Finite Group Databasebeta

← #110710

Subgroups — #110710

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21407C211yesmaximal, characteristic
2#20481C211yesmaximal, characteristic
3C2×#1531C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5C2×#1302C211yesmaximal, characteristic
6C2×#1295C211yesmaximal, characteristic
7#8017C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#97C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993C412yescomplemented
16#1960C412yescomplemented
17#1892—22no—
18#1771—22no—
19#1763—22no—
20#1752—21no—
21C8×C22⋊C4—21no—
22#1746—21no—
23#1726—21no—
24#1531—22no—
25#1531C414yescomplemented
26C2×#330—21no—
27C2×#330—21no—
28#1518—22no—
29#1518C414yescomplemented
30C2×#328—21no—
31C2×#328—21no—
32#1302—22no—
33#1295—22no—
34#1294—22no—
35C23×C8—21no—
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431D412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×#103—24no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8D411yescharacteristic
68C2×C22⋊C8D411yescharacteristic
69C2×#97—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—22no—
78C2×#97—21no—
79C2×#97C2×C412yes—
80C2×#97C2×C412yes—
81C2×#97D412yes—
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#383—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#382C2×C412yes—
91#382C2×C412yes—
92#381—21no—
93#380D412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—21no—
96#376—21no—
97C2×#95—21no—
98C2×#95—21no—
99#340—24no—
100#336—22no—
101#336—22no—
102#330—24no—
103#330—24no—
104#328—24no—
105#328—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—44no—
120C22×C8—44no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8C4212yes—
131C4⋊Q8C4212yes—
132#125—22no—
133#125—22no—
134#125—41no—
135#125—41no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152#117—22no—
153#117—22no—
154#117—41no—
155#117—41no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C22⋊C414yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—41no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186#103—48no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8C22⋊C412yes—
192C22⋊C8C22⋊C412yes—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—21no—
199#97—21no—
200#97—44no—
201#97—41no—
202#97—41no—
203#97—41no—
204#97C22⋊C412yes—
205#97C22⋊C412yes—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—44no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95—41no—
217C24#12111yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12711yescharacteristic
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—82no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42C2×C4211yescharacteristic
294C42C2×C22⋊C411yescharacteristic
295C42#11711yescharacteristic
296C42#11711yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42#12311yescharacteristic
307C42#12411yescharacteristic
308C42#12611yescharacteristic
309C23—22no—
310C23—22no—
311C23#38611yescharacteristic
312C23#38811yescharacteristic
313C23#39011yescharacteristic
314C23#47112yes—
315C23#47312yes—
316Q8—44no—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
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—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—164no—
351C2×C4C4×C22⋊C411yescharacteristic
352C2×C4C4×C22⋊C411yescharacteristic
353C2×C4C4×C22⋊C411yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#39011yescharacteristic
362C2×C4#39011yescharacteristic
363C8—24no—
364C8—42no—
365C8—42no—
366C8—42no—
367C8—44no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22#143711yescharacteristic
376C22#188412yes—
377C22#190612yes—
378C22#190812yes—
379C22#193012yes—
380C22#197711yescharacteristic
381C22#197911yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—82no—
395C4—82no—
396C4—84no—
397C4—162no—
398C2—21no—
399C2—22no—
400C2—21no—
401C2#943114yes—
402C2#1094211yescharacteristic
403C2#1094511yescharacteristic
404C2#1095911yescharacteristic
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. #1960C4semidirect
25. #1531C4semidirect
29. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#214072
#204812
C2×#15312
C2×#15182
C2×#13022
C2×#12952
#80172