Finite Group Databasebeta

← #792801

Subgroups — #792801

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2346C211yesmaximal, characteristic
3#17046C212yesmaximal
4#15799C214yesmaximal, complemented
5#15650C214yesmaximal
6#15197C211yesmaximal, characteristic, complemented
7#15194C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic
9#15122C212yesmaximal, complemented
10#14894C212yesmaximal
11#14133C212yesmaximal, complemented
12#14129C212yesmaximal
13#14006C212yesmaximal, complemented
14#13963C212yesmaximal, complemented
15#13875C212yesmaximal
16C4×#397C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×#117C2211yescharacteristic
19#2571—28no—
20C2×#400C2214yes—
21C2×#399C2212yes—
22C2×#397C2211yescharacteristic
23#2346—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#387C2214yes—
33C2×#384C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#382C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×#376C2212yes—
46C2×#376C2212yes—
47#1445C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1443C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2212yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2211yescharacteristic
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#400—216no—
120#399—44no—
121#397—24no—
122#389—28no—
123#389—24no—
124#389—28no—
125#389—28no—
126#389—28no—
127#389—24no—
128#388—28no—
129#388—28no—
130#387—48no—
131#384—24no—
132#382—28no—
133#382—28no—
134#382—28no—
135#380—24no—
136#380—28no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4D418yes—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—24no—
143#376—28no—
144#376—28no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C25C2411yescharacteristic
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4#4114yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201#117—232no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4#4114yes—
240C2×C4⋊C4#4118yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42C2×D418yes—
282C2×C42C2×D414yes—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95C2×D418yes—
316#95#4114yes—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24C2×#4112yes—
328C24C2511yescharacteristic
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4118yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C22×C42−1+414yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—216no—
432C4⋊C4—416no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—88no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—84no—
449C22⋊C4—88no—
450C42—216no—
451C42—28no—
452C42—216no—
453C42—416no—
454C42—48no—
455C42—48no—
456C42—44no—
457C42—48no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C23×D411yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4112yes—
485C23C22×#4111yescharacteristic
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×2−1+412yes—
488C23C2×2−1+411yescharacteristic
489C23#58511yescharacteristic
490C23#58512yes—
491C23#58512yes—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—416no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
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—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4#52514yes—
582C2×C4#53214yes—
583C2×C4#53614yes—
584C2×C4#54714yes—
585C2×C4#55218yes—
586C2×C4#55218yes—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22—81no—
597C22—82no—
598C22C2×#52511yescharacteristic
599C22C2×#53211yescharacteristic
600C22C2×#53611yescharacteristic
601C22C2×#54711yescharacteristic
602C22C2×#55212yes—
603C22C2×#55212yes—
604C22#347511yescharacteristic
605C22#347512yes—
606C22#348311yescharacteristic
607C22#348711yescharacteristic
608C22#348811yescharacteristic
609C22#349312yes—
610C22#350011yescharacteristic
611C22#351111yescharacteristic
612C22#351312yes—
613C22#351512yes—
614C22#351811yescharacteristic
615C22#351812yes—
616C22#352112yes—
617C22#352412yes—
618C22#354712yes—
619C22#356312yes—
620C22#356512yes—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—44no—
626C4—48no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—84no—
631C4—82no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#261118yes—
637C2—82no—
638C2C2×#261111yescharacteristic
639C2#3585311yescharacteristic
640C2#3601911yescharacteristic
641C2#3618511yescharacteristic
642C2#3641111yescharacteristic
643C2#3891911yescharacteristic
644C2#4296511yescharacteristic
645C2#4501712yes—
646C2#4528412yes—
647C2#5929812yes—
648C2#5950912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2571C2semidirect
4. #15799C2semidirect
6. #15197C2semidirect
7. #15194C2semidirect
9. #15122C2semidirect
11. #14133C2semidirect
13. #14006C2semidirect
14. #13963C2semidirect
17. #13810C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#23462
#170462
#157992
#156502
#151972
#151942
#151352
#151222
#148942
#141332
#141292
#140062
#139632
#138752
C4×#3972
#138102