Finite Group Databasebeta

← #7572978

Subgroups — #7572978

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59668C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#32515C212yesmaximal
4#32130C212yesmaximal
5#29319C211yesmaximal, characteristic
6#29178C211yesmaximal, characteristic
7#28835C211yesmaximal, characteristic
8#28791C211yesmaximal, characteristic
9#28006C212yesmaximal
10#27313C212yesmaximal
11#27305C212yesmaximal
12#26876C212yesmaximal
13#26511C211yesmaximal, characteristic
14#26508C211yesmaximal, characteristic
15#26108C211yesmaximal, characteristic
16#26056C212yesmaximal
17#26046C211yesmaximal, characteristic
18#23258C211yesmaximal, characteristic
19#22971C212yesmaximal
20#22971C211yesmaximal, characteristic
21#22965C211yesmaximal, characteristic
22#21616C211yesmaximal, characteristic
23C4×#475C211yesmaximal, characteristic
24C2×#477C2212yes—
25C2×#477C2211yescharacteristic
26C2×#475C2211yescharacteristic
27#2859—21no—
28#2859C2211yescharacteristic
29#2806—22no—
30#2584—21no—
31#2571—21no—
32#2571C2211yescharacteristic
33#2552—21no—
34#2552C2211yescharacteristic
35#2552C2211yescharacteristic
36#2551C2211yescharacteristic
37#2551C2212yes—
38#2550—21no—
39#2546—21no—
40#2508—22no—
41#2508C2212yes—
42#2507C2212yes—
43#2507C2212yes—
44#2506—22no—
45#2506C2212yes—
46#2451—22no—
47#2370—21no—
48#2369C2211yescharacteristic
49#2369C2211yescharacteristic
50#2369C2212yes—
51#2351C2212yes—
52Q8×C4⋊C4C2212yes—
53Q8×C4⋊C4C2211yescharacteristic
54Q8×C4⋊C4C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125—21no—
58C4×#125C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2212yes—
61#2099C2212yes—
62#2099C2212yes—
63#2088C2212yes—
64#2088C2212yes—
65#2088C2212yes—
66#2087C2212yes—
67#2086C2212yes—
68#2086C2212yes—
69#2085C2212yes—
70#2084C2212yes—
71#2084C2212yes—
72#2083C2212yes—
73#2058C2212yes—
74#2057C2212yes—
75#2057C2212yes—
76#1990C2212yes—
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1949C2212yes—
82#1948C2211yescharacteristic
83#1948C2212yes—
84#1948C2211yescharacteristic
85#1948C2211yescharacteristic
86#1948C2211yescharacteristic
87#1947C2211yescharacteristic
88#1947C2211yescharacteristic
89#1943C2212yes—
90#1943C2212yes—
91#1943C2212yes—
92#1943C2212yes—
93#1942C2212yes—
94#1941C2212yes—
95#1940C2212yes—
96#1939C2211yescharacteristic
97#1939C2211yescharacteristic
98#1921C2212yes—
99#1919C2211yescharacteristic
100#1919C2211yescharacteristic
101#1919C2212yes—
102#1918C2212yes—
103#1918C2212yes—
104#1918C2212yes—
105#1918C2212yes—
106#1918C2212yes—
107#1917C2212yes—
108#1916C2212yes—
109#1865C2211yescharacteristic
110#1865C2211yescharacteristic
111#1865C2212yes—
112#1865C2212yes—
113#1865C2212yes—
114#1863C2212yes—
115#1863C2212yes—
116#1847C2212yes—
117#1845C2211yescharacteristic
118#1834C2211yescharacteristic
119#1803C2211yescharacteristic
120#1803C2211yescharacteristic
121#1803C2212yes—
122#1776C2212yes—
123C4×#106C2211yescharacteristic
124C4×#106C2211yescharacteristic
125#1770C2212yes—
126#1770C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128#1767C2211yescharacteristic
129C4×C4⋊C8C2211yescharacteristic
130#1763C2212yes—
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#103C2212yes—
135C2×C4⋊Q8—22no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2312yes—
139C2×C4⋊Q8C2311yescharacteristic
140C2×#125—21no—
141C2×#125—21no—
142C2×#125—22no—
143C2×#125—21no—
144C2×#125D411yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2312yes—
149C2×C4×Q8C2312yes—
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2311yescharacteristic
152#477—28no—
153#477Q818yescomplemented
154#475—24no—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2312yes—
184#400—22no—
185#400—22no—
186#400—22no—
187#398—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#395—22no—
197#395—22no—
198#391—21no—
199#391—21no—
200#391—22no—
201#391—22no—
202#391—22no—
203#391—22no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—22no—
208#389—21no—
209#389—22no—
210#389—22no—
211#389—21no—
212#389—21no—
213#389—22no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—22no—
219#389—22no—
220#389—22no—
221#389—22no—
222#389—22no—
223#389—22no—
224#389D411yescharacteristic
225#389D411yescharacteristic
226#389C2312yes—
227#389C2311yescharacteristic
228#389C2311yescharacteristic
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387D412yes—
244#387D412yes—
245#384—21no—
246#384—21no—
247#384—22no—
248#384C2311yescharacteristic
249#384C2311yescharacteristic
250#384C2312yes—
251#384C2311yescharacteristic
252#384C2312yes—
253#384C2311yescharacteristic
254#382—21no—
255#382—21no—
256#382—22no—
257#382—22no—
258#382D411yescharacteristic
259#382C2312yes—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—22no—
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2311yescharacteristic
274C4×C4⋊C4C2311yescharacteristic
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284C43—21no—
285C43C2311yescharacteristic
286#340C2311yescharacteristic
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2311yescharacteristic
292#340C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2311yescharacteristic
307#340C2311yescharacteristic
308#336C2312yes—
309#336C2312yes—
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2411yescharacteristic
321C22×C8C2411yescharacteristic
322C22×C8C2412yes—
323C22×C8C2412yes—
324C22×C8C2411yescharacteristic
325C4⋊Q8—22no—
326C4⋊Q8—24no—
327C4⋊Q8—22no—
328C4⋊Q8—42no—
329C4⋊Q8—41no—
330C4⋊Q8—82no—
331C4⋊Q8—84no—
332C4⋊Q8C2×Q814yes—
333#125—24no—
334#125—44no—
335#125—82no—
336#125—84no—
337#125—82no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—24no—
341C4×Q8—24no—
342C4×Q8—22no—
343C4×Q8—42no—
344C4×Q8—41no—
345C4×Q8—42no—
346C4×Q8—41no—
347C4×Q8—82no—
348C4×Q8—82no—
349C4×Q8C2×Q814yes—
350C2×C4⋊C4—21no—
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—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
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—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2412yes—
401C2×C4⋊C4C2411yescharacteristic
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2411yescharacteristic
420C2×C42C2412yes—
421C2×C42C2411yescharacteristic
422C2×C42C2412yes—
423C2×C42C2411yescharacteristic
424#107—24no—
425#107—24no—
426#107—42no—
427#107—42no—
428#107C2×Q818yes—
429#106—22no—
430#106—22no—
431#106—42no—
432#106—41no—
433#106—42no—
434#106—41no—
435#106—42no—
436C4⋊C8—22no—
437C4⋊C8—24no—
438C4⋊C8—22no—
439C4⋊C8—42no—
440C4⋊C8—41no—
441C4⋊C8C2×Q814yes—
442#103—22no—
443#103—24no—
444#103—22no—
445#103—22no—
446#103C2×Q818yes—
447#103#4114yes—
448#97—42no—
449#97—42no—
450C4×C8—22no—
451C4×C8—22no—
452C4×C8—42no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—21no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—21no—
470#95—22no—
471#95—21no—
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—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—42no—
487C2×Q8—44no—
488C2×Q8—42no—
489C2×Q8—81no—
490C2×Q8—81no—
491C2×Q8—81no—
492C2×Q8—82no—
493C2×Q8—82no—
494C2×Q8C22×Q812yes—
495C2×Q8C2×#4112yes—
496C2×Q82−1+412yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C2511yescharacteristic
520C2×C8—21no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8—22no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8C22×Q814yes—
537C2×C8C2×#4112yes—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—44no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—42no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—81no—
580C4⋊C4—82no—
581C4⋊C4—81no—
582C4⋊C4—81no—
583C4⋊C4—81no—
584C4⋊C4—81no—
585C4⋊C4—81no—
586C4⋊C4—81no—
587C4⋊C4—82no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4#13714yes—
598C4⋊C4C22×Q814yes—
599C4⋊C4C22×Q812yes—
600C4⋊C4C2×#4112yes—
601C42—24no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—21no—
607C42—21no—
608C42—22no—
609C42—22no—
610C42—21no—
611C42—22no—
612C42—21no—
613C42—41no—
614C42—44no—
615C42—42no—
616C42—42no—
617C42—44no—
618C42—42no—
619C42—42no—
620C42—42no—
621C42—42no—
622C42—42no—
623C42—82no—
624C42—82no—
625C42—82no—
626C42C22×Q812yes—
627C422−1+412yes—
628C23C23×D411yescharacteristic
629Q8—22no—
630Q8—22no—
631Q8—22no—
632Q8—84no—
633Q8—84no—
634Q8—164no—
635Q8#55714yes—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4D4×Q812yes—
697C2×C4D4×Q812yes—
698C2×C4C2×#13714yes—
699C2×C4C2×#13712yes—
700C2×C4#57612yes—
701C2×C4#57712yes—
702C2×C4C23×Q811yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4#58511yescharacteristic
707C2×C4#58511yescharacteristic
708C8—24no—
709C8—22no—
710C8—41no—
711C8—42no—
712C8—41no—
713C8—41no—
714C22C2×D4×Q811yescharacteristic
715C22D4×#4111yescharacteristic
716C22#348911yescharacteristic
717C22C22×#13711yescharacteristic
718C22C2×#57611yescharacteristic
719C22C2×#57711yescharacteristic
720C22#358611yescharacteristic
721C4—24no—
722C4—24no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—41no—
731C4—42no—
732C4—41no—
733C4—42no—
734C4—81no—
735C4—82no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4#338714yes—
743C4C2×#55711yescharacteristic
744C4#350411yescharacteristic
745C4#351511yescharacteristic
746C4#356311yescharacteristic
747C2#5915411yescharacteristic
748C2C2×#338711yescharacteristic
749C2#6138111yescharacteristic
750C2#6145711yescharacteristic
751C2#6149811yescharacteristic
752C2#6214911yescharacteristic
753C2#6219111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
153. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#596682
#364472
#325152
#321302
#293192
#291782
#288352
#287912
#280062
#273132
#273052
#268762
#265112
#265082
#261082
#260562
#260462
#232582
#229712
#229652
#216162
C4×#4752