Finite Group Databasebeta

← #616130

Subgroups — #616130

Order 512.

2993 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2852C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#15213C212yesmaximal, complemented
4#15175C212yesmaximal, complemented
5#15092C211yesmaximal, characteristic, complemented
6#15091C211yesmaximal, characteristic
7#15080C212yesmaximal, complemented
8#15078C212yesmaximal, complemented
9#14843C211yesmaximal, characteristic, complemented
10#14777C211yesmaximal, characteristic
11#14696C212yesmaximal
12#14695C212yesmaximal, complemented
13#14250C212yesmaximal, complemented
14#14231C212yesmaximal, complemented
15#14166C212yesmaximal, complemented
16#14044C212yesmaximal, complemented
17#14020C212yesmaximal
18#13952C212yesmaximal, complemented
19#13753C211yesmaximal, characteristic, complemented
20C22×#126C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C23×C42C2212yes—
25#2852—28no—
26C2×#400C2212yes—
27C2×#399C2212yes—
28C2×#388C2211yescharacteristic
29C2×#388C2211yescharacteristic
30C2×#388C2211yescharacteristic
31C2×#388C2214yes—
32C2×#387C2211yescharacteristic
33C2×#387C2211yescharacteristic
34C2×#387C2211yescharacteristic
35C2×#387C2214yes—
36C2×#386C2211yescharacteristic
37C2×#386C2211yescharacteristic
38C2×#386C2211yescharacteristic
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2211yescharacteristic
42C2×#385C2211yescharacteristic
43C2×#385C2211yescharacteristic
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#383C2211yescharacteristic
47C2×C4×C4⋊C4—28no—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2212yes—
54C22×#95C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1440C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2212yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2214yes—
84#1434C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C22⋊#95C2212yes—
88C22⋊#95C2212yes—
89C22⋊#95C2212yes—
90C22⋊#95C2211yescharacteristic
91C22⋊#95C2211yescharacteristic
92C4×#95C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2312yes—
99C2×#126—24no—
100C2×#126—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C42—28no—
131C22×C42—28no—
132C22×C42—48no—
133C22×C42D418yes—
134C22×C42C2314yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141#400—28no—
142#399—28no—
143#388—24no—
144#388—24no—
145#388—24no—
146#388—48no—
147#387—24no—
148#387—24no—
149#387—24no—
150#387—48no—
151#386—24no—
152#386—24no—
153#386—24no—
154#386—28no—
155#386—28no—
156#385—24no—
157#385—24no—
158#385—24no—
159#385—28no—
160#385—28no—
161#383—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—28no—
167#376—28no—
168C2×#95—28no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C25C2411yescharacteristic
203C25C2411yescharacteristic
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
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—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4118yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#126—416no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4#4118yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4#4118yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—232no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—432no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42#4118yes—
329#95—28no—
330#95—28no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24C22×D412yes—
376C24C2×#4114yes—
377C24C2×#4114yes—
378C242−1+412yes—
379C242−1+412yes—
380C24C2511yescharacteristic
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C22×C4—232no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4118yes—
487C22×C4C2×#4118yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)14yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—416no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—88no—
506C22⋊C4—216no—
507C22⋊C4—416no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—88no—
519C42—28no—
520C42—28no—
521C42—28no—
522C42—416no—
523C42—416no—
524C42—416no—
525C42#123116yes—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23#52514yes—
556C23#53214yes—
557C23#53614yes—
558C23#55218yes—
559C23#55618yes—
560C23C23×D411yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C22×#4112yes—
564C23C22×#4112yes—
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23#58512yes—
569D4—88no—
570C2×C4—232no—
571C2×C4—232no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—432no—
603C2×C4—432no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4C2×#12314yes—
656C2×C4C2×#12314yes—
657C2×C4C2×#12314yes—
658C2×C4#52514yes—
659C2×C4#53514yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—216no—
666C22—216no—
667C22—24no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—84no—
675C22—84no—
676C22—81no—
677C22#260418yes—
678C22C22×#12311yescharacteristic
679C22C2×#52511yescharacteristic
680C22C2×#52511yescharacteristic
681C22C2×#53211yescharacteristic
682C22#344414yes—
683C22C2×#53511yescharacteristic
684C22C2×#53611yescharacteristic
685C22#344911yescharacteristic
686C22C2×#55212yes—
687C22C2×#55612yes—
688C22#347514yes—
689C22#348114yes—
690C22#349212yes—
691C22#349212yes—
692C22#350514yes—
693C22#351612yes—
694C22#351612yes—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—24no—
699C4—24no—
700C4—48no—
701C4—48no—
702C4—48no—
703C4—42no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—84no—
711C4—81no—
712C2—28no—
713C2—161no—
714C2#3481011yescharacteristic
715C2C2×#260411yescharacteristic
716C2#3496511yescharacteristic
717C2#3510211yescharacteristic
718C2#3522711yescharacteristic
719C2#3524511yescharacteristic
720C2#3633614yes—
721C2#3633714yes—
722C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
3. #15213C2semidirect
4. #15175C2semidirect
5. #15092C2semidirect
7. #15080C2semidirect
8. #15078C2semidirect
9. #14843C2semidirect
12. #14695C2semidirect
13. #14250C2semidirect
14. #14231C2semidirect
15. #14166C2semidirect
16. #14044C2semidirect
18. #13952C2semidirect
19. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28522
C22×C4×C4⋊C42
#152132
#151752
#150922
#150912
#150802
#150782
#148432
#147772
#146962
#146952
#142502
#142312
#141662
#140442
#140202
#139522
#137532