Finite Group Databasebeta

← #107307

Subgroups — #107307

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21404C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#8073C211yesmaximal, characteristic
5#8069C211yesmaximal, characteristic
6#8064C211yesmaximal, characteristic
7#8050C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2985—22no—
11#2984—22no—
12#2983—22no—
13#2962C414yescomplemented
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#1993C412yescomplemented
23#1960C412yescomplemented
24#1894—22no—
25C4×#103—22no—
26#1515C414yescomplemented
27#1511—22no—
28#1503—22no—
29#1300—22no—
30#1300—22no—
31#1299—22no—
32#1297—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8D412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8D412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#446—28no—
52#446—28no—
53#446—28no—
54#446—24no—
55C8×Q8—24no—
56#433—28no—
57#432—24no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432D412yes—
65#432D412yes—
66#431—24no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#424—24no—
72#423—21no—
73#423D412yes—
74C2×C4⋊C8C2×C412yes—
75C2×#103—24no—
76#407—21no—
77#407D412yes—
78C2×C22⋊C8—21no—
79#405—24no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16C2×C412yes—
86C2×#97—21no—
87C2×#97C2×C412yes—
88C2×C4×C8—21no—
89#383—21no—
90#383C2×C412yes—
91#382—21no—
92#382C2×C412yes—
93#381—21no—
94#381—21no—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101#376—21no—
102#376—21no—
103#340—24no—
104#328—28no—
105C22×Q8—21no—
106C22×Q8C22⋊C412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—44no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16C22⋊C414yes—
115C22×C8—22no—
116C22×C8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4212yes—
122C4⋊Q8C4212yes—
123C4⋊Q8C4⋊C412yes—
124C4⋊Q8C4⋊C412yes—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8C22⋊C414yes—
136C22⋊Q8C4⋊C414yes—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8C22⋊C414yes—
144C4×Q8C4⋊C414yes—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C22⋊C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—28no—
176C4⋊C8—28no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183#103—48no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190#97—28no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197C4×C8—28no—
198C4×C8—22no—
199C4×C8—41no—
200C4×C8—41no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209#95—41no—
210C24#12111yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—82no—
216C2×Q8#9514yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242M16—44no—
243M16—44no—
244M16—44no—
245M16—82no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—44no—
265C4⋊C4—82no—
266C4⋊C4#9514yes—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C42—24no—
272C42—24no—
273C42—82no—
274C42—82no—
275C42C2×C4211yescharacteristic
276C42C2×C22⋊C411yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42C2×C4⋊C411yescharacteristic
279C42C2×C4⋊C411yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42#12011yescharacteristic
286C42#12011yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42#12311yescharacteristic
290C42#12311yescharacteristic
291C23—22no—
292C23—41no—
293C23—41no—
294C23#38011yescharacteristic
295C23#38611yescharacteristic
296C23#38611yescharacteristic
297Q8—44no—
298Q8—44no—
299Q8—82no—
300Q8—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
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—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C2×#9511yescharacteristic
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4#37911yescharacteristic
333C2×C4#38011yescharacteristic
334C2×C4#38111yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38511yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C8—44no—
342C8—44no—
343C8—44no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—82no—
354C22C22⋊#9511yescharacteristic
355C22#188412yes—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—44no—
369C4—162no—
370C4—162no—
371C4#179612yes—
372C2—22no—
373C2—41no—
374C2—41no—
375C2#1075111yescharacteristic
376C2#1077711yescharacteristic
377C2#1077811yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2962C4semidirect
22. #1993C4semidirect
23. #1960C4semidirect
26. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#214042
#175622
#80732
#80692
#80642
#80502