Finite Group Databasebeta

← #4306411

Subgroups — #4306411

Order 512.

5250 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2397C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic, complemented
3C2×#2361C213yesmaximal, complemented
4C2×#2358C213yesmaximal, complemented
5#15172C211yesmaximal, characteristic, complemented
6#15084C213yesmaximal, complemented
7#14977C213yesmaximal, complemented
8#14910C211yesmaximal, characteristic, complemented
9#14807C213yesmaximal, complemented
10#14796C213yesmaximal, complemented
11#14777C213yesmaximal, complemented
12#14757C211yesmaximal, characteristic, complemented
13#14716C213yesmaximal, complemented
14C4×#399C211yesmaximal, characteristic, complemented
15#13809C211yesmaximal, characteristic, complemented
16C22×#121C2213yes—
17C22×#120C2213yes—
18C22×C4×D4C2213yes—
19C22×C4×D4C2213yes—
20C22×C4×D4C2213yescomplemented
21C23×C4⋊C4C2213yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2213yes—
24#2397—28no—
25C2×#399—28no—
26C2×#399C2211yescharacteristic
27C2×#399C2213yes—
28C2×#399C2213yes—
29C2×#399C2211yescharacteristic
30C2×#399C2213yescomplemented
31C2×#399C2213yescomplemented
32C2×#392C2213yes—
33C2×#392C2213yes—
34C2×#392C2211yescharacteristic, complemented
35#2361—224no—
36#2358—224no—
37C2×#390C2213yes—
38C2×#390C2213yescomplemented
39C2×#389C2213yescomplemented
40C2×#388C2213yes—
41C2×#388C2213yes—
42C2×#388C2216yescomplemented
43C2×#386C2213yes—
44C2×#386C2213yes—
45C2×#386C2213yes—
46C2×#386C2216yes—
47C2×#385C2213yes—
48C2×#385C2213yes—
49C2×#385C2216yescomplemented
50C2×C4×C4⋊C4C2213yescomplemented
51C2×C4×C22⋊C4C2213yes—
52C2×C4×C22⋊C4C2213yes—
53C2×#376C2211yescharacteristic, complemented
54C22×#95C2211yescharacteristic
55#1445C2213yescomplemented
56#1445C2213yescomplemented
57#1444C2211yescharacteristic, complemented
58#1444C2216yescomplemented
59#1439C2213yes—
60#1439C2213yes—
61#1439C2216yescomplemented
62#1439C2213yescomplemented
63#1438C2216yes—
64#1438C2216yes—
65#1438C2213yescomplemented
66#1434C2213yescomplemented
67#1434C2213yescomplemented
68C22⋊#95C2213yes—
69C4×#95C2211yescharacteristic, complemented
70C26C2311yescharacteristic
71C23×D4C2313yes—
72C23×D4C2313yes—
73C23×D4C2313yes—
74C23×D4C2313yes—
75C24×C4C2313yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2313yes—
79C24×C4C2313yes—
80C2×#121—224no—
81C2×#121—212no—
82C2×#120—212no—
83C2×#120—212no—
84C2×#120—212no—
85C2×C4×D4—212no—
86C2×C4×D4—212no—
87C2×C4×D4—224no—
88C2×C4×D4—212no—
89C2×C4×D4—212no—
90C2×C4×D4—212no—
91C2×C4×D4—46no—
92C2×C4×D4—46no—
93C22×C4⋊C4—212no—
94C22×C4⋊C4—212no—
95C22×C4⋊C4—46no—
96C22×C4⋊C4C2313yes—
97C22×C4⋊C4C2313yes—
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2316yes—
100C22×C4⋊C4C2313yes—
101C22×C4⋊C4C2313yes—
102C22×C4⋊C4C2313yes—
103C22×C22⋊C4—212no—
104C22×C22⋊C4—212no—
105C22×C22⋊C4—212no—
106C22×C22⋊C4—46no—
107C22×C22⋊C4C2313yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2316yes—
116C22×C22⋊C4C2316yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2313yes—
119C22×C22⋊C4C2313yes—
120C22×C22⋊C4C2313yes—
121C22×C22⋊C4C2316yes—
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2313yes—
125C22×C42C2313yes—
126C22×C42C2313yes—
127#399—212no—
128#399—212no—
129#399—212no—
130#399—24no—
131#399—46no—
132#399D418yescomplemented
133#392—212no—
134#392—24no—
135#392—46no—
136#390—212no—
137#390—212no—
138#389—212no—
139#388—212no—
140#388—212no—
141#388—224no—
142#386—212no—
143#386—224no—
144#386—212no—
145#386—212no—
146#385—212no—
147#385—212no—
148#385—224no—
149C4×C4⋊C4—212no—
150C4×C22⋊C4—212no—
151C4×C22⋊C4D4124yescomplemented
152#376—24no—
153C2×#95—24no—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2316yes—
158C2×#95C2316yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2316yes—
164C2×#95C2311yescharacteristic
165C2×#95C2313yes—
166C2×#95C2313yes—
167C25—22no—
168C25—22no—
169C25—26no—
170C25—26no—
171C25—43no—
172C25—43no—
173C25#4114yes—
174C25C2411yescharacteristic
175C25C2413yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×D4—26no—
179C22×D4—212no—
180C22×D4—26no—
181C22×D4—26no—
182C22×D4—26no—
183C22×D4—212no—
184C22×D4—212no—
185C22×D4—26no—
186C22×D4—26no—
187C22×D4—26no—
188C22×D4—26no—
189C22×D4—26no—
190C22×D4—26no—
191C22×D4—26no—
192C22×D4—26no—
193C22×D4—26no—
194C22×D4—43no—
195C22×D4—46no—
196C22×D4—43no—
197C22×D4—46no—
198C22×D4—46no—
199C22×D4#41112yes—
200C23×C4—212no—
201C23×C4—26no—
202C23×C4—26no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—26no—
206C23×C4—212no—
207C23×C4—26no—
208C23×C4—26no—
209C23×C4—26no—
210C23×C4—26no—
211C23×C4—212no—
212C23×C4—26no—
213C23×C4—46no—
214C23×C4—43no—
215C23×C4—46no—
216C23×C4—43no—
217C23×C4—46no—
218C23×C4—43no—
219C23×C4—43no—
220C23×C4C2×D4112yes—
221C23×C4C2411yescharacteristic
222C23×C4C2413yes—
223C23×C4C2411yescharacteristic
224C23×C4C2413yes—
225C23×C4C2413yes—
226C23×C4C2413yes—
227C23×C4C2413yes—
228C23×C4C2413yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2413yes—
232#121—296no—
233#120—296no—
234C4×D4—448no—
235C4×D4—448no—
236C4×D4—448no—
237C2×C4⋊C4—26no—
238C2×C4⋊C4—26no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4—26no—
241C2×C4⋊C4—212no—
242C2×C4⋊C4—26no—
243C2×C4⋊C4—212no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—26no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—212no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—26no—
251C2×C4⋊C4—448no—
252C2×C4⋊C4—46no—
253C2×C4⋊C4—43no—
254C2×C4⋊C4—46no—
255C2×C4⋊C4C2×D4112yes—
256C2×C22⋊C4—26no—
257C2×C22⋊C4—26no—
258C2×C22⋊C4—26no—
259C2×C22⋊C4—26no—
260C2×C22⋊C4—212no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4—26no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—212no—
265C2×C22⋊C4—26no—
266C2×C22⋊C4—26no—
267C2×C22⋊C4—212no—
268C2×C22⋊C4—26no—
269C2×C22⋊C4—26no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—26no—
276C2×C22⋊C4—26no—
277C2×C22⋊C4—26no—
278C2×C22⋊C4—212no—
279C2×C22⋊C4—26no—
280C2×C22⋊C4—26no—
281C2×C22⋊C4—212no—
282C2×C22⋊C4—212no—
283C2×C22⋊C4—26no—
284C2×C22⋊C4—26no—
285C2×C22⋊C4—26no—
286C2×C22⋊C4—212no—
287C2×C22⋊C4—26no—
288C2×C22⋊C4—448no—
289C2×C22⋊C4—43no—
290C2×C22⋊C4—43no—
291C2×C22⋊C4—46no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—46no—
294C2×C22⋊C4—46no—
295C2×C22⋊C4—46no—
296C2×C22⋊C4C2×D4112yes—
297C2×C22⋊C4C2×D4112yes—
298C2×C22⋊C4C2×D4112yes—
299C2×C42—26no—
300C2×C42—26no—
301C2×C42—26no—
302C2×C42—212no—
303C2×C42—46no—
304C2×C42C2×D4112yes—
305C2×C42C2×D4112yes—
306#95—26no—
307#95—26no—
308#95—26no—
309#95—26no—
310#95—212no—
311#95—26no—
312#95—26no—
313#95—212no—
314#95—26no—
315#95—46no—
316#95C2×D414yes—
317#95C2×D4112yes—
318#95C2×D4112yes—
319C24—21no—
320C24—23no—
321C24—21no—
322C24—23no—
323C24—23no—
324C24—23no—
325C24—23no—
326C24—23no—
327C24—28no—
328C24—21no—
329C24—23no—
330C24—26no—
331C24—23no—
332C24—23no—
333C24—26no—
334C24—23no—
335C24—23no—
336C24—23no—
337C24—23no—
338C24—26no—
339C24—23no—
340C24—21no—
341C24—26no—
342C24—23no—
343C24—23no—
344C24—224no—
345C24—412no—
346C24—412no—
347C24—412no—
348C24—412no—
349C24—412no—
350C24—412no—
351C24—44no—
352C24—86no—
353C24—86no—
354C24—812no—
355C24C22×D412yes—
356C24C22×D416yes—
357C24C2×#4112yes—
358C24C2×#4116yes—
359C24C2×#4112yes—
360C24C2×#4112yes—
361C24C2×#4116yes—
362C242−1+412yes—
363C24C2511yescharacteristic
364C2×D4—224no—
365C2×D4—224no—
366C2×D4—224no—
367C2×D4—248no—
368C2×D4—224no—
369C2×D4—412no—
370C2×D4—424no—
371C2×D4—424no—
372C2×D4—412no—
373C2×D4—412no—
374C2×D4—412no—
375C2×D4—412no—
376C2×D4—412no—
377C2×D4—424no—
378C2×D4—412no—
379C2×D4—412no—
380C2×D4—412no—
381C2×D4—412no—
382C2×D4—86no—
383C2×D4—86no—
384C2×D4—812no—
385C2×D4—812no—
386C22×C4—21no—
387C22×C4—26no—
388C22×C4—23no—
389C22×C4—21no—
390C22×C4—23no—
391C22×C4—26no—
392C22×C4—224no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—26no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—26no—
399C22×C4—26no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—26no—
406C22×C4—26no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—26no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—26no—
414C22×C4—26no—
415C22×C4—224no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—26no—
420C22×C4—21no—
421C22×C4—23no—
422C22×C4—23no—
423C22×C4—23no—
424C22×C4—26no—
425C22×C4—21no—
426C22×C4—23no—
427C22×C4—412no—
428C22×C4—412no—
429C22×C4—424no—
430C22×C4—424no—
431C22×C4—424no—
432C22×C4—412no—
433C22×C4—424no—
434C22×C4—412no—
435C22×C4—412no—
436C22×C4—412no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—412no—
440C22×C4—86no—
441C22×C4—812no—
442C22×C4—86no—
443C22×C4—86no—
444C22×C4—812no—
445C22×C4—86no—
446C22×C4C22×D416yes—
447C22×C4C22×D416yes—
448C22×C4C22×D416yes—
449C22×C4C22×D416yes—
450C22×C4C22×D416yes—
451C22×C4C22×D416yes—
452C22×C4C22×D416yes—
453C22×C4C22×D416yes—
454C22×C4C2×#4116yes—
455C22×C4H(5,𝔽2)16yes—
456C4⋊C4—224no—
457C4⋊C4—224no—
458C4⋊C4—424no—
459C4⋊C4—412no—
460C4⋊C4—412no—
461C4⋊C4—412no—
462C4⋊C4—86no—
463C22⋊C4—224no—
464C22⋊C4—224no—
465C22⋊C4—248no—
466C22⋊C4—224no—
467C22⋊C4—224no—
468C22⋊C4—412no—
469C22⋊C4—424no—
470C22⋊C4—412no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—412no—
475C22⋊C4—412no—
476C22⋊C4—412no—
477C22⋊C4—86no—
478C22⋊C4—812no—
479C22⋊C4#127116yes—
480C42—224no—
481C42—412no—
482C42—412no—
483C23—26no—
484C23—22no—
485C23—26no—
486C23—26no—
487C23—26no—
488C23—26no—
489C23—26no—
490C23—212no—
491C23—26no—
492C23—26no—
493C23—26no—
494C23—26no—
495C23—26no—
496C23—26no—
497C23—26no—
498C23—212no—
499C23—26no—
500C23—26no—
501C23—22no—
502C23—26no—
503C23—26no—
504C23—212no—
505C23—26no—
506C23—448no—
507C23—43no—
508C23—41no—
509C23—43no—
510C23—43no—
511C23—46no—
512C23—416no—
513C23—46no—
514C23—43no—
515C23—43no—
516C23—41no—
517C23—43no—
518C23—43no—
519C23—43no—
520C23—46no—
521C23—43no—
522C23—46no—
523C23—43no—
524C23—43no—
525C23—46no—
526C23—43no—
527C23—43no—
528C23—46no—
529C23—43no—
530C23—46no—
531C23—43no—
532C23—43no—
533C23—43no—
534C23—46no—
535C23—46no—
536C23—46no—
537C23—46no—
538C23—46no—
539C23—43no—
540C23—824no—
541C23—824no—
542C23—88no—
543C23—824no—
544C23—88no—
545C23#525112yes—
546C23#525112yes—
547C23C2×#12714yes—
548C23#54814yes—
549C23C23×D413yes—
550C23C23×D411yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4113yes—
553C23C2×H(5,𝔽2)13yes—
554C23C2×2−1+411yescharacteristic
555C23#58513yes—
556D4—448no—
557D4—448no—
558D4—448no—
559D4—824no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—212no—
563C2×C4—26no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—26no—
567C2×C4—212no—
568C2×C4—26no—
569C2×C4—26no—
570C2×C4—26no—
571C2×C4—26no—
572C2×C4—212no—
573C2×C4—26no—
574C2×C4—26no—
575C2×C4—26no—
576C2×C4—212no—
577C2×C4—26no—
578C2×C4—26no—
579C2×C4—26no—
580C2×C4—26no—
581C2×C4—448no—
582C2×C4—46no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—46no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—46no—
591C2×C4—43no—
592C2×C4—46no—
593C2×C4—46no—
594C2×C4—46no—
595C2×C4—43no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—43no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—46no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—46no—
605C2×C4—43no—
606C2×C4—43no—
607C2×C4—824no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—824no—
611C2×C4C2×#12714yes—
612C2×C4C2×#12714yes—
613C2×C4D42112yes—
614C22—212no—
615C22—212no—
616C22—212no—
617C22—24no—
618C22—212no—
619C22—24no—
620C22—212no—
621C22—46no—
622C22—46no—
623C22—42no—
624C22—46no—
625C22—42no—
626C22—42no—
627C22—46no—
628C22—46no—
629C22—46no—
630C22—46no—
631C22—412no—
632C22—412no—
633C22—46no—
634C22—42no—
635C22—832no—
636C22—86no—
637C22—86no—
638C22—83no—
639C22—83no—
640C22—83no—
641C22—81no—
642C22—83no—
643C22—83no—
644C22—83no—
645C22—81no—
646C22#239718yescomplemented
647C22C2×#52513yes—
648C22C2×#52513yes—
649C22C22×#12711yescharacteristic
650C22C2×D4213yes—
651C22C2×#54811yescharacteristic
652C22D4×#4113yes—
653C22#348513yes—
654C22#348913yes—
655C22#349813yes—
656C22#350513yes—
657C22#351316yes—
658C22#352013yes—
659C4—212no—
660C4—24no—
661C4—24no—
662C4—46no—
663C4—46no—
664C4—46no—
665C4—46no—
666C4—46no—
667C4—81no—
668C4—83no—
669C4—81no—
670C2—28no—
671C2—44no—
672C2—86no—
673C2—82no—
674C2C2×#239711yescharacteristic
675C2#3584013yes—
676C2#3599013yes—
677C2#3606011yescharacteristic
678C2#3609713yes—
679C2#3609913yes—
680C2D4×#12711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2397C2semidirect
2. C22×#399C2semidirect
3. C2×#2361C2semidirect
4. C2×#2358C2semidirect
5. #15172C2semidirect
6. #15084C2semidirect
7. #14977C2semidirect
8. #14910C2semidirect
9. #14807C2semidirect
10. #14796C2semidirect
11. #14777C2semidirect
12. #14757C2semidirect
13. #14716C2semidirect
14. C4×#399C2semidirect
15. #13809C2semidirect
20. C22×C4×D4C22semidirect
30. C2×#399C22semidirect
31. C2×#399C22semidirect
34. C2×#392C22semidirect
38. C2×#390C22semidirect
39. C2×#389C22semidirect
42. C2×#388C22semidirect
49. C2×#385C22semidirect
50. C2×C4×C4⋊C4C22semidirect
53. C2×#376C22semidirect
55. #1445C22semidirect
56. #1445C22semidirect
57. #1444C22semidirect
58. #1444C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1438C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
69. C4×#95C22semidirect
132. #399D4semidirect
151. C4×C22⋊C4D4semidirect
646. C22#2397semidirect

Maximal subgroups

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

subgroupindex
C2×#23972
C22×#3992
C2×#23612
C2×#23582
#151722
#150842
#149772
#149102
#148072
#147962
#147772
#147572
#147162
C4×#3992
#138092