Finite Group Databasebeta

← #516366

Subgroups — #516366

Order 512.

3754 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2C22×#388C211yesmaximal, characteristic
3C2×#2273C211yesmaximal, characteristic
4#13966C211yesmaximal, characteristic, complemented
5#13965C211yesmaximal, characteristic, complemented
6#13948C214yesmaximal, complemented
7#13915C214yesmaximal, complemented
8#13891C214yesmaximal, complemented
9#13881C211yesmaximal, characteristic, complemented
10#13880C211yesmaximal, characteristic, complemented
11#13877C212yesmaximal
12#13877C212yesmaximal
13#13876C214yesmaximal, complemented
14#13782C212yesmaximal, complemented
15C4×#385C212yesmaximal, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2312—28no—
23C2×#388—28no—
24C2×#388C2212yes—
25C2×#388C2214yes—
26C2×#388C2212yes—
27C2×#388C2214yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#384C2212yescomplemented
35C2×#382C2212yescomplemented
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2214yes—
39C2×#380C2214yes—
40C2×#379C2212yes—
41#2273C4116yescomplemented
42C2×C4×C4⋊C4C2214yescomplemented
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2214yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1444C2214yescomplemented
52#1444C2212yescomplemented
53#1444C2212yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1442C2212yescomplemented
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C4⋊#95C2214yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2214yescomplemented
78C26C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2×C4116yes—
105C22×C22⋊C4C2×C418yes—
106C22×C22⋊C4D418yes—
107C22×C22⋊C4Q818yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42—24no—
122C22×C42—44no—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#388—28no—
130#388—28no—
131#388—28no—
132#388—28no—
133#388—216no—
134#388—216no—
135#385—28no—
136#385—28no—
137#385—28no—
138#385—28no—
139#384—44no—
140#382—44no—
141#380—24no—
142#380—24no—
143#380—48no—
144#380—48no—
145#379—28no—
146C4×C4⋊C4—48no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4C2×C418yes—
149C4×C22⋊C4C2×C4116yes—
150C4×C22⋊C4C2×C418yes—
151C4×C22⋊C4C2×C4132yescomplemented
152C4×C22⋊C4C2×C4132yescomplemented
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25—24no—
178C25—24no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25C22×C414yes—
184C25C2×D414yes—
185C25C2×Q814yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C22×C4116yes—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q814yes—
220C23×C4C2×Q814yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
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—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—264no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—416no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C4⋊C4164yescomplemented
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C2×D418yes—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×Q818yes—
297C2×C22⋊C4C2×Q818yes—
298C2×C22⋊C4#4118yes—
299C2×C22⋊C4#4118yes—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—416no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C22×C418yes—
315C2×C42C22×C4116yes—
316C2×C42C22×C418yes—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C22×C418yes—
337#95C22×C4116yes—
338#95C22×C418yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—216no—
352C24—28no—
353C24—28no—
354C24—216no—
355C24—44no—
356C24—44no—
357C24—48no—
358C24—48no—
359C24—44no—
360C24—44no—
361C24—48no—
362C24—48no—
363C24—88no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24C2×C4⋊C4116yes—
368C24C23×C414yes—
369C24C23×C412yes—
370C24C22×D414yes—
371C24C22×D412yes—
372C24C22×Q814yes—
373C24C22×Q812yes—
374C24C2×#4114yes—
375C24C2×#4114yes—
376C24C2511yescharacteristic
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—232no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C2×C4⋊C4116yes—
463C22×C4C2×C4⋊C4116yes—
464C22×C4C23×C418yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C414yes—
468C22×C4C23×C418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q814yes—
475C22×C4C2×#4118yes—
476C22×C4C2×#4118yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)14yes—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—232no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4C2×C4⋊C4132yes—
500C22⋊C4C2×C4⋊C4132yes—
501C22⋊C4#117132yes—
502C42—48no—
503C42—416no—
504C42—48no—
505C42—416no—
506C42—84no—
507C42—84no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—232no—
535C23—416no—
536C23—416no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—88no—
546C23—88no—
547C23—832no—
548C23—88no—
549C23—88no—
550C23—832no—
551C23C22×C4⋊C418yes—
552C23C22×C4⋊C414yes—
553C23C2×#11718yes—
554C23#53214yes—
555C23#53214yes—
556C23#53318yes—
557C23C24×C411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×Q811yescharacteristic
560C23C22×#4112yes—
561C23C22×#4112yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)12yes—
564C23#58512yes—
565C23#58512yes—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—832no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4C22×C4⋊C418yes—
639C2×C4C22×C4⋊C414yes—
640C2×C4C22×C4⋊C414yes—
641C2×C4C2×#117116yes—
642C22—24no—
643C22—24no—
644C22—216no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—216no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—44no—
654C22—44no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—44no—
659C22—48no—
660C22—48no—
661C22—832no—
662C22#228518yes—
663C22C23×C4⋊C411yescharacteristic
664C22C22×#11712yes—
665C22C4×H(5,𝔽2)12yes—
666C22C4×H(5,𝔽2)12yes—
667C22C2×#53211yescharacteristic
668C22C2×#53211yescharacteristic
669C22C2×#53312yes—
670C22#348411yescharacteristic
671C22#348411yescharacteristic
672C22#348511yescharacteristic
673C22#348511yescharacteristic
674C22#350112yes—
675C22#350112yes—
676C22#351314yes—
677C22#351414yes—
678C22#351914yes—
679C22#352114yes—
680C4—28no—
681C4—24no—
682C4—24no—
683C4—216no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C2—28no—
694C2—48no—
695C2C2×#228511yescharacteristic
696C2#3424811yescharacteristic
697C2#3424811yescharacteristic
698C2#3426114yes—
699C2#3539411yescharacteristic
700C2#3539511yescharacteristic
701C2#3540214yes—
702C2#3546611yescharacteristic
703C2#3546711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13966C2semidirect
5. #13965C2semidirect
6. #13948C2semidirect
7. #13915C2semidirect
8. #13891C2semidirect
9. #13881C2semidirect
10. #13880C2semidirect
13. #13876C2semidirect
14. #13782C2semidirect
15. C4×#385C2semidirect
34. C2×#384C22semidirect
35. C2×#382C22semidirect
41. #2273C4semidirect
42. C2×C4×C4⋊C4C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
77. C4×#95C22semidirect
151. C4×C22⋊C4C2×C4semidirect
152. C4×C22⋊C4C2×C4semidirect
288. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
C22×#3882
C2×#22732
#139662
#139652
#139482
#139152
#138912
#138812
#138802
#138772
#138762
#137822
C4×#3852