Finite Group Databasebeta

← #6591961

Subgroups — #6591961

Order 512.

1823 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32958C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27682C212yesmaximal, complemented
8#26531C212yesmaximal, complemented
9#26413C214yesmaximal, complemented
10#26111C212yesmaximal, complemented
11#25954C214yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#22707C212yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22417C212yesmaximal
16C4×#476C211yesmaximal, characteristic, complemented
17#20645C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#477C2212yes—
21C2×#476C2211yescharacteristic
22C2×#474C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25C2×#416C2214yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2571C2212yes—
29#2563—21no—
30#2538—21no—
31#2497—22no—
32#2492—22no—
33#2439C2214yes—
34#2369—21no—
35#2365C2211yescharacteristic
36#2362C2212yes—
37#2361—21no—
38#2359—22no—
39#2355—21no—
40#2354C2212yes—
41#2354C2211yescharacteristic
42D4×C4⋊C4C2211yescharacteristic
43#2344—21no—
44#2339C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C4×#123—22no—
49C4×#121—21no—
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54#2101C2214yes—
55#2097C2214yes—
56#2087C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2024C2214yes—
60#1988C2212yes—
61#1987C2211yescharacteristic
62#1981C2212yes—
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1920C2212yes—
70#1892C2214yes—
71#1873C2214yes—
72#1865C2214yes—
73#1864C2212yes—
74#1862C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1850C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1808C2214yes—
81#1805C2214yes—
82#1804C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1788C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1735C2214yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#547—24no—
98#547Q818yescomplemented
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—21no—
108C4×#41—28no—
109C4×#41—44no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117C2312yes—
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C42—22no—
128C22×C42—21no—
129C22×C42C2312yes—
130#477—44no—
131#476—24no—
132#474—44no—
133C8×D4—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8—44no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145#416—48no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#400—24no—
156#396—24no—
157#393—24no—
158#391—21no—
159#391—22no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—21no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#389C2312yes—
174#388—22no—
175#388—24no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382C2314yes—
194#381—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C2×#95—22no—
226C43C2311yescharacteristic
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41C2×Q818yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271C4⋊Q8C2×Q814yes—
272#123—216no—
273C22⋊Q8—24no—
274C22⋊Q8—84no—
275C22⋊Q8C2×Q818yes—
276#121—28no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—42no—
288C4×D4—41no—
289C4×D4—42no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4C2×Q814yes—
293C4×D4C2×Q814yes—
294C4×D4#4114yes—
295C4×D4#4114yes—
296#117—24no—
297#117—48no—
298#117—42no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—48no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—44no—
366#107—44no—
367#106—24no—
368#106—42no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—44no—
374C4⋊C8—42no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379#102—24no—
380#102—24no—
381#102—28no—
382C22⋊C8—28no—
383C4×C8—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D412yes—
404#41—216no—
405#41—48no—
406#41—48no—
407#41—48no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8C22×Q814yes—
420C2×D4—28no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4C22⋊Q814yes—
429C2×D4C22×Q812yes—
430C2×D4C2×#4112yes—
431C2×D4H(5,𝔽2)12yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—82no—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q814yes—
475C22×C4C2×#4114yes—
476C22×C4C2511yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—88no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C22⋊Q814yes—
520C4⋊C4C22×Q812yes—
521C4⋊C4C2×#4112yes—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4C22⋊Q818yes—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—84no—
551C42#13514yes—
552C42C22×Q812yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C42H(5,𝔽2)12yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C2×C22⋊Q814yes—
570C23C23×D411yescharacteristic
571Q8—24no—
572Q8—24no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576Q8—164no—
577D4—28no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—416no—
582D4#55414yes—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4C2×C22⋊Q812yes—
636C2×C4C2×C22⋊Q814yes—
637C2×C4C2×C22⋊Q812yes—
638C2×C4#54612yes—
639C2×C4#54612yes—
640C2×C4C2×#13512yes—
641C2×C4C2×#13512yes—
642C2×C4C2×#13512yes—
643C2×C4#57512yes—
644C2×C4#57812yes—
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4#58511yescharacteristic
651C8—42no—
652C8—44no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22C22×C22⋊Q811yescharacteristic
669C22C2×#54611yescharacteristic
670C22#348811yescharacteristic
671C22C22×#13511yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57811yescharacteristic
674C22#358511yescharacteristic
675C4—24no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—84no—
690C4#307614yes—
691C4#330214yes—
692C4C2×#55411yescharacteristic
693C4#347511yescharacteristic
694C4#350111yescharacteristic
695C4#351911yescharacteristic
696C2—44no—
697C2—44no—
698C2#3615411yescharacteristic
699C2C2×#307611yescharacteristic
700C2#6049811yescharacteristic
701C2C2×#330211yescharacteristic
702C2#6116611yescharacteristic
703C2#6194911yescharacteristic
704C2#6198511yescharacteristic
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. #33195C2semidirect
4. #32958C2semidirect
5. #29141C2semidirect
6. #28284C2semidirect
7. #27682C2semidirect
8. #26531C2semidirect
9. #26413C2semidirect
10. #26111C2semidirect
11. #25954C2semidirect
12. #24837C2semidirect
13. #22707C2semidirect
14. D4×C4⋊C8C2semidirect
16. C4×#476C2semidirect
17. #20645C2semidirect
98. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5472
#331952
#329582
#291412
#282842
#276822
#265312
#264132
#261112
#259542
#248372
#227072
D4×C4⋊C82
#224172
C4×#4762
#206452