Finite Group Databasebeta

← #6497352

Subgroups — #6497352

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33290C211yesmaximal, characteristic, complemented
5#33287C211yesmaximal, characteristic, complemented
6#33278C211yesmaximal, characteristic, complemented
7#30229C211yesmaximal, characteristic, complemented
8#30229C211yesmaximal, characteristic, complemented
9#30206C211yesmaximal, characteristic, complemented
10#30205C211yesmaximal, characteristic, complemented
11#27828C214yesmaximal, complemented
12#27742C214yesmaximal, complemented
13#27723C214yesmaximal
14#27705C214yesmaximal
15#21864C211yesmaximal, characteristic
16#21834C211yesmaximal, characteristic
17#21409C211yesmaximal, characteristic
18#21324C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2862C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2642—21no—
33#2642—21no—
34#2629—21no—
35#2628—21no—
36#2570—21no—
37#2570C2212yes—
38#2568—21no—
39#2568—21no—
40#2568C2212yes—
41#2568C2212yes—
42#2550—21no—
43#2550C2212yes—
44#2503—21no—
45#2503—21no—
46#2503C2212yes—
47#2503C2212yes—
48#2441—21no—
49#2441C2212yes—
50#2439—21no—
51#2439C2212yes—
52#2331—21no—
53#2330—21no—
54#2310—21no—
55#2310—21no—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yes—
60#2101C2214yes—
61#2101C2214yes—
62#2101C2214yes—
63#2074C2214yes—
64#2072C2214yes—
65#2026C2214yes—
66#2024C2214yes—
67#1993C2214yes—
68#1993C2214yes—
69#1991C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1987C2214yes—
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1935C2214yes—
88#1935C2214yes—
89#1933C2214yes—
90#1933C2214yes—
91#1894C2214yes—
92#1873C2214yes—
93#1836C2214yes—
94#1835C2214yes—
95#1824C2214yes—
96#1805C2214yes—
97#1771C2212yes—
98#1771C2212yes—
99#1763C2214yes—
100#1753C2212yes—
101#1753C2212yes—
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1736C2214yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108C22×M16C2312yes—
109#533—24no—
110#533—24no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—24no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#125—21no—
124C2×#125—21no—
125C2×#125—21no—
126C2×#125C2312yes—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117D411yescharacteristic
134C2×#117D411yescharacteristic
135C2×#117C2312yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139#431—24no—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2312yes—
146C4×M16—24no—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#396—22no—
169#396—22no—
170#395—22no—
171#393—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—22no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#384—21no—
213#384—21no—
214#384C2312yes—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#382—21no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#382C2312yes—
232#381—22no—
233#381—22no—
234#381—22no—
235#381—22no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4C2311yescharacteristic
251C4×C22⋊C4C2311yescharacteristic
252#376D411yescharacteristic
253#376D411yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—44no—
278C22×Q8—44no—
279C22×Q8—44no—
280C22×Q8C2412yes—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×M16—24no—
286C2×M16—24no—
287C2×M16—42no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—82no—
307#125—44no—
308#125—44no—
309#125—44no—
310#125—82no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—44no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—44no—
317#117—42no—
318#117—42no—
319#117—44no—
320#117—44no—
321#117—82no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42#4114yes—
383C2×C42#4114yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C4⋊C8—24no—
391#103—28no—
392#103—44no—
393#103—44no—
394#103—44no—
395C22⋊C8—24no—
396#97—24no—
397#97—24no—
398#97—42no—
399C4×C8—24no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—42no—
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#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439C24C22×D411yescharacteristic
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8—816no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#12414yes—
478C22×C4#12414yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2511yescharacteristic
495M16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C82−1+414yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—44no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—44no—
568C42—82no—
569C42—82no—
570C42#12414yes—
571C42#12414yes—
572C42C2×#4112yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23C2×#12412yes—
583C23C23×D411yescharacteristic
584Q8—44no—
585Q8—44no—
586Q8—88no—
587Q8—84no—
588Q8—84no—
589Q8—84no—
590Q8—164no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—88no—
650C2×C4C2×#12412yes—
651C2×C4C2×#12412yes—
652C2×C4C2×#12412yes—
653C2×C4C2×#12412yes—
654C2×C4C2×#12412yes—
655C2×C4#57512yes—
656C2×C4#57512yes—
657C2×C4#57712yes—
658C2×C4#57812yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×2−1+412yes—
662C2×C4#58512yes—
663C8—24no—
664C8—42no—
665C8—42no—
666C8—42no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22C22×#12411yescharacteristic
675C22#349911yescharacteristic
676C22#350011yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57711yescharacteristic
680C22C2×#57811yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#313714yes—
701C4#314014yes—
702C4#346112yes—
703C4#349512yes—
704C2—42no—
705C2#3531211yescharacteristic
706C2C2×#313711yescharacteristic
707C2C2×#314011yescharacteristic
708C2#6248211yescharacteristic
709C2#6248311yescharacteristic
710C2#6254411yescharacteristic
711C2#6254511yescharacteristic
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. #33293C2semidirect
4. #33290C2semidirect
5. #33287C2semidirect
6. #33278C2semidirect
7. #30229C2semidirect
8. #30229C2semidirect
9. #30206C2semidirect
10. #30205C2semidirect
11. #27828C2semidirect
12. #27742C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#352942
#332932
#332902
#332872
#332782
#302292
#302062
#302052
#278282
#277422
#277232
#277052
#218642
#218342
#214092
#213242
#204722