Finite Group Databasebeta

← #6593616

Subgroups — #6593616

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32955C212yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28283C211yesmaximal, characteristic, complemented
7#27631C212yesmaximal, complemented
8#27533C214yesmaximal, complemented
9#27045C214yesmaximal, complemented
10#26297C212yesmaximal, complemented
11#25886C212yesmaximal
12#24703C212yesmaximal, complemented
13#23217C212yesmaximal, complemented
14#22963C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17#20737C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#487C2212yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#474C2211yescharacteristic
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563C2212yes—
29#2538—21no—
30#2497C2214yes—
31#2492—22no—
32#2439—22no—
33#2369C2212yes—
34#2365C2211yescharacteristic
35#2362C2212yes—
36#2361—21no—
37#2359—22no—
38#2355—21no—
39#2354—21no—
40#2354C2211yescharacteristic
41D4×C4⋊C4C2211yescharacteristic
42#2344—21no—
43#2339C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C4×#123—22no—
48C4×#121—21no—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53#2077C2214yes—
54#2075C2214yes—
55#2072C2214yes—
56#2066C2214yes—
57#2035C2214yes—
58#2035C2214yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1987C2212yes—
62#1981C2212yes—
63#1954C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1947C2211yescharacteristic
68#1941C2214yes—
69#1933C2214yes—
70#1928C2214yes—
71#1923C2214yes—
72#1919C2214yes—
73#1918C2212yes—
74#1864C2214yes—
75#1863C2214yes—
76#1862C2212yes—
77#1854C2214yes—
78#1842C2211yescharacteristic
79#1825C2214yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2212yes—
86#1797C2212yes—
87#1788C2214yes—
88#1788C2214yes—
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#102C2214yes—
94C4×#102C2212yes—
95C22×#41C2312yes—
96#547—24no—
97#547—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C4×#41—28no—
108C4×#41—44no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C42—22no—
127C22×C42—21no—
128C22×C42C2312yes—
129#487—28no—
130#476—28no—
131#475—44no—
132#474—24no—
133#431—44no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—24no—
155#396—24no—
156#393—24no—
157#391—21no—
158#391—22no—
159#391—24no—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#390—21no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#388—24no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380C2314yes—
198#380C2314yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C2×#95—22no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41—42no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2412yes—
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—82no—
266C4⋊Q8—82no—
267C4⋊Q8#4114yes—
268#123—216no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—84no—
272#121—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—41no—
287C4×D4—42no—
288C4×D4—48no—
289C4×D4—48no—
290C4×D4#4114yes—
291C4×D4#4114yes—
292#117—24no—
293#117—48no—
294#117—42no—
295#117—84no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—48no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—42no—
362#106—28no—
363#106—44no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—42no—
369#103—28no—
370#103—28no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—28no—
375#102—24no—
376C22⋊C8—44no—
377C22⋊C8—44no—
378C4×C8—24no—
379C4×C8—44no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D412yes—
400#41—48no—
401#41—48no—
402#41—48no—
403#41—48no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8C2×#4114yes—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4C2×#4112yes—
426C2×D4C2×#4112yes—
427C2×D42−1+412yes—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—81no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—81no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4#13714yes—
515C4⋊C4C2×#4112yes—
516C4⋊C4C2×#4112yes—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—82no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—44no—
545C42—84no—
546C42#13514yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C42H(5,𝔽2)12yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C23×D411yescharacteristic
566Q8—24no—
567Q8—24no—
568Q8—42no—
569Q8—42no—
570Q8—84no—
571Q8—164no—
572D4—22no—
573D4—22no—
574D4—22no—
575D4—416no—
576D4—44no—
577D4#55614yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—84no—
629C2×C4—88no—
630C2×C4#52512yes—
631C2×C4#52512yes—
632C2×C4#54612yes—
633C2×C4#54612yes—
634C2×C4C2×#13512yes—
635C2×C4C2×#13512yes—
636C2×C4C2×#13512yes—
637C2×C4C2×#13714yes—
638C2×C4C2×#13712yes—
639C2×C4#57712yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4#58511yescharacteristic
646C8—42no—
647C8—44no—
648C8—42no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22C2×#52511yescharacteristic
664C22C2×#54611yescharacteristic
665C22#348911yescharacteristic
666C22C22×#13511yescharacteristic
667C22C22×#13711yescharacteristic
668C22C2×#57711yescharacteristic
669C22#358511yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—24no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—41no—
680C4—82no—
681C4—81no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4#310714yes—
686C4#330314yes—
687C4C2×#55611yescharacteristic
688C4#347511yescharacteristic
689C4#350611yescharacteristic
690C4#352011yescharacteristic
691C2—44no—
692C2—82no—
693C2#3616111yescharacteristic
694C2C2×#310711yescharacteristic
695C2#6059311yescharacteristic
696C2C2×#330311yescharacteristic
697C2#6116211yescharacteristic
698C2#6209811yescharacteristic
699C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
1. #36194C2semidirect
3. #33197C2semidirect
4. #32955C2semidirect
5. #28717C2semidirect
6. #28283C2semidirect
7. #27631C2semidirect
8. #27533C2semidirect
9. #27045C2semidirect
10. #26297C2semidirect
12. #24703C2semidirect
13. #23217C2semidirect
14. #22963C2semidirect
15. #22795C2semidirect
16. C4×#474C2semidirect
17. #20737C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5472
#331972
#329552
#287172
#282832
#276312
#275332
#270452
#262972
#258862
#247032
#232172
#229632
#227952
C4×#4742
#207372