Finite Group Databasebeta

← #108102

Subgroups — #108102

Order 512.

827 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34365C211yesmaximal, characteristic
2#21811C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#8050C211yesmaximal, characteristic
5#7978C211yesmaximal, characteristic
6#7951C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2381—22no—
13#2369—22no—
14#2344—22no—
15#2344—22no—
16#2331—22no—
17#2331C414yescomplemented
18#2310—22no—
19#2283C2211yescharacteristic
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#1981—22no—
23#1960C414yescomplemented
24#1787—22no—
25#1558—22no—
26#1548—22no—
27#1548C414yescomplemented
28#1297—22no—
29#1288—22no—
30#1285—22no—
31#1284—22no—
32C23×Q8D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q812yes—
45C2×#117—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#423—21no—
62#423C2×C412yes—
63C2×C4⋊C8—21no—
64C2×#103—28no—
65#407—21no—
66#407—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C2×#97C2×C412yes—
71#391—28no—
72#391—28no—
73#389—24no—
74#389—24no—
75#387—28no—
76#387—28no—
77#383—21no—
78#383—21no—
79#383D412yes—
80#382—21no—
81#382C2×C412yes—
82#382C2×C412yes—
83#381—24no—
84#381—21no—
85#381—21no—
86#381—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—21no—
96C4×C22⋊C4D412yes—
97C4×C22⋊C4D412yes—
98#376—21no—
99#376—21no—
100#376C2×C412yes—
101#333—28no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—84no—
106C22×Q8C22⋊C414yes—
107C22×Q8C22⋊C414yes—
108C22×Q8C4⋊C414yes—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C22×C8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8C4214yes—
129C4⋊Q8C22⋊C414yes—
130C4⋊Q8C4⋊C414yes—
131C4⋊Q8C4⋊C414yes—
132#117—88no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—41no—
152C2×C42—22no—
153C2×C42—28no—
154C2×C42—41no—
155C2×C42C22⋊C414yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174#103—88no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181#97—22no—
182#97—22no—
183#97—41no—
184#97—41no—
185C4×C8—22no—
186C4×C8—41no—
187#95—28no—
188#95—28no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95—41no—
200#95C22⋊C414yes—
201C24#12011yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—48no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—84no—
212C2×Q8#9518yes—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241M16—82no—
242M16—82no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—88no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—82no—
262C42—44no—
263C42—44no—
264C42—82no—
265C42—88no—
266C42C2×C4211yescharacteristic
267C42C2×C22⋊C411yescharacteristic
268C42C2×C4⋊C411yescharacteristic
269C42C2×C4⋊C411yescharacteristic
270C42#11711yescharacteristic
271C42#11711yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42#12311yescharacteristic
281C42#12311yescharacteristic
282C23—22no—
283C23—22no—
284C23—41no—
285C23#37911yescharacteristic
286C23#38511yescharacteristic
287C23#38611yescharacteristic
288Q8—48no—
289Q8—84no—
290Q8—84no—
291Q8—84no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—28no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×#9511yescharacteristic
325C2×C4C4×C22⋊C411yescharacteristic
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4#38011yescharacteristic
328C2×C4#38011yescharacteristic
329C2×C4#38111yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38611yescharacteristic
336C8—82no—
337C8—82no—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22C22⋊#9511yescharacteristic
348C22#179612yes—
349C22#191112yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—48no—
362C4—84no—
363C4—84no—
364C4—162no—
365C4#190512yes—
366C2—22no—
367C2—22no—
368C2—41no—
369C2#1074011yescharacteristic
370C2#1077711yescharacteristic
371C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2331C4semidirect
23. #1960C4semidirect
27. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#343652
#218112
#177462
#80502
#79782
#79512
#79442