Finite Group Databasebeta

← #7604751

Subgroups — #7604751

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59668C211yesmaximal, characteristic
2#59443C211yesmaximal, characteristic
3#32311C212yesmaximal
4#30808C212yesmaximal
5#29276C211yesmaximal, characteristic
6#29038C212yesmaximal
7#28803C212yesmaximal
8#28206C211yesmaximal, characteristic
9#28039C212yesmaximal
10#27823C211yesmaximal, characteristic
11#27737C211yesmaximal, characteristic
12#27373C212yesmaximal
13#27131C212yesmaximal
14#26088C211yesmaximal, characteristic
15#26042C211yesmaximal, characteristic
16#25454C211yesmaximal, characteristic
17#25454C211yesmaximal, characteristic
18#23077C212yesmaximal
19#23069C211yesmaximal, characteristic
20#22971C211yesmaximal, characteristic
21C4×#498C211yesmaximal, characteristic
22#21673C211yesmaximal, characteristic
23C4×#487C211yesmaximal, characteristic
24C2×#498C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#477C2211yescharacteristic
27C2×#477C2211yescharacteristic
28C2×#475C2212yes—
29#2806—22no—
30#2776—22no—
31#2584—21no—
32#2570C2212yes—
33#2568C2211yescharacteristic
34#2568C2211yescharacteristic
35#2565—21no—
36#2552C2211yescharacteristic
37#2551C2212yes—
38#2551C2211yescharacteristic
39#2550—21no—
40#2546—21no—
41#2533—22no—
42#2514—21no—
43#2514C2212yes—
44#2511—21no—
45#2507C2212yes—
46#2507C2212yes—
47#2451—22no—
48#2446C2212yes—
49#2370—21no—
50#2369C2211yescharacteristic
51#2369C2212yes—
52#2369C2211yescharacteristic
53#2356—21no—
54#2356C2212yes—
55#2355C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#125C2211yescharacteristic
62#2101C2212yes—
63#2101C2212yes—
64#2100C2212yes—
65#2096C2212yes—
66#2088C2212yes—
67#2085C2212yes—
68#2085C2212yes—
69#2083C2212yes—
70#2072C2212yes—
71#2066C2212yes—
72#2066C2212yes—
73#2066C2212yes—
74#2058C2212yes—
75#2058C2212yes—
76#2043C2212yes—
77#2024C2212yes—
78#1993C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1956C2211yescharacteristic
85#1949C2212yes—
86#1949C2212yes—
87#1948C2211yescharacteristic
88#1948C2211yescharacteristic
89#1948C2212yes—
90#1943C2212yes—
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#1928C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1920C2212yes—
108#1920C2212yes—
109#1919C2212yes—
110#1918C2212yes—
111#1918C2212yes—
112#1916C2212yes—
113#1916C2212yes—
114#1865C2212yes—
115#1865C2212yes—
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⋊Q8—22no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2312yes—
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×#125—21no—
143C2×#125D411yescharacteristic
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#477—24no—
153#477—24no—
154#475—28no—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
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#401—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—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—21no—
209#391—21no—
210#391—22no—
211#391—22no—
212#391—22no—
213#391—22no—
214#391C2311yescharacteristic
215#391C2311yescharacteristic
216#391C2312yes—
217#391C2312yes—
218#391C2312yes—
219#389—21no—
220#389—22no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—22no—
225#389—21no—
226#389—21no—
227#389—21no—
228#389—21no—
229#389—21no—
230#389—22no—
231#389—22no—
232#389—22no—
233#389D411yescharacteristic
234#389D411yescharacteristic
235#389C2312yes—
236#389C2312yes—
237#389C2311yescharacteristic
238#389C2311yescharacteristic
239#389C2312yes—
240#389C2312yes—
241#389C2312yes—
242#389C2312yes—
243#389C2312yes—
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#384C2311yescharacteristic
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#384C2312yes—
262#383—22no—
263#383—22no—
264#382—21no—
265#382D411yescharacteristic
266#382C2312yes—
267#382C2312yes—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
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⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
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#340C2311yescharacteristic
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
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×Q8—22no—
321C22×Q8C2411yescharacteristic
322C22×Q8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2412yes—
325C22×C8C2412yes—
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C4⋊Q8—24no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—41no—
336C4⋊Q8—42no—
337C4⋊Q8—41no—
338C4⋊Q8—82no—
339C4⋊Q8—84no—
340C4⋊Q8#4114yes—
341#125—24no—
342#125—82no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—42no—
349C4×Q8—44no—
350C4×Q8—82no—
351C4×Q8—82no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
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—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2412yes—
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2412yes—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—21no—
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×C42C2412yes—
416C2×C42C2411yescharacteristic
417#107—24no—
418#107—24no—
419#106—24no—
420#106—24no—
421#106—24no—
422#106—24no—
423#106—42no—
424#106—42no—
425C4⋊C8—24no—
426C4⋊C8—24no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429#103—24no—
430#103—22no—
431#103—22no—
432#103—22no—
433#103—22no—
434#103—22no—
435#103—22no—
436#103—24no—
437#103—24no—
438#103—22no—
439#103—42no—
440#103#4114yes—
441#97—24no—
442#97—42no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8—22no—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—41no—
449#95—21no—
450#95—22no—
451#95—22no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—22no—
456#95—22no—
457#95—21no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—21no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—21no—
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—44no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—81no—
504C2×Q8—81no—
505C2×Q8—81no—
506C2×Q8—82no—
507C2×Q8—82no—
508C2×Q8C2×#4112yes—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C2511yescharacteristic
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8C2×#4112yes—
548C2×C82−1+412yes—
549C2×C82−1+412yes—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—21no—
576C4⋊C4—42no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—44no—
587C4⋊C4—42no—
588C4⋊C4—44no—
589C4⋊C4—42no—
590C4⋊C4—81no—
591C4⋊C4—81no—
592C4⋊C4—81no—
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—41no—
634Q8—41no—
635Q8—41no—
636Q8—41no—
637Q8—41no—
638Q8—41no—
639Q8—84no—
640Q8—84no—
641Q8—84no—
642Q8—164no—
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—21no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—42no—
702C2×C4#54812yes—
703C2×C4#54812yes—
704C2×C4C2×#13712yes—
705C2×C4C2×#13712yes—
706C2×C4C2×#13712yes—
707C2×C4#57712yes—
708C2×C4#57712yes—
709C2×C4C22×#4111yescharacteristic
710C2×C4C22×#4111yescharacteristic
711C2×C4C2×2−1+411yescharacteristic
712C2×C4C2×2−1+411yescharacteristic
713C2×C4C2×2−1+411yescharacteristic
714C2×C4#58511yescharacteristic
715C8—22no—
716C8—22no—
717C8—22no—
718C8—42no—
719C8—41no—
720C8—42no—
721C22C2×#54811yescharacteristic
722C22D4×#4111yescharacteristic
723C22#349911yescharacteristic
724C22C22×#13711yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57711yescharacteristic
727C22#358411yescharacteristic
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—41no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—44no—
740C4—82no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—82no—
746C4—82no—
747C4—81no—
748C4—81no—
749C4#336914yes—
750C4#348211yescharacteristic
751C4#350911yescharacteristic
752C4#351611yescharacteristic
753C4#356311yescharacteristic
754C2#5965611yescharacteristic
755C2C2×#336911yescharacteristic
756C2#6132611yescharacteristic
757C2#6144011yescharacteristic
758C2#6148111yescharacteristic
759C2#6247811yescharacteristic
760C2#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
#596682
#594432
#323112
#308082
#292762
#290382
#288032
#282062
#280392
#278232
#277372
#273732
#271312
#260882
#260422
#254542
#230772
#230692
#229712
C4×#4982
#216732
C4×#4872