Finite Group Databasebeta

← #5025102

Subgroups — #5025102

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2660C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic
4C2×#2381C211yesmaximal, characteristic, complemented
5#15786C214yesmaximal, complemented
6#15769C214yesmaximal, complemented
7#15739C214yesmaximal, complemented
8#15648C214yesmaximal, complemented
9#15500C214yesmaximal, complemented
10#15321C214yesmaximal
11#14289C211yesmaximal, characteristic, complemented
12#14164C211yesmaximal, characteristic, complemented
13#13959C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#124C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2660—28no—
19#2633—28no—
20#2547—28no—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#394C2212yes—
30#2381—28no—
31C2×#391C2214yes—
32C2×#391C2212yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#383C2211yescharacteristic
39C2×#383C2211yescharacteristic
40C2×#382C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1445C2212yes—
45#1445C2212yes—
46#1445C2214yes—
47#1445C2212yes—
48#1444C2212yes—
49#1442C2214yes—
50#1442C2212yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8D418yes—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#124—28no—
81C2×#124—44no—
82C2×#124—44no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#399—28no—
111#398—28no—
112#398—28no—
113#398—28no—
114#397—216no—
115#397—28no—
116#396—48no—
117#395—216no—
118#394—28no—
119#391—28no—
120#391—216no—
121#388—48no—
122#387—216no—
123#387—216no—
124#385—216no—
125#384—28no—
126#383—24no—
127#383D418yes—
128#382—28no—
129#380—216no—
130C4×C4⋊C4—24no—
131#376—24no—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25C2412yes—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8C2×D418yes—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—42no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—232no—
189#125—416no—
190#124—432no—
191C4×D4—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×D418yes—
229C2×C4⋊C4C2×D418yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—28no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42C2×D414yes—
261C2×C42C2×D414yes—
262C2×C42#4114yes—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95C2×D418yes—
287#95C2×D418yes—
288#95C2×D418yes—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24H(5,𝔽2)14yes—
300C24C2511yescharacteristic
301C2×Q8—216no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×D4—28no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—44no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
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—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—416no—
368C22×C4—48no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4C22×D418yes—
373C22×C4C22×D418yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C42−1+414yes—
384C22×C42−1+414yes—
385C4⋊C4—216no—
386C4⋊C4—216no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C42—216no—
406C42—48no—
407C42—88no—
408C42#121116yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23C23×D411yescharacteristic
433C23C23×D411yescharacteristic
434C23C22×#4111yescharacteristic
435C23C2×H(5,𝔽2)12yes—
436C23C2×2−1+412yes—
437C23C2×2−1+412yes—
438C23#58512yes—
439C23#58514yes—
440Q8—816no—
441D4—416no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—816no—
507C2×C4C2×#12114yes—
508C2×C4C2×#12114yes—
509C2×C4C2×#12114yes—
510C22—44no—
511C22—48no—
512C22—44no—
513C22—48no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—84no—
518C22—82no—
519C22—82no—
520C22—84no—
521C22C22×#12111yescharacteristic
522C22#348411yescharacteristic
523C22#348511yescharacteristic
524C22#348612yes—
525C22#348712yes—
526C22#348812yes—
527C22#349412yes—
528C22#349912yes—
529C22#350012yes—
530C22#350911yescharacteristic
531C22#350911yescharacteristic
532C22#351012yes—
533C22#353514yes—
534C22#355914yes—
535C22#356414yes—
536C22#356514yes—
537C4—24no—
538C4—24no—
539C4—24no—
540C4—44no—
541C4—44no—
542C4—48no—
543C4—48no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4—84no—
548C4—82no—
549C2—84no—
550C2#3551411yescharacteristic
551C2#3566711yescharacteristic
552C2#3567711yescharacteristic
553C2#5071912yes—
554C2#5087014yes—
555C2#5109012yes—
556C2#5200612yes—
557C2#5205612yes—
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
2. C2×#2660C2semidirect
4. C2×#2381C2semidirect
5. #15786C2semidirect
6. #15769C2semidirect
7. #15739C2semidirect
8. #15648C2semidirect
9. #15500C2semidirect
11. #14289C2semidirect
12. #14164C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#26602
C2×#26332
C2×#23812
#157862
#157692
#157392
#156482
#155002
#153212
#142892
#141642
#139592