Finite Group Databasebeta

← #125555

Subgroups — #125555

Order 512.

400 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 72 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#15121C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7132C212yesmaximal
5#7122C212yesmaximal
6C2×#389C411yescharacteristic
7C2×#376C2211yescharacteristic
8#1817—21no—
9#1817—21no—
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1444C411yescharacteristic
13#1442—21no—
14#1442C2211yescharacteristic
15#1441—21no—
16#1436—21no—
17#1434—21no—
18#1434C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D411yescharacteristic
29C22×C42D411yescharacteristic
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32#389—42no—
33#376—42no—
34C2×#95—21no—
35C2×#95—21no—
36C2×#95—21no—
37C2×#95—21no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95C2×C411yescharacteristic
45C2×#95C2×C411yescharacteristic
46C2×#95C2×C411yescharacteristic
47C2×#95D411yescharacteristic
48C2×#95D411yescharacteristic
49C2×#95C2311yescharacteristic
50#336—42no—
51#336—42no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C22×C8—42no—
64C22×C8—42no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—81no—
74C2×C4⋊C4—81no—
75C2×C4⋊C4—81no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—41no—
83C2×C42—41no—
84C2×C42—41no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—42no—
88C22⋊C8—42no—
89#95—22no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—41no—
94#95—41no—
95#95—42no—
96#95—42no—
97#95—42no—
98#95—42no—
99#95—42no—
100#95—42no—
101#95—42no—
102#95—81no—
103#95—81no—
104#95—81no—
105C24C2×C22⋊C411yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4#9912yes—
168C22×C4#13112yes—
169C22×C4#13512yes—
170C22×C4#13712yes—
171C2×C8—82no—
172C2×C8—82no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8—82no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—82no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C42—44no—
183C42—44no—
184C42—44no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23#40811yescharacteristic
190C23C2×#9911yescharacteristic
191C23#41311yescharacteristic
192C23#41611yescharacteristic
193C23#41911yescharacteristic
194C23#42111yescharacteristic
195C23#42111yescharacteristic
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C8—162no—
262C8—162no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22#146811yescharacteristic
276C22#148211yescharacteristic
277C22#149711yescharacteristic
278C22#151111yescharacteristic
279C22#151611yescharacteristic
280C22#153511yescharacteristic
281C22#155411yescharacteristic
282C22#212011yescharacteristic
283C22#212311yescharacteristic
284C22#212611yescharacteristic
285C22#213411yescharacteristic
286C4—42no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—82no—
295C4—161no—
296C4—161no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2—21no—
301C2#772111yescharacteristic
302C2#1154811yescharacteristic
303C2#1155011yescharacteristic
304C2#1181011yescharacteristic
305C2#1181811yescharacteristic
306C2#1184611yescharacteristic
307C2#1184711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#151212
#86992
#71322
#71222