Finite Group Databasebeta

← #6627605

Subgroups — #6627605

Order 512.

1679 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2C4×#554C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#33024C212yesmaximal
5#29356C212yesmaximal, complemented
6#29274C212yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28280C211yesmaximal, characteristic, complemented
9#27574C214yesmaximal, complemented
10#27154C214yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24327C212yesmaximal, complemented
13#23067C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22758C212yesmaximal, complemented
16#21613C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#554C2211yescharacteristic
20C2×#476C2212yes—
21C2×#474C2212yes—
22C2×#459C2211yescharacteristic
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2858C2212yes—
26#2547—21no—
27#2545—22no—
28#2538—21no—
29#2538C2212yes—
30#2512—22no—
31#2512C2214yes—
32#2370C2212yes—
33#2369C2211yescharacteristic
34#2368—22no—
35#2360—21no—
36#2355C2211yescharacteristic
37#2354—21no—
38#2354C2212yes—
39#2349C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41#2345—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×#127C2211yescharacteristic
45C4×#125C2212yes—
46C4×C22⋊Q8—22no—
47C4×C22⋊Q8—21no—
48C42×Q8C2211yescharacteristic
49C42×D4C2212yes—
50C42×D4C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52#2101C2214yes—
53#2099C2214yes—
54#2073C2214yes—
55#2065C2214yes—
56#2059C2214yes—
57#2025C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1936C2214yes—
70#1936C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1919C2214yes—
77#1917C2214yes—
78#1879C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1802C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×D8C2312yes—
96#556—48no—
97#554—24no—
98#554—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#126—24no—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C42—21no—
128C22×C42—22no—
129#476—28no—
130#474—28no—
131#459—24no—
132C4×D8—28no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#397—24no—
151#396—24no—
152#396—24no—
153#391—21no—
154#391—21no—
155#391—22no—
156#391—22no—
157#390—21no—
158#390—21no—
159#390—22no—
160#390—22no—
161#390C2312yes—
162#390C2314yes—
163#390C2314yes—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382C2312yes—
189#380—21no—
190#380—22no—
191#380—22no—
192#380—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C2×#95—22no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C2×D8—24no—
249C2×D8—24no—
250C2×D8—24no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yes—
260#127—22no—
261#127—22no—
262#126—416no—
263#125—24no—
264#125—24no—
265#124—48no—
266#124—84no—
267C22⋊Q8—28no—
268C22⋊Q8—216no—
269C22⋊Q8—48no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—42no—
274C4×Q8—48no—
275C4×Q8—82no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—42no—
285C4×D4—41no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4#4114yes—
289C4×D4#4114yes—
290#117—48no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—84no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—28no—
358#106—28no—
359C4⋊C8—22no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—42no—
364#102—24no—
365#102—28no—
366#102—24no—
367#102—24no—
368#102—28no—
369#102—42no—
370#102—44no—
371#97—44no—
372C4×C8—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×#4112yes—
416C2×D4C2×#4112yes—
417C2×D42−1+412yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—82no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454D8—48no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4C2×#4114yes—
498C4⋊C4C2×#4112yes—
499C4⋊C4C2×#4112yes—
500C22⋊C4—28no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—41no—
523C42—42no—
524C42—42no—
525C42—48no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—82no—
531C42—82no—
532C42AGL(1,ℤ/8)14yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C422−1+412yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C23×D411yescharacteristic
552Q8—44no—
553Q8—84no—
554Q8—88no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—84no—
563D4—84no—
564D4#55614yescomplemented
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4#52512yes—
614C2×C4#52512yes—
615C2×C4#54812yes—
616C2×C4#54812yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4#57512yes—
621C2×C4#57612yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4#58511yescharacteristic
628C8—42no—
629C8—44no—
630C8—42no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C2×#52511yescharacteristic
646C22C2×#54811yescharacteristic
647C22#348911yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57611yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—41no—
662C4—42no—
663C4—84no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4#311514yes—
668C4#336614yes—
669C4C2×#55611yescharacteristic
670C4#348011yescharacteristic
671C4#350711yescharacteristic
672C4#352311yescharacteristic
673C2—44no—
674C2—162no—
675C2#3657911yescharacteristic
676C2C2×#311511yescharacteristic
677C2#6059611yescharacteristic
678C2C2×#336611yescharacteristic
679C2#6132311yescharacteristic
680C2#6210711yescharacteristic
681C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. C4×#554C2semidirect
3. #33195C2semidirect
5. #29356C2semidirect
6. #29274C2semidirect
7. #29219C2semidirect
8. #28280C2semidirect
9. #27574C2semidirect
10. #27154C2semidirect
11. #24993C2semidirect
12. #24327C2semidirect
13. #23067C2semidirect
14. #22825C2semidirect
15. #22758C2semidirect
16. #21613C2semidirect
564. D4#556semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×#5542
#331952
#330242
#293562
#292742
#292192
#282802
#275742
#271542
#249932
#243272
#230672
#228252
#227582
#216132
C4×#4592