Finite Group Databasebeta

← #6548944

Subgroups — #6548944

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39180C211yesmaximal, characteristic
2#34491C211yesmaximal, characteristic
3#30005C212yesmaximal
4#29997C211yesmaximal, characteristic
5#29292C212yesmaximal
6#29290C211yesmaximal, characteristic
7#29189C212yesmaximal
8#29178C211yesmaximal, characteristic
9#26508C211yesmaximal, characteristic
10#26506C212yesmaximal
11#26455C212yesmaximal
12#26445C211yesmaximal, characteristic
13#26104C212yesmaximal
14#26092C211yesmaximal, characteristic
15#26031C212yesmaximal
16#26023C211yesmaximal, characteristic
17#23792C212yesmaximal
18#23776C211yesmaximal, characteristic
19#22739C211yesmaximal, characteristic
20#22728C211yesmaximal, characteristic
21#22721C211yesmaximal, characteristic
22#22525C211yesmaximal, characteristic
23Q8×#103C211yesmaximal, characteristic
24C2×Q82C2211yescharacteristic
25C2×#446C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27#2859C2211yescharacteristic
28#2755—22no—
29#2726—22no—
30#2647C2212yes—
31#2584—21no—
32#2582—21no—
33#2552C2212yes—
34#2552C2211yescharacteristic
35#2552C2211yescharacteristic
36#2550C2212yes—
37#2506—21no—
38#2453—21no—
39#2381—22no—
40#2369C2211yescharacteristic
41#2365—21no—
42#2365C2212yes—
43#2356—21no—
44#2356C2212yes—
45#2355C2211yescharacteristic
46#2355C2211yescharacteristic
47#2355C2211yescharacteristic
48Q8×C4⋊C4C2211yescharacteristic
49#2331—21no—
50#2331C2212yes—
51#2330C2212yes—
52#2327—22no—
53#2320—21no—
54#2320C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2295C2211yescharacteristic
58#2295C2211yescharacteristic
59#2099C2211yescharacteristic
60#2099C2212yes—
61#2096C2211yescharacteristic
62#2095C2212yes—
63#2088C2211yescharacteristic
64#2088C2212yes—
65#2088C2212yes—
66#2088C2211yescharacteristic
67#2087C2212yes—
68#2087C2212yes—
69#2086C2212yes—
70#2086C2212yes—
71#2086C2212yes—
72#2086C2212yes—
73#2068C2212yes—
74#2068C2212yes—
75#2068C2212yes—
76#2066C2212yes—
77#2066C2211yescharacteristic
78#2066C2211yescharacteristic
79#2059C2212yes—
80#2058C2211yescharacteristic
81#2026C2212yes—
82#2024C2211yescharacteristic
83#1993C2212yes—
84#1990C2212yes—
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1989C2212yes—
88#1989C2212yes—
89#1988C2212yes—
90#1987C2211yescharacteristic
91#1987C2212yes—
92#1987C2212yes—
93#1987C2211yescharacteristic
94#1949C2212yes—
95#1949C2212yes—
96#1949C2212yes—
97#1948C2211yescharacteristic
98#1948C2212yes—
99#1947C2211yescharacteristic
100#1947C2212yes—
101#1947C2212yes—
102#1947C2211yescharacteristic
103#1947C2211yescharacteristic
104#1947C2212yes—
105#1940C2212yes—
106#1939C2211yescharacteristic
107#1939C2212yes—
108#1939C2211yescharacteristic
109#1921C2212yes—
110#1920C2212yes—
111#1919C2212yes—
112#1918C2212yes—
113#1918C2212yes—
114#1918C2212yes—
115#1847C2212yes—
116#1845C2211yescharacteristic
117#1843C2212yes—
118#1842C2211yescharacteristic
119#1835C2212yes—
120#1834C2211yescharacteristic
121#1805C2212yes—
122#1803C2211yescharacteristic
123#1803C2211yescharacteristic
124#1803C2212yes—
125#1776C2212yes—
126C4×#107C2212yes—
127C4×#106C2211yescharacteristic
128C4×#106C2211yescharacteristic
129#1768C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2212yes—
134C4×#103C2211yescharacteristic
135Q82—24no—
136Q82Q818yescomplemented
137C2×C4⋊Q8—22no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2312yes—
142C2×#125—21no—
143C2×#125—22no—
144C2×#125—21no—
145C2×#125C2312yes—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8—22no—
149C2×C4×Q8—21no—
150C2×C4×Q8—21no—
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2311yescharacteristic
154#446—24no—
155C8×Q8—24no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106C2311yescharacteristic
162C2×#106C2311yescharacteristic
163C2×#106C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2312yes—
181C2×#103C2311yescharacteristic
182C2×#97C2312yes—
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2312yes—
185C2×C4×C8C2311yescharacteristic
186#400—22no—
187#400—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#395—22no—
196#391—21no—
197#391—21no—
198#391—22no—
199#391—21no—
200#391—21no—
201#391—22no—
202#391—22no—
203#391—22no—
204#391—22no—
205#391D411yescharacteristic
206#391D411yescharacteristic
207#391C2311yescharacteristic
208#391C2312yes—
209#391C2311yescharacteristic
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—22no—
215#389—22no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—22no—
221#389—22no—
222#389—22no—
223#389D411yescharacteristic
224#389D411yescharacteristic
225#389C2312yes—
226#389C2311yescharacteristic
227#389C2311yescharacteristic
228#389C2311yescharacteristic
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2311yescharacteristic
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—21no—
240#387—21no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387D412yes—
251#387D412yes—
252#384—21no—
253#384C2311yescharacteristic
254#384C2311yescharacteristic
255#384C2311yescharacteristic
256#384C2311yescharacteristic
257#384C2312yes—
258#384C2312yes—
259#383—22no—
260#383—22no—
261#382—21no—
262#382—21no—
263#382—22no—
264#382C2312yes—
265#382C2312yes—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4C2311yescharacteristic
274C4×C4⋊C4C2311yescharacteristic
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284#376—21no—
285#376—22no—
286#376C2312yes—
287C43C2311yescharacteristic
288#340C2311yescharacteristic
289#340C2311yescharacteristic
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#340C2311yescharacteristic
311#340C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2312yes—
314#336C2311yescharacteristic
315#336C2312yes—
316C22×Q8—22no—
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8C2×D411yescharacteristic
320C22×Q8C2411yescharacteristic
321C22×C8C2411yescharacteristic
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2412yes—
325C22×C8C2412yes—
326C22×C8C2411yescharacteristic
327C4⋊Q8—216no—
328C4⋊Q8—28no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—24no—
332C4⋊Q8—41no—
333C4⋊Q8C2×Q814yes—
334C4⋊Q8C2×Q818yes—
335#125—24no—
336#125—42no—
337#125—82no—
338#125—82no—
339#125—84no—
340C4×Q8—28no—
341C4×Q8—22no—
342C4×Q8—24no—
343C4×Q8—82no—
344C4×Q8—82no—
345C4×Q8—84no—
346C4×Q8—82no—
347C4×Q8C2×Q814yes—
348C4×Q8C2×Q818yes—
349C4×Q8C2×Q814yes—
350C4×Q8#4114yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2411yescharacteristic
397C2×C42—21no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2412yes—
417#107—24no—
418#107—42no—
419#107—42no—
420#107—42no—
421#107—42no—
422#106—22no—
423#106—22no—
424#106—42no—
425#106—41no—
426#106—41no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—24no—
430C4⋊C8—22no—
431C4⋊C8—24no—
432C4⋊C8—41no—
433C4⋊C8—42no—
434C4⋊C8—42no—
435C4⋊C8—42no—
436C4⋊C8—42no—
437#103—22no—
438#103—22no—
439#103—42no—
440#103—41no—
441#103—41no—
442#103—42no—
443#97—24no—
444C4×C8—22no—
445C4×C8—24no—
446C4×C8—22no—
447#95—21no—
448#95—21no—
449#95—22no—
450#95—22no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—21no—
458#95—22no—
459#95—22no—
460#95—21no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—21no—
467#95—21no—
468#95—22no—
469#95—22no—
470#95—21no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95C2×D412yes—
477#95C2×D412yes—
478#95C2×D412yes—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483C2×Q8—24no—
484C2×Q8—28no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—81no—
495C2×Q8—82no—
496C2×Q8—81no—
497C2×Q8—81no—
498C2×Q8—82no—
499C2×Q8C22⋊Q814yes—
500C2×Q8C22×Q812yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C2511yescharacteristic
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—22no—
529C2×C8—21no—
530C2×C8—22no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—21no—
536C2×C8—22no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—21no—
541C2×C82−1+412yes—
542C4⋊C4—28no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—28no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—81no—
579C4⋊C4—81no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4—82no—
583C4⋊C4—81no—
584C4⋊C4—81no—
585C4⋊C4—81no—
586C4⋊C4—81no—
587C4⋊C4—81no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4C22⋊Q818yes—
595C4⋊C4C22⋊Q814yes—
596C4⋊C4C22×Q812yes—
597C4⋊C4C22×Q814yes—
598C4⋊C42−1+412yes—
599C42—28no—
600C42—24no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—21no—
605C42—22no—
606C42—21no—
607C42—21no—
608C42—22no—
609C42—21no—
610C42—41no—
611C42—44no—
612C42—42no—
613C42—42no—
614C42—42no—
615C42—82no—
616C42—82no—
617C42—82no—
618C42—82no—
619C42—82no—
620C42C22×Q814yes—
621C42C22×Q812yes—
622C42C2×#4112yes—
623C42C2×#4114yes—
624C23C23×D411yescharacteristic
625Q8—28no—
626Q8—22no—
627Q8—22no—
628Q8—41no—
629Q8—41no—
630Q8—84no—
631Q8—88no—
632Q8—84no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—42no—
692C2×C4C2×C22⋊Q814yes—
693C2×C4C2×C22⋊Q812yes—
694C2×C4C2×C22⋊Q814yes—
695C2×C4C2×C22⋊Q812yes—
696C2×C4#57612yes—
697C2×C4#57712yes—
698C2×C4#57812yes—
699C2×C4C23×Q811yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C2×C4C2×2−1+411yescharacteristic
703C2×C4#58511yescharacteristic
704C2×C4#58511yescharacteristic
705C8—22no—
706C8—41no—
707C8—41no—
708C8—41no—
709C8—42no—
710C8—42no—
711C22C22×C22⋊Q811yescharacteristic
712C22#348911yescharacteristic
713C22#349711yescharacteristic
714C22C2×#57611yescharacteristic
715C22C2×#57711yescharacteristic
716C22C2×#57811yescharacteristic
717C22#358611yescharacteristic
718C4—24no—
719C4—22no—
720C4—24no—
721C4—22no—
722C4—44no—
723C4—42no—
724C4—41no—
725C4—41no—
726C4—42no—
727C4—42no—
728C4—41no—
729C4—42no—
730C4—42no—
731C4—82no—
732C4—81no—
733C4—81no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4#308214yes—
740C4#350211yescharacteristic
741C4#350211yescharacteristic
742C4#352311yescharacteristic
743C4#352411yescharacteristic
744C2#3578611yescharacteristic
745C2C2×#308211yescharacteristic
746C2#6049411yescharacteristic
747C2#6218911yescharacteristic
748C2#6220611yescharacteristic
749C2#6238111yescharacteristic
750C2#6240611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
136. Q82Q8semidirect

Maximal subgroups

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

subgroupindex
#391802
#344912
#300052
#299972
#292922
#292902
#291892
#291782
#265082
#265062
#264552
#264452
#261042
#260922
#260312
#260232
#237922
#237762
#227392
#227282
#227212
#225252
Q8×#1032