Finite Group Databasebeta

← #7604736

Subgroups — #7604736

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59666C211yesmaximal, characteristic
2#59443C211yesmaximal, characteristic
3#32315C212yesmaximal
4#30814C212yesmaximal
5#29276C211yesmaximal, characteristic
6#29009C212yesmaximal
7#28794C212yesmaximal
8#28205C211yesmaximal, characteristic
9#28033C212yesmaximal
10#27816C211yesmaximal, characteristic
11#27730C211yesmaximal, characteristic
12#27366C212yesmaximal
13#27134C212yesmaximal
14#26093C211yesmaximal, characteristic
15#26047C211yesmaximal, characteristic
16#25456C211yesmaximal, characteristic
17#25456C211yesmaximal, characteristic
18#23064C212yesmaximal
19#23062C211yesmaximal, characteristic
20#22964C211yesmaximal, characteristic
21C4×#500C211yesmaximal, characteristic
22C8×#125C211yesmaximal, characteristic
23C4×#487C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#479C2212yes—
27C2×#475C2211yescharacteristic
28C2×#475C2211yescharacteristic
29#2802—22no—
30#2776—22no—
31#2582—21no—
32#2581—21no—
33#2571C2211yescharacteristic
34#2570C2212yes—
35#2568C2211yescharacteristic
36#2568C2211yescharacteristic
37#2565—21no—
38#2565—21no—
39#2552C2211yescharacteristic
40#2550C2212yes—
41#2533—22no—
42#2520—22no—
43#2514—21no—
44#2514C2212yes—
45#2511—21no—
46#2508C2212yes—
47#2507C2212yes—
48#2446C2212yes—
49#2370C2212yes—
50#2365C2211yescharacteristic
51#2365C2211yescharacteristic
52#2356—21no—
53#2356C2212yes—
54#2355C2211yescharacteristic
55#2351—21no—
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#125C2211yescharacteristic
62#2101C2212yes—
63#2101C2212yes—
64#2100C2212yes—
65#2097C2212yes—
66#2086C2212yes—
67#2085C2212yes—
68#2085C2212yes—
69#2084C2212yes—
70#2074C2212yes—
71#2068C2212yes—
72#2068C2212yes—
73#2068C2212yes—
74#2059C2212yes—
75#2058C2212yes—
76#2043C2212yes—
77#2026C2212yes—
78#1991C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1956C2211yescharacteristic
85#1949C2212yes—
86#1949C2212yes—
87#1949C2212yes—
88#1948C2211yescharacteristic
89#1948C2211yescharacteristic
90#1942C2212yes—
91#1942C2212yes—
92#1941C2212yes—
93#1939C2211yescharacteristic
94#1939C2211yescharacteristic
95#1933C2212yes—
96#1933C2211yescharacteristic
97#1933C2212yes—
98#1933C2212yes—
99#1933C2211yescharacteristic
100#1933C2211yescharacteristic
101#1933C2211yescharacteristic
102#1928C2212yes—
103#1928C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1920C2212yes—
108#1920C2212yes—
109#1920C2212yes—
110#1919C2212yes—
111#1918C2212yes—
112#1916C2212yes—
113#1916C2212yes—
114#1865C2212yes—
115#1863C2212yes—
116#1845C2211yescharacteristic
117#1835C2212yes—
118#1834C2211yescharacteristic
119#1831C2211yescharacteristic
120#1803C2211yescharacteristic
121#1803C2211yescharacteristic
122C4×#107C2212yes—
123C4×#107C2212yes—
124C4×#106C2212yes—
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2212yes—
127C8×C4⋊C4C2212yes—
128C8×C4⋊C4C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2212yes—
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C42×C8C2211yescharacteristic
135C2×C4⋊Q8C2311yescharacteristic
136C2×C4⋊Q8C2311yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125D411yescharacteristic
140C2×#125C2312yes—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2312yes—
149#500—24no—
150#487—24no—
151#479—28no—
152#475—24no—
153#475—24no—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#103C2311yescharacteristic
176C2×#103C2312yes—
177C2×#103C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2312yes—
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2312yes—
185C2×C4×C8C2311yescharacteristic
186#401—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#395—22no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391C2311yescharacteristic
213#391C2311yescharacteristic
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#389—21no—
218#389—21no—
219#389—22no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—21no—
225#389—22no—
226#389—22no—
227#389—22no—
228#389D411yescharacteristic
229#389D411yescharacteristic
230#389C2312yes—
231#389C2312yes—
232#389C2311yescharacteristic
233#389C2311yescharacteristic
234#389C2312yes—
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#387—22no—
240#387—22no—
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#387D412yes—
254#387D412yes—
255#384—21no—
256#384—21no—
257#384C2311yescharacteristic
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#383—22no—
262#383—22no—
263#382—21no—
264#382—21no—
265#382—22no—
266#382D411yescharacteristic
267#382C2312yes—
268#382C2312yes—
269#382C2312yes—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—22no—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2312yes—
289#376—22no—
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#340C2312yes—
310#340C2311yescharacteristic
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—
339C4×Q8—24no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—42no—
345C4×Q8—44no—
346C4×Q8—44no—
347C4×Q8—82no—
348C4×Q8—82no—
349C4×Q8—82no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2412yes—
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414#107—24no—
415#107—24no—
416#107—24no—
417#107—42no—
418#106—24no—
419#106—24no—
420#106—24no—
421#106—42no—
422C4⋊C8—24no—
423C4⋊C8—24no—
424C4⋊C8—22no—
425C4⋊C8—22no—
426#103—24no—
427#103—22no—
428#103—22no—
429#103—22no—
430#103—22no—
431#103—22no—
432#103—22no—
433#103—24no—
434#103—24no—
435#103—22no—
436#103—42no—
437#103#4114yes—
438C4×C8—22no—
439C4×C8—22no—
440C4×C8—24no—
441C4×C8—22no—
442C4×C8—22no—
443C4×C8—22no—
444C4×C8—42no—
445C4×C8—41no—
446#95—21no—
447#95—22no—
448#95—22no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—21no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—21no—
463#95—21no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
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#95C2×D412yes—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483#95C2×D412yes—
484C2×Q8—24no—
485C2×Q8—21no—
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—42no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—81no—
504C2×Q8C2×#4112yes—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C2511yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8C2×#4112yes—
544C2×C82−1+412yes—
545C2×C82−1+412yes—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—44no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—44no—
586C4⋊C4—42no—
587C4⋊C4—81no—
588C4⋊C4—81no—
589C4⋊C4—81no—
590C4⋊C4—81no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4C2×#4114yes—
600C4⋊C4C2×#4112yes—
601C42—22no—
602C42—22no—
603C42—21no—
604C42—22no—
605C42—21no—
606C42—21no—
607C42—22no—
608C42—22no—
609C42—22no—
610C42—21no—
611C42—41no—
612C42—44no—
613C42—42no—
614C42—42no—
615C42—44no—
616C42—44no—
617C42—42no—
618C42—42no—
619C42—82no—
620C42—82no—
621C42—82no—
622C42#13514yes—
623C42C2×#4112yes—
624C422−1+412yes—
625C23C23×D411yescharacteristic
626Q8—22no—
627Q8—22no—
628Q8—48no—
629Q8—41no—
630Q8—41no—
631Q8—41no—
632Q8—41no—
633Q8—41no—
634Q8—41no—
635Q8—84no—
636Q8—84no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4#54812yes—
697C2×C4#54812yes—
698C2×C4C2×#13512yes—
699C2×C4C2×#13512yes—
700C2×C4C2×#13512yes—
701C2×C4#57712yes—
702C2×C4#57812yes—
703C2×C4C22×#4111yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4C2×2−1+411yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C2×C4#58511yescharacteristic
709C8—22no—
710C8—22no—
711C8—22no—
712C8—42no—
713C8—41no—
714C8—42no—
715C22C2×#54811yescharacteristic
716C22D4×#4111yescharacteristic
717C22#349911yescharacteristic
718C22C22×#13511yescharacteristic
719C22C2×#57711yescharacteristic
720C22C2×#57811yescharacteristic
721C22#358411yescharacteristic
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—42no—
727C4—42no—
728C4—41no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—82no—
739C4—81no—
740C4—81no—
741C4—82no—
742C4—82no—
743C4#336014yes—
744C4#348211yescharacteristic
745C4#350911yescharacteristic
746C4#351611yescharacteristic
747C4#356311yescharacteristic
748C2#5965611yescharacteristic
749C2C2×#336011yescharacteristic
750C2#6132811yescharacteristic
751C2#6143011yescharacteristic
752C2#6149711yescharacteristic
753C2#6247311yescharacteristic
754C2#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
#594432
#323152
#308142
#292762
#290092
#287942
#282052
#280332
#278162
#277302
#273662
#271342
#260932
#260472
#254562
#230642
#230622
#229642
C4×#5002
C8×#1252
C4×#4872