Finite Group Databasebeta

← #556018

Subgroups — #556018

Order 512.

2960 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#14684C212yesmaximal, complemented
3#14432C214yesmaximal, complemented
4#14431C214yesmaximal, complemented
5#14376C214yesmaximal, complemented
6#14375C214yesmaximal, complemented
7#14351C212yesmaximal, complemented
8#13966C211yesmaximal, characteristic, complemented
9#13956C211yesmaximal, characteristic
10#13953C211yesmaximal, characteristic, complemented
11#13947C212yesmaximal, complemented
12#13944C212yesmaximal, complemented
13#13937C211yesmaximal, characteristic
14#13796C211yesmaximal, characteristic, complemented
15#13780C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2297C4116yescomplemented
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#387C2214yes—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#381C2211yescharacteristic
30C2×#381C2214yes—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37C22×#95C2211yescharacteristic
38#1446C2212yes—
39#1445C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2212yes—
50#1438C2212yes—
51#1438C2212yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58#1434C2212yes—
59#1434C2212yes—
60#1434C2212yes—
61#1434C2214yes—
62#1434C2214yes—
63#1434C2214yes—
64#1434C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C22×#41—24no—
77C22×#41—24no—
78C22×#41—44no—
79C22×#41—44no—
80C22×#41C2×C418yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#117—44no—
91C2×#117C2×C418yescomplemented
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42—44no—
108C22×C42C2×C418yescomplemented
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#391—44no—
117#391C2×C4116yescomplemented
118#390—44no—
119#390C2×C4116yes—
120#387—48no—
121#386—44no—
122#386—44no—
123#386C2×C4116yes—
124#386C2×C4116yescomplemented
125#381—24no—
126#381—48no—
127#380—24no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4C2×C418yes—
130C4×C22⋊C4C2×C418yes—
131C4×C22⋊C4C2×C418yescomplemented
132C4×C22⋊C4C2×C418yescomplemented
133C2×#95—24no—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C25C2411yescharacteristic
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C2×#41—232no—
166C2×#41—416no—
167C2×#41—416no—
168C2×#41—432no—
169C2×#41—432no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—42no—
174C22×Q8C22×C414yes—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4C22×C414yes—
188C22×D4C22×C414yes—
189C22×D4C22×C414yes—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C22×C418yes—
212C23×C4C22×C414yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#117—416no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22×C418yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C22×C418yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—416no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283C2×C42C22×C418yes—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
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#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24C23×C412yes—
331C24C23×C412yes—
332C24C23×C412yes—
333C24C2×#4112yes—
334C24C2×#4112yes—
335C24H(5,𝔽2)14yes—
336C24C2511yescharacteristic
337#41—4128no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
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—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4#117116yes—
438C22×C4C23×C412yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C4⋊C4—216no—
452C4⋊C4—84no—
453C4⋊C4—88no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—44no—
467C42—44no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—84no—
472C42—88no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C2×#11714yes—
520C23#51814yes—
521C23C24×C411yescharacteristic
522C23C22×#4111yescharacteristic
523C23C22×#4111yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23#58514yes—
527C23#58512yes—
528C23#58512yes—
529Q8—416no—
530D4—416no—
531D4—416no—
532D4—416no—
533C2×C4—232no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—432no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
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—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4C2×#11714yes—
604C2×C4C2×#11714yes—
605C2×C4#51814yes—
606C22—24no—
607C22—28no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—48no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22C22×#11711yescharacteristic
631C22C2×#51811yescharacteristic
632C22C2×#51811yescharacteristic
633C22#342812yes—
634C22#342812yes—
635C22#345512yes—
636C22#345512yes—
637C22#349012yes—
638C22#349012yes—
639C22#351011yescharacteristic
640C22#351011yescharacteristic
641C22#351212yes—
642C22#353314yes—
643C22#353314yes—
644C22#356214yes—
645C22#356214yes—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—84no—
655C4—84no—
656C4—84no—
657C4—84no—
658C2—44no—
659C2—44no—
660C2—82no—
661C2#3405611yescharacteristic
662C2#3417611yescharacteristic
663C2#3418911yescharacteristic
664C2#3439612yes—
665C2#3440212yes—
666C2#3682014yes—
667C2#3682112yes—
668C2#3682212yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14684C2semidirect
3. #14432C2semidirect
4. #14431C2semidirect
5. #14376C2semidirect
6. #14375C2semidirect
7. #14351C2semidirect
8. #13966C2semidirect
10. #13953C2semidirect
11. #13947C2semidirect
12. #13944C2semidirect
14. #13796C2semidirect
15. #13780C2semidirect
19. #2297C4semidirect
91. C2×#117C2×C4semidirect
108. C22×C42C2×C4semidirect
117. #391C2×C4semidirect
124. #386C2×C4semidirect
131. C4×C22⋊C4C2×C4semidirect
132. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#146842
#144322
#144312
#143762
#143752
#143512
#139662
#139562
#139532
#139472
#139442
#139372
#137962
#137802