Finite Group Databasebeta

← #776122

Subgroups — #776122

Order 512.

3668 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2441C212yesmaximal
3C2×#2362C211yesmaximal, characteristic
4#16511C212yesmaximal
5#16104C214yesmaximal, complemented
6#15846C214yesmaximal
7#15158C211yesmaximal, characteristic, complemented
8#15143C211yesmaximal, characteristic, complemented
9#15128C212yesmaximal, complemented
10#15125C211yesmaximal, characteristic
11#14877C212yesmaximal
12#14174C212yesmaximal, complemented
13#14054C212yesmaximal, complemented
14#13976C212yesmaximal, complemented
15#13969C212yesmaximal
16#13935C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×#125C2214yes—
23C22×#125C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C22×#117C2211yescharacteristic
30#2441—216no—
31C2×#399C2214yes—
32C2×#396C2214yes—
33C2×#395C2212yes—
34C2×#395C2212yes—
35C2×#395C2212yes—
36C2×#393C2211yescharacteristic
37C2×#393C2212yes—
38#2362—28no—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#389C2214yes—
42C2×#389C2211yescharacteristic
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#387C2212yes—
48C2×#387C2214yes—
49C2×#384C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55#1446C2212yes—
56#1446C2211yescharacteristic
57#1445C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1442C2212yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2212yes—
70#1439C2211yescharacteristic
71#1439C2212yes—
72#1439C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#557—2128no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—28no—
96C2×#125—216no—
97C2×#125—216no—
98C2×#125—216no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125D4116yes—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8Q818yescomplemented
105C2×C4×Q8—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—216no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—24no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8D418yes—
114C2×#117—24no—
115C2×#117—28no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C42C2312yes—
142C22×C42C2314yes—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149#399—216no—
150#396—216no—
151#395—28no—
152#395—44no—
153#395—44no—
154#393—24no—
155#393—44no—
156#391—24no—
157#391—28no—
158#389—216no—
159#389—28no—
160#389—24no—
161#389—28no—
162#388—28no—
163#388—28no—
164#387—28no—
165#387—216no—
166#384—28no—
167#380—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—28no—
170C4×C22⋊C4—24no—
171C4×C22⋊C4—28no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2311yescharacteristic
200C25C2411yescharacteristic
201C22×Q8—24no—
202C22×Q8—28no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8—44no—
213C22×Q8C2×D414yes—
214C22×Q8C2×Q814yes—
215C22×Q8#4114yes—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4C2×Q814yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—264no—
238C4⋊Q8—232no—
239#125—2128no—
240#125—264no—
241C22⋊Q8—232no—
242C4×Q8—264no—
243C4×Q8—264no—
244C4×Q8—232no—
245C4×Q8—232no—
246#117—416no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×D4116yes—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×Q818yes—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42C2×D418yes—
334C2×C42C2×D414yes—
335C2×C42C2×D418yes—
336#95—22no—
337#95—24no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—24no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95#4114yes—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—22no—
372C24—21no—
373C24—22no—
374C24C22×Q812yes—
375C24C2511yescharacteristic
376C2×Q8—216no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—216no—
381C2×Q8—216no—
382C2×Q8—48no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8C22⋊Q8116yes—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—216no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×Q814yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q812yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C22×C42−1+414yes—
489C22×C42−1+414yes—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—232no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—232no—
496C4⋊C4—216no—
497C4⋊C4—232no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C4⋊C4#120116yes—
508C22⋊C4—216no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—88no—
515C42—232no—
516C42—216no—
517C42—216no—
518C42—28no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—48no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C23×D411yescharacteristic
547C23C23×D412yes—
548C23C23×Q811yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+412yes—
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+412yes—
555C23#58512yes—
556C23#58511yescharacteristic
557Q8—232no—
558Q8—416no—
559Q8—432no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—416no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4C2×#12018yes—
648C2×C4C2×#12014yes—
649C2×C4C2×C22⋊Q818yes—
650C2×C4C2×C22⋊Q814yes—
651C2×C4#53614yes—
652C2×C4#54818yes—
653C2×C4D4×Q818yes—
654C2×C4#55014yes—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—82no—
661C22—82no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22C22×#12011yescharacteristic
667C22C22×C22⋊Q811yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#54812yes—
670C22C2×D4×Q812yes—
671C22C2×#55011yescharacteristic
672C22D4×#4112yes—
673C22Q8×#4111yescharacteristic
674C22#348211yescharacteristic
675C22#348711yescharacteristic
676C22#348812yes—
677C22#348912yes—
678C22#348911yescharacteristic
679C22#349911yescharacteristic
680C22#349912yes—
681C22#350212yes—
682C22#350311yescharacteristic
683C22#350912yes—
684C22#351612yes—
685C22#351711yescharacteristic
686C22#355212yes—
687C22#356312yes—
688C22#356512yes—
689C4—24no—
690C4—28no—
691C4—28no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—48no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—84no—
702C4—81no—
703C4—82no—
704C4#240918yes—
705C2—82no—
706C2C2×#240911yescharacteristic
707C2#3584511yescharacteristic
708C2#3608511yescharacteristic
709C2#3642911yescharacteristic
710C2#3644511yescharacteristic
711C2#3974411yescharacteristic
712C2#4288811yescharacteristic
713C2#4527212yes—
714C2#4531312yes—
715C2#5915212yes—
716C2#5965612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
5. #16104C2semidirect
7. #15158C2semidirect
8. #15143C2semidirect
9. #15128C2semidirect
12. #14174C2semidirect
13. #14054C2semidirect
14. #13976C2semidirect
16. #13935C2semidirect
104. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#24412
C2×#23622
#165112
#161042
#158462
#151582
#151432
#151282
#151252
#148772
#141742
#140542
#139762
#139692
#139352
C4×#3932