Finite Group Databasebeta

← #125231

Subgroups — #125231

Order 512.

432 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 78 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#15085C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7132C212yesmaximal
5#7104C212yesmaximal
6C2×#400C411yescharacteristic
7C2×#398C411yescharacteristic
8C2×#382C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1817—21no—
11C4×C22⋊C8—21no—
12C2×#324—21no—
13#1445C2211yescharacteristic
14#1445C2211yescharacteristic
15#1443—21no—
16#1441—21no—
17#1436—21no—
18#1434—21no—
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42D411yescharacteristic
30C22×C42D411yescharacteristic
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C4×C8—41no—
34#400—42no—
35#398—42no—
36#382—41no—
37#382—41no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95C2311yescharacteristic
55#336—41no—
56#336—41no—
57#336—41no—
58#324—24no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C22×C8—42no—
71C22×C8—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—41no—
95C2×C42—42no—
96C2×C42—42no—
97C22⋊C8—24no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C4×C8—82no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—41no—
113#95—41no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118C24C2×C22⋊C411yescharacteristic
119C22×C4—21no—
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—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
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—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#9911yescharacteristic
179C22×C4#9911yescharacteristic
180C22×C4#10012yes—
181C22×C4#10412yes—
182C22×C4#13112yes—
183C22×C4#13512yes—
184C22×C4#13512yes—
185C2×C8—44no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—81no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23#40811yescharacteristic
207C23C2×#9911yescharacteristic
208C23C2×#10011yescharacteristic
209C23#41611yescharacteristic
210C23#41611yescharacteristic
211C23C2×#10411yescharacteristic
212C23#42111yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
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—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4#35212yes—
281C8—84no—
282C8—162no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22#147111yescharacteristic
296C22#148211yescharacteristic
297C22#149411yescharacteristic
298C22#150111yescharacteristic
299C22#151611yescharacteristic
300C22#152711yescharacteristic
301C22#154311yescharacteristic
302C22C2×#35211yescharacteristic
303C22#212311yescharacteristic
304C22#212611yescharacteristic
305C22#213311yescharacteristic
306C4—22no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—42no—
313C4—82no—
314C4—81no—
315C4—161no—
316C4—161no—
317C4—161no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2—21no—
322C2#767011yescharacteristic
323C2#1155811yescharacteristic
324C2#1156511yescharacteristic
325C2#1172711yescharacteristic
326C2#1174011yescharacteristic
327C2#1184611yescharacteristic
328C2#1185011yescharacteristic
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×C4×C4⋊C4 ◁ 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
#150852
#87012
C4×#3242
#71322
#71042