Finite Group Databasebeta

← #7605784

Subgroups — #7605784

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59670C212yesmaximal
2#32132C212yesmaximal
3#32131C212yesmaximal
4#29278C212yesmaximal
5#28832C211yesmaximal, characteristic
6#28813C211yesmaximal, characteristic
7#28812C211yesmaximal, characteristic
8#28812C211yesmaximal, characteristic
9#27995C211yesmaximal, characteristic
10#27995C211yesmaximal, characteristic
11#27994C211yesmaximal, characteristic
12#27993C211yesmaximal, characteristic
13#26869C212yesmaximal
14#26859C212yesmaximal
15#26515C212yesmaximal
16#26448C212yesmaximal
17#23160C211yesmaximal, characteristic
18#23160C211yesmaximal, characteristic
19#23160C211yesmaximal, characteristic
20#23157C211yesmaximal, characteristic
21C4×#499C211yesmaximal, characteristic
22C4×#500C211yesmaximal, characteristic
23C8×#125C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#500C2211yescharacteristic
26C2×#499C2211yescharacteristic
27C2×#499C2211yescharacteristic
28#2755—22no—
29#2713—22no—
30#2590—22no—
31#2588—22no—
32#2585—22no—
33#2584—22no—
34#2570C2212yes—
35#2551C2212yes—
36#2550C2212yes—
37#2550C2212yes—
38#2509C2212yes—
39#2507C2212yes—
40#2507C2212yes—
41#2506C2212yes—
42#2462—22no—
43#2370C2212yes—
44#2370C2212yes—
45#2370C2212yes—
46#2365—22no—
47#2365C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×#125C2212yes—
50C4×#125C2211yescharacteristic
51#2101C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2088C2212yes—
56#2087C2212yes—
57#2087C2212yes—
58#2086C2212yes—
59#2085C2212yes—
60#2084C2212yes—
61#2059C2212yes—
62#2059C2212yes—
63#2058C2212yes—
64#2058C2212yes—
65#2057C2212yes—
66#2057C2212yes—
67#1991C2212yes—
68#1990C2212yes—
69#1989C2212yes—
70#1989C2212yes—
71#1989C2212yes—
72#1988C2212yes—
73#1949C2212yes—
74#1949C2212yes—
75#1949C2212yes—
76#1947C2212yes—
77#1942C2211yescharacteristic
78#1942C2211yescharacteristic
79#1942C2211yescharacteristic
80#1942C2211yescharacteristic
81#1942C2211yescharacteristic
82#1942C2211yescharacteristic
83#1942C2211yescharacteristic
84#1942C2211yescharacteristic
85#1942C2211yescharacteristic
86#1942C2211yescharacteristic
87#1942C2211yescharacteristic
88#1942C2211yescharacteristic
89#1942C2212yes—
90#1941C2212yes—
91#1939C2211yescharacteristic
92#1939C2211yescharacteristic
93#1921C2212yes—
94#1921C2212yes—
95#1920C2212yes—
96#1920C2212yes—
97#1920C2212yes—
98#1919C2212yes—
99#1919C2212yes—
100#1919C2212yes—
101#1919C2212yes—
102#1918C2212yes—
103#1918C2212yes—
104#1918C2212yes—
105#1916C2211yescharacteristic
106#1916C2211yescharacteristic
107#1916C2211yescharacteristic
108#1916C2211yescharacteristic
109#1846C2211yescharacteristic
110#1846C2211yescharacteristic
111#1845C2211yescharacteristic
112#1845C2211yescharacteristic
113#1831C2211yescharacteristic
114C4×#107C2212yes—
115C4×#107C2211yescharacteristic
116C4×#107C2211yescharacteristic
117C4×#107C2211yescharacteristic
118C4×#107C2211yescharacteristic
119C4×#107C2211yescharacteristic
120C4×#107C2211yescharacteristic
121C4×#107C2211yescharacteristic
122C4×#107C2211yescharacteristic
123C4×#106C2212yes—
124C8×C4⋊C4C2211yescharacteristic
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C8×C4⋊C4C2211yescharacteristic
130C42×C8C2211yescharacteristic
131C2×C4⋊Q8—22no—
132C2×C4⋊Q8C2312yes—
133C2×C4⋊Q8C2312yes—
134C2×#125—22no—
135C2×#125D411yescharacteristic
136C2×#125C2312yes—
137C2×#125C2312yes—
138C2×C4×Q8—22no—
139C2×C4×Q8—22no—
140#500—24no—
141#500—24no—
142#499—24no—
143#499Q818yescomplemented
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#107C2311yescharacteristic
148C2×#107C2311yescharacteristic
149C2×#107C2311yescharacteristic
150C2×#107C2311yescharacteristic
151C2×#107C2311yescharacteristic
152C2×#107C2312yes—
153C2×#107C2311yescharacteristic
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176#400—22no—
177#400—22no—
178#400—22no—
179#400—22no—
180#398—22no—
181#398—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#395—22no—
189#395—22no—
190#395—22no—
191#395—22no—
192#391—22no—
193#391—22no—
194#391—22no—
195#391—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389—22no—
206#389D411yescharacteristic
207#389D411yescharacteristic
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387D411yescharacteristic
237#387D411yescharacteristic
238#387D411yescharacteristic
239#387D411yescharacteristic
240#384C2312yes—
241#384C2312yes—
242#382D411yescharacteristic
243#382C2312yes—
244#382C2312yes—
245C4×C4⋊C4—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4—22no—
248C4×C4⋊C4—22no—
249C4×C4⋊C4—22no—
250C4×C4⋊C4—22no—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2311yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C4⋊C4C2311yescharacteristic
263C4×C4⋊C4C2311yescharacteristic
264C4×C4⋊C4C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#340C2312yes—
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#336C2311yescharacteristic
287#336C2311yescharacteristic
288#336C2311yescharacteristic
289#336C2311yescharacteristic
290#336C2311yescharacteristic
291#336C2311yescharacteristic
292#336C2311yescharacteristic
293#336C2311yescharacteristic
294#336C2311yescharacteristic
295#336C2311yescharacteristic
296#336C2311yescharacteristic
297#336C2311yescharacteristic
298C22×Q8—22no—
299C22×Q8—22no—
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C22×C8C2411yescharacteristic
308C4⋊Q8—24no—
309C4⋊Q8—24no—
310C4⋊Q8—42no—
311C4⋊Q8—42no—
312C4⋊Q8—48no—
313#125—24no—
314#125—24no—
315#125—24no—
316#125—42no—
317#125—84no—
318#125C2×Q818yes—
319C4×Q8—84no—
320C4×Q8—84no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—22no—
383#107—22no—
384#107—22no—
385#107—24no—
386#107—22no—
387#107—22no—
388#107—24no—
389#107—24no—
390#107—42no—
391#107C2×Q814yes—
392#107C2×Q814yes—
393#107#4114yes—
394#106—24no—
395#106—42no—
396#106—42no—
397#106C2×Q818yes—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—41no—
403C4⋊C8—41no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—41no—
413C4×C8C2×Q814yes—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—82no—
455C2×Q8—82no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C22×Q812yes—
499C2×C8C22×Q812yes—
500C2×C8C2×#4112yes—
501C2×C82−1+412yes—
502C2×C82−1+412yes—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4#13514yes—
550C4⋊C4C22×Q814yes—
551C4⋊C4C22×Q814yes—
552C4⋊C4C2×#4114yes—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—41no—
569C42—44no—
570C42—44no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42—82no—
576C42—82no—
577C42C22×Q812yes—
578C422−1+412yes—
579C23C23×D411yescharacteristic
580Q8—88no—
581Q8—88no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4D4×Q812yes—
642C2×C4D4×Q812yes—
643C2×C4C2×#13512yes—
644C2×C4C2×#13512yes—
645C2×C4C2×#13512yes—
646C2×C4#57612yes—
647C2×C4#57812yes—
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58511yescharacteristic
654C8—22no—
655C8—22no—
656C8—41no—
657C8—41no—
658C8—41no—
659C8—41no—
660C8—41no—
661C8#55714yes—
662C22C2×D4×Q811yescharacteristic
663C22D4×#4111yescharacteristic
664C22#349911yescharacteristic
665C22C22×#13511yescharacteristic
666C22C2×#57611yescharacteristic
667C22C2×#57811yescharacteristic
668C22#358411yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—44no—
683C4—44no—
684C4—41no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4C2×#55711yescharacteristic
692C4#350211yescharacteristic
693C4#352311yescharacteristic
694C4#355511yescharacteristic
695C2#5966711yescharacteristic
696C2Q8×#13511yescharacteristic
697C2#6137311yescharacteristic
698C2#6143111yescharacteristic
699C2#6147311yescharacteristic
700C2#6247311yescharacteristic
701C2#6249211yescharacteristic
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
143. #499Q8semidirect

Maximal subgroups

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

subgroupindex
#596702
#321322
#321312
#292782
#288322
#288132
#288122
#279952
#279942
#279932
#268692
#268592
#265152
#264482
#231602
#231572
C4×#4992
C4×#5002
C8×#1252