Finite Group Databasebeta

← #515976

Subgroups — #515976

Order 512.

4234 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14178C214yesmaximal, complemented
5#14177C214yesmaximal, complemented
6#14128C211yesmaximal, characteristic, complemented
7#14118C214yesmaximal, complemented
8#14039C212yesmaximal, complemented
9#14032C214yesmaximal, complemented
10#13991C212yesmaximal
11#13958C211yesmaximal, characteristic, complemented
12#13868C211yesmaximal, characteristic
13#13782C212yesmaximal, complemented
14#13776C211yesmaximal, characteristic, complemented
15C4×#379C212yesmaximal
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2482C4116yescomplemented
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yescomplemented
29C2×#385—28no—
30C2×#385C2214yes—
31C2×#385C2212yes—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#381C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×#379C2212yes—
37C2×#379C2212yes—
38#2274—28no—
39C2×C4×C4⋊C4C2212yescomplemented
40C2×C4×C4⋊C4C2212yescomplemented
41C2×C4×C4⋊C4C2212yescomplemented
42C2×C4×C4⋊C4C2212yescomplemented
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2214yes—
47C2×C4×C22⋊C4C2214yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2212yescomplemented
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2212yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2214yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—28no—
94C2×#117C2×C418yescomplemented
95C22×C4⋊C4—28no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2×C418yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131#399C2×C4116yescomplemented
132#397C2×C4116yescomplemented
133#388—48no—
134#388C2×C4132yescomplemented
135#387—48no—
136#385—216no—
137#385—216no—
138#385—28no—
139#385C2×C4116yescomplemented
140#381—24no—
141#381C2×C418yescomplemented
142#379—28no—
143#379C2×C4116yes—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—48no—
152C4×C22⋊C4—48no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25—22no—
180C25—22no—
181C25—24no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—42no—
186C25—42no—
187C25C22×C414yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—416no—
217C23×C4—464no—
218C23×C4—416no—
219C23×C4—416no—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4#4114yes—
226C23×C4#4118yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240#117—232no—
241#117—232no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
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—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C22×C414yescomplemented
265C2×C4⋊C4C22×C414yescomplemented
266C2×C4⋊C4C22×C4116yescomplemented
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—232no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4C22×C418yes—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C4116yes—
302C2×C22⋊C4C22×C4116yes—
303C2×C22⋊C4#4118yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C418yescomplemented
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
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#95C22×C418yescomplemented
343#95C22×C418yescomplemented
344#95C22×C4116yescomplemented
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—28no—
350C24—28no—
351C24—216no—
352C24—28no—
353C24—216no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—216no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—416no—
368C24—44no—
369C24—44no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24C23×C414yes—
375C24C23×C412yes—
376C24C22×D412yes—
377C24C2×#4112yes—
378C24C2×#4114yes—
379C24C2×#4114yes—
380C24H(5,𝔽2)14yes—
381C242−1+414yes—
382C24C2511yescharacteristic
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—232no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—4128no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C4×D4116yes—
468C22×C4#123116yes—
469C22×C4C23×C418yes—
470C22×C4C23×C414yes—
471C22×C4C23×C414yes—
472C22×C4C23×C414yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—88no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—232no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C42—216no—
507C42—216no—
508C42—84no—
509C42—84no—
510C42—88no—
511C42—84no—
512C42—84no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—232no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—464no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—464no—
543C23—416no—
544C23—416no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—416no—
556C23C2×C4×D414yes—
557C23C4×#4118yes—
558C23#52018yes—
559C23C2×#12314yes—
560C23#52514yes—
561C23#52918yes—
562C23C24×C411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×2−1+412yes—
570C23#58512yes—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—232no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—416no—
603C2×C4—464no—
604C2×C4—416no—
605C2×C4—416no—
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—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4C2×C4×D414yes—
645C2×C4C2×C4×D414yes—
646C2×C4C2×#12318yes—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—216no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—216no—
655C22—28no—
656C22—216no—
657C22—464no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22C4×#12318yes—
668C22C22×C4×D411yescharacteristic
669C22C2×C4×#4112yes—
670C22C2×#52012yes—
671C22C4×H(5,𝔽2)12yes—
672C22C22×#12311yescharacteristic
673C22C2×#52511yescharacteristic
674C22C2×#52912yes—
675C22#348411yescharacteristic
676C22#348411yescharacteristic
677C22#348611yescharacteristic
678C22#348711yescharacteristic
679C22#349014yes—
680C22#349212yes—
681C22#349514yes—
682C22#350612yes—
683C22#351314yes—
684C22#352014yes—
685C4—28no—
686C4—24no—
687C4—24no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—48no—
696C4—82no—
697C4—84no—
698C2—28no—
699C2—48no—
700C2C2×C4×#12311yescharacteristic
701C2#3420011yescharacteristic
702C2#3426511yescharacteristic
703C2#3429614yes—
704C2#3535011yescharacteristic
705C2#3535611yescharacteristic
706C2#3535711yescharacteristic
707C2#3537214yes—
708C2#3567211yescharacteristic
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
3. C2×#2274C2semidirect
4. #14178C2semidirect
5. #14177C2semidirect
6. #14128C2semidirect
7. #14118C2semidirect
8. #14039C2semidirect
9. #14032C2semidirect
11. #13958C2semidirect
13. #13782C2semidirect
14. #13776C2semidirect
23. #2482C4semidirect
28. C2×#387C22semidirect
39. C2×C4×C4⋊C4C22semidirect
40. C2×C4×C4⋊C4C22semidirect
41. C2×C4×C4⋊C4C22semidirect
42. C2×C4×C4⋊C4C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
94. C2×#117C2×C4semidirect
131. #399C2×C4semidirect
132. #397C2×C4semidirect
134. #388C2×C4semidirect
139. #385C2×C4semidirect
141. #381C2×C4semidirect
264. C2×C4⋊C4C22×C4semidirect
265. C2×C4⋊C4C22×C4semidirect
266. C2×C4⋊C4C22×C4semidirect
320. C2×C42C22×C4semidirect
342. #95C22×C4semidirect
343. #95C22×C4semidirect
344. #95C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C22×#3852
C2×#22742
#141782
#141772
#141282
#141182
#140392
#140322
#139912
#139582
#138682
#137822
#137762
C4×#3792