Finite Group Databasebeta

← #776727

Subgroups — #776727

Order 512.

3534 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2453C212yesmaximal
3C2×#2365C211yesmaximal, characteristic
4#16560C212yesmaximal
5#16104C214yesmaximal
6#15847C214yesmaximal
7#15201C211yesmaximal, characteristic
8#15191C211yesmaximal, characteristic
9#15128C212yesmaximal
10#15128C211yesmaximal, characteristic
11#14878C212yesmaximal
12#14174C212yesmaximal
13#14054C212yesmaximal
14#14024C212yesmaximal
15#14017C212yesmaximal
16#13935C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2214yes—
24C22×#125C2212yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2212yes—
29C22×C4×Q8C2211yescharacteristic
30#2453—216no—
31C2×#400C2214yes—
32C2×#398C2214yes—
33C2×#395C2211yescharacteristic
34C2×#395C2212yes—
35#2365—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2211yescharacteristic
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#387C2212yes—
46C2×#387C2212yes—
47C2×#387C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#1446C2212yes—
53#1446C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1446C2212yes—
57#1445C2214yes—
58#1444C2214yes—
59#1444C2212yes—
60#1444C2211yescharacteristic
61#1444C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2211yescharacteristic
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2312yes—
90C23×Q8C2311yescharacteristic
91#557—2128no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—216no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8Q818yescomplemented
99C2×#125—216no—
100C2×#125—216no—
101C2×#125—216no—
102C2×#125—28no—
103C2×#125—28no—
104C2×#125D4116yes—
105C2×C4×Q8—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—216no—
108C2×C4×Q8—28no—
109C2×C4×Q8—24no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8—24no—
114C2×C4×Q8—28no—
115C2×C4×Q8D418yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145#400—216no—
146#398—216no—
147#395—24no—
148#395—28no—
149#391—24no—
150#391—28no—
151#391—28no—
152#391—24no—
153#391—28no—
154#389—28no—
155#389—28no—
156#389—216no—
157#389—28no—
158#387—28no—
159#387—216no—
160#387—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C22×Q8—24no—
198C22×Q8—28no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
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×Q8C2×Q814yes—
216C22×Q8#4114yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—264no—
235C4⋊Q8—232no—
236C4⋊Q8—232no—
237#125—2128no—
238#125—264no—
239C4×Q8—264no—
240C4×Q8—264no—
241C4×Q8—232no—
242C4×Q8—232no—
243C4×Q8—416no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
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—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D4116yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D4116yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C4⋊C4C2×Q818yes—
297C2×C4⋊C4#4114yes—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42C2×D418yes—
318C2×C42C2×D418yes—
319C2×C42C2×D414yes—
320C2×C42C2×Q814yes—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—22no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—44no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—42no—
351#95—44no—
352#95—44no—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—416no—
363C2×Q8—48no—
364C2×Q8—416no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8C22⋊Q8116yes—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4C22×D414yes—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q814yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C42−1+414yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C22×C42−1+414yes—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—232no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—232no—
479C4⋊C4—216no—
480C4⋊C4—232no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—44no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4#120116yes—
493C42—232no—
494C42—216no—
495C42—216no—
496C42—28no—
497C42—28no—
498C42—48no—
499C42—48no—
500C42—44no—
501C42—48no—
502C23C23×D411yescharacteristic
503C23C23×D412yes—
504C23C23×Q811yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58511yescharacteristic
513Q8—232no—
514Q8—432no—
515Q8—416no—
516Q8—416no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4C2×#12018yes—
608C2×C4C2×#12014yes—
609C2×C4C2×C22⋊Q818yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4#53614yes—
612C2×C4#54814yes—
613C2×C4#54818yes—
614C2×C4D4×Q818yes—
615C2×C4#55714yescomplemented
616C22C22×#12011yescharacteristic
617C22C22×C22⋊Q811yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#54811yescharacteristic
620C22C2×#54812yes—
621C22C2×D4×Q812yes—
622C22D4×#4112yes—
623C22C2×#55711yescharacteristic
624C22#348011yescharacteristic
625C22#348711yescharacteristic
626C22#348912yes—
627C22#348911yescharacteristic
628C22#348912yes—
629C22#349911yescharacteristic
630C22#349912yes—
631C22#350212yes—
632C22#350211yescharacteristic
633C22#351612yes—
634C22#352312yes—
635C22#352311yescharacteristic
636C22#355512yes—
637C22#355612yes—
638C22#355712yes—
639C4—24no—
640C4—28no—
641C4—24no—
642C4—28no—
643C4—42no—
644C4—42no—
645C4—48no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—81no—
655C4#240918yes—
656C2C2×#240911yescharacteristic
657C2#3584511yescharacteristic
658C2#3608511yescharacteristic
659C2#3642811yescharacteristic
660C2#3657711yescharacteristic
661C2#3917011yescharacteristic
662C2#4288411yescharacteristic
663C2#4527712yes—
664C2#4534312yes—
665C2#5908512yes—
666C2#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
98. C2×C4⋊Q8Q8semidirect
615. 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×#24532
C2×#23652
#165602
#161042
#158472
#152012
#151912
#151282
#148782
#141742
#140542
#140242
#140172
#139352
C4×#3952