Finite Group Databasebeta

← #6631389

Subgroups — #6631389

Order 512.

1459 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34627C212yesmaximal
2#32794C211yesmaximal, characteristic, complemented
3#32780C212yesmaximal, complemented
4#32768C211yesmaximal, characteristic, complemented
5#32752C214yesmaximal, complemented
6#32733C214yesmaximal, complemented
7#32714C214yesmaximal
8#23765C214yesmaximal, complemented
9#23709C214yesmaximal, complemented
10#23577C214yesmaximal, complemented
11#20719C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#441C2214yes—
15C2×#432C2212yes—
16#2638—22no—
17#2638—22no—
18#2634—22no—
19#2633—22no—
20#2628C2212yes—
21#2628C2212yes—
22#2625C2214yes—
23#2621C2212yes—
24#2621C2212yes—
25#2616C2214yes—
26#2399—22no—
27#2398—22no—
28#2395—22no—
29#2395—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32#2283C2212yes—
33#2101C2214yes—
34#2097C2214yes—
35#2088C2214yes—
36#2085C2214yes—
37#2077C2214yes—
38#2077C2214yes—
39#2076C2214yes—
40#2074C2214yes—
41#2068C2214yes—
42#2066C2214yes—
43#2058C2214yes—
44#2056C2214yes—
45#2043C2214yes—
46#2036C2214yes—
47#2036C2214yes—
48#2035C2214yes—
49#2035C2214yes—
50#2035C2214yes—
51#2035C2214yes—
52#2024C2214yes—
53#2016C2214yes—
54#2001C2214yes—
55#2001C2214yes—
56#2001C2214yes—
57#1818C2211yescharacteristic
58#1818C2211yescharacteristic
59#1798C2214yes—
60#1780C2212yes—
61#1776C2214yes—
62#1763C2214yes—
63#1763C2214yes—
64#1749C2214yes—
65C22×#41C2311yescharacteristic
66C22×Q16C2314yes—
67#533—44no—
68#533—44no—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×#125—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C2×C22⋊Q8C2314yes—
76C4×#41—44no—
77C2×C4×Q8—22no—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2314yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4D412yes—
83C2×#117—22no—
84C2×#117C2312yes—
85C2×#117C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88#441—48no—
89#432—44no—
90C2×#107C2314yes—
91C2×#106C2314yes—
92C2×#106C2314yes—
93C2×C4⋊C8C2314yes—
94C2×#103C2314yes—
95C2×#103C2314yes—
96C2×#103C2314yes—
97C2×#103C2314yes—
98C2×#103C2314yes—
99C2×#103C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2312yes—
104C2×#97C2314yes—
105#400—24no—
106#400—22no—
107#400—22no—
108#399—22no—
109#399—22no—
110#398—22no—
111#398—22no—
112#398—22no—
113#398—22no—
114#398—24no—
115#397—22no—
116#397—22no—
117#397—22no—
118#397—22no—
119#397—24no—
120#397—24no—
121#395—24no—
122#395—22no—
123#395—22no—
124#394—24no—
125#394—24no—
126#394—24no—
127#393—22no—
128#393—22no—
129#391C2314yes—
130#391C2314yes—
131#389C2314yes—
132#389C2314yes—
133#387—24no—
134#387—24no—
135#385—24no—
136#385—24no—
137#383—22no—
138#382C2312yes—
139#382C2312yes—
140#381—22no—
141#380C2314yes—
142#380C2314yes—
143#380C2314yes—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4C2314yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150#376—22no—
151#340C2314yes—
152#340C2314yes—
153#340C2314yes—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#336C2314yes—
158#336C2314yes—
159C2×#41—42no—
160C2×#41—44no—
161C22×Q8—22no—
162C22×Q8C2×D411yescharacteristic
163C22×Q8C2414yes—
164C22×D4C2×D412yes—
165C22×D4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C2×Q16—44no—
172C2×Q16—44no—
173C2×Q16—44no—
174C22×C8C2414yes—
175C22×C8C2414yes—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180#125—48no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—48no—
188C4×Q8—28no—
189C4×Q8—28no—
190C4×Q8—44no—
191C4×Q8—48no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—44no—
195#117—42no—
196#117—42no—
197#117—42no—
198#117—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C22⋊C4C2×D411yescharacteristic
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2412yes—
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235#107—44no—
236#106—44no—
237#106—44no—
238C4⋊C8—44no—
239#103—44no—
240#103—44no—
241#103—44no—
242#103—44no—
243#103—44no—
244#103—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249#97—28no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264C24C22×D412yes—
265C24C22×D411yescharacteristic
266#41—416no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—44no—
276C2×Q8—84no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C2511yescharacteristic
305Q16—416no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—84no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—48no—
347C42—48no—
348C42H(5,𝔽2)14yes—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—21no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23C23×D411yescharacteristic
358Q8—44no—
359Q8—44no—
360Q8—44no—
361Q8—44no—
362Q8—44no—
363Q8—88no—
364D4—48no—
365D4—44no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4#53414yes—
397C2×C4C2×H(5,𝔽2)12yes—
398C2×C4#58514yes—
399C8—44no—
400C8—44no—
401C22—42no—
402C22—41no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22C2×#53411yescharacteristic
410C22#344912yes—
411C22#358614yes—
412C4—24no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—84no—
419C4—84no—
420C4—84no—
421C4—84no—
422C4#352914yes—
423C2—82no—
424C2—81no—
425C2#3665211yescharacteristic
426C2#6075412yes—
427C2#6097514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32794C2semidirect
3. #32780C2semidirect
4. #32768C2semidirect
5. #32752C2semidirect
6. #32733C2semidirect
8. #23765C2semidirect
9. #23709C2semidirect
10. #23577C2semidirect

Maximal subgroups

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

subgroupindex
#346272
#327942
#327802
#327682
#327522
#327332
#327142
#237652
#237092
#235772
#207192