Finite Group Databasebeta

← #613529

Subgroups — #613529

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#14936C212yesmaximal
4#14935C212yesmaximal
5#14862C214yesmaximal, complemented
6#14804C214yesmaximal, complemented
7#14697C212yesmaximal
8#14695C212yesmaximal
9#14685C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14054C212yesmaximal, complemented
12#14029C212yesmaximal, complemented
13#14005C212yesmaximal, complemented
14#13982C212yesmaximal, complemented
15#13907C211yesmaximal, characteristic
16#13880C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2212yes—
23#2442—28no—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2214yes—
31C2×#387C2214yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#384C2212yes—
37C2×#382C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4—28no—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2212yes—
52#1445C2214yes—
53#1445C2214yes—
54#1445C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77#1434C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#125—24no—
91C2×#125—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42—24no—
119C22×C42—24no—
120C22×C42—28no—
121C22×C42—48no—
122C22×C42Q818yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#400—28no—
131#398—28no—
132#389—24no—
133#389—24no—
134#389—28no—
135#389—28no—
136#389—48no—
137#387—48no—
138#385—28no—
139#385—28no—
140#385—44no—
141#385—44no—
142#384—28no—
143#382—24no—
144#381—24no—
145#381—24no—
146#381—44no—
147#380—28no—
148#380—24no—
149#380—24no—
150#380—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—216no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95—28no—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C25C2411yescharacteristic
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q814yes—
214C23×C4C2×Q814yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#125—416no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4#4118yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4#4114yes—
290C2×C22⋊C4#4114yes—
291C2×C42—232no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—432no—
310C2×C42C2×Q818yes—
311C2×C42C2×Q818yes—
312#95—28no—
313#95—28no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24C22×D412yes—
348C24C22×Q812yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24H(5,𝔽2)12yes—
353C242−1+412yes—
354C24C2511yescharacteristic
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—216no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C22⋊Q8116yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q814yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+414yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—216no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4#123116yes—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—216no—
497C42—216no—
498C42—416no—
499C42—416no—
500C42—44no—
501C42—44no—
502C42—48no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23C2×C22⋊Q814yes—
520C23#53314yes—
521C23#54414yes—
522C23#54714yes—
523C23#55014yes—
524C23#55214yes—
525C23#55314yes—
526C23C23×D411yescharacteristic
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4112yes—
530C23C22×#4112yes—
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+412yes—
537C2×C4—232no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—432no—
569C2×C4—432no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4C2×#12318yes—
629C2×C4C2×#12314yes—
630C2×C4#52514yes—
631C2×C4#52918yes—
632C2×C4#53318yes—
633C2×C4#54418yes—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—28no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—24no—
644C22#244118yes—
645C22C22×C22⋊Q811yescharacteristic
646C22C22×#12311yescharacteristic
647C22C2×#52511yescharacteristic
648C22C2×#52912yes—
649C22C2×#53312yes—
650C22C2×#53311yescharacteristic
651C22C2×#54412yes—
652C22C2×#54411yescharacteristic
653C22C2×#54711yescharacteristic
654C22C2×#55011yescharacteristic
655C22C2×#55211yescharacteristic
656C22C2×#55311yescharacteristic
657C22#348112yes—
658C22#348112yes—
659C22#348212yes—
660C22#348212yes—
661C22#348411yescharacteristic
662C22#348611yescharacteristic
663C22#349212yes—
664C22#351912yes—
665C22#352012yes—
666C22#352212yes—
667C22#352312yes—
668C4—24no—
669C4—28no—
670C4—28no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—48no—
676C4—81no—
677C4—81no—
678C4—82no—
679C4—84no—
680C4—81no—
681C4—81no—
682C4—82no—
683C4#285518yes—
684C2—28no—
685C2C2×#244111yescharacteristic
686C2#3496612yes—
687C2#3522411yescharacteristic
688C2#3523912yes—
689C2#3623711yescharacteristic
690C2#3625612yes—
691C2#3642311yescharacteristic
692C2#3648311yescharacteristic
693C2#3648812yes—
694C2#3652911yescharacteristic
695C2C2×#285511yescharacteristic
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
5. #14862C2semidirect
6. #14804C2semidirect
11. #14054C2semidirect
12. #14029C2semidirect
13. #14005C2semidirect
14. #13982C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×C4×C4⋊C42
#149362
#149352
#148622
#148042
#146972
#146952
#146852
#141302
#140542
#140292
#140052
#139822
#139072
#138802
C4×#3812