Finite Group Databasebeta

← #664415

Subgroups — #664415

Order 512.

2790 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3#15809C214yesmaximal, complemented
4#15699C214yesmaximal, complemented
5#15187C212yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#14934C212yesmaximal
8#14843C212yesmaximal
9#14685C211yesmaximal, characteristic
10#14139C212yesmaximal
11#14053C212yesmaximal, complemented
12#13972C212yesmaximal
13#13933C212yesmaximal, complemented
14#13924C211yesmaximal, characteristic
15#13924C211yesmaximal, characteristic
16C22⋊C4×C4⋊C4C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#2440—28no—
25C2×#400C2214yes—
26C2×#398C2212yes—
27C2×#395C2212yes—
28C2×#391C2214yes—
29C2×#389—28no—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2212yes—
47#1446C2214yes—
48#1445C2214yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—48no—
102C22×C4⋊C4D418yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42—24no—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133#400—48no—
134#398—28no—
135#395—28no—
136#391—48no—
137#389—24no—
138#389—24no—
139#389—28no—
140#389—216no—
141#389—28no—
142#385—28no—
143#384—24no—
144#381—24no—
145#381—44no—
146#381Q818yescomplemented
147#380—24no—
148#380—24no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—44no—
153C2×#95—28no—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—42no—
191C22×Q8—44no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4C2×Q814yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236C22⋊Q8—432no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—432no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C2×Q814yes—
294C2×C22⋊C4C2×Q814yes—
295C2×C22⋊C4#4114yes—
296C2×C42—22no—
297C2×C42—22no—
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—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—416no—
311C2×C42—44no—
312C2×C42—44no—
313#95—28no—
314#95—28no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—42no—
329#95—42no—
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#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×Q818yes—
344#95C2×Q818yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22×D412yes—
350C24C22×Q812yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24C2×#4112yes—
354C24H(5,𝔽2)12yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—48no—
360C2×Q8—84no—
361C2×Q8—88no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4#124116yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q814yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4118yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+414yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—84no—
495C4⋊C4#123116yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C42—44no—
503C42—48no—
504C42—48no—
505C42—44no—
506C42—48no—
507C42—88no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23C2×#12414yes—
525C23#53314yes—
526C23#54214yes—
527C23#54614yes—
528C23D4×Q814yes—
529C23#55414yes—
530C23#55614yes—
531C23C23×D411yescharacteristic
532C23C23×Q811yescharacteristic
533C23C22×#4112yes—
534C23C22×#4111yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542Q8—816no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—232no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—432no—
573C2×C4—416no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4C2×#12318yes—
634C2×C4C2×#12314yes—
635C2×C4C2×#12414yes—
636C2×C4C2×#12414yes—
637C2×C4#53614yes—
638C2×C4#55218yes—
639C2×C4#55218yes—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22#249218yes—
651C22C22×#12311yescharacteristic
652C22C22×#12411yescharacteristic
653C22C2×#53311yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#54211yescharacteristic
656C22C2×#54611yescharacteristic
657C22C2×D4×Q811yescharacteristic
658C22C2×#55212yes—
659C22C2×#55212yes—
660C22C2×#55411yescharacteristic
661C22C2×#55611yescharacteristic
662C22Q8×#4112yes—
663C22#348112yes—
664C22#348411yescharacteristic
665C22#348711yescharacteristic
666C22#349412yes—
667C22#349412yes—
668C22#350312yes—
669C22#351512yes—
670C22#351612yes—
671C22#351912yes—
672C22#355212yes—
673C22#355912yes—
674C4—28no—
675C4—24no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—84no—
685C4—82no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4#250318yes—
690C2—28no—
691C2C2×#249211yescharacteristic
692C2C2×#250311yescharacteristic
693C2#3558411yescharacteristic
694C2#3623611yescharacteristic
695C2#3628012yes—
696C2Q8×#12311yescharacteristic
697C2#3806812yes—
698C2#3890711yescharacteristic
699C2#3959211yescharacteristic
700C2#5924012yes—
701C2#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. #15809C2semidirect
4. #15699C2semidirect
5. #15187C2semidirect
6. #15146C2semidirect
11. #14053C2semidirect
13. #13933C2semidirect
146. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C22×#3892
#158092
#156992
#151872
#151462
#149342
#148432
#146852
#141392
#140532
#139722
#139332
#139242
C22⋊C4×C4⋊C42
C4×#3812