Finite Group Databasebeta

← #5486498

Subgroups — #5486498

Order 512.

1872 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17308C211yesmaximal, characteristic, complemented
2#17127C214yesmaximal, complemented
3#16480C214yesmaximal, complemented
4#16438C214yesmaximal
5#15917C214yesmaximal, complemented
6#15791C214yesmaximal, complemented
7#15762C211yesmaximal, characteristic, complemented
8#15761C211yesmaximal, characteristic
9#15761C211yesmaximal, characteristic
10#15095C214yesmaximal, complemented
11#14663C211yesmaximal, characteristic, complemented
12#14662C211yesmaximal, characteristic
13#14161C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C2×#400C2214yes—
18C2×#397C2214yes—
19C2×#395C2212yes—
20C2×#395C2212yes—
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#393C2212yes—
24C2×#393C2212yes—
25C2×#388C2214yes—
26C2×#383C2212yes—
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2212yes—
29#1446C2211yescharacteristic
30#1446C2211yescharacteristic
31#1446C2212yes—
32#1446C2211yescharacteristic
33#1446C2211yescharacteristic
34#1446C2212yes—
35#1445C2214yes—
36#1445C2214yes—
37#1444C2214yes—
38#1444C2214yes—
39#1443C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1437C2212yes—
57#1437C2214yes—
58#1437C2212yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66#1434C2214yes—
67C23×Q8C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—44no—
74C2×C4⋊Q8—44no—
75C2×#124—24no—
76C2×#124—44no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2314yes—
97#400—48no—
98#397—48no—
99#395—44no—
100#395—44no—
101#394—44no—
102#394—44no—
103#393—44no—
104#393—44no—
105#388—48no—
106#383—44no—
107#382—44no—
108C4×C4⋊C4—44no—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C25C2412yes—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8#4114yes—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2411yescharacteristic
162C23×C4C2411yescharacteristic
163C23×C4C2411yescharacteristic
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C4⋊Q8—416no—
168C4⋊Q8—416no—
169#124—416no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—28no—
225C2×C42—44no—
226C2×C42—44no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—44no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24C2511yescharacteristic
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—82no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×D4—44no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—88no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)14yes—
344C22×C4H(5,𝔽2)14yes—
345C22×C4H(5,𝔽2)14yes—
346C22×C42−1+418yes—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—416no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—416no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C42—28no—
364C42—28no—
365C42—28no—
366C42—88no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23C22×#4111yescharacteristic
389C23C2×H(5,𝔽2)12yes—
390C23C2×H(5,𝔽2)12yes—
391C23C2×H(5,𝔽2)12yes—
392C23C2×2−1+414yes—
393C23#58514yes—
394Q8—88no—
395Q8—88no—
396Q8—88no—
397D4—88no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4#56318yes—
463C22—84no—
464C22—82no—
465C22—82no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—82no—
470C22—84no—
471C22—84no—
472C22#349011yescharacteristic
473C22#349011yescharacteristic
474C22#349011yescharacteristic
475C22#351614yes—
476C22C2×#56312yes—
477C22#354114yes—
478C22#354412yes—
479C22#354512yes—
480C22#354512yes—
481C22#354914yes—
482C22#355012yes—
483C22#355112yes—
484C22#355414yes—
485C22#355914yes—
486C4—28no—
487C4—44no—
488C4—44no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—84no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4—84no—
497C4—84no—
498C2—162no—
499C2#4634812yes—
500C2#4756311yescharacteristic
501C2#5184012yes—
502C2#5198812yes—
503C2#5310914yes—
504C2#5627914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17308C2semidirect
2. #17127C2semidirect
3. #16480C2semidirect
5. #15917C2semidirect
6. #15791C2semidirect
7. #15762C2semidirect
10. #15095C2semidirect
11. #14663C2semidirect
13. #14161C2semidirect

Maximal subgroups

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

subgroupindex
#173082
#171272
#164802
#164382
#159172
#157912
#157622
#157612
#150952
#146632
#146622
#141612