Finite Group Databasebeta

← #776141

Subgroups — #776141

Order 512.

3506 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2552C212yesmaximal
3C2×#2453C211yesmaximal, characteristic
4#16507C212yesmaximal
5#15850C214yesmaximal
6#15848C214yesmaximal
7#15158C211yesmaximal, characteristic
8#15143C212yesmaximal
9#15143C211yesmaximal, characteristic
10#15128C211yesmaximal, characteristic
11#14880C212yesmaximal
12#14186C212yesmaximal
13#14054C212yesmaximal
14#14017C212yesmaximal
15#13976C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×#125C2211yescharacteristic
23C22×#125C2214yes—
24C22×#125C2212yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29#2552—216no—
30#2453—28no—
31C2×#400C2212yes—
32C2×#398C2214yes—
33C2×#398C2214yes—
34C2×#398C2212yes—
35C2×#395C2211yescharacteristic
36C2×#395C2212yes—
37C2×#395C2212yes—
38C2×#395C2212yes—
39C2×#395C2212yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2211yescharacteristic
46C2×#389C2212yes—
47C2×#389C2214yes—
48C2×#389C2211yescharacteristic
49C2×#387C2212yes—
50C2×#384C2212yes—
51C2×#384C2214yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55#1446C2212yes—
56#1446C2211yescharacteristic
57#1445C2214yes—
58#1444C2212yes—
59#1444C2211yescharacteristic
60#1444C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2211yescharacteristic
68#1441C2212yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90#557—2128no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—28no—
95C2×#125—216no—
96C2×#125—216no—
97C2×#125—216no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—24no—
101C2×#125—28no—
102C2×#125D4116yes—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—216no—
106C2×C4×Q8—28no—
107C2×C4×Q8—28no—
108C2×C4×Q8—24no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8D418yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2314yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#400—28no—
145#398—28no—
146#398—216no—
147#398—216no—
148#395—24no—
149#395—28no—
150#395—44no—
151#395—44no—
152#395—44no—
153#391—24no—
154#391—24no—
155#391Q8116yescomplemented
156#389—24no—
157#389—28no—
158#389—216no—
159#389—28no—
160#389—24no—
161#389—28no—
162#387—28no—
163#384—28no—
164#384—216no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—28no—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2311yescharacteristic
198C22×Q8—24no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—42no—
210C22×Q8—44no—
211C22×Q8C2×D414yes—
212C22×Q8C2×Q818yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C4⋊Q8—264no—
231C4⋊Q8—232no—
232#125—2128no—
233#125—264no—
234#125—416no—
235C4×Q8—264no—
236C4×Q8—264no—
237C4×Q8—232no—
238C4×Q8—232no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D4116yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D4116yes—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×Q818yes—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—22no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42C2×D418yes—
324C2×C42C2×Q818yes—
325#95—22no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—42no—
346#95—44no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×Q8116yes—
353#95C2×Q8116yes—
354#95#4114yes—
355C24C2511yescharacteristic
356C2×Q8—216no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—416no—
363C2×Q8—48no—
364C2×Q8—416no—
365C2×Q8—48no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—48no—
370C2×Q8C4⋊Q8116yes—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D418yes—
453C22×C4C22×D414yes—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q818yes—
462C22×C4C22×Q818yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C22×C42−1+414yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—232no—
478C4⋊C4—216no—
479C4⋊C4—232no—
480C4⋊C4—216no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C4⋊C4#120116yes—
496C42—232no—
497C42—216no—
498C42—216no—
499C42—28no—
500C42—216no—
501C42—416no—
502C42—44no—
503C42—44no—
504C23C23×D411yescharacteristic
505C23C23×D412yes—
506C23C23×Q812yes—
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+412yes—
512C23C2×2−1+411yescharacteristic
513C23#58512yes—
514C23#58511yescharacteristic
515Q8—232no—
516Q8—432no—
517Q8—416no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4C2×#12018yes—
611C2×C4C2×#12014yes—
612C2×C4C2×C4⋊Q818yes—
613C2×C4C2×C4⋊Q814yes—
614C2×C4#53614yes—
615C2×C4D4×Q818yes—
616C2×C4D4×Q818yes—
617C2×C4#55014yes—
618C2×C4#55718yescomplemented
619C22C22×#12011yescharacteristic
620C22C22×C4⋊Q811yescharacteristic
621C22C2×#53611yescharacteristic
622C22C2×D4×Q812yes—
623C22C2×D4×Q812yes—
624C22C2×#55011yescharacteristic
625C22D4×#4112yes—
626C22C2×#55712yes—
627C22#348711yescharacteristic
628C22#348812yes—
629C22#348912yes—
630C22#348911yescharacteristic
631C22#349912yes—
632C22#349911yescharacteristic
633C22#350212yes—
634C22#350312yes—
635C22#350412yes—
636C22#351511yescharacteristic
637C22#352311yescharacteristic
638C22#355211yescharacteristic
639C22#355411yescharacteristic
640C22#355512yes—
641C22#356312yes—
642C4—28no—
643C4—24no—
644C4—28no—
645C4—28no—
646C4—48no—
647C4—44no—
648C4—48no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4#243418yes—
659C2C2×#243411yescharacteristic
660C2#3595711yescharacteristic
661C2#3608511yescharacteristic
662C2#3642911yescharacteristic
663C2#3659311yescharacteristic
664C2#3913911yescharacteristic
665C2#4287911yescharacteristic
666C2#4527212yes—
667C2#4542412yes—
668C2#5915412yes—
669C2#5966712yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
155. #391Q8semidirect
618. C2×C4#557semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#25522
C2×#24532
#165072
#158502
#158482
#151582
#151432
#151282
#148802
#141862
#140542
#140172
#139762
C4×#3952
#138172