Finite Group Databasebeta

← #110695

Subgroups — #110695

Order 512.

604 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#21870C211yesmaximal, characteristic
2#20537C211yesmaximal, characteristic
3C2×#1537C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#8044C211yesmaximal, characteristic
6C2×#1290C211yesmaximal, characteristic
7C2×#1287C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2285C2211yescharacteristic
15#1993C412yes—
16#1960C412yes—
17#1873—22no—
18#1803—22no—
19#1771—22no—
20#1770—21no—
21C8×C4⋊C4—21no—
22#1750—22no—
23#1537—22no—
24#1537C414yescomplemented
25C2×#330—22no—
26#1481—22no—
27#1481C414yescomplemented
28C2×Q8⋊C8—21no—
29C2×Q8⋊C8—21no—
30#1296—22no—
31#1290—22no—
32#1287—22no—
33C23×C8—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117D412yes—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#432C2×C412yes—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2×C412yes—
78C2×#103—24no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×#97—21no—
84C2×#97—22no—
85C2×#97C2×C412yes—
86C2×#97D412yes—
87C2×C4×C8—22no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90#384D412yes—
91#382C2×C412yes—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#340—24no—
97#336—22no—
98#336—22no—
99#330—28no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4111yescharacteristic
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—82no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4212yes—
123C4⋊Q8C4212yes—
124#125—22no—
125#125—22no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148#117—82no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186#103—84no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—82no—
201#97C22⋊C414yes—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—82no—
205C4×C8C22⋊C412yes—
206C4×C8C22⋊C412yes—
207#95—22no—
208#95—22no—
209C24#12311yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12611yescharacteristic
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—82no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42C2×C4211yescharacteristic
285C42C2×C22⋊C411yescharacteristic
286C42C2×C22⋊C411yescharacteristic
287C42#11711yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C42#12411yescharacteristic
299C42#12711yescharacteristic
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23#38611yescharacteristic
305C23#38811yescharacteristic
306C23#38811yescharacteristic
307Q8—44no—
308Q8—44no—
309Q8—82no—
310Q8—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
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—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4C4×C22⋊C411yescharacteristic
341C2×C4C4×C22⋊C411yescharacteristic
342C2×C4C4×C22⋊C411yescharacteristic
343C2×C4#38611yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#39011yescharacteristic
350C2×C4#39011yescharacteristic
351C2×C4#39011yescharacteristic
352C2×C4#47112yes—
353C2×C4#47312yes—
354C8—42no—
355C8—42no—
356C8—44no—
357C8—82no—
358C8—82no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—42no—
367C22—42no—
368C22#143711yescharacteristic
369C22#188112yes—
370C22#190812yes—
371C22#191111yescharacteristic
372C22#191211yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—84no—
384C4—82no—
385C4—82no—
386C4#193012yes—
387C4#197711yescharacteristic
388C4#197911yescharacteristic
389C2—41no—
390C2—41no—
391C2#937312yes—
392C2#937412yes—
393C2#1094211yescharacteristic
394C2#1094911yescharacteristic
395C2#1095611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
24. #1537C4semidirect
27. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#218702
#205372
C2×#15372
C2×#14812
#80442
C2×#12902
C2×#12872