Finite Group Databasebeta

← #107907

Subgroups — #107907

Order 512.

789 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34221C211yesmaximal, characteristic
2#17718C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#8072C211yesmaximal, characteristic
5#8013C211yesmaximal, characteristic
6#7944C211yesmaximal, characteristic
7#7857C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2371—22no—
13#2368—22no—
14#2356—22no—
15#2343—22no—
16#2331—21no—
17#2327—21no—
18#2320C412yescomplemented
19C4×C22⋊Q8—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#2276C412yescomplemented
23C2×#376C2211yescharacteristic
24#1554C414yescomplemented
25#1548—21no—
26#1546—21no—
27#1542—22no—
28#1518C414yescomplemented
29#1515—22no—
30#1508—22no—
31#1300—22no—
32#1294—22no—
33#1284—22no—
34#1274—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8D412yes—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×#117—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8—21no—
69#407—21no—
70#407—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C2×C4×C8—21no—
75#391—24no—
76#391—22no—
77#389—22no—
78#387—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#385—24no—
84#383—21no—
85#383—22no—
86#383—21no—
87#382—21no—
88#382—22no—
89#382C2×C412yes—
90#381—21no—
91#381—21no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2×C412yes—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4D412yes—
103#376—21no—
104#376—21no—
105#376—22no—
106#376—21no—
107#376C2×C411yescharacteristic
108#376C2×C411yescharacteristic
109#376D412yes—
110#376D412yes—
111C2×#95—21no—
112C2×#95D412yes—
113#333—24no—
114#332—24no—
115#328—28no—
116C22×Q8—21no—
117C22×Q8C22⋊C412yes—
118C23×C4—22no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—41no—
123C2×M16—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C22⋊C412yes—
130C4⋊Q8C22⋊C412yes—
131C4⋊Q8C4⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133#125—22no—
134#125—22no—
135#125—22no—
136#125—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8C22⋊C414yes—
144C22⋊Q8C4⋊C414yes—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—48no—
152C4×Q8C4214yes—
153C4×Q8C4⋊C414yes—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—24no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—44no—
178C2×C42—41no—
179C2×C42C22⋊C414yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—22no—
205#97—41no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8—41no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228C24#12011yescharacteristic
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—44no—
233C2×Q8—82no—
234C2×Q8#9514yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265M16—82no—
266M16—82no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—82no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4#9514yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—44no—
290C22⋊C4—82no—
291C42—24no—
292C42—24no—
293C42—24no—
294C42—44no—
295C42—42no—
296C42—44no—
297C42—48no—
298C42—82no—
299C42—84no—
300C42C2×C4211yescharacteristic
301C42C2×C22⋊C411yescharacteristic
302C42C2×C4⋊C411yescharacteristic
303C42C2×C4⋊C411yescharacteristic
304C42#11711yescharacteristic
305C42#11711yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C23—22no—
317C23—22no—
318C23—41no—
319C23#37911yescharacteristic
320C23#38511yescharacteristic
321C23#38611yescharacteristic
322Q8—44no—
323Q8—44no—
324Q8—82no—
325Q8—82no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4C2×#9511yescharacteristic
363C2×C4C4×C22⋊C411yescharacteristic
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4#38011yescharacteristic
366C2×C4#38011yescharacteristic
367C2×C4#38111yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38611yescharacteristic
372C2×C4#38611yescharacteristic
373C2×C4#38611yescharacteristic
374C8—82no—
375C8—82no—
376C8—82no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22C22⋊#9511yescharacteristic
386C22#179412yes—
387C22#190512yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—44no—
400C4—44no—
401C4—44no—
402C4—84no—
403C4—82no—
404C4—162no—
405C4#182712yes—
406C2—22no—
407C2—22no—
408C2—41no—
409C2#1075811yescharacteristic
410C2#1077911yescharacteristic
411C2#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
18. #2320C4semidirect
22. #2276C4semidirect
24. #1554C4semidirect
28. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#342212
#177182
#175842
#80722
#80132
#79442
#78572