Finite Group Databasebeta

← #6494443

Subgroups — #6494443

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33311C211yesmaximal, characteristic, complemented
5#33263C211yesmaximal, characteristic, complemented
6#33258C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30307C211yesmaximal, characteristic, complemented
9#30307C211yesmaximal, characteristic, complemented
10#30304C211yesmaximal, characteristic, complemented
11#28839C214yesmaximal, complemented
12#28820C214yesmaximal, complemented
13#26079C214yesmaximal
14#26073C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21982C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic
18#20800C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#501C2214yes—
21C2×#499C2214yes—
22C2×#431C2211yescharacteristic
23C2×#428C2214yes—
24C2×C4×M16C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2647—21no—
34#2647—21no—
35#2629—21no—
36#2628—21no—
37#2571—21no—
38#2571C2212yes—
39#2570—21no—
40#2570C2212yes—
41#2552—21no—
42#2552C2212yes—
43#2550—21no—
44#2550C2212yes—
45#2442—21no—
46#2442—21no—
47#2442C2212yes—
48#2442C2212yes—
49#2441—21no—
50#2441C2212yes—
51#2439—21no—
52#2439C2212yes—
53#2331—21no—
54#2330—21no—
55#2317—21no—
56#2316—21no—
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#2085C2214yes—
63#2085C2214yes—
64#2085C2214yes—
65#2083C2214yes—
66#2059C2214yes—
67#2059C2214yes—
68#2059C2214yes—
69#2057C2214yes—
70#1949C2214yes—
71#1947C2214yes—
72#1944C2214yes—
73#1943C2214yes—
74#1942C2214yes—
75#1942C2214yes—
76#1854C2214yes—
77#1852C2214yes—
78#1847C2214yes—
79#1847C2214yes—
80#1846C2214yes—
81#1845C2211yescharacteristic
82#1845C2211yescharacteristic
83#1845C2211yescharacteristic
84#1845C2211yescharacteristic
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1809C2214yes—
98#1776C2214yes—
99#1771C2212yes—
100#1771C2212yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#117—21no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C2×#117C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127#501—48no—
128#499—48no—
129#431Q818yescomplemented
130#428—48no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C4×M16Q818yescomplemented
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#400—22no—
154#400—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#396—22no—
166#396—22no—
167#395—22no—
168#393—22no—
169#391—22no—
170#391—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#385—22no—
208#385—22no—
209#384—22no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384C2312yes—
215#384C2312yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#382—22no—
225#382—21no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382—21no—
230#382—21no—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#382C2312yes—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—22no—
240#380—22no—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257#376D411yescharacteristic
258#376D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—22no—
272C23×C4—22no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×M16—24no—
277C2×M16—24no—
278C2×M16C2×Q818yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—82no—
286#125—44no—
287#125—44no—
288#125—44no—
289#125—44no—
290#125—82no—
291#125—82no—
292#125—82no—
293#117—42no—
294#117—42no—
295#117—44no—
296#117—44no—
297#117—82no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×Q814yes—
370C2×C42C2×Q814yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—28no—
378#107—28no—
379#107—44no—
380#107—44no—
381#107—44no—
382#107—44no—
383#106—28no—
384#106—28no—
385#106—44no—
386#106—44no—
387C4⋊C8C2×Q818yes—
388C22⋊C8C2×Q818yes—
389#97—24no—
390#97—24no—
391#97C2×Q818yes—
392C4×C8—24no—
393C4×C8—24no—
394C4×C8C2×Q814yes—
395C4×C8C2×Q814yes—
396C4×C8C2×Q814yes—
397C4×C8C2×Q814yes—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426C24C22×D411yescharacteristic
427C2×Q8—44no—
428C2×Q8—82no—
429C2×Q8—82no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C4⋊Q814yes—
448C22×C4C4⋊Q814yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C2511yescharacteristic
465M16—216no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8C22×Q814yes—
473C2×C8C22×Q814yes—
474C2×C8C22×Q814yes—
475C2×C8C22×Q814yes—
476C2×C8H(5,𝔽2)14yes—
477C2×C82−1+414yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—44no—
546C42—82no—
547C42—82no—
548C42C4⋊Q814yes—
549C42C4⋊Q814yes—
550C42C22×Q812yes—
551C42C22×Q812yes—
552C42C22×Q812yes—
553C42C22×Q812yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23C2×C4⋊Q812yes—
561C23C23×D411yescharacteristic
562Q8—164no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—164no—
622C2×C4C2×C4⋊Q812yes—
623C2×C4C2×C4⋊Q812yes—
624C2×C4C2×C4⋊Q812yes—
625C2×C4C2×C4⋊Q812yes—
626C2×C4C2×C4⋊Q812yes—
627C2×C4#57512yes—
628C2×C4#57512yes—
629C2×C4#57612yes—
630C2×C4#57812yes—
631C2×C4C23×Q811yescharacteristic
632C2×C4C23×Q811yescharacteristic
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×2−1+412yes—
635C8—24no—
636C8—24no—
637C8—24no—
638C8#54418yes—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22C22×C4⋊Q811yescharacteristic
647C22#349811yescharacteristic
648C22#349911yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57611yescharacteristic
652C22C2×#57811yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#316414yes—
673C4#316614yes—
674C4C2×#54412yes—
675C4#350312yes—
676C2—42no—
677C2#3527511yescharacteristic
678C2C2×#316411yescharacteristic
679C2C2×#316611yescharacteristic
680C2#6242511yescharacteristic
681C2#6242611yescharacteristic
682C2#6248111yescharacteristic
683C2#6248211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33311C2semidirect
5. #33263C2semidirect
6. #33258C2semidirect
7. #30307C2semidirect
8. #30307C2semidirect
9. #30307C2semidirect
10. #30304C2semidirect
11. #28839C2semidirect
12. #28820C2semidirect
129. #431Q8semidirect
143. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#352862
#333162
#333112
#332632
#332582
#303072
#303042
#288392
#288202
#260792
#260732
#219842
#219822
#208242
#208002
#204722