Finite Group Databasebeta

← #7604752

Subgroups — #7604752

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59666C211yesmaximal, characteristic
2#59445C211yesmaximal, characteristic
3#32312C212yesmaximal
4#30806C212yesmaximal
5#29277C211yesmaximal, characteristic
6#29038C212yesmaximal
7#28802C212yesmaximal
8#28205C211yesmaximal, characteristic
9#28042C212yesmaximal
10#27824C211yesmaximal, characteristic
11#27738C211yesmaximal, characteristic
12#27359C212yesmaximal
13#27137C212yesmaximal
14#26089C211yesmaximal, characteristic
15#26043C211yesmaximal, characteristic
16#25453C211yesmaximal, characteristic
17#25453C211yesmaximal, characteristic
18#23078C212yesmaximal
19#23068C211yesmaximal, characteristic
20#22970C211yesmaximal, characteristic
21C4×#498C211yesmaximal, characteristic
22#21673C211yesmaximal, characteristic
23C4×#487C211yesmaximal, characteristic
24C2×#498C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#479C2212yes—
27C2×#475C2211yescharacteristic
28C2×#475C2211yescharacteristic
29#2802—22no—
30#2774—22no—
31#2582—21no—
32#2581—21no—
33#2571C2211yescharacteristic
34#2570—21no—
35#2567C2211yescharacteristic
36#2567C2211yescharacteristic
37#2565—21no—
38#2552C2212yes—
39#2552C2211yescharacteristic
40#2550C2212yes—
41#2520—22no—
42#2511C2212yes—
43#2508—21no—
44#2508C2212yes—
45#2507C2212yes—
46#2506—21no—
47#2462—22no—
48#2446C2212yes—
49#2370—21no—
50#2370C2212yes—
51#2365C2211yescharacteristic
52#2365C2211yescharacteristic
53#2355C2211yescharacteristic
54#2355C2212yes—
55#2351—21no—
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#125C2211yescharacteristic
62#2100C2212yes—
63#2099C2212yes—
64#2099C2212yes—
65#2097C2212yes—
66#2088C2212yes—
67#2084C2212yes—
68#2084C2212yes—
69#2084C2212yes—
70#2074C2212yes—
71#2066C2212yes—
72#2066C2212yes—
73#2066C2212yes—
74#2059C2212yes—
75#2057C2212yes—
76#2043C2212yes—
77#2026C2212yes—
78#1992C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1987C2211yescharacteristic
83#1987C2211yescharacteristic
84#1956C2211yescharacteristic
85#1949C2212yes—
86#1948C2212yes—
87#1948C2212yes—
88#1947C2211yescharacteristic
89#1947C2211yescharacteristic
90#1944C2212yes—
91#1941C2212yes—
92#1941C2212yes—
93#1939C2211yescharacteristic
94#1939C2211yescharacteristic
95#1935C2212yes—
96#1935C2212yes—
97#1935C2212yes—
98#1933C2211yescharacteristic
99#1933C2211yescharacteristic
100#1933C2211yescharacteristic
101#1933C2211yescharacteristic
102#1928C2212yes—
103#1927C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1921C2212yes—
108#1921C2212yes—
109#1920C2212yes—
110#1920C2212yes—
111#1919C2212yes—
112#1916C2212yes—
113#1916C2212yes—
114#1865C2212yes—
115#1863C2212yes—
116#1844C2211yescharacteristic
117#1836C2212yes—
118#1834C2211yescharacteristic
119#1831C2211yescharacteristic
120#1803C2211yescharacteristic
121#1803C2211yescharacteristic
122#1776C2212yes—
123C4×#106C2212yes—
124C4×#106C2212yes—
125#1771C2212yes—
126#1771C2212yes—
127#1770C2211yescharacteristic
128#1770C2211yescharacteristic
129#1763C2212yes—
130C4×#103C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C42×C8C2211yescharacteristic
135C2×C4⋊Q8C2311yescharacteristic
136C2×C4⋊Q8C2311yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125—22no—
140C2×#125D411yescharacteristic
141C2×#125C2312yes—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2312yes—
150#498—24no—
151#487—24no—
152#479—28no—
153#475—24no—
154#475—24no—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186C2×C4×C8C2311yescharacteristic
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#395—22no—
201#395—22no—
202#395—22no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391C2311yescharacteristic
210#391C2311yescharacteristic
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—22no—
221#389—22no—
222#389—21no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389—22no—
228#389—22no—
229#389—22no—
230#389D411yescharacteristic
231#389D411yescharacteristic
232#389C2312yes—
233#389C2312yes—
234#389C2311yescharacteristic
235#389C2311yescharacteristic
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#389C2312yes—
240#389C2312yes—
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#387—22no—
251#387—22no—
252#387—22no—
253#387—22no—
254#387D412yes—
255#387D412yes—
256#384—21no—
257#384—21no—
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#384C2311yescharacteristic
262#384C2312yes—
263#384C2312yes—
264#382—21no—
265#382—21no—
266#382—22no—
267#382—22no—
268#382D411yescharacteristic
269#382C2312yes—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—22no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—21no—
278C4×C4⋊C4—22no—
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2311yescharacteristic
288C4×C4⋊C4C2312yes—
289C4×C4⋊C4C2312yes—
290C43C2311yescharacteristic
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2312yes—
307#340C2312yes—
308#340C2312yes—
309#340C2311yescharacteristic
310#340C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2312yes—
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8C2411yescharacteristic
321C22×Q8C2411yescharacteristic
322C22×C8C2411yescharacteristic
323C22×C8C2412yes—
324C22×C8C2412yes—
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—41no—
334C4⋊Q8—41no—
335C4⋊Q8#4114yes—
336#125—24no—
337#125—24no—
338#125—42no—
339#125—84no—
340C4×Q8—24no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—42no—
346C4×Q8—44no—
347C4×Q8—82no—
348C4×Q8—82no—
349C4×Q8—82no—
350C4×Q8—82no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2412yes—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417C2×C42C2412yes—
418C2×C42C2411yescharacteristic
419#107—24no—
420#107—42no—
421#106—24no—
422#106—24no—
423#106—24no—
424#106—24no—
425#106—24no—
426#106—42no—
427C4⋊C8—24no—
428C4⋊C8—24no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431#103—24no—
432#103—22no—
433#103—22no—
434#103—22no—
435#103—22no—
436#103—22no—
437#103—22no—
438#103—24no—
439#103—24no—
440#103—22no—
441#103—42no—
442#103#4114yes—
443#97—24no—
444#97—42no—
445C4×C8—22no—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—22no—
449C4×C8—22no—
450C4×C8—41no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
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—22no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483#95C2×D412yes—
484#95C2×D412yes—
485C2×Q8—24no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—81no—
503C2×Q8—81no—
504C2×Q8—81no—
505C2×Q8C2×#4112yes—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C2511yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8C2×#4112yes—
545C2×C82−1+412yes—
546C2×C82−1+412yes—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—42no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—44no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—44no—
587C4⋊C4—42no—
588C4⋊C4—81no—
589C4⋊C4—81no—
590C4⋊C4—81no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4C2×#4114yes—
605C4⋊C4C2×#4112yes—
606C42—22no—
607C42—22no—
608C42—21no—
609C42—22no—
610C42—21no—
611C42—21no—
612C42—22no—
613C42—22no—
614C42—21no—
615C42—22no—
616C42—41no—
617C42—44no—
618C42—42no—
619C42—42no—
620C42—44no—
621C42—42no—
622C42—42no—
623C42—82no—
624C42—82no—
625C42—82no—
626C42—82no—
627C42#13714yes—
628C42C2×#4112yes—
629C422−1+412yes—
630C23C23×D411yescharacteristic
631Q8—22no—
632Q8—22no—
633Q8—48no—
634Q8—41no—
635Q8—41no—
636Q8—41no—
637Q8—41no—
638Q8—41no—
639Q8—41no—
640Q8—84no—
641Q8—84no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—42no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—42no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—42no—
700C2×C4—41no—
701C2×C4#54812yes—
702C2×C4#54812yes—
703C2×C4C2×#13712yes—
704C2×C4C2×#13712yes—
705C2×C4C2×#13712yes—
706C2×C4#57712yes—
707C2×C4#57712yes—
708C2×C4C22×#4111yescharacteristic
709C2×C4C22×#4111yescharacteristic
710C2×C4C2×2−1+411yescharacteristic
711C2×C4C2×2−1+411yescharacteristic
712C2×C4C2×2−1+411yescharacteristic
713C2×C4#58511yescharacteristic
714C8—22no—
715C8—22no—
716C8—22no—
717C8—42no—
718C8—41no—
719C8—42no—
720C22C2×#54811yescharacteristic
721C22D4×#4111yescharacteristic
722C22#349911yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57711yescharacteristic
725C22C2×#57711yescharacteristic
726C22#358411yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—41no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—44no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—82no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—82no—
748C4#337214yes—
749C4#348211yescharacteristic
750C4#350911yescharacteristic
751C4#351611yescharacteristic
752C4#356311yescharacteristic
753C2#5965611yescharacteristic
754C2C2×#337211yescharacteristic
755C2#6132711yescharacteristic
756C2#6144011yescharacteristic
757C2#6148311yescharacteristic
758C2#6247811yescharacteristic
759C2#6249411yescharacteristic
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

Maximal subgroups

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

subgroupindex
#596662
#594452
#323122
#308062
#292772
#290382
#288022
#282052
#280422
#278242
#277382
#273592
#271372
#260892
#260432
#254532
#230782
#230682
#229702
C4×#4982
#216732
C4×#4872