Finite Group Databasebeta

← #6595478

Subgroups — #6595478

Order 512.

1437 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33212C211yesmaximal, characteristic, complemented
5#33209C211yesmaximal, characteristic, complemented
6#29220C211yesmaximal, characteristic, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28715C214yesmaximal
9#27878C211yesmaximal, characteristic, complemented
10#27877C211yesmaximal, characteristic, complemented
11#27875C211yesmaximal, characteristic, complemented
12#27875C211yesmaximal, characteristic, complemented
13#27845C214yesmaximal, complemented
14#27841C214yesmaximal, complemented
15#23169C214yesmaximal, complemented
16#21726C211yesmaximal, characteristic, complemented
17#20824C211yesmaximal, characteristic, complemented
18#20816C211yesmaximal, characteristic, complemented
19C4×#435C211yesmaximal, characteristic
20C2×#501C2214yes—
21C2×#500C2211yescharacteristic
22C2×#435C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2571—21no—
32#2563C2212yes—
33#2562C2212yes—
34#2538—21no—
35#2514—21no—
36#2509—21no—
37#2497C2212yes—
38#2497C2212yes—
39#2495C2212yes—
40#2494—21no—
41#2494C2212yes—
42#2492—21no—
43#2492—21no—
44#2485—21no—
45#2439—21no—
46#2439—21no—
47#2421—21no—
48#2415—21no—
49#2370C2212yes—
50#2369C2212yes—
51#2362—21no—
52#2355—21no—
53#2354—21no—
54#2347—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2101C2214yes—
63#2099C2214yes—
64#2077C2214yes—
65#2077C2214yes—
66#2076C2214yes—
67#2075C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1989C2214yes—
75#1988C2214yes—
76#1944C2214yes—
77#1943C2214yes—
78#1942C2214yes—
79#1921C2214yes—
80#1920C2214yes—
81#1920C2214yes—
82#1918C2214yes—
83#1847C2214yes—
84#1845C2211yescharacteristic
85#1845C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1809C2214yes—
95#1808C2214yes—
96#1807C2214yes—
97#1776C2214yes—
98#1776C2214yes—
99C4×#106C2214yes—
100#1771C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102#1753C2212yes—
103#1752C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×M16C2312yes—
108C23×C8C2312yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#125C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#501—216no—
131#500—24no—
132#435—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C4×M16—24no—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×C4×C8—24no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2312yes—
151C2×C4×C8C2311yescharacteristic
152#401—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#400—22no—
157#397—22no—
158#397—22no—
159#396—22no—
160#396—22no—
161#394—22no—
162#393—22no—
163#393—22no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391—22no—
172#390—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#383—22no—
212#382—22no—
213#382—21no—
214#382—21no—
215#382—21no—
216#382—21no—
217#382C2312yes—
218#380—21no—
219#380—21no—
220#380—21no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380C2312yes—
227#380C2312yes—
228#380C2312yes—
229#380C2312yes—
230#380C2312yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×D4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—24no—
282C2×M16—24no—
283C2×M16—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—28no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298#125—28no—
299C22⋊Q8—44no—
300C22⋊Q8—82no—
301C22⋊Q8—82no—
302C22⋊Q8—82no—
303C4×Q8—44no—
304C4×Q8—82no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—42no—
308#117—44no—
309#117—82no—
310#117—82no—
311#117—82no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C42—22no—
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×C42—22no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—28no—
388#106—28no—
389#106—28no—
390#106—28no—
391C4⋊C8—24no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394#97—24no—
395#97—24no—
396#97#4118yes—
397C4×C8—24no—
398C4×C8—22no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8#4114yes—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
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#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—41no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4#13714yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485M16—48no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C8C2×#4114yes—
496C2×C8C2×#4114yes—
497C2×C8C2×#4114yes—
498C2×C82−1+414yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—21no—
563C42—21no—
564C42—44no—
565C42—44no—
566C42—82no—
567C42—82no—
568C42—82no—
569C42—82no—
570C42C2×#4114yes—
571C42C2×#4112yes—
572C422−1+412yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23C2×#13712yes—
587C23C23×D411yescharacteristic
588Q8—84no—
589Q8—84no—
590Q8—164no—
591D4—44no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
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—44no—
621C2×C4—44no—
622C2×C4—44no—
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×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4#53212yes—
652C2×C4#53212yes—
653C2×C4#54712yes—
654C2×C4#54712yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4#57512yes—
658C2×C4#57612yes—
659C2×C4#57712yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4112yes—
662C2×C4C2×2−1+412yes—
663C2×C4C2×2−1+411yescharacteristic
664C8—24no—
665C8—42no—
666C8—42no—
667C8#55618yes—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22C2×#53211yescharacteristic
683C22C2×#54711yescharacteristic
684C22#349911yescharacteristic
685C22C22×#13711yescharacteristic
686C22C2×#57511yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C4—22no—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#334414yes—
708C4#334714yes—
709C4C2×#55612yes—
710C4#350912yes—
711C2—42no—
712C2—81no—
713C2#3618811yescharacteristic
714C2#6070111yescharacteristic
715C2#6072211yescharacteristic
716C2C2×#334411yescharacteristic
717C2C2×#334711yescharacteristic
718C2#6248011yescharacteristic
719C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33212C2semidirect
5. #33209C2semidirect
6. #29220C2semidirect
7. #29219C2semidirect
9. #27878C2semidirect
10. #27877C2semidirect
11. #27875C2semidirect
12. #27875C2semidirect
13. #27845C2semidirect
14. #27841C2semidirect
15. #23169C2semidirect
16. #21726C2semidirect
17. #20824C2semidirect
18. #20816C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#361942
#333162
#332122
#332092
#292202
#292192
#287152
#278782
#278772
#278752
#278452
#278412
#231692
#217262
#208242
#208162
C4×#4352