Finite Group Databasebeta

← #110675

Subgroups — #110675

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21840C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3#20544C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5#8021C211yesmaximal, characteristic
6C2×#1290C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993C414yescomplemented
16#1988—22no—
17#1988C414yescomplemented
18#1987—22no—
19#1823—22no—
20#1771—22no—
21#1768—22no—
22#1752—21no—
23C8×C22⋊C4—21no—
24C2×#340—21no—
25C2×#340—21no—
26#1558—22no—
27#1558C414yescomplemented
28C2×#333—22no—
29#1294—22no—
30#1290—22no—
31#1285—22no—
32C23×Q8—21no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42—21no—
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—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×C4⋊C8D412yes—
72C2×C4⋊C8D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×#97C2×C412yes—
80C2×C4×C8—22no—
81#383—21no—
82#382C2×C412yes—
83#381—21no—
84#380D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—21no—
91#376—21no—
92C2×#95—21no—
93#340—24no—
94#340—24no—
95#340—28no—
96#336—22no—
97#336—22no—
98#333—28no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—82no—
106C22×Q8—82no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—41no—
121C22×C8C22⋊C412yes—
122C22×C8C22⋊C412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8C4214yes—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42—82no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—41no—
195#97—82no—
196C4×C8—82no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—41no—
203C24#12111yescharacteristic
204C2×Q8—24no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—82no—
215C2×Q8—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4#12711yescharacteristic
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—81no—
262C2×C8—81no—
263C2×C8—82no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—42no—
279C42—42no—
280C42—82no—
281C42—82no—
282C42C2×C4211yescharacteristic
283C42C2×C22⋊C411yescharacteristic
284C42#11711yescharacteristic
285C42#11711yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12311yescharacteristic
296C42#12411yescharacteristic
297C42#12611yescharacteristic
298C23—22no—
299C23—22no—
300C23#38611yescharacteristic
301C23#38811yescharacteristic
302C23#39011yescharacteristic
303C23#47112yes—
304C23#47212yes—
305Q8—84no—
306Q8—84no—
307Q8—84no—
308Q8—84no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—88no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4C4×C22⋊C411yescharacteristic
343C2×C4C4×C22⋊C411yescharacteristic
344C2×C4C4×C22⋊C411yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#39011yescharacteristic
353C2×C4#39011yescharacteristic
354C2×C4#45512yes—
355C2×C4#45612yes—
356C8—44no—
357C8—42no—
358C8—42no—
359C8—82no—
360C8—82no—
361C22—24no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22#143711yescharacteristic
368C22#188212yes—
369C22#190712yes—
370C22#190912yes—
371C22#193112yes—
372C22#197711yescharacteristic
373C22#197811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4#188212yes—
388C4#188511yescharacteristic
389C4#188611yescharacteristic
390C2—21no—
391C2—22no—
392C2—21no—
393C2#937514yes—
394C2#1094311yescharacteristic
395C2#1094711yescharacteristic
396C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
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. #1993C4semidirect
17. #1988C4semidirect
27. #1558C4semidirect

Maximal subgroups

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

subgroupindex
#218402
C2×#19882
#205442
C2×#15582
#80212
C2×#12902
C2×#12852