Finite Group Databasebeta

← #669707

Subgroups — #669707

Order 512.

2710 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3#16075C214yesmaximal, complemented
4#15414C214yesmaximal, complemented
5#15187C212yesmaximal, complemented
6#15137C212yesmaximal, complemented
7#14936C212yesmaximal
8#14821C212yesmaximal
9#14685C211yesmaximal, characteristic
10#14132C212yesmaximal, complemented
11#14130C211yesmaximal, characteristic
12#14039C212yesmaximal
13#14029C212yesmaximal, complemented
14#13992C212yesmaximal
15#13924C211yesmaximal, characteristic
16#13880C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#398C2214yes—
23#2345—28no—
24C2×#389—28no—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2211yescharacteristic
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2214yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—48no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42—24no—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128#398—48no—
129#389—24no—
130#389—24no—
131#389—28no—
132#389—216no—
133#389—28no—
134#389—28no—
135#387—28no—
136#387—28no—
137#387—48no—
138#385—28no—
139#385—44no—
140#385—44no—
141#384—28no—
142#381—24no—
143#381—44no—
144#381Q818yes—
145#380—28no—
146#380—24no—
147#380—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—24no—
153C2×#95—28no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2×Q814yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C2×C4⋊C4—232no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—432no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D414yes—
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—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×Q814yes—
289C2×C22⋊C4C2×Q814yes—
290C2×C22⋊C4#4114yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—416no—
307C2×C42—44no—
308C2×C42—44no—
309#95—28no—
310#95—28no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×Q818yes—
340#95C2×Q818yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24C22×D412yes—
346C24C22×Q812yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24H(5,𝔽2)12yes—
351C242−1+412yes—
352C24C2511yescharacteristic
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
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—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—216no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4#123116yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q814yes—
462C22×C4C22×Q814yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4118yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+414yes—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—416no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C42—48no—
494C42—48no—
495C42—44no—
496C42—44no—
497C42—48no—
498C42—84no—
499C42—84no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23C2×#12314yes—
517C23#53314yes—
518C23#54214yes—
519C23D4×Q814yes—
520C23#55014yes—
521C23#55214yes—
522C23#55414yes—
523C23C23×D411yescharacteristic
524C23C23×Q811yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C22×#4111yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+412yes—
532C23C2×2−1+411yescharacteristic
533C23#58512yes—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—232no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
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—432no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—416no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—816no—
625C2×C4C2×#12314yes—
626C2×C4C2×#12314yes—
627C2×C4#52514yes—
628C2×C4#53614yes—
629C2×C4#55218yes—
630C2×C4#55218yes—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22#242318yes—
642C22C22×#12311yescharacteristic
643C22C2×#52511yescharacteristic
644C22C2×#53311yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#54211yescharacteristic
647C22C2×D4×Q811yescharacteristic
648C22C2×#55011yescharacteristic
649C22C2×#55211yescharacteristic
650C22C2×#55212yes—
651C22C2×#55212yes—
652C22C2×#55411yescharacteristic
653C22Q8×#4112yes—
654C22#348112yes—
655C22#348411yescharacteristic
656C22#348511yescharacteristic
657C22#349412yes—
658C22#349412yes—
659C22#350312yes—
660C22#351512yes—
661C22#351612yes—
662C22#351912yes—
663C22#355212yes—
664C22#355912yes—
665C4—24no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—48no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—81no—
676C4—82no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4#248418yes—
681C2—28no—
682C2C2×#242311yescharacteristic
683C2C2×#248411yescharacteristic
684C2#3554011yescharacteristic
685C2#3628012yes—
686C2#3652911yescharacteristic
687C2#3654711yescharacteristic
688C2#3806812yes—
689C2#3913411yescharacteristic
690C2#3959211yescharacteristic
691C2#5925412yes—
692C2#5929112yes—
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
3. #16075C2semidirect
4. #15414C2semidirect
5. #15187C2semidirect
6. #15137C2semidirect
10. #14132C2semidirect
13. #14029C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C22×#3892
#160752
#154142
#151872
#151372
#149362
#148212
#146852
#141322
#141302
#140392
#140292
#139922
#139242
#138802
C4×#3812