Finite Group Databasebeta

← #108111

Subgroups — #108111

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34403C211yesmaximal, characteristic
2#17596C211yesmaximal, characteristic
3#17491C211yesmaximal, characteristic
4#8069C211yesmaximal, characteristic
5#7951C211yesmaximal, characteristic
6#7944C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2378—22no—
13#2370—22no—
14#2356—22no—
15#2347—21no—
16#2346—21no—
17#2332—22no—
18#2331C412yescomplemented
19#2331C412yescomplemented
20#2320—22no—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24#1520—22no—
25#1515—22no—
26#1506C414yescomplemented
27#1497C414yescomplemented
28#1487—22no—
29#1485—22no—
30#1300—22no—
31#1285—22no—
32#1284—22no—
33#1284—21no—
34#1284—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8D412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8D412yes—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#432C2×C412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#423—21no—
65#423—21no—
66C2×C4⋊C8—21no—
67#407—21no—
68#407—21no—
69C2×C22⋊C8—21no—
70C4×M16—21no—
71C4×M16—21no—
72C2×#97—21no—
73#391—24no—
74#391—24no—
75#389—24no—
76#389—22no—
77#389—22no—
78#388—24no—
79#388—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#383—22no—
85#383—21no—
86#383—21no—
87#383—21no—
88#382—22no—
89#382—21no—
90#382—21no—
91#382C2×C412yes—
92#381—21no—
93#381—21no—
94#381D412yes—
95#380—21no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108C4×C22⋊C4D411yescharacteristic
109#376—22no—
110#376C2×C412yes—
111#376C2×C412yes—
112#376D412yes—
113#328—28no—
114Q8⋊C8—28no—
115C22×Q8—21no—
116C22×Q8C22⋊C412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—41no—
121C2×M16—41no—
122C22×C8—41no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129C4⋊Q8C4⋊C412yes—
130C4⋊Q8C4⋊C412yes—
131#125—22no—
132#125—22no—
133#125—22no—
134#125—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8C22⋊C414yes—
142C22⋊Q8C4⋊C414yes—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8C22⋊C414yes—
150C4×Q8C4⋊C414yes—
151#117—22no—
152#117—22no—
153#117—22no—
154#117—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#97—22no—
203#97—22no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—41no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224#95C22⋊C414yes—
225C24#12011yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8#9514yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259M16—82no—
260M16—82no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—82no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4#9514yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—81no—
285C22⋊C4—82no—
286C22⋊C4—81no—
287C22⋊C4—82no—
288C42—24no—
289C42—24no—
290C42—42no—
291C42—42no—
292C42—82no—
293C42—84no—
294C42—84no—
295C42—82no—
296C42C2×C4211yescharacteristic
297C42C2×C22⋊C411yescharacteristic
298C42C2×C4⋊C411yescharacteristic
299C42C2×C4⋊C411yescharacteristic
300C42#11711yescharacteristic
301C42#11711yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42#12311yescharacteristic
311C42#12311yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23#37911yescharacteristic
316C23#38511yescharacteristic
317C23#38611yescharacteristic
318Q8—44no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
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—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4C2×#9511yescharacteristic
358C2×C4C4×C22⋊C411yescharacteristic
359C2×C4C4×C22⋊C411yescharacteristic
360C2×C4#38011yescharacteristic
361C2×C4#38011yescharacteristic
362C2×C4#38111yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C8—82no—
370C8—82no—
371C8—82no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—82no—
381C22C22⋊#9511yescharacteristic
382C22#179612yes—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—44no—
397C4—82no—
398C4—84no—
399C4—82no—
400C4—162no—
401C2—22no—
402C2—41no—
403C2—41no—
404C2#1074311yescharacteristic
405C2#1077711yescharacteristic
406C2#1078311yescharacteristic
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. #2331C4semidirect
19. #2331C4semidirect
26. #1506C4semidirect
27. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#344032
#175962
#174912
#80692
#79512
#79442