Finite Group Databasebeta

← #140884

Subgroups — #140884

Order 512.

393 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 102 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#33215C211yesmaximal, characteristic
2#11724C212yesmaximal
3#11723C212yesmaximal
4#10867C211yesmaximal, characteristic
5#8156C211yesmaximal, characteristic
6C2×#500C411yescharacteristic
7C22×#108—21no—
8C2×#405C2211yescharacteristic
9#2628C2212yes—
10#1993C412yes—
11#1944—21no—
12#1942—21no—
13#1846C411yescharacteristic
14#1809—21no—
15#1808—21no—
16#1742C2212yes—
17C2×#338—21no—
18#1458C2211yescharacteristic
19#1458C2211yescharacteristic
20#1450—21no—
21#1354—24no—
22#1353—24no—
23#1281—21no—
24C22×M16—22no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×#117C2311yescharacteristic
29#500—41no—
30#500—41no—
31#431—21no—
32#431—21no—
33C2×#108—24no—
34C2×#108—41no—
35C2×#108—41no—
36C2×#108—42no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#107—21no—
40C2×#106—21no—
41#405—41no—
42#405—41no—
43C2×#97C2×C411yescharacteristic
44C2×#97D411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#400—22no—
48#399—22no—
49#395—22no—
50#393—22no—
51#389—22no—
52#382C2×C412yes—
53#380—22no—
54#343—22no—
55#343—22no—
56#340—22no—
57#340—22no—
58#338—42no—
59#324D412yes—
60#324D412yes—
61#323—21no—
62#323—21no—
63#322—41no—
64C82—41no—
65C22×Q8—22no—
66C23×C4C2×D411yescharacteristic
67C2×M16—24no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16SD1612yes—
76C2×M16Q1612yes—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8SD1612yes—
86C22×C8Q1612yes—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4C2×D411yescharacteristic
96C2×C4⋊C4C2×D411yescharacteristic
97C2×C22⋊C4—42no—
98C2×C22⋊C4#4111yescharacteristic
99C2×C22⋊C4#4111yescharacteristic
100C2×C42C22×C411yescharacteristic
101C2×C42C2×D411yescharacteristic
102#108—44no—
103#108—44no—
104#107—42no—
105#107—41no—
106#107—41no—
107#107—42no—
108#106—42no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#97—41no—
114#97—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—41no—
120C4×C8—41no—
121#95—22no—
122#95—22no—
123#95—42no—
124#95—42no—
125C24#12411yescharacteristic
126C2×Q8—82no—
127C2×Q8—162no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C4×D411yescharacteristic
136C22×C4C4×D411yescharacteristic
137C22×C4#12111yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12711yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141C22×C4C2×SD1611yescharacteristic
142C22×C4C2×Q1611yescharacteristic
143C22×C4C2×Q1611yescharacteristic
144M16—42no—
145M16—42no—
146M16—44no—
147M16—44no—
148M16—82no—
149M16—82no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8#9911yescharacteristic
166C2×C8#9911yescharacteristic
167C2×C8#10312yes—
168C2×C8#10312yes—
169C2×C8#10312yes—
170C2×C8#10312yes—
171C2×C8#10412yes—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—82no—
179C4⋊C4—162no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—162no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23#39011yescharacteristic
192C23#48711yescharacteristic
193C23#48711yescharacteristic
194Q8—164no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4C2×#9911yescharacteristic
220C2×C4#41011yescharacteristic
221C2×C4C2×#10311yescharacteristic
222C2×C4C2×#10311yescharacteristic
223C2×C4C2×#10411yescharacteristic
224C2×C4#42111yescharacteristic
225C2×C4#49211yescharacteristic
226C2×C4#49411yescharacteristic
227C8—22no—
228C8—22no—
229C8—22no—
230C8—84no—
231C8—82no—
232C8—82no—
233C8#32812yes—
234C8#32812yes—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#195611yescharacteristic
245C22#196411yescharacteristic
246C22#198011yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—162no—
254C4—162no—
255C4C2×#32811yescharacteristic
256C4#150311yescharacteristic
257C4#151811yescharacteristic
258C4#152011yescharacteristic
259C2—21no—
260C2—21no—
261C2—41no—
262C2—41no—
263C2#967911yescharacteristic
264C2#1323311yescharacteristic
265C2#1323411yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332152
#117242
#117232
#108672
#81562