Finite Group Databasebeta

← #6667449

Subgroups — #6667449

Order 512.

1265 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#32854C214yesmaximal, complemented
4#29946C214yesmaximal
5#29945C214yesmaximal
6#29878C211yesmaximal, characteristic
7#29877C211yesmaximal, characteristic
8#29877C211yesmaximal, characteristic
9#29876C211yesmaximal, characteristic
10#29292C214yesmaximal, complemented
11#29220C211yesmaximal, characteristic, complemented
12#29220C211yesmaximal, characteristic, complemented
13#29219C211yesmaximal, characteristic, complemented
14#29218C211yesmaximal, characteristic, complemented
15#28689C211yesmaximal, characteristic, complemented
16#28688C211yesmaximal, characteristic, complemented
17#28688C211yesmaximal, characteristic, complemented
18#28687C211yesmaximal, characteristic, complemented
19#20809C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2647C2212yes—
35#2647C2212yes—
36#2629C2212yes—
37#2628C2212yes—
38#2511—21no—
39#2511—21no—
40#2507—21no—
41#2507—21no—
42#2497—21no—
43#2497—21no—
44#2492—21no—
45#2492—21no—
46#2451—21no—
47#2451—21no—
48#2446—21no—
49#2446—21no—
50#2429—21no—
51#2429—21no—
52#2423—21no—
53#2423—21no—
54#2331C2212yes—
55#2330C2212yes—
56#2314C2212yes—
57#2314C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2101C2214yes—
64#2101C2214yes—
65#2100C2214yes—
66#2099C2214yes—
67#2088C2214yes—
68#2088C2214yes—
69#2087C2214yes—
70#2086C2214yes—
71#2085C2214yes—
72#2085C2214yes—
73#2084C2214yes—
74#2083C2214yes—
75#2077C2214yes—
76#2077C2214yes—
77#2076C2214yes—
78#2075C2214yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1989C2214yes—
88#1989C2214yes—
89#1988C2214yes—
90#1987C2214yes—
91#1949C2214yes—
92#1949C2214yes—
93#1948C2214yes—
94#1947C2214yes—
95#1843C2212yes—
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1832C2212yes—
101#1798C2212yes—
102#1798C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#500—42no—
129#500—42no—
130#498—42no—
131#498—42no—
132#431—42no—
133#431—42no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#396—22no—
163#396—22no—
164#395—22no—
165#393—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#391—22no—
179#391—22no—
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—22no—
193#389—22no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384—21no—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#384C2312yes—
224#384C2312yes—
225#382C2312yes—
226#382C2312yes—
227#382C2312yes—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235#380C2312yes—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252#376C2311yescharacteristic
253#376C2311yescharacteristic
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—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—28no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—82no—
294C4⋊Q8—82no—
295C4⋊Q8—82no—
296C4⋊Q8—82no—
297#125—28no—
298C22⋊Q8—82no—
299C22⋊Q8—82no—
300C22⋊Q8—82no—
301C22⋊Q8—82no—
302C4×Q8—82no—
303C4×Q8—82no—
304C4×Q8—82no—
305C4×Q8—82no—
306#117—24no—
307#117—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
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—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
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⋊C4—22no—
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
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×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—44no—
380#107—44no—
381#106—44no—
382#106—44no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387C4⋊C8—42no—
388C4⋊C8—42no—
389C22⋊C8—42no—
390C22⋊C8—42no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—22no—
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#95C2×D411yescharacteristic
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425C24C22×D411yescharacteristic
426C2×Q8—44no—
427C2×Q8—42no—
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—82no—
440C2×Q8—82no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
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—22no—
458C22×C4—22no—
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×C4C22×D411yescharacteristic
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
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—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
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—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
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—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—22no—
551C42—22no—
552C42—82no—
553C42—82no—
554C42—82no—
555C42—82no—
556C42H(5,𝔽2)12yes—
557C422−1+414yes—
558C422−1+414yes—
559C422−1+412yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C23×D411yescharacteristic
568Q8—84no—
569Q8—84no—
570Q8—84no—
571Q8—84no—
572Q8—164no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—44no—
603C2×C4—44no—
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—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#53612yes—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4#53612yes—
634C2×C4#53712yes—
635C2×C4#53712yes—
636C2×C4#57612yes—
637C2×C4#57712yes—
638C2×C4#57712yes—
639C2×C4#57812yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4C2×2−1+412yes—
643C2×C4C2×2−1+411yescharacteristic
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22C2×#53611yescharacteristic
656C22C2×#53611yescharacteristic
657C22C2×#53711yescharacteristic
658C22C2×#57611yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57711yescharacteristic
661C22C2×#57811yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#325714yes—
682C4#325814yes—
683C4#355012yes—
684C4#355612yes—
685C2—81no—
686C2#3764111yescharacteristic
687C2#6087711yescharacteristic
688C2#6087811yescharacteristic
689C2#6088011yescharacteristic
690C2#6088111yescharacteristic
691C2C2×#325711yescharacteristic
692C2C2×#325811yescharacteristic
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. #32854C2semidirect
10. #29292C2semidirect
11. #29220C2semidirect
12. #29220C2semidirect
13. #29219C2semidirect
14. #29218C2semidirect
15. #28689C2semidirect
16. #28688C2semidirect
17. #28688C2semidirect
18. #28687C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#352832
#328542
#299462
#299452
#298782
#298772
#298762
#292922
#292202
#292192
#292182
#286892
#286882
#286872
#208092