—
229C4—162no—
230C4#272711yescharacteristic
231C4#326211yescharacteristic
232C4#335511yescharacteristic
233C4#340211yescharacteristic
234C2#3252411yescharacteristic
235C2#3382211yescharacteristic
236C2#3382711yescharacteristic
237C2#3391612yes—
238C2#3396912yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ 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
#280122
#135292
#134752
#134502
#132402
#131782
#131672
#131602
#131332