Finite Group Databasebeta

← #140692

Subgroups — #140692

Order 512.

375 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 88 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#21156C211yesmaximal, characteristic
2#11454C212yesmaximal
3#11450C212yesmaximal
4#8602C211yesmaximal, characteristic
5#8157C211yesmaximal, characteristic
6C2×#427C411yescharacteristic
7C22×#107C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2317C2212yes—
10#1944—21no—
11#1944—21no—
12#1942—21no—
13#1942—21no—
14#1736C412yes—
15#1564—21no—
16#1458C2212yes—
17#1458C2211yescharacteristic
18#1458C2211yescharacteristic
19#1452—21no—
20#1326—24no—
21#1325—24no—
22#1281—21no—
23#1275—21no—
24C22×M16C2×C411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C2×#117C2×C412yes—
27C2×#117C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29#431—21no—
30#431—21no—
31#431—22no—
32#427—24no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—41no—
37C2×#107—41no—
38C2×#106—21no—
39#405—41no—
40#405—41no—
41C2×#97D411yescharacteristic
42C2×#97D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#389—22no—
46#389—22no—
47#385—22no—
48#385—22no—
49#384—22no—
50#382—22no—
51#340—22no—
52#340—22no—
53#335—41no—
54#334—41no—
55#324—21no—
56#324—22no—
57#324D412yes—
58#323—21no—
59#323—22no—
60#323D412yes—
61#322—41no—
62#322—41no—
63C23×C4—22no—
64C23×C4C22×C411yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—41no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—22no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76#117—22no—
77#117—22no—
78#117—22no—
79#117—41no—
80#117—41no—
81#117M1614yes—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C4⋊C4#4111yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4C2×D411yescharacteristic
93C2×C22⋊C4#4111yescharacteristic
94C2×C42—22no—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97#107—24no—
98#107—42no—
99#107—41no—
100#107—41no—
101#106—24no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—42no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—42no—
108#97—41no—
109#97—41no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8—42no—
117#95—42no—
118#95—42no—
119C24C4×D411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—82no—
127C22×C4—162no—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C4×D411yescharacteristic
130C22×C4#12111yescharacteristic
131C22×C4#12111yescharacteristic
132C22×C4#12411yescharacteristic
133C22×C4#12711yescharacteristic
134C22×C4C2×M1612yes—
135C22×C4#13112yes—
136M16—44no—
137C2×C8—24no—
138C2×C8—24no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—82no—
151C2×C8#9911yescharacteristic
152C2×C8#9911yescharacteristic
153C2×C8#10112yes—
154C2×C8#10112yes—
155C4⋊C4—24no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—162no—
163C4⋊C4—162no—
164C22⋊C4—24no—
165C22⋊C4—41no—
166C22⋊C4—41no—
167C22⋊C4—42no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—24no—
173C42—42no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23#39011yescharacteristic
179C23#43512yes—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—164no—
201C2×C4#40711yescharacteristic
202C2×C4#40811yescharacteristic
203C2×C4C2×#9911yescharacteristic
204C2×C4#41011yescharacteristic
205C2×C4C2×#10111yescharacteristic
206C2×C4C2×#10111yescharacteristic
207C2×C4#43512yes—
208C8—21no—
209C8—21no—
210C8—22no—
211C8—22no—
212C8—41no—
213C8—82no—
214C8—82no—
215C8—162no—
216C22—21no—
217C22—21no—
218C22—22no—
219C22—42no—
220C22—42no—
221C22—41no—
222C22—41no—
223C22#195511yescharacteristic
224C22#196411yescharacteristic
225C22#196611yescharacteristic
226C4—22no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—84no—
232C4—162no—
233C4#146311yescharacteristic
234C4#146711yescharacteristic
235C4#147011yescharacteristic
236C4#147211yescharacteristic
237C2—21no—
238C2—21no—
239C2—42no—
240C2#965411yescharacteristic
241C2#1323111yescharacteristic
242C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
#211562
#114542
#114502
#86022
#81572