Finite Group Databasebeta

← #6468068

Subgroups — #6468068

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2C4×#536C211yesmaximal, characteristic
3#29153C214yesmaximal, complemented
4#29151C214yesmaximal, complemented
5#29150C212yesmaximal, complemented
6#29148C212yesmaximal, complemented
7#29141C212yesmaximal, complemented
8#29140C212yesmaximal, complemented
9#23458C212yesmaximal
10#23457C212yesmaximal, complemented
11#21205C212yesmaximal
12#20763C212yesmaximal, complemented
13#20648C211yesmaximal, characteristic
14#20647C211yesmaximal, characteristic, complemented
15#20645C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17#20431C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#536C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#417C2212yes—
25C2×#419C2212yes—
26C2×#416C2212yes—
27C22×#103C2212yes—
28C2×#405C2212yes—
29#2868C2212yes—
30#2859C2212yes—
31#2660C2214yes—
32#2552—21no—
33#2552—21no—
34#2551—21no—
35#2551—21no—
36#2440—22no—
37#2440—22no—
38#2327C2212yes—
39#2326C2212yes—
40#2323—21no—
41#2323—21no—
42#2322—22no—
43#2317C2211yescharacteristic
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×#123—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2026C2214yes—
56#2026C2214yes—
57#2026C2214yes—
58#2025C2214yes—
59#2024C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#1960C2214yes—
64#1959C2214yes—
65#1958C2214yes—
66#1956C2214yes—
67#1836C2214yes—
68#1836C2214yes—
69#1835C2214yes—
70#1834C2214yes—
71#1805C2212yes—
72#1805C2214yes—
73#1804C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76#1803C2214yes—
77#1803C2211yescharacteristic
78#1803C2211yescharacteristic
79#1803C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1768C2214yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—44no—
100#544—44no—
101#536—24no—
102#536Q818yescomplemented
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#125—22no—
109C2×#125C2314yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C4×#41—24no—
116C4×#41—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117C2311yescharacteristic
131C2×#117C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C42—21no—
140C22×C42—21no—
141C22×C42C2311yescharacteristic
142#423—24no—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149#419—44no—
150#417—44no—
151#416—44no—
152C2×#103—44no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163#405—28no—
164C2×#97C2314yes—
165C2×C4×C8C2314yes—
166#398—24no—
167#398—24no—
168#395—24no—
169#395—24no—
170#391—21no—
171#391—21no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—22no—
195#385—22no—
196#385—24no—
197#385—22no—
198#385—22no—
199#384—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2314yes—
209#382C2312yes—
210#382C2312yes—
211#382C2314yes—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229#376C2312yes—
230C43C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41C2×Q814yes—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8C2×Q814yes—
257C22×Q8C2412yes—
258C22×Q8C2411yescharacteristic
259C22×D4—21no—
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C4⋊Q8—24no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—48no—
283C4⋊Q8—84no—
284C4⋊Q8—84no—
285C4⋊Q8C2×Q818yes—
286#125—44no—
287#125—44no—
288#125—48no—
289#124—24no—
290#124C2×Q818yes—
291#123—216no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—416no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—44no—
310C4×D4—44no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—24no—
316#117—24no—
317#117—44no—
318#117C2×Q814yes—
319#117#4114yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4—82no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—44no—
377C2×C42—44no—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42#4114yes—
385C2×C42#4114yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—22no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395#103—44no—
396#103—44no—
397#103—42no—
398#103—42no—
399#103—44no—
400#103—44no—
401#103—44no—
402#102—42no—
403#102—42no—
404#102—44no—
405#97—28no—
406C4×C8—28no—
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—24no—
418#95—24no—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423C24—21no—
424C24C22×D411yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—84no—
448C2×Q8—84no—
449C2×Q8C22×Q814yes—
450C2×Q8C22×Q812yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C22×Q812yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4#12414yes—
491C22×C4#12414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×Q812yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4C2511yescharacteristic
508M16—48no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—48no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—84no—
555C4⋊C4—84no—
556C4⋊C4C22⋊Q814yes—
557C4⋊C4C22⋊Q814yes—
558C22⋊C4—24no—
559C22⋊C4—28no—
560C22⋊C4—28no—
561C22⋊C4—42no—
562C22⋊C4—44no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4C22⋊Q818yes—
568C42—24no—
569C42—24no—
570C42—22no—
571C42—22no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42#12414yes—
581C42#12414yes—
582C42C22×Q814yes—
583C42C2×#4114yes—
584C42C2×#4112yes—
585C42C2×#4112yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—22no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23C2×C22⋊Q812yes—
600C23C2×#12412yes—
601C23#53512yes—
602C23C23×D411yescharacteristic
603Q8—24no—
604Q8—24no—
605Q8—22no—
606Q8—22no—
607Q8—48no—
608Q8—42no—
609Q8—42no—
610Q8—41no—
611Q8—41no—
612Q8—168no—
613D4—22no—
614D4—22no—
615D4—48no—
616D4—41no—
617D4—41no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—164no—
666C2×C4C2×C22⋊Q814yes—
667C2×C4C2×C22⋊Q814yes—
668C2×C4C2×C22⋊Q812yes—
669C2×C4C2×#12412yes—
670C2×C4C2×#12412yes—
671C2×C4C2×#12412yes—
672C2×C4C2×#12412yes—
673C2×C4C2×#12412yes—
674C2×C4#53512yes—
675C2×C4#55414yes—
676C2×C4#55414yes—
677C2×C4#55514yes—
678C2×C4#55514yes—
679C2×C4#57712yes—
680C2×C4#57812yes—
681C2×C4C23×Q811yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×H(5,𝔽2)11yescharacteristic
687C8—42no—
688C8—42no—
689C8—44no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22#256314yes—
704C22C22×C22⋊Q811yescharacteristic
705C22C22×#12411yescharacteristic
706C22C2×#53511yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57811yescharacteristic
709C22#358511yescharacteristic
710C22#358611yescharacteristic
711C4—24no—
712C4—24no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—84no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—82no—
725C4—81no—
726C4#256314yes—
727C4#313714yes—
728C4C2×#55411yescharacteristic
729C4C2×#55411yescharacteristic
730C4C2×#55511yescharacteristic
731C4C2×#55511yescharacteristic
732C2—24no—
733C2—82no—
734C2C2×#256311yescharacteristic
735C2#6049711yescharacteristic
736C2#6049811yescharacteristic
737C2C2×#313711yescharacteristic
738C2#6062111yescharacteristic
739C2#6083311yescharacteristic
740C2#6083711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35275C2semidirect
3. #29153C2semidirect
4. #29151C2semidirect
5. #29150C2semidirect
6. #29148C2semidirect
7. #29141C2semidirect
8. #29140C2semidirect
10. #23457C2semidirect
12. #20763C2semidirect
14. #20647C2semidirect
16. C2×#1803C2semidirect
17. #20431C2semidirect
102. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
C4×#5362
#291532
#291512
#291502
#291482
#291412
#291402
#234582
#234572
#212052
#207632
#206482
#206472
#206452
C2×#18032
#204312