Finite Group Databasebeta

← #640986

Subgroups — #640986

Order 512.

3332 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2417C212yesmaximal
3C2×#2621C211yesmaximal, characteristic, complemented
4C2×#2290C211yesmaximal, characteristic
5C22×#382C211yesmaximal, characteristic, complemented
6#16518C214yesmaximal, complemented
7#16517C214yesmaximal, complemented
8#15237C212yesmaximal, complemented
9#15236C212yesmaximal, complemented
10#14766C214yesmaximal, complemented
11#14735C214yesmaximal
12#14340C211yesmaximal, characteristic, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14274C211yesmaximal, characteristic, complemented
15#14139C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2857—28no—
24#2621—28no—
25#2417—216no—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#397C2214yes—
29C2×#396C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36#2290—28no—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382—28no—
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2214yes—
47C2×#382C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2214yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C22×#95C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1444C2214yes—
59#1443C2218yes—
60#1442C2218yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2218yes—
64#1441C2218yes—
65#1440C2218yes—
66#1439C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C22⋊#95C2218yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2314yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—44no—
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4D418yes—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42—24no—
115C22×C42—24no—
116C22×C42D418yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#398—216no—
124#398—216no—
125#397—216no—
126#396—216no—
127#391—28no—
128#391—28no—
129#391—28no—
130#391—44no—
131#389—216no—
132#389—28no—
133#388—28no—
134#387—28no—
135#386—216no—
136#386—28no—
137#385—28no—
138#384—24no—
139#382—28no—
140#382—24no—
141#382—216no—
142#382D418yescomplemented
143#380—216no—
144#380—28no—
145#380—216no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—24no—
149#376—24no—
150C2×#95—216no—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—28no—
180C22×D4C2×D414yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—216no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—48no—
200C23×C4C2×D414yes—
201C23×C4C2×D418yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4⋊Q8—416no—
219C22⋊Q8—432no—
220C22⋊Q8—432no—
221C4×D4—232no—
222#117—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D414yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—232no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D418yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297#95—216no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—216no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—216no—
310#95—48no—
311#95—48no—
312#95—48no—
313#95C2×D4116yes—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—24no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—24no—
326C24—24no—
327C24C22×D412yes—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C2×#4112yes—
331C242−1+418yes—
332C24C2511yescharacteristic
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×D4—28no—
341C2×D4—216no—
342C2×D4—48no—
343C2×D4—48no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—216no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—432no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—816no—
420C22×C4#121116yes—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D418yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+412yes—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—88no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—88no—
459C42—216no—
460C42—28no—
461C42—28no—
462C42—44no—
463C42—44no—
464C42#121116yes—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23C2×#12114yes—
496C23#53618yes—
497C23#53618yes—
498C23#54118yes—
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C23×D411yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×2−1+414yes—
506C23C2×2−1+411yescharacteristic
507C23#58514yes—
508Q8—816no—
509D4—416no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—216no—
532C2×C4—24no—
533C2×C4—432no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—832no—
576C2×C4C2×#12118yes—
577C2×C4C2×#12114yes—
578C2×C4C2×#12114yes—
579C2×C4C2×#12114yes—
580C2×C4#53714yes—
581C2×C4D4214yes—
582C2×C4#54714yes—
583C2×C4#54814yes—
584C2×C4#55314yes—
585C22—24no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—24no—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22#262118yescomplemented
603C22C22×#12111yescharacteristic
604C22C22×#12111yescharacteristic
605C22C2×#53612yes—
606C22C2×#53612yes—
607C22C2×#53711yescharacteristic
608C22#344912yes—
609C22C2×#54112yes—
610C22C2×D4211yescharacteristic
611C22C2×#54711yescharacteristic
612C22C2×#54811yescharacteristic
613C22C2×#55311yescharacteristic
614C22#348914yes—
615C22#349514yes—
616C22#350012yes—
617C22#351212yes—
618C22#356114yes—
619C22#356314yes—
620C4—24no—
621C4—28no—
622C4—42no—
623C4—48no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—84no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4#241618yes—
633C2—28no—
634C2—82no—
635C2C2×#262111yescharacteristic
636C2C2×#241611yescharacteristic
637C2#3530612yes—
638C2#3532811yescharacteristic
639C2#3755512yes—
640C2#3767612yes—
641C2#3770512yes—
642C2#3786512yes—
643C2#3789412yes—
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
1. C2×#2857C2semidirect
3. C2×#2621C2semidirect
5. C22×#382C2semidirect
6. #16518C2semidirect
7. #16517C2semidirect
8. #15237C2semidirect
9. #15236C2semidirect
10. #14766C2semidirect
12. #14340C2semidirect
13. #14277C2semidirect
14. #14274C2semidirect
15. #14139C2semidirect
142. #382D4semidirect
602. C22#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#24172
C2×#26212
C2×#22902
C22×#3822
#165182
#165172
#152372
#152362
#147662
#147352
#143402
#142772
#142742
#141392