Finite Group Databasebeta

← #6555968

Subgroups — #6555968

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36600C211yesmaximal, characteristic
2#34335C211yesmaximal, characteristic
3#29989C212yesmaximal
4#29988C211yesmaximal, characteristic
5#29275C211yesmaximal, characteristic
6#29274C212yesmaximal
7#29212C212yesmaximal
8#29207C211yesmaximal, characteristic
9#26520C212yesmaximal
10#26519C212yesmaximal
11#26512C211yesmaximal, characteristic
12#26508C211yesmaximal, characteristic
13#26101C212yesmaximal
14#26100C212yesmaximal
15#26092C211yesmaximal, characteristic
16#26088C211yesmaximal, characteristic
17#23785C212yesmaximal
18#23779C211yesmaximal, characteristic
19#22736C211yesmaximal, characteristic
20#22729C211yesmaximal, characteristic
21#22728C211yesmaximal, characteristic
22#22525C211yesmaximal, characteristic
23#22525C211yesmaximal, characteristic
24C2×Q82C2211yescharacteristic
25C2×#446C2211yescharacteristic
26C2×#446C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2212yes—
29#2585—22no—
30#2585—22no—
31#2570C2212yes—
32#2570C2212yes—
33#2552C2211yescharacteristic
34#2552C2211yescharacteristic
35#2453—21no—
36#2453—21no—
37#2370—21no—
38#2370—21no—
39#2369C2211yescharacteristic
40#2369C2212yes—
41#2365—22no—
42#2365—21no—
43#2355C2211yescharacteristic
44#2355C2212yes—
45#2355C2211yescharacteristic
46#2355C2211yescharacteristic
47#2351—21no—
48Q8×C4⋊C4C2211yescharacteristic
49#2331—22no—
50#2331—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×#125C2212yes—
54C4×#125C2211yescharacteristic
55#2295C2211yescharacteristic
56#2295C2212yes—
57#2276—21no—
58C42×Q8C2211yescharacteristic
59#2101C2212yes—
60#2101C2211yescharacteristic
61#2101C2212yes—
62#2100C2212yes—
63#2099C2212yes—
64#2097C2212yes—
65#2097C2211yescharacteristic
66#2096C2212yes—
67#2095C2212yes—
68#2088C2212yes—
69#2088C2211yescharacteristic
70#2087C2212yes—
71#2086C2211yescharacteristic
72#2068C2211yescharacteristic
73#2068C2212yes—
74#2066C2211yescharacteristic
75#2066C2212yes—
76#2059C2212yes—
77#2059C2212yes—
78#2059C2211yescharacteristic
79#2059C2212yes—
80#2059C2212yes—
81#2026C2212yes—
82#2026C2211yescharacteristic
83#1993C2212yes—
84#1993C2212yes—
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1988C2212yes—
88#1988C2212yes—
89#1987C2211yescharacteristic
90#1987C2211yescharacteristic
91#1949C2212yes—
92#1949C2212yes—
93#1949C2212yes—
94#1949C2212yes—
95#1948C2212yes—
96#1948C2212yes—
97#1948C2211yescharacteristic
98#1948C2212yes—
99#1948C2211yescharacteristic
100#1947C2212yes—
101#1947C2212yes—
102#1947C2211yescharacteristic
103#1947C2212yes—
104#1947C2211yescharacteristic
105#1940C2212yes—
106#1940C2212yes—
107#1939C2211yescharacteristic
108#1939C2211yescharacteristic
109#1921C2212yes—
110#1921C2212yes—
111#1920C2212yes—
112#1919C2212yes—
113#1919C2212yes—
114#1918C2212yes—
115#1847C2212yes—
116#1847C2212yes—
117#1845C2211yescharacteristic
118#1844C2211yescharacteristic
119#1831C2211yescharacteristic
120#1831C2212yes—
121#1805C2212yes—
122#1805C2212yes—
123#1803C2211yescharacteristic
124#1803C2211yescharacteristic
125#1776C2212yes—
126#1776C2212yes—
127C4×#107C2211yescharacteristic
128C4×#106C2211yescharacteristic
129#1767C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132#1763C2212yes—
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135Q82—24no—
136Q82Q818yes—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8—22no—
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2312yes—
141C2×#125—21no—
142C2×#125—21no—
143C2×#125C2312yes—
144C2×#125C2311yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—22no—
148C2×C4×Q8—21no—
149C2×C4×Q8D411yescharacteristic
150C2×C4×Q8D411yescharacteristic
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2311yescharacteristic
154#446—24no—
155#446—24no—
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#107C2311yescharacteristic
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#106C2311yescharacteristic
163C2×#106C2312yes—
164C2×#106C2311yescharacteristic
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#97C2312yes—
183C2×#97C2312yes—
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186#400—22no—
187#400—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#391—21no—
193#391—21no—
194#391—22no—
195#391—21no—
196#391—22no—
197#391—22no—
198#391—22no—
199#391C2312yes—
200#391C2311yescharacteristic
201#391C2311yescharacteristic
202#389—21no—
203#389—22no—
204#389—21no—
205#389—22no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—22no—
210#389—21no—
211#389—21no—
212#389—22no—
213#389—22no—
214#389—22no—
215#389—22no—
216#389—21no—
217#389—21no—
218#389C2311yescharacteristic
219#389C2311yescharacteristic
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2311yescharacteristic
224#389C2311yescharacteristic
225#389C2312yes—
226#387—22no—
227#387—22no—
228#387—21no—
229#387—21no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—21no—
234#387—22no—
235#387—21no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#384C2311yescharacteristic
244#384C2312yes—
245#384C2312yes—
246#384C2312yes—
247#384C2311yescharacteristic
248#384C2311yescharacteristic
249#382—21no—
250#382—22no—
251#382C2311yescharacteristic
252#382C2312yes—
253#382C2312yes—
254#382C2312yes—
255C4×C4⋊C4—22no—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4—21no—
259C4×C4⋊C4—22no—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—22no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—22no—
269C4×C4⋊C4D412yes—
270C4×C4⋊C4D412yes—
271C4×C4⋊C4D411yescharacteristic
272C4×C4⋊C4D411yescharacteristic
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284#376—21no—
285#376—22no—
286C43C2311yescharacteristic
287C43C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2311yescharacteristic
308#340C2312yes—
309#340C2311yescharacteristic
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316C22×Q8—22no—
317C22×Q8—21no—
318C22×Q8C2×D411yescharacteristic
319C22×Q8C2411yescharacteristic
320C22×C8C2411yescharacteristic
321C22×C8C2411yescharacteristic
322C22×C8C2412yes—
323C22×C8C2412yes—
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C4⋊Q8—216no—
327C4⋊Q8—28no—
328C4⋊Q8—22no—
329C4⋊Q8—24no—
330C4⋊Q8C2×Q814yes—
331C4⋊Q8C2×Q818yes—
332#125—42no—
333#125—41no—
334#125—42no—
335#125—41no—
336#125—44no—
337#125—44no—
338C4×Q8—28no—
339C4×Q8—24no—
340C4×Q8—22no—
341C4×Q8—24no—
342C4×Q8—42no—
343C4×Q8—44no—
344C4×Q8—84no—
345C4×Q8—82no—
346C4×Q8C2×Q814yes—
347C4×Q8C2×Q818yes—
348C4×Q8C2×Q814yes—
349C4×Q8#4114yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
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—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D412yes—
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D412yes—
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D412yes—
412C2×C42C2×D412yes—
413C2×C42C2×D411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2412yes—
417C2×C42C2412yes—
418C2×C42C2411yescharacteristic
419#107—22no—
420#107—42no—
421#107—41no—
422#107—42no—
423#107—42no—
424#107—42no—
425#106—22no—
426#106—41no—
427#106—42no—
428#106—42no—
429#106—42no—
430#106—42no—
431C4⋊C8—22no—
432C4⋊C8—24no—
433C4⋊C8—24no—
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—22no—
437C4⋊C8—24no—
438C4⋊C8—24no—
439#103—22no—
440#103—22no—
441#103—41no—
442#103—42no—
443#103—42no—
444#103—41no—
445#97—24no—
446#97—24no—
447C4×C8—22no—
448C4×C8—22no—
449#95—21no—
450#95—21no—
451#95—22no—
452#95—21no—
453#95—22no—
454#95—21no—
455#95—22no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—21no—
464#95—21no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—21no—
471#95—21no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—21no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
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—82no—
493C2×Q8—81no—
494C2×Q8—82no—
495C2×Q8—81no—
496C2×Q8C22⋊Q814yes—
497C2×Q8C22×Q812yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C2511yescharacteristic
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—21no—
537C2×C82−1+412yes—
538C2×C82−1+412yes—
539C4⋊C4—28no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—28no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—42no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—42no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—81no—
581C4⋊C4—82no—
582C4⋊C4—81no—
583C4⋊C4—81no—
584C4⋊C4—82no—
585C4⋊C4—81no—
586C4⋊C4—82no—
587C4⋊C4—81no—
588C4⋊C4—82no—
589C4⋊C4—81no—
590C4⋊C4C22⋊Q818yes—
591C4⋊C4C22⋊Q814yes—
592C4⋊C4#13714yes—
593C4⋊C4C22×Q812yes—
594C4⋊C4C22×Q814yes—
595C42—28no—
596C42—24no—
597C42—24no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—21no—
603C42—21no—
604C42—21no—
605C42—21no—
606C42—42no—
607C42—41no—
608C42—42no—
609C42—42no—
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—88no—
631Q8—84no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4C2×C22⋊Q814yes—
692C2×C4C2×C22⋊Q812yes—
693C2×C4C2×C22⋊Q814yes—
694C2×C4C2×C22⋊Q812yes—
695C2×C4C2×#13714yes—
696C2×C4C2×#13712yes—
697C2×C4#57512yes—
698C2×C4C23×Q811yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C2×C4#58511yescharacteristic
703C2×C4#58511yescharacteristic
704C8—22no—
705C8—22no—
706C8—41no—
707C8—41no—
708C8—42no—
709C8—42no—
710C22C22×C22⋊Q811yescharacteristic
711C22#348711yescharacteristic
712C22#349911yescharacteristic
713C22C22×#13711yescharacteristic
714C22C2×#57511yescharacteristic
715C22#358411yescharacteristic
716C22#358511yescharacteristic
717C4—24no—
718C4—24no—
719C4—24no—
720C4—22no—
721C4—42no—
722C4—41no—
723C4—42no—
724C4—41no—
725C4—44no—
726C4—42no—
727C4—41no—
728C4—41no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4—81no—
733C4—82no—
734C4—82no—
735C4—81no—
736C4—82no—
737C4—81no—
738C4#307814yes—
739C4#350211yescharacteristic
740C4#350211yescharacteristic
741C4#351611yescharacteristic
742C4#351611yescharacteristic
743C2#3584511yescharacteristic
744C2C2×#307811yescharacteristic
745C2#6049611yescharacteristic
746C2#6189611yescharacteristic
747C2#6190911yescharacteristic
748C2#6248511yescharacteristic
749C2#6250411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#366002
#343352
#299892
#299882
#292752
#292742
#292122
#292072
#265202
#265192
#265122
#265082
#261012
#261002
#260922
#260882
#237852
#237792
#227362
#227292
#227282
#225252