Finite Group Databasebeta

← #723306

Subgroups — #723306

Order 512.

2722 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#17057C212yesmaximal, complemented
4#16960C212yesmaximal, complemented
5#15807C214yesmaximal, complemented
6#15545C214yesmaximal, complemented
7#15106C212yesmaximal, complemented
8#15014C212yesmaximal, complemented
9#14736C214yesmaximal, complemented
10#14353C211yesmaximal, characteristic, complemented
11#13975C214yesmaximal
12#13965C211yesmaximal, characteristic, complemented
13#13924C211yesmaximal, characteristic, complemented
14C2×#1445C211yesmaximal, characteristic, complemented
15#13784C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19#2487—28no—
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#398C2214yes—
23#2349—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#382C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—28no—
87C2×C4×D4—48no—
88C2×C4×D4D418yes—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116#400—28no—
117#399—28no—
118#398—48no—
119#389—28no—
120#389—28no—
121#386—28no—
122#386Q8116yes—
123#385—28no—
124#385—28no—
125#385—28no—
126#385—48no—
127#385—48no—
128#382—24no—
129#380—24no—
130#380—24no—
131#380—28no—
132#380—24no—
133#380—48no—
134C4×C4⋊C4—24no—
135C4×C22⋊C4—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—48no—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4C2×D414yes—
174C22×D4C2×Q818yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×Q818yes—
201C23×C4#4114yes—
202C23×C4#4114yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4×D4—232no—
217C4×D4—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D414yes—
272C2×C22⋊C4C2×D414yes—
273C2×C22⋊C4C2×Q818yes—
274C2×C22⋊C4C2×Q8116yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C2×D414yes—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C2×Q8116yes—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C22×Q814yes—
322C24C22×Q814yes—
323C24C2×#4112yes—
324C24C2×#4112yes—
325C24H(5,𝔽2)14yes—
326C24C2511yescharacteristic
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—216no—
330C2×D4—216no—
331C2×D4C4⋊Q8116yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
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—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4#123116yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×Q818yes—
423C22×C4C22×Q818yes—
424C22×C4C22×Q814yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C4⋊C4—28no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—28no—
444C22⋊C4—216no—
445C22⋊C4—28no—
446C22⋊C4—416no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—88no—
451C42—28no—
452C42—48no—
453C42—84no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—28no—
464C23—28no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—44no—
479C23—44no—
480C23C2×#12314yes—
481C23C2×C4⋊Q814yes—
482C23C2×C4⋊Q814yes—
483C23#53514yes—
484C23#55418yes—
485C23#55418yes—
486C23C23×D411yescharacteristic
487C23C23×Q812yes—
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)12yes—
492C23#58512yes—
493C23#58514yes—
494D4—232no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—416no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—88no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4—816no—
568C2×C4C2×#12314yes—
569C2×C4C2×#12314yes—
570C2×C4C2×C4⋊Q814yes—
571C22—28no—
572C22—24no—
573C22—28no—
574C22—24no—
575C22—28no—
576C22—216no—
577C22—28no—
578C22—28no—
579C22—24no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—48no—
585C22—44no—
586C22#250218yes—
587C22C22×#12311yescharacteristic
588C22C22×C4⋊Q811yescharacteristic
589C22C2×#53511yescharacteristic
590C22#346012yes—
591C22C2×#55412yes—
592C22C2×#55412yes—
593C22#348512yes—
594C22#348812yes—
595C22#349112yes—
596C22#349112yes—
597C22#350114yes—
598C22#350414yes—
599C22#350811yescharacteristic
600C22#351011yescharacteristic
601C22#353414yes—
602C22#356014yes—
603C4—24no—
604C4—24no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—48no—
610C4—48no—
611C4—84no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—82no—
616C2—28no—
617C2—44no—
618C2C2×#250211yescharacteristic
619C2#3525411yescharacteristic
620C2#3544811yescharacteristic
621C2#3595012yes—
622C2#3967014yes—
623C2#4141012yes—
624C2#4351212yes—
625C2#4446312yes—
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. #17057C2semidirect
4. #16960C2semidirect
5. #15807C2semidirect
6. #15545C2semidirect
7. #15106C2semidirect
8. #15014C2semidirect
9. #14736C2semidirect
10. #14353C2semidirect
12. #13965C2semidirect
13. #13924C2semidirect
14. C2×#1445C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#23492
#170572
#169602
#158072
#155452
#151062
#150142
#147362
#143532
#139752
#139652
#139242
C2×#14452
#137842