Finite Group Databasebeta

← #4453169

Subgroups — #4453169

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#393C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic, complemented
3#16452C214yesmaximal, complemented
4#16375C214yesmaximal, complemented
5#16353C214yesmaximal, complemented
6#16327C214yesmaximal, complemented
7#14936C211yesmaximal, characteristic, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14918C212yesmaximal
10#14874C212yesmaximal, complemented
11#14734C211yesmaximal, characteristic
12#14717C211yesmaximal, characteristic
13#14349C211yesmaximal, characteristic, complemented
14#13992C212yesmaximal, complemented
15#13991C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#399C2212yes—
22C2×#395C2214yescomplemented
23C2×#395C2214yescomplemented
24C2×#393—28no—
25C2×#393C2214yes—
26C2×#393C2214yes—
27C2×#393C2212yes—
28#2329—28no—
29C2×#391C2212yescomplemented
30C2×#389C2212yescomplemented
31C2×#388C2214yes—
32C2×#387C2212yescomplemented
33C2×#387C2212yescomplemented
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#379C2212yes—
41C2×#379C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2214yescomplemented
47#1446C2214yescomplemented
48#1446C2212yescomplemented
49#1445C2212yescomplemented
50#1445C2214yescomplemented
51#1444C2212yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1443C2214yescomplemented
55#1442C2214yescomplemented
56#1442C2212yescomplemented
57#1442C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#399—44no—
121#395—48no—
122#395—48no—
123#393—216no—
124#393—216no—
125#393—44no—
126#391—28no—
127#389—28no—
128#388—48no—
129#387—28no—
130#387—28no—
131#385—28no—
132#385—44no—
133#385—44no—
134#381—28no—
135#381—44no—
136#380—24no—
137#379—28no—
138#379—44no—
139C4×C22⋊C4—28no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95D418yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25—22no—
172C25—24no—
173C25—24no—
174C25—24no—
175C25—22no—
176C25—22no—
177C25—42no—
178C25—42no—
179C25—42no—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—28no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—28no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—464no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C22⋊Q8—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
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—42no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—416no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4#4118yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—44no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—28no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—416no—
343C24—416no—
344C24—48no—
345C24—48no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—48no—
353C24—48no—
354C24—88no—
355C24—88no—
356C24—84no—
357C24—84no—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24H(5,𝔽2)14yes—
361C242−1+414yes—
362C242−1+412yes—
363C242−1+412yes—
364C24C2511yescharacteristic
365C2×Q8—28no—
366C2×Q8—416no—
367C2×Q8—416no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—4128no—
373C2×Q8—416no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—88no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)14yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—84no—
491C42—84no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—416no—
510C23—416no—
511C23—416no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—832no—
531C23—832no—
532C23—88no—
533C23—88no—
534C23—88no—
535C23—88no—
536C23#52514yes—
537C23#53514yes—
538C23#53614yes—
539C23#54218yes—
540C23#54218yes—
541C23C23×D411yescharacteristic
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+412yes—
549C23#58512yes—
550Q8—464no—
551Q8—88no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
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—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4—88no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—216no—
631C22—216no—
632C22—24no—
633C22—44no—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—44no—
638C22—832no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22#263218yes—
646C22C2×#52511yescharacteristic
647C22C2×#53511yescharacteristic
648C22C2×#53611yescharacteristic
649C22C2×#54212yes—
650C22C2×#54212yes—
651C22#348411yescharacteristic
652C22#348411yescharacteristic
653C22#348511yescharacteristic
654C22#348611yescharacteristic
655C22#351414yes—
656C22#352014yes—
657C22#353214yes—
658C22#353514yes—
659C22#353812yes—
660C22#353812yes—
661C22#356112yes—
662C22#356112yes—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4—84no—
675C4—82no—
676C2—28no—
677C2—84no—
678C2C2×#263211yescharacteristic
679C2#3535711yescharacteristic
680C2#3545111yescharacteristic
681C2#3670311yescharacteristic
682C2#3670811yescharacteristic
683C2#3696611yescharacteristic
684C2#3711811yescharacteristic
685C2#4084314yes—
686C2#4089214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2329C2semidirect
3. #16452C2semidirect
4. #16375C2semidirect
5. #16353C2semidirect
6. #16327C2semidirect
7. #14936C2semidirect
8. #14934C2semidirect
10. #14874C2semidirect
13. #14349C2semidirect
14. #13992C2semidirect
22. C2×#395C22semidirect
23. C2×#395C22semidirect
29. C2×#391C22semidirect
30. C2×#389C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
46. #1446C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1443C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
69. #1436C22semidirect
70. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3932
C2×#23292
#164522
#163752
#163532
#163272
#149362
#149342
#149182
#148742
#147342
#147172
#143492
#139922
#139912