Finite Group Databasebeta

← #792625

Subgroups — #792625

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 774 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2345C211yesmaximal, characteristic, complemented
3#17026C212yesmaximal, complemented
4#15664C214yesmaximal, complemented
5#15476C214yesmaximal, complemented
6#15181C211yesmaximal, characteristic, complemented
7#15143C211yesmaximal, characteristic, complemented
8#15125C211yesmaximal, characteristic, complemented
9#15021C212yesmaximal, complemented
10#15017C212yesmaximal, complemented
11#14228C212yesmaximal
12#13993C212yesmaximal, complemented
13#13906C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15#13872C212yesmaximal, complemented
16C4×#392C211yesmaximal, characteristic
17#13812C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2563—28no—
23C2×#395C2214yes—
24C2×#393C2214yes—
25C2×#393C2212yes—
26C2×#392C2211yescharacteristic
27#2345—28no—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2214yes—
36C2×#384C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1446C2212yes—
52#1444C2211yescharacteristic
53#1444C2212yes—
54#1444C2212yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2211yescharacteristic
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2211yescharacteristic
71#1438C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—44no—
97C2×C4×D4—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134#395—48no—
135#393—216no—
136#393—44no—
137#392Q818yescomplemented
138#390—28no—
139#390—48no—
140#389—24no—
141#389—24no—
142#388—48no—
143#387—28no—
144#387—28no—
145#386—48no—
146#384—44no—
147#382—28no—
148#380—28no—
149#380—28no—
150#380—24no—
151#380—28no—
152#380—28no—
153#380—44no—
154#380—44no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4D418yescomplemented
158C4×C22⋊C4—24no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—28no—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—24no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—44no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4C2×Q818yes—
205C22×D4C2×Q814yes—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C4⋊Q8—416no—
239C4×D4—432no—
240C4×D4—416no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4114yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×Q814yes—
316C2×C22⋊C4C2×Q818yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42C2×D418yes—
330C2×C42#4118yes—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95C2×D418yes—
356#95C2×Q814yes—
357C24—21no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×Q814yes—
380C24C22×Q812yes—
381C24C2×#4112yes—
382C24H(5,𝔽2)12yes—
383C24C2511yescharacteristic
384C2×Q8—48no—
385C2×Q8—84no—
386C2×Q8—84no—
387C2×Q8—88no—
388C2×Q8—88no—
389C2×D4—216no—
390C2×D4—28no—
391C2×D4—216no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—48no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×Q812yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q814yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4114yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)14yes—
502C22×C42−1+412yes—
503C4⋊C4—216no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—88no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C22⋊C4—28no—
518C22⋊C4—216no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C42—28no—
531C42—216no—
532C42—48no—
533C42—44no—
534C42—88no—
535C42—84no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23#55414yes—
594C23C23×D411yescharacteristic
595C23C23×Q811yescharacteristic
596C23C22×#4111yescharacteristic
597C23C22×#4111yescharacteristic
598C23C22×#4112yes—
599C23C2×H(5,𝔽2)12yes—
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×H(5,𝔽2)12yes—
602C23C2×2−1+411yescharacteristic
603C23#58511yescharacteristic
604C23#58512yes—
605Q8—816no—
606D4—432no—
607D4—416no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—416no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—816no—
688C2×C4—816no—
689C2×C4#52514yes—
690C2×C4#53214yes—
691C2×C4#53514yes—
692C2×C4D4×Q814yes—
693C2×C4#55118yes—
694C2×C4#55418yes—
695C2×C4#55518yes—
696C22—28no—
697C22—24no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22C2×#52511yescharacteristic
723C22C2×#53211yescharacteristic
724C22C2×#53511yescharacteristic
725C22C2×D4×Q811yescharacteristic
726C22C2×#55112yes—
727C22C2×#55411yescharacteristic
728C22C2×#55412yes—
729C22C2×#55512yes—
730C22Q8×#4111yescharacteristic
731C22#348511yescharacteristic
732C22#348811yescharacteristic
733C22#349312yes—
734C22#349412yes—
735C22#349412yes—
736C22#349811yescharacteristic
737C22#350112yes—
738C22#350711yescharacteristic
739C22#351112yes—
740C22#351412yes—
741C22#352211yescharacteristic
742C22#353712yes—
743C22#354112yes—
744C22#354612yes—
745C4—28no—
746C4—24no—
747C4—42no—
748C4—44no—
749C4—44no—
750C4—44no—
751C4—42no—
752C4—44no—
753C4—42no—
754C4—82no—
755C4—84no—
756C4—81no—
757C4—82no—
758C4—84no—
759C4#261018yes—
760C2—44no—
761C2—84no—
762C2C2×#261011yescharacteristic
763C2#3602211yescharacteristic
764C2#3606211yescharacteristic
765C2#3654411yescharacteristic
766C2#3655911yescharacteristic
767C2#3959411yescharacteristic
768C2#4298611yescharacteristic
769C2#4512612yes—
770C2#4520812yes—
771C2#5919912yes—
772C2#5923012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2563C2semidirect
2. C2×#2345C2semidirect
3. #17026C2semidirect
4. #15664C2semidirect
5. #15476C2semidirect
6. #15181C2semidirect
7. #15143C2semidirect
8. #15125C2semidirect
9. #15021C2semidirect
10. #15017C2semidirect
12. #13993C2semidirect
13. #13906C2semidirect
14. #13876C2semidirect
15. #13872C2semidirect
17. #13812C2semidirect
137. #392Q8semidirect
157. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23452
#170262
#156642
#154762
#151812
#151432
#151252
#150212
#150172
#142282
#139932
#139062
#138762
#138722
C4×#3922
#138122