Finite Group Databasebeta

← #207279

Subgroups — #207279

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3#13807C211yesmaximal, characteristic
4#9913C214yesmaximal, complemented
5#9910C214yesmaximal, complemented
6#8019C212yesmaximal
7#8015C212yesmaximal
8C2×#407C2212yes—
9#2851C2214yes—
10#2646—21no—
11#2646—21no—
12#2569—24no—
13#2442C414yescomplemented
14#2442C414yescomplemented
15#2315—22no—
16C2×#381C2211yescharacteristic
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1851—24no—
21#1797C2214yes—
22#1437—22no—
23#1434—21no—
24#1434C2212yes—
25C4×#95—21no—
26C4×#95C2212yes—
27#1294C2218yes—
28#1294C2214yes—
29#1294C2214yes—
30C24×C4C2311yescharacteristic
31C22×M16—22no—
32C2×#125C2×C414yes—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4—22no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42—21no—
40C22×C42—22no—
41C22×C42C2312yes—
42C2×C4⋊C8—24no—
43#407—24no—
44#407—24no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2314yes—
47#400—24no—
48#400—24no—
49#398—24no—
50#398—24no—
51#389C2×C414yes—
52#389C2×C414yes—
53#387—24no—
54#387—24no—
55#385—24no—
56#385—24no—
57#385—24no—
58#385—24no—
59#382C2×C414yes—
60#381—22no—
61#381Q814yes—
62#380—22no—
63#380—22no—
64#380—24no—
65#380—24no—
66#380C2×C414yes—
67#380C2314yes—
68C4×C22⋊C4—22no—
69C4×C22⋊C4C2×C414yes—
70C4×C22⋊C4D414yes—
71C4×C22⋊C4Q814yes—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79C2×#95C2311yescharacteristic
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82#336—24no—
83C25C2×D411yescharacteristic
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—82no—
90C23×C4C22×C412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×Q812yes—
96C23×C4C2×Q812yes—
97C23×C4#4112yes—
98C23×C4#4112yes—
99C23×C4C2411yescharacteristic
100C2×M16—44no—
101C2×M16—82no—
102C2×M16—82no—
103C22×C8—24no—
104#125—84no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C22×C414yes—
114C2×C4⋊C4C22×C414yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4C22⋊C414yes—
127C2×C22⋊C4C22⋊C414yes—
128C2×C42—24no—
129C2×C42—44no—
130C2×C42—44no—
131C2×C42—82no—
132C2×C42—82no—
133C2×C42—82no—
134C2×C42C22×C414yes—
135C2×C42C2×D414yes—
136C4⋊C8—84no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—44no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—21no—
148#95—21no—
149#95—22no—
150#95—44no—
151#95—44no—
152#95—82no—
153#95C22⋊C414yes—
154#95C22⋊C414yes—
155#95C2×Q812yes—
156#95C2×Q812yes—
157#95C2×Q814yes—
158C24—22no—
159C24—22no—
160C24—41no—
161C24—41no—
162C24—42no—
163C24C2×C22⋊C412yes—
164C24C22⋊Q812yes—
165C24C22⋊Q812yes—
166C24C22×D411yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4C2×C22⋊C414yes—
211C22×C4C2×C22⋊C414yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C4×Q814yes—
214C22×C4C4×Q814yes—
215C22×C4C22⋊Q812yes—
216C22×C4C22⋊Q812yes—
217C22×C4#12414yes—
218C22×C4C4⋊Q814yes—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×Q811yescharacteristic
222C22×C4C22×Q811yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C22×C4C2×#4111yescharacteristic
225M16—88no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—48no—
240C22⋊C4—48no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C42—44no—
245C42—44no—
246C42—84no—
247C42—88no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—82no—
265C23—82no—
266C23—82no—
267C23—82no—
268C23#39114yes—
269C23C22×C22⋊C411yescharacteristic
270C23C2×C22⋊Q811yescharacteristic
271C23C2×C22⋊Q811yescharacteristic
272C23#57512yes—
273C23#57512yes—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—164no—
322C2×C4—164no—
323C2×C4#39114yes—
324C2×C4C2×C4×Q811yescharacteristic
325C2×C4C2×C4×Q811yescharacteristic
326C2×C4C2×#12411yescharacteristic
327C2×C4C2×C4⋊Q811yescharacteristic
328C8—84no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—82no—
345C22—82no—
346C22—84no—
347C22C2×#39111yescharacteristic
348C22#288312yes—
349C22#289612yes—
350C22#290111yescharacteristic
351C22#290111yescharacteristic
352C22#308011yescharacteristic
353C22#308011yescharacteristic
354C22#308012yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—24no—
360C4—44no—
361C4—44no—
362C4—44no—
363C4—82no—
364C4—84no—
365C4—84no—
366C4—84no—
367C2—21no—
368C2—21no—
369C2—22no—
370C2—44no—
371C2#1762312yes—
372C2#1762512yes—
373C2#2131411yescharacteristic
374C2#2133011yescharacteristic
375C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9910C2semidirect
13. #2442C4semidirect
14. #2442C4semidirect

Maximal subgroups

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

subgroupindex
#350202
#219342
#138072
#99132
#99102
#80192
#80152