Finite Group Databasebeta

← #107981

Subgroups — #107981

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34389C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17715C211yesmaximal, characteristic
4#8025C211yesmaximal, characteristic
5#7975C211yesmaximal, characteristic
6#7945C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2381—21no—
13#2381—21no—
14#2368—22no—
15#2356—22no—
16#2343—22no—
17#2336C414yescomplemented
18#2329—22no—
19#2320—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C2×#383C2211yescharacteristic
23#1995—22no—
24#1986C414yescomplemented
25#1826—22no—
26#1552—22no—
27#1549—21no—
28#1546—21no—
29#1544C414yescomplemented
30#1294—22no—
31#1288—22no—
32#1284—22no—
33#1274—22no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125C2×C412yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117D412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431C2×C412yes—
63#423—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66#407—21no—
67#407—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71C2×C4×C8—21no—
72#391—24no—
73#391—24no—
74#389—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#385—24no—
82#383—22no—
83#383—21no—
84#383—21no—
85#383—21no—
86#383—21no—
87#383C2×C412yes—
88#383D412yes—
89#382—22no—
90#382—21no—
91#382—21no—
92#381—21no—
93#381—21no—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4D412yes—
103#376—22no—
104#376—21no—
105#376D412yes—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95D412yes—
109#339—28no—
110#332—24no—
111#332—24no—
112C22×Q8—21no—
113C22×Q8—21no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—22no—
119C2×M16—41no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C22⋊C412yes—
126C4⋊Q8C22⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129#125—22no—
130#125—22no—
131#125C4214yes—
132#125C4⋊C414yes—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—22no—
150#117—22no—
151#117C22⋊C414yes—
152#117C4⋊C414yes—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42C22⋊C414yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—41no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—41no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95C22⋊C414yes—
219C24#12011yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—82no—
225C2×Q8—81no—
226C2×Q8—81no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256M16—44no—
257M16—82no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—82no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—82no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—82no—
276C4⋊C4#9514yes—
277C4⋊C4#9514yes—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C22⋊C4—82no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—44no—
286C42—84no—
287C42—84no—
288C42C2×C4211yescharacteristic
289C42C2×C22⋊C411yescharacteristic
290C42C2×C4⋊C411yescharacteristic
291C42C2×C4⋊C411yescharacteristic
292C42#11711yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42#12311yescharacteristic
303C42#12311yescharacteristic
304C23—22no—
305C23—22no—
306C23—41no—
307C23#37911yescharacteristic
308C23#38511yescharacteristic
309C23#38611yescharacteristic
310Q8—82no—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—164no—
347C2×C4C2×#9511yescharacteristic
348C2×C4C4×C22⋊C411yescharacteristic
349C2×C4C4×C22⋊C411yescharacteristic
350C2×C4#38011yescharacteristic
351C2×C4#38011yescharacteristic
352C2×C4#38111yescharacteristic
353C2×C4#38511yescharacteristic
354C2×C4#38511yescharacteristic
355C2×C4#38511yescharacteristic
356C2×C4#38611yescharacteristic
357C2×C4#38611yescharacteristic
358C2×C4#38611yescharacteristic
359C8—82no—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—44no—
370C22C22⋊#9511yescharacteristic
371C22#179412yes—
372C22#191212yes—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—44no—
386C4—44no—
387C4—84no—
388C4—84no—
389C4—162no—
390C2—22no—
391C2—22no—
392C2—41no—
393C2#1074111yescharacteristic
394C2#1077911yescharacteristic
395C2#1079211yescharacteristic
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. #2336C4semidirect
24. #1986C4semidirect
29. #1544C4semidirect

Maximal subgroups

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

subgroupindex
#343892
#218452
#177152
#80252
#79752
#79452
#78582