Finite Group Databasebeta

← #110640

Subgroups — #110640

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21394C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#13797C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1988C412yescomplemented
16#1986—22no—
17#1986C414yescomplemented
18#1981C412yescomplemented
19#1872—22no—
20#1763—22no—
21C2×#339—22no—
22#1506—22no—
23#1506C414yescomplemented
24C2×#328—22no—
25C4⋊#95—22no—
26#1434—22no—
27#1434—22no—
28#1300—22no—
29#1285—22no—
30#1279—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D412yes—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2×C412yes—
64C2×#103—24no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4C2×C412yes—
80C4×C4⋊C4C2×C412yes—
81C4×C4⋊C4D412yes—
82C4×C4⋊C4D412yes—
83#376—21no—
84#376—21no—
85#376C2×C412yes—
86#376D412yes—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95D412yes—
93#340—24no—
94#339—28no—
95#328—28no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—24no—
105C2×M16—22no—
106C2×M16—42no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4212yes—
118C4⋊Q8C4212yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—44no—
162C2×C42—41no—
163C2×C42—44no—
164C2×C42—82no—
165C2×C42C22⋊C414yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—41no—
182#103—48no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191#97—41no—
192#97—41no—
193C4×C8—22no—
194C4×C8—41no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—84no—
203#95—84no—
204#95C22⋊C414yes—
205C24#12011yescharacteristic
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—44no—
211C2×Q8—81no—
212C2×Q8—81no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255M16—24no—
256M16—44no—
257M16—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—81no—
264C2×C8—81no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—44no—
285C42—44no—
286C42—44no—
287C42—42no—
288C42—42no—
289C42—82no—
290C42—82no—
291C42—82no—
292C42C2×C4211yescharacteristic
293C42#11711yescharacteristic
294C42#11711yescharacteristic
295C42#11711yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42#12411yescharacteristic
305C42#12611yescharacteristic
306C42#12611yescharacteristic
307C42#12711yescharacteristic
308C23—22no—
309C23—21no—
310C23—21no—
311C23#38611yescharacteristic
312C23#38611yescharacteristic
313C23#38611yescharacteristic
314C23#45512yes—
315C23#45612yes—
316Q8—44no—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C4×C22⋊C411yescharacteristic
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#38811yescharacteristic
372C2×C4#39011yescharacteristic
373C2×C4#39011yescharacteristic
374C2×C4#39011yescharacteristic
375C2×C4#42114yes—
376C8—44no—
377C8—82no—
378C8—82no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22#143711yescharacteristic
388C22#188212yes—
389C22#188212yes—
390C22#188511yescharacteristic
391C22#188611yescharacteristic
392C22#189712yes—
393C22#190411yescharacteristic
394C22#190511yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—84no—
406C4—84no—
407C4—84no—
408C4—82no—
409C4—82no—
410C4#175912yes—
411C4#196312yes—
412C2—21no—
413C2—22no—
414C2—21no—
415C2#932712yes—
416C2#932812yes—
417C2#1092411yescharacteristic
418C2#1095411yescharacteristic
419C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1988C4semidirect
17. #1986C4semidirect
18. #1981C4semidirect
23. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#213942
C2×#19862
C2×#15062
#137972
C2×#13002
C2×#12852
#79042