Finite Group Databasebeta

← #6595405

Subgroups — #6595405

Order 512.

1545 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33176C211yesmaximal, characteristic, complemented
5#33172C211yesmaximal, characteristic, complemented
6#29144C211yesmaximal, characteristic, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28238C214yesmaximal
9#27892C211yesmaximal, characteristic, complemented
10#27892C211yesmaximal, characteristic, complemented
11#27889C211yesmaximal, characteristic, complemented
12#27889C211yesmaximal, characteristic, complemented
13#27797C214yesmaximal, complemented
14#27793C214yesmaximal, complemented
15#22818C214yesmaximal, complemented
16#21691C211yesmaximal, characteristic, complemented
17#20824C211yesmaximal, characteristic, complemented
18#20816C211yesmaximal, characteristic, complemented
19C4×#435C211yesmaximal, characteristic
20C2×#501C2214yes—
21C2×#494C2211yescharacteristic
22C2×#470C2214yes—
23C2×#467C2214yes—
24C2×#441C2214yes—
25C2×#435C2212yes—
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2571—21no—
35#2563—21no—
36#2538C2212yes—
37#2536C2212yes—
38#2508—21no—
39#2508—21no—
40#2497—21no—
41#2497—21no—
42#2495—21no—
43#2495—21no—
44#2492C2212yes—
45#2492C2212yes—
46#2489C2212yes—
47#2489C2212yes—
48#2485—21no—
49#2485—21no—
50#2439—21no—
51#2439—21no—
52#2369—21no—
53#2356C2212yes—
54#2355C2212yes—
55#2354—21no—
56#2346—21no—
57#2346—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60#2285C2211yescharacteristic
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63#2285C2211yescharacteristic
64C42×D4C2211yescharacteristic
65#2036C2214yes—
66#2036C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#2026C2214yes—
70#2024C2214yes—
71#1971C2214yes—
72#1971C2214yes—
73#1969C2214yes—
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1847C2214yes—
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1809C2214yes—
95#1807C2214yes—
96#1805C2214yes—
97#1803C2214yes—
98#1776C2214yes—
99C4×#106C2214yes—
100#1771C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102#1753C2212yes—
103#1752C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#125C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#501—48no—
132#494—24no—
133#470—48no—
134#467—48no—
135#441—216no—
136#435—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C4×M16—24no—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×C4×C8—24no—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2311yescharacteristic
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#398—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#396—22no—
170#394—22no—
171#393—22no—
172#393—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#390—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2312yes—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#383—22no—
219#382—22no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—22no—
225#382—22no—
226#382C2312yes—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380C2312yes—
240#380C2312yes—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4C2311yescharacteristic
264C4×C22⋊C4C2311yescharacteristic
265#376—22no—
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—22no—
275C22×Q8C2412yes—
276C22×Q8C2412yes—
277C22×D4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×Q16—28no—
285C2×Q16—28no—
286C2×Q16—28no—
287C2×M16—24no—
288C2×M16—24no—
289C2×M16—24no—
290C22×C8—24no—
291C22×C8—24no—
292C22×C8—24no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—82no—
304#125—44no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C4×Q8—28no—
310C4×Q8—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—42no—
314#117—44no—
315#117—44no—
316#117—82no—
317#117—82no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
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—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
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⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—28no—
392#107—44no—
393#106—28no—
394#106—28no—
395#106—44no—
396C4⋊C8—24no—
397#103—28no—
398#103—28no—
399#103—44no—
400#103—44no—
401#103—44no—
402#103—44no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405#97—24no—
406#97—24no—
407#97#4118yes—
408C4×C8—24no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8#4114yes—
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#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#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441C24C22×D411yescharacteristic
442C24C22×D411yescharacteristic
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—44no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—41no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
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—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4#13714yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498Q16—416no—
499M16—48no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—48no—
509C2×C8C2×#4114yes—
510C2×C8C2×#4114yes—
511C2×C8C2×#4114yes—
512C2×C8H(5,𝔽2)14yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—41no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—21no—
575C42—21no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—82no—
581C42—82no—
582C42C2×#4114yes—
583C42C2×#4112yes—
584C422−1+412yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23C2×#13712yes—
599C23C23×D411yescharacteristic
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8—164no—
605D4—44no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—28no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
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—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4#53212yes—
666C2×C4#53212yes—
667C2×C4#54712yes—
668C2×C4#54712yes—
669C2×C4C2×#13712yes—
670C2×C4C2×#13712yes—
671C2×C4#57512yes—
672C2×C4#57712yes—
673C2×C4#57812yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4112yes—
676C2×C4C2×H(5,𝔽2)12yes—
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—42no—
680C8—42no—
681C8#55518yes—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22C2×#53211yescharacteristic
697C22C2×#54711yescharacteristic
698C22#349811yescharacteristic
699C22C22×#13711yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57811yescharacteristic
703C4—22no—
704C4—22no—
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4#334414yes—
722C4#334914yes—
723C4C2×#55512yes—
724C4#351112yes—
725C2—42no—
726C2—81no—
727C2#3618711yescharacteristic
728C2#6070111yescharacteristic
729C2#6072311yescharacteristic
730C2C2×#334411yescharacteristic
731C2C2×#334911yescharacteristic
732C2#6242411yescharacteristic
733C2#6242711yescharacteristic
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. #33176C2semidirect
5. #33172C2semidirect
6. #29144C2semidirect
7. #29141C2semidirect
9. #27892C2semidirect
10. #27892C2semidirect
11. #27889C2semidirect
12. #27889C2semidirect
13. #27797C2semidirect
14. #27793C2semidirect
15. #22818C2semidirect
16. #21691C2semidirect
17. #20824C2semidirect
18. #20816C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361942
#333162
#331762
#331722
#291442
#291412
#282382
#278922
#278892
#277972
#277932
#228182
#216912
#208242
#208162
C4×#4352