Finite Group Databasebeta

← #6497931

Subgroups — #6497931

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic
3#33314C211yesmaximal, characteristic, complemented
4#33263C211yesmaximal, characteristic, complemented
5#33209C211yesmaximal, characteristic
6#33207C211yesmaximal, characteristic
7#30229C211yesmaximal, characteristic, complemented
8#30228C211yesmaximal, characteristic, complemented
9#29878C211yesmaximal, characteristic
10#29877C211yesmaximal, characteristic
11#28009C214yesmaximal, complemented
12#28006C214yesmaximal, complemented
13#27848C214yesmaximal
14#27845C214yesmaximal
15#22247C211yesmaximal, characteristic, complemented
16#22246C211yesmaximal, characteristic, complemented
17#21727C211yesmaximal, characteristic, complemented
18#21726C211yesmaximal, characteristic, complemented
19#20473C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2647—21no—
35#2645—21no—
36#2629—21no—
37#2625—21no—
38#2514—21no—
39#2509C2212yes—
40#2508C2212yes—
41#2507C2212yes—
42#2507C2212yes—
43#2497C2212yes—
44#2497C2212yes—
45#2495C2212yes—
46#2494C2212yes—
47#2489—21no—
48#2453—21no—
49#2446—21no—
50#2446—21no—
51#2429—21no—
52#2429—21no—
53#2421—21no—
54#2330—21no—
55#2323—21no—
56#2320—21no—
57#2314—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2101C2214yes—
64#2101C2214yes—
65#2099C2214yes—
66#2099C2214yes—
67#2088C2214yes—
68#2087C2214yes—
69#2077C2214yes—
70#2076C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1944C2214yes—
80#1944C2214yes—
81#1943C2214yes—
82#1943C2214yes—
83#1942C2214yes—
84#1941C2214yes—
85#1920C2214yes—
86#1919C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1847C2214yes—
90#1847C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1808C2214yes—
96#1807C2214yes—
97#1776C2214yes—
98#1776C2214yes—
99#1770C2212yes—
100#1770C2212yes—
101#1752C2212yes—
102#1752C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C23×C8C2312yes—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C42C2311yescharacteristic
127#500—24no—
128#500—24no—
129#498—24no—
130#498—24no—
131#431—24no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×#97C2312yes—
144C2×C4×C8—24no—
145C2×C4×C8C2312yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#400—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#397—22no—
164#396—22no—
165#396—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#389—22no—
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—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#384—21no—
217#384—21no—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#384C2312yes—
224#383—22no—
225#382—21no—
226#382—21no—
227#382—22no—
228#382C2312yes—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380C2312yes—
235#380C2312yes—
236#380C2312yes—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4C2311yescharacteristic
253C4×C22⋊C4C2311yescharacteristic
254#376D411yescharacteristic
255#376D411yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
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—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C22×C8—24no—
280C22×C8—24no—
281C22×C8—24no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—82no—
295C4⋊Q8—82no—
296#125—44no—
297#125—82no—
298C22⋊Q8—44no—
299C22⋊Q8—82no—
300C22⋊Q8—82no—
301C22⋊Q8—82no—
302C4×Q8—44no—
303C4×Q8—82no—
304C4×Q8—82no—
305C4×Q8—82no—
306#117—24no—
307#117—42no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
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—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
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—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
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×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—28no—
380#107—28no—
381#106—28no—
382#106—28no—
383#106—28no—
384C4⋊C8—24no—
385C22⋊C8—24no—
386#97—24no—
387#97—24no—
388#97—24no—
389#97—24no—
390C4×C8—24no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8#4114yes—
394C4×C8#4114yes—
395#95—22no—
396#95—22no—
397#95—22no—
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#95—22no—
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427C24C22×D411yescharacteristic
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4#13714yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—48no—
481C2×C8C2×#4114yes—
482C2×C8C2×#4114yes—
483C2×C82−1+414yes—
484C2×C82−1+414yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
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—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—44no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23C2×#13712yes—
566C23C23×D411yescharacteristic
567Q8—84no—
568Q8—84no—
569Q8—84no—
570Q8—84no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—28no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—44no—
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—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4#53212yes—
632C2×C4#53212yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57512yes—
636C2×C4#57612yes—
637C2×C4#57712yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×2−1+412yes—
641C2×C4C2×2−1+412yes—
642C8—24no—
643C8—42no—
644C8—42no—
645C8#54118yes—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22C2×#53211yescharacteristic
654C22#349911yescharacteristic
655C22#349911yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57611yescharacteristic
659C22C2×#57711yescharacteristic
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#317914yes—
680C4#318114yes—
681C4C2×#54112yes—
682C4#350912yes—
683C2—42no—
684C2#3531911yescharacteristic
685C2C2×#317911yescharacteristic
686C2C2×#318111yescharacteristic
687C2#6247811yescharacteristic
688C2#6248011yescharacteristic
689C2#6248111yescharacteristic
690C2#6248311yescharacteristic
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. #33314C2semidirect
4. #33263C2semidirect
7. #30229C2semidirect
8. #30228C2semidirect
11. #28009C2semidirect
12. #28006C2semidirect
15. #22247C2semidirect
16. #22246C2semidirect
17. #21727C2semidirect
18. #21726C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352842
#333142
#332632
#332092
#332072
#302292
#302282
#298782
#298772
#280092
#280062
#278482
#278452
#222472
#222462
#217272
#217262
#204732