Finite Group Databasebeta

← #723307

Subgroups — #723307

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#16694C212yesmaximal, complemented
4#16365C212yesmaximal, complemented
5#15698C214yesmaximal, complemented
6#15378C214yesmaximal, complemented
7#15104C212yesmaximal, complemented
8#14914C212yesmaximal, complemented
9#14736C214yesmaximal, complemented
10#14353C211yesmaximal, characteristic, complemented
11#13975C214yesmaximal
12#13965C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14C2×#1446C211yesmaximal, characteristic, complemented
15#13784C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20#2409—28no—
21C2×#398C2214yes—
22C2×#395C2212yes—
23C2×#393C2212yes—
24#2349—28no—
25C2×#391C2212yes—
26C2×#391C2214yes—
27C2×#389C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#382C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2214yes—
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1446C2212yes—
50#1446C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—48no—
96C2×C4×D4D418yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119#398—48no—
120#395—28no—
121#393—28no—
122#391—28no—
123#391—48no—
124#389—28no—
125#386—28no—
126#386Q8116yescomplemented
127#385—28no—
128#385—28no—
129#385—28no—
130#385—48no—
131#385—48no—
132#382—24no—
133#380—24no—
134#380—24no—
135#380—24no—
136#380—48no—
137C4×C4⋊C4—24no—
138C4×C22⋊C4—24no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95—48no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—432no—
184C22×Q8—432no—
185C22×Q8—432no—
186C22×Q8—416no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4C2×D414yes—
191C22×D4C2×Q818yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×Q818yes—
218C23×C4#4114yes—
219C23×C4#4114yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—432no—
234C4×D4—232no—
235C4×D4—432no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×Q818yes—
285C2×C22⋊C4C2×Q8116yescomplemented
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×Q8116yescomplemented
321C24—24no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C22×D412yes—
331C24C22×D412yes—
332C24C22×Q814yes—
333C24C22×Q814yes—
334C24C2×#4112yes—
335C24C2×#4112yes—
336C24H(5,𝔽2)14yes—
337C24C2511yescharacteristic
338C2×Q8—216no—
339C2×Q8—216no—
340C2×Q8—416no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—4128no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—216no—
352C2×D4—216no—
353C2×D4C4⋊Q8116yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4#123116yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×Q818yes—
444C22×C4C22×Q818yes—
445C22×C4C22×Q814yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)18yes—
451C22×C42−1+414yes—
452C4⋊C4—28no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—28no—
464C22⋊C4—216no—
465C22⋊C4—28no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C42—28no—
472C42—48no—
473C42—84no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—22no—
483C23—28no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—44no—
499C23—44no—
500C23C2×#12314yes—
501C23C2×C4⋊Q814yes—
502C23C2×C4⋊Q814yes—
503C23#53514yes—
504C23#55418yes—
505C23#55418yes—
506C23C23×D411yescharacteristic
507C23C23×Q812yes—
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)14yes—
512C23C2×2−1+412yes—
513C23#58512yes—
514Q8—432no—
515Q8—432no—
516D4—232no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—416no—
543C2×C4—432no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—816no—
588C2×C4—816no—
589C2×C4—816no—
590C2×C4C2×#12314yes—
591C2×C4C2×#12314yes—
592C2×C4C2×C4⋊Q814yes—
593C22—24no—
594C22—28no—
595C22—28no—
596C22—28no—
597C22—28no—
598C22—24no—
599C22—216no—
600C22—28no—
601C22—24no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—48no—
608C22#250218yes—
609C22C22×#12311yescharacteristic
610C22C22×C4⋊Q811yescharacteristic
611C22C2×#53511yescharacteristic
612C22#346112yes—
613C22C2×#55412yes—
614C22C2×#55412yes—
615C22#348412yes—
616C22#348912yes—
617C22#349012yes—
618C22#349012yes—
619C22#350114yes—
620C22#350314yes—
621C22#350911yescharacteristic
622C22#351011yescharacteristic
623C22#353614yes—
624C22#355814yes—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—48no—
629C4—42no—
630C4—48no—
631C4—44no—
632C4—42no—
633C4—48no—
634C4—82no—
635C4—84no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—44no—
640C2C2×#250211yescharacteristic
641C2#3530711yescharacteristic
642C2#3535011yescharacteristic
643C2#3592812yes—
644C2#3966814yes—
645C2#4059312yes—
646C2#4308112yes—
647C2#4396112yes—
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
3. #16694C2semidirect
4. #16365C2semidirect
5. #15698C2semidirect
6. #15378C2semidirect
7. #15104C2semidirect
8. #14914C2semidirect
9. #14736C2semidirect
10. #14353C2semidirect
12. #13965C2semidirect
13. #13922C2semidirect
14. C2×#1446C2semidirect
126. #386Q8semidirect
285. C2×C22⋊C4C2×Q8semidirect
320. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23492
#166942
#163652
#156982
#153782
#151042
#149142
#147362
#143532
#139752
#139652
#139222
C2×#14462
#137842