Finite Group Databasebeta

← #522233

Subgroups — #522233

Order 512.

2862 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3C22×#382C211yesmaximal, characteristic
4#14400C214yesmaximal, complemented
5#14394C214yesmaximal, complemented
6#14265C212yesmaximal
7#14139C211yesmaximal, characteristic
8#14130C211yesmaximal, characteristic
9#14006C212yesmaximal, complemented
10#14003C212yesmaximal, complemented
11#13992C212yesmaximal
12#13982C212yesmaximal, complemented
13#13978C212yesmaximal, complemented
14#13972C212yesmaximal
15C22⋊C4×C4⋊C4C211yesmaximal, characteristic
16#13782C212yesmaximal
17C4×#381C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855C4116yescomplemented
24#2360—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#382—28no—
38C2×#382C2214yes—
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C22×#95C2212yes—
52C22×#95C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73#1434C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76#1434C2212yes—
77#1434C2214yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2×C418yes—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42—24no—
120C22×C42—24no—
121C22×C42C2×C418yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130#391—24no—
131#391—44no—
132#391C2×C4116yescomplemented
133#389—24no—
134#389—28no—
135#389—44no—
136#389C2×C4116yescomplemented
137#387—28no—
138#387C2×C4132yescomplemented
139#385—28no—
140#385—28no—
141#385C2×C4116yes—
142#382—28no—
143#382—28no—
144#382—216no—
145#381—24no—
146#381—24no—
147#381C2×C4116yes—
148#380C2×C418yes—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—28no—
155C2×#95—28no—
156C2×#95—28no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8C22×C414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D418yes—
222C23×C4#4114yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C22⋊Q8—232no—
240C4×Q8—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
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—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C22×C418yes—
311C2×C42C22×C418yes—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—24no—
318#95—22no—
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—44no—
330#95—42no—
331#95—44no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C22×C418yes—
342#95C22×C4116yes—
343#95C22×C4116yes—
344#95C22×C4116yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C23×C412yes—
350C24C22×D412yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24C2×#4112yes—
354C242−1+412yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—216no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—48no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—216no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
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—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C4×D4116yes—
463C22×C4C23×C414yes—
464C22×C4C23×C418yes—
465C22×C4C23×C414yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+414yes—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—216no—
500C42—216no—
501C42—48no—
502C42—44no—
503C42—44no—
504C42—48no—
505C42—84no—
506C42—88no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23C2×C4×D414yes—
524C23C4×#4114yes—
525C23#51914yes—
526C23#54714yes—
527C23#54814yes—
528C23#55214yes—
529C23#55614yes—
530C23C24×C411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+412yes—
539C23#58512yes—
540C23#58512yes—
541Q8—416no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—232no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4C2×C4×D414yes—
634C2×C4C2×C4×D414yes—
635C2×C4#52018yes—
636C2×C4#53414yes—
637C2×C4#53614yes—
638C2×C4#53918yes—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22#236218yes—
650C22C22×C4×D411yescharacteristic
651C22C2×C4×#4111yescharacteristic
652C22C2×#51911yescharacteristic
653C22C2×#52012yes—
654C22#342812yes—
655C22C2×#53411yescharacteristic
656C22C2×#53611yescharacteristic
657C22C2×#53912yes—
658C22C2×#54711yescharacteristic
659C22C2×#54811yescharacteristic
660C22C2×#55211yescharacteristic
661C22C2×#55611yescharacteristic
662C22#348511yescharacteristic
663C22#348711yescharacteristic
664C22#349312yes—
665C22#351412yes—
666C22#351412yes—
667C22#351512yes—
668C22#351512yes—
669C22#356112yes—
670C22#356112yes—
671C22#356312yes—
672C22#356312yes—
673C4—24no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—48no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—84no—
687C4—82no—
688C4#232218yes—
689C2—28no—
690C2C2×#232211yescharacteristic
691C2C2×#236211yescharacteristic
692C2#3420111yescharacteristic
693C2#3422912yes—
694C2#3439912yes—
695C2#3891811yescharacteristic
696C2#3891911yescharacteristic
697C2#3892912yes—
698C2#3912211yescharacteristic
699C2#3912311yescharacteristic
700C2#3915512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14400C2semidirect
5. #14394C2semidirect
9. #14006C2semidirect
10. #14003C2semidirect
12. #13982C2semidirect
13. #13978C2semidirect
23. #2855C4semidirect
132. #391C2×C4semidirect
136. #389C2×C4semidirect
138. #387C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#23602
C22×#3822
#144002
#143942
#142652
#141392
#141302
#140062
#140032
#139922
#139822
#139782
#139722
C22⋊C4×C4⋊C42
#137822
C4×#3812