Finite Group Databasebeta

← #7104841

Subgroups — #7104841

Order 512.

1253 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59256C211yesmaximal, characteristic
2#39623C211yesmaximal, characteristic
3#32570C212yesmaximal, complemented
4#32338C212yesmaximal, complemented
5#32337C212yesmaximal, complemented
6#32248C212yesmaximal, complemented
7#29165C211yesmaximal, characteristic
8#28801C212yesmaximal, complemented
9#28039C212yesmaximal, complemented
10#27968C211yesmaximal, characteristic
11#27722C212yesmaximal
12#27377C212yesmaximal, complemented
13#26280C212yesmaximal, complemented
14#26011C212yesmaximal
15#24590C212yesmaximal
16#23657C211yesmaximal, characteristic
17#23242C211yesmaximal, characteristic
18#22621C212yesmaximal
19#22084C211yesmaximal, characteristic
20C2×#483C2211yescharacteristic
21C2×#428C2212yes—
22#2857C2211yescharacteristic
23#2806—21no—
24#2792—22no—
25#2770—22no—
26#2734—21no—
27#2659—21no—
28#2582C2211yescharacteristic
29#2571C2211yescharacteristic
30#2570C2212yes—
31#2567C2212yes—
32#2551C2212yes—
33#2528—22no—
34#2514C2212yes—
35#2508—21no—
36#2507C2212yes—
37#2507C2212yes—
38#2499C2212yes—
39#2494—21no—
40#2440—22no—
41#2440C2212yes—
42#2421—21no—
43#2356—21no—
44#2355C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2346C2212yes—
47#2345C2212yes—
48#2329—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53#2101C2212yes—
54#2100C2212yes—
55#2099C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2099C2211yescharacteristic
59#2096C2211yescharacteristic
60#2088C2212yes—
61#2087C2212yes—
62#2085C2212yes—
63#2085C2212yes—
64#2085C2212yes—
65#2084C2212yes—
66#2084C2212yes—
67#2083C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2066C2212yes—
71#2057C2212yes—
72#2057C2211yescharacteristic
73#2024C2212yes—
74#2024C2211yescharacteristic
75#1993C2212yes—
76#1990C2212yes—
77#1987C2212yes—
78#1984C2212yes—
79#1949C2212yes—
80#1948C2212yes—
81#1947C2212yes—
82#1944C2212yes—
83#1943C2212yes—
84#1942C2212yes—
85#1935C2212yes—
86#1935C2212yes—
87#1933C2212yes—
88#1928C2212yes—
89#1927C2212yes—
90#1921C2212yes—
91#1920C2212yes—
92#1920C2212yes—
93#1920C2212yes—
94#1920C2212yes—
95#1919C2212yes—
96#1918C2212yes—
97#1918C2212yes—
98#1917C2212yes—
99#1917C2212yes—
100#1894C2212yes—
101#1872C2211yescharacteristic
102#1863C2212yes—
103#1863C2212yes—
104#1852C2211yescharacteristic
105#1851C2212yes—
106#1836C2212yes—
107#1836C2212yes—
108#1825C2212yes—
109#1825C2212yes—
110#1823C2211yescharacteristic
111#1809C2212yes—
112#1776C2212yes—
113C4×#106C2212yes—
114#1771C2212yes—
115#1753C2212yes—
116#1749C2211yescharacteristic
117#1736C2212yes—
118#1728C2211yescharacteristic
119C22×M16C2312yes—
120C2×C4⋊Q8—22no—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8D411yescharacteristic
126C2×C22⋊Q8C2311yescharacteristic
127C2×C4×Q8—21no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#483—24no—
136#428—28no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#401—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#397—22no—
168#397—22no—
169#395—22no—
170#395—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#391—22no—
175#391—21no—
176#391—21no—
177#391D411yescharacteristic
178#391C2312yes—
179#391C2312yes—
180#391C2311yescharacteristic
181#389—21no—
182#389—21no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389D411yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2311yescharacteristic
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—21no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387D412yes—
216#385—21no—
217#385—22no—
218#385D412yes—
219#384—22no—
220#384C2312yes—
221#384C2311yescharacteristic
222#384C2312yes—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#383—22no—
226#382—21no—
227#382—21no—
228#382—22no—
229#382C2312yes—
230#382C2312yes—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—22no—
235#380D411yescharacteristic
236#380C2311yescharacteristic
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—21no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4C2312yes—
252#376—22no—
253C2×#95—21no—
254#340C2312yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#336C2312yes—
275#336C2311yescharacteristic
276#336C2312yes—
277#336C2312yes—
278#336C2311yescharacteristic
279#336C2312yes—
280#336C2312yes—
281#336C2312yes—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8C2×D411yescharacteristic
285C22×Q8C2411yescharacteristic
286C23×C4—21no—
287C23×C4—21no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—42no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—42no—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—84no—
302C22⋊Q8—24no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—84no—
306C22⋊Q8—82no—
307C4×Q8—42no—
308C4×Q8—42no—
309C4×Q8—44no—
310#117—24no—
311#117—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4#4114yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—42no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377#107—24no—
378#107—24no—
379#107—42no—
380#107—42no—
381#106—24no—
382#106—24no—
383#106—24no—
384#106—24no—
385C4⋊C8—24no—
386C4⋊C8—42no—
387#103—24no—
388#103—24no—
389#103—42no—
390#103—42no—
391C22⋊C8—22no—
392C22⋊C8—22no—
393#97—24no—
394#97—42no—
395C4×C8—42no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—21no—
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#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428C24C22×D411yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—82no—
437C2×Q8—81no—
438C2×Q8—81no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8#13714yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C82−1+414yes—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×#4114yes—
539C22⋊C4—24no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—82no—
558C42—82no—
559C42—82no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#54812yes—
567C23C23×D411yescharacteristic
568Q8—44no—
569Q8—42no—
570Q8—42no—
571Q8—162no—
572Q8—164no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—82no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—162no—
633C2×C4#54812yes—
634C2×C4C2×#13714yes—
635C2×C4C2×#13712yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×2−1+412yes—
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58512yes—
640C8—24no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×#54811yescharacteristic
652C22#348511yescharacteristic
653C22#349911yescharacteristic
654C22C22×#13711yescharacteristic
655C22#358511yescharacteristic
656C22#358511yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—24no—
660C4—22no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—42no—
667C4—44no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4#349512yes—
678C4#356312yes—
679C2—42no—
680C2#4444811yescharacteristic
681C2#6131911yescharacteristic
682C2#6133311yescharacteristic
683C2#6170511yescharacteristic
684C2#6177811yescharacteristic
685C2#6249911yescharacteristic
686C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #32570C2semidirect
4. #32338C2semidirect
5. #32337C2semidirect
6. #32248C2semidirect
8. #28801C2semidirect
9. #28039C2semidirect
12. #27377C2semidirect
13. #26280C2semidirect

Maximal subgroups

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

subgroupindex
#592562
#396232
#325702
#323382
#323372
#322482
#291652
#288012
#280392
#279682
#277222
#273772
#262802
#260112
#245902
#236572
#232422
#226212
#220842