Finite Group Databasebeta

← #7604735

Subgroups — #7604735

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59668C211yesmaximal, characteristic
2#59445C211yesmaximal, characteristic
3#32316C212yesmaximal
4#30816C212yesmaximal
5#29277C211yesmaximal, characteristic
6#29009C212yesmaximal
7#28789C212yesmaximal
8#28206C211yesmaximal, characteristic
9#28036C212yesmaximal
10#27815C211yesmaximal, characteristic
11#27729C211yesmaximal, characteristic
12#27364C212yesmaximal
13#27132C212yesmaximal
14#26092C211yesmaximal, characteristic
15#26046C211yesmaximal, characteristic
16#25455C211yesmaximal, characteristic
17#25455C211yesmaximal, characteristic
18#23063C211yesmaximal, characteristic
19#23063C212yesmaximal
20#22965C211yesmaximal, characteristic
21C4×#500C211yesmaximal, characteristic
22C8×#125C211yesmaximal, characteristic
23C4×#487C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#477C2211yescharacteristic
27C2×#477C2212yes—
28C2×#477C2211yescharacteristic
29#2806—22no—
30#2774—22no—
31#2584—21no—
32#2570—21no—
33#2567C2211yescharacteristic
34#2567C2211yescharacteristic
35#2552C2211yescharacteristic
36#2552C2212yes—
37#2551C2212yes—
38#2551C2211yescharacteristic
39#2550—21no—
40#2546—21no—
41#2511C2212yes—
42#2508—21no—
43#2507C2212yes—
44#2507C2212yes—
45#2506—21no—
46#2462—22no—
47#2451—22no—
48#2446C2212yes—
49#2370—21no—
50#2370—21no—
51#2369C2211yescharacteristic
52#2369C2212yes—
53#2369C2211yescharacteristic
54#2355C2212yes—
55#2355C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#125C2211yescharacteristic
62#2100C2212yes—
63#2100C2212yes—
64#2100C2212yes—
65#2095C2212yes—
66#2087C2212yes—
67#2083C2212yes—
68#2083C2212yes—
69#2083C2212yes—
70#2072C2212yes—
71#2068C2212yes—
72#2068C2212yes—
73#2068C2212yes—
74#2057C2212yes—
75#2057C2212yes—
76#2043C2212yes—
77#2024C2212yes—
78#1990C2212yes—
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1987C2211yescharacteristic
83#1987C2211yescharacteristic
84#1956C2211yescharacteristic
85#1947C2212yes—
86#1947C2211yescharacteristic
87#1947C2211yescharacteristic
88#1947C2212yes—
89#1947C2212yes—
90#1942C2212yes—
91#1942C2212yes—
92#1942C2212yes—
93#1939C2211yescharacteristic
94#1939C2211yescharacteristic
95#1933C2211yescharacteristic
96#1933C2212yes—
97#1933C2212yes—
98#1933C2212yes—
99#1933C2211yescharacteristic
100#1933C2211yescharacteristic
101#1933C2211yescharacteristic
102#1928C2212yes—
103#1928C2212yes—
104#1928C2212yes—
105#1928C2212yes—
106#1928C2212yes—
107#1921C2212yes—
108#1921C2212yes—
109#1921C2212yes—
110#1921C2212yes—
111#1921C2212yes—
112#1916C2212yes—
113#1916C2212yes—
114#1863C2212yes—
115#1863C2212yes—
116#1845C2211yescharacteristic
117#1834C2211yescharacteristic
118#1834C2212yes—
119#1831C2211yescharacteristic
120#1803C2211yescharacteristic
121#1803C2211yescharacteristic
122C4×#107C2212yes—
123C4×#107C2212yes—
124C4×#107C2212yes—
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2212yes—
127C8×C4⋊C4C2212yes—
128C8×C4⋊C4C2211yescharacteristic
129C4×#103C2212yes—
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×#125—22no—
144C2×#125D411yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2312yes—
151#500—24no—
152#487—24no—
153#477—24no—
154#477—24no—
155#477—28no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2312yes—
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186C2×C4×C8C2312yes—
187C2×C4×C8C2311yescharacteristic
188#400—22no—
189#400—22no—
190#400—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#395—22no—
200#395—22no—
201#395—22no—
202#395—22no—
203#395—22no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—22no—
208#391—22no—
209#391—22no—
210#391—22no—
211#391C2311yescharacteristic
212#391C2311yescharacteristic
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—22no—
222#389—21no—
223#389—21no—
224#389—22no—
225#389—21no—
226#389—22no—
227#389—21no—
228#389—21no—
229#389—21no—
230#389—21no—
231#389—21no—
232#389—22no—
233#389—22no—
234#389—22no—
235#389D411yescharacteristic
236#389D411yescharacteristic
237#389C2312yes—
238#389C2312yes—
239#389C2311yescharacteristic
240#389C2311yescharacteristic
241#389C2312yes—
242#389C2312yes—
243#389C2312yes—
244#389C2312yes—
245#389C2312yes—
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#387—22no—
255#387D412yes—
256#387D412yes—
257#384—21no—
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#384C2311yescharacteristic
262#384C2312yes—
263#384C2312yes—
264#384C2312yes—
265#382—21no—
266#382—22no—
267#382D411yescharacteristic
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—22no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—22no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—21no—
278C4×C4⋊C4—21no—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2311yescharacteristic
288C4×C4⋊C4C2312yes—
289C4×C4⋊C4C2312yes—
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#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×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—42no—
336C4⋊Q8—41no—
337C4⋊Q8—41no—
338C4⋊Q8—82no—
339C4⋊Q8—84no—
340C4⋊Q8#4114yes—
341#125—82no—
342#125—84no—
343#125Q1618yes—
344C4×Q8—24no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—42no—
350C4×Q8—82no—
351C4×Q8—82no—
352C4×Q8—82no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
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—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
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—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D411yescharacteristic
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2411yescharacteristic
402C2×C4⋊C4C2412yes—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—22no—
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2412yes—
418C2×C42C2411yescharacteristic
419C2×C42C2411yescharacteristic
420C2×C42C2412yes—
421C2×C42C2411yescharacteristic
422#107—24no—
423#107—24no—
424#107—24no—
425#107—24no—
426#107—24no—
427#107—24no—
428#107—42no—
429#107—42no—
430C4⋊C8—24no—
431C4⋊C8—24no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434#103—24no—
435#103—22no—
436#103—22no—
437#103—22no—
438#103—22no—
439#103—22no—
440#103—22no—
441#103—24no—
442#103—24no—
443#103—22no—
444#103—42no—
445#103#4114yes—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—24no—
449C4×C8—22no—
450C4×C8—22no—
451C4×C8—22no—
452C4×C8—42no—
453C4×C8—41no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—22no—
459#95—21no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—21no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—21no—
473#95—22no—
474#95—21no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483#95C2×D412yes—
484#95C2×D412yes—
485#95C2×D412yes—
486C2×Q8—24no—
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—21no—
498C2×Q8—42no—
499C2×Q8—44no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—81no—
503C2×Q8—81no—
504C2×Q8—81no—
505C2×Q8—81no—
506C2×Q8—81no—
507C2×Q8—82no—
508C2×Q8—82no—
509C2×Q8C2×#4112yes—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C2511yescharacteristic
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8C2×#4112yes—
549C2×C82−1+412yes—
550C2×C82−1+412yes—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—22no—
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—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—81no—
588C4⋊C4—81no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—81no—
593C4⋊C4—81no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4C2×Q1618yes—
606C4⋊C4C2×Q1618yes—
607C4⋊C4C2×Q1614yes—
608C4⋊C4C2×Q1614yes—
609C4⋊C4C2×#4114yes—
610C4⋊C4C2×#4112yes—
611C42—22no—
612C42—22no—
613C42—21no—
614C42—22no—
615C42—21no—
616C42—22no—
617C42—21no—
618C42—22no—
619C42—22no—
620C42—21no—
621C42—41no—
622C42—44no—
623C42—42no—
624C42—42no—
625C42—42no—
626C42—42no—
627C42—82no—
628C42—82no—
629C42—82no—
630C42—82no—
631C42—82no—
632C42C2×Q1614yes—
633C42C2×#4112yes—
634C422−1+412yes—
635C23C23×D411yescharacteristic
636Q8—22no—
637Q8—22no—
638Q8—41no—
639Q8—41no—
640Q8—41no—
641Q8—41no—
642Q8—41no—
643Q8—41no—
644Q8—84no—
645Q8—84no—
646Q8—84no—
647Q8—164no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4#54812yes—
706C2×C4#54812yes—
707C2×C4C22×Q1612yes—
708C2×C4C22×Q1612yes—
709C2×C4C22×Q1614yes—
710C2×C4C22×Q1614yes—
711C2×C4C22×Q1612yes—
712C2×C4#57612yes—
713C2×C4#57712yes—
714C2×C4C22×#4111yescharacteristic
715C2×C4C22×#4111yescharacteristic
716C2×C4C2×2−1+411yescharacteristic
717C2×C4C2×2−1+411yescharacteristic
718C2×C4C2×2−1+411yescharacteristic
719C2×C4#58511yescharacteristic
720C8—22no—
721C8—22no—
722C8—22no—
723C8—42no—
724C8—41no—
725C8—42no—
726C22C2×#54811yescharacteristic
727C22D4×#4111yescharacteristic
728C22#349911yescharacteristic
729C22C23×Q1611yescharacteristic
730C22C2×#57611yescharacteristic
731C22C2×#57711yescharacteristic
732C22#358411yescharacteristic
733C4—22no—
734C4—24no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—24no—
739C4—42no—
740C4—41no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—44no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4—82no—
749C4—81no—
750C4—82no—
751C4—81no—
752C4—81no—
753C4—82no—
754C4#336314yes—
755C4#348211yescharacteristic
756C4#350911yescharacteristic
757C4#351611yescharacteristic
758C4#356311yescharacteristic
759C2#5965611yescharacteristic
760C2C2×#336311yescharacteristic
761C2#6132511yescharacteristic
762C2Q16×#4111yescharacteristic
763C2#6149811yescharacteristic
764C2#6247211yescharacteristic
765C2#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
#594452
#323162
#308162
#292772
#290092
#287892
#282062
#280362
#278152
#277292
#273642
#271322
#260922
#260462
#254552
#230632
#229652
C4×#5002
C8×#1252
C4×#4872