Finite Group Databasebeta

← #125549

Subgroups — #125549

Order 512.

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

normal subgroups only

This group has 416 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#15119C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7122C212yesmaximal
6C2×#391C411yescharacteristic
7C2×#376C2211yescharacteristic
8#1817—21no—
9#1817—21no—
10#1446C411yescharacteristic
11#1446C2211yescharacteristic
12#1445—21no—
13#1444C411yescharacteristic
14#1443—21no—
15#1442C2211yescharacteristic
16#1436—21no—
17#1434—21no—
18#1434C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C42C2×C411yescharacteristic
27C22×C42D411yescharacteristic
28C22×C42D411yescharacteristic
29C2×C22⋊C8—22no—
30C2×C22⋊C8—22no—
31#391—42no—
32#376—42no—
33C2×#95—21no—
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—
52C22×Q8—22no—
53C22×Q8—42no—
54C22×Q8—41no—
55C22×Q8—41no—
56C22×Q8—81no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C22×C8—42no—
69C22×C8—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—81no—
74C2×C4⋊C4—81no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—41no—
82C2×C42—41no—
83C2×C42—41no—
84C22⋊C8—42no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—42no—
88#95—22no—
89#95—41no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—41no—
94#95—42no—
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
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—82no—
109C2×Q8—84no—
110C2×Q8—84no—
111C2×Q8—82no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
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—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—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4#9912yes—
174C22×C4#13112yes—
175C22×C4#13512yes—
176C22×C4#13712yes—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C4⋊C4—84no—
184C4⋊C4—82no—
185C42—44no—
186C42—44no—
187C42—44no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23#40811yescharacteristic
193C23C2×#9911yescharacteristic
194C23#41311yescharacteristic
195C23#41611yescharacteristic
196C23#41911yescharacteristic
197C23#42111yescharacteristic
198C23#42111yescharacteristic
199Q8—84no—
200Q8—84no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
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—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—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C8—162no—
267C8—162no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22#146811yescharacteristic
281C22#148211yescharacteristic
282C22#149711yescharacteristic
283C22#151111yescharacteristic
284C22#151611yescharacteristic
285C22#153511yescharacteristic
286C22#155411yescharacteristic
287C22#212011yescharacteristic
288C22#212211yescharacteristic
289C22#212911yescharacteristic
290C22#213211yescharacteristic
291C4—42no—
292C4—42no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—82no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—161no—
301C4—161no—
302C2—21no—
303C2—21no—
304C2—21no—
305C2—21no—
306C2#772111yescharacteristic
307C2#1154211yescharacteristic
308C2#1155611yescharacteristic
309C2#1181211yescharacteristic
310C2#1181611yescharacteristic
311C2#1184211yescharacteristic
312C2#1185111yescharacteristic
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
#151192
#86992
#71312
#71222