Finite Group Databasebeta

← #825726

Subgroups — #825726

Order 512.

2746 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C212yesmaximal
2C2×#2546C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#16603C212yesmaximal
5#15438C214yesmaximal
6#15333C214yesmaximal
7#15158C211yesmaximal, characteristic
8#15143C211yesmaximal, characteristic
9#15128C211yesmaximal, characteristic
10#14881C212yesmaximal
11#14872C212yesmaximal
12#14053C212yesmaximal
13#14046C212yesmaximal
14#14005C212yesmaximal
15#13910C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23#2567—216no—
24#2546—28no—
25C2×#398C2214yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2211yescharacteristic
31#2365—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2214yes—
35C2×#389C2211yescharacteristic
36C2×#389C2214yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#384C2214yes—
47C2×#384C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#1446C2212yes—
56#1446C2214yes—
57#1446C2211yescharacteristic
58#1444C2211yescharacteristic
59#1444C2212yes—
60#1444C2214yes—
61#1444C2212yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2211yescharacteristic
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8Q818yescomplemented
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—44no—
101C2×C4×Q8—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#398—216no—
135#395—28no—
136#395—28no—
137#395—28no—
138#395—44no—
139#395D418yes—
140#391—28no—
141#391—216no—
142#391—28no—
143#389—216no—
144#389—24no—
145#389—28no—
146#389—28no—
147#389—24no—
148#389—216no—
149#389—44no—
150#389—44no—
151#389Q8116yescomplemented
152#387—28no—
153#387—28no—
154#384—216no—
155#384—44no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—44no—
162C4×C4⋊C4D418yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8C2×D414yes—
209C22×Q8C2×Q818yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C4⋊Q8—232no—
228C4⋊Q8—432no—
229C4⋊Q8—416no—
230C4×Q8—432no—
231C4×Q8—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×Q818yes—
291C2×C4⋊C4C2×Q814yes—
292C2×C4⋊C4C2×Q8116yes—
293C2×C4⋊C4C2×Q818yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C4⋊C4#4114yes—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D418yes—
320C2×C42C2×Q814yes—
321C2×C42C2×Q818yes—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—42no—
340#95—44no—
341#95—42no—
342#95—44no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C2×D414yes—
349#95C2×D418yes—
350#95C2×D418yes—
351#95C2×Q8116yes—
352C24C2511yescharacteristic
353C2×Q8—216no—
354C2×Q8—28no—
355C2×Q8—216no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—88no—
369C2×Q8—88no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q818yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q814yes—
462C22×C4C22×Q818yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q812yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q814yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+414yes—
472C22×C42−1+412yes—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—232no—
477C4⋊C4—216no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4C22⋊Q8116yes—
495C4⋊C4C4⋊Q8116yes—
496C42—28no—
497C42—28no—
498C42—216no—
499C42—216no—
500C42—416no—
501C42—48no—
502C42—48no—
503C42—44no—
504C42—48no—
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C23×Q811yescharacteristic
508C23C23×Q812yes—
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×2−1+411yescharacteristic
513C23C2×2−1+412yes—
514C23#58511yescharacteristic
515C23#58512yes—
516Q8—432no—
517Q8—416no—
518Q8—816no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
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—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4C2×C22⋊Q818yes—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C4⋊Q818yes—
615C2×C4C2×C4⋊Q814yes—
616C2×C4#53514yes—
617C2×C4D4214yes—
618C2×C4D4×Q818yes—
619C2×C4D4×Q814yes—
620C2×C4#55418yes—
621C2×C4Q8218yes—
622C22C22×C22⋊Q811yescharacteristic
623C22C22×C4⋊Q811yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×D4211yescharacteristic
626C22C2×D4×Q812yes—
627C22C2×D4×Q811yescharacteristic
628C22D4×#4111yescharacteristic
629C22C2×#55412yes—
630C22C2×Q8212yes—
631C22#348711yescharacteristic
632C22#348911yescharacteristic
633C22#349612yes—
634C22#349911yescharacteristic
635C22#350111yescharacteristic
636C22#350212yes—
637C22#350212yes—
638C22#350312yes—
639C22#350412yes—
640C22#351612yes—
641C22#352011yescharacteristic
642C22#354112yes—
643C22#354212yes—
644C22#355012yes—
645C4—28no—
646C4—24no—
647C4—28no—
648C4—28no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—48no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—84no—
661C4#256718yescomplemented
662C2C2×#256711yescharacteristic
663C2#3584511yescharacteristic
664C2#3595711yescharacteristic
665C2D4×C22⋊Q811yescharacteristic
666C2D4×C4⋊Q811yescharacteristic
667C2#3933911yescharacteristic
668C2#4291111yescharacteristic
669C2#4495012yes—
670C2#4499712yes—
671C2#5983112yes—
672C2#5983212yes—
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
96. C2×C4⋊Q8Q8semidirect
151. #389Q8semidirect
661. C4#2567semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25462
C2×#23652
#166032
#154382
#153332
#151582
#151432
#151282
#148812
#148722
#140532
#140462
#140052
#139102
C4×#3952
#138212