Finite Group Databasebeta

← #110689

Subgroups — #110689

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21842C211yesmaximal, characteristic
2#20477C211yesmaximal, characteristic
3C2×#1515C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5#8025C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1287C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×#383C2211yescharacteristic
15#1993C412yescomplemented
16#1993C412yescomplemented
17#1989—22no—
18#1825—22no—
19#1771—22no—
20#1753—22no—
21#1746—21no—
22#1726—21no—
23#1515—22no—
24#1515C414yescomplemented
25C2×#328—22no—
26#1485—22no—
27#1485C414yescomplemented
28C2×Q8⋊C8—21no—
29C2×Q8⋊C8—21no—
30#1294—22no—
31#1288—22no—
32#1287—22no—
33C22×M16—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—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8D412yes—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C4×M16D412yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—22no—
76C2×#97C2×C412yes—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81#383—21no—
82#383—21no—
83#383—21no—
84#383—21no—
85#383C2×C412yes—
86#382C2×C412yes—
87#381—21no—
88#380D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—21no—
91#376—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95#340—24no—
96#340—24no—
97#336—24no—
98#328—28no—
99Q8⋊C8—24no—
100Q8⋊C8—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—44no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4212yes—
122C4⋊Q8C4212yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C22⋊C414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8C22⋊C414yes—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—44no—
189C4×C8—22no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—44no—
193C4×C8—41no—
194C4×C8C22⋊C412yes—
195C4×C8C22⋊C412yes—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—41no—
206C24#12111yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—44no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—82no—
213C22×C4—24no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12711yescharacteristic
241M16—44no—
242M16—44no—
243C2×C8—24no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42C2×C4211yescharacteristic
275C42C2×C22⋊C411yescharacteristic
276C42#11711yescharacteristic
277C42#11711yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12311yescharacteristic
288C42#12411yescharacteristic
289C42#12611yescharacteristic
290C23—22no—
291C23—22no—
292C23—22no—
293C23#38611yescharacteristic
294C23#38811yescharacteristic
295C23#39011yescharacteristic
296Q8—44no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—164no—
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4#39011yescharacteristic
341C8—24no—
342C8—44no—
343C8—42no—
344C8—82no—
345C8—82no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22#143711yescharacteristic
354C22#188112yes—
355C22#190612yes—
356C22#190912yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—84no—
370C4—82no—
371C4—82no—
372C4—162no—
373C2—41no—
374C2—41no—
375C2#936914yes—
376C2#1094511yescharacteristic
377C2#1094711yescharacteristic
378C2#1095611yescharacteristic
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
16. #1993C4semidirect
24. #1515C4semidirect
27. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218422
#204772
C2×#15152
C2×#14852
#80252
C2×#12882
C2×#12872