Finite Group Databasebeta

← #947127

Subgroups — #947127

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#17195C214yesmaximal, complemented
4#16452C214yesmaximal, complemented
5#16366C214yesmaximal, complemented
6#15796C214yesmaximal, complemented
7#15239C211yesmaximal, characteristic, complemented
8#15234C211yesmaximal, characteristic, complemented
9#15230C211yesmaximal, characteristic, complemented
10#15224C211yesmaximal, characteristic
11#15224C211yesmaximal, characteristic
12#15153C211yesmaximal, characteristic, complemented
13#14282C211yesmaximal, characteristic, complemented
14#14260C211yesmaximal, characteristic
15#14051C211yesmaximal, characteristic, complemented
16#14002C211yesmaximal, characteristic
17#13979C211yesmaximal, characteristic
18#13906C211yesmaximal, characteristic
19#13846C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2570—28no—
24#2452—28no—
25C2×#400C2212yes—
26C2×#400C2214yes—
27C2×#398C2212yes—
28C2×#396C2212yes—
29C2×#393C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#385C2212yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1446C2214yes—
55#1445C2212yes—
56#1445C2214yes—
57#1445C2214yes—
58#1443C2212yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1437C2214yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87#1434C2214yes—
88C22⋊#95C2214yes—
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#400—28no—
139#400—48no—
140#398—28no—
141#396—28no—
142#393—28no—
143#390—24no—
144#390—24no—
145#390—24no—
146#390—24no—
147#390—24no—
148#390—42no—
149#389—24no—
150#389—24no—
151#389—24no—
152#389—24no—
153#387—28no—
154#387—28no—
155#385—28no—
156#384—24no—
157#382—24no—
158#382—24no—
159#382—24no—
160#382—42no—
161#380—48no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—42no—
164C4×C4⋊C4D418yes—
165#376—42no—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C25C2412yes—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—42no—
205C22×Q8—44no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4#4114yes—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—416no—
237C22⋊Q8—432no—
238C4×D4—416no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4#4114yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24C2×#4114yes—
373C24C2511yescharacteristic
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—84no—
377C2×Q8—88no—
378C2×Q8—88no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C22×C42−1+412yes—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—28no—
518C42—28no—
519C42—28no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—84no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C23×D411yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C2×H(5,𝔽2)12yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23C2×2−1+411yescharacteristic
556C23#58512yes—
557C23#58512yes—
558C23#58512yes—
559Q8—816no—
560D4—416no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
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—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—816no—
654C2×C4#53514yes—
655C2×C4#53614yes—
656C2×C4#53914yes—
657C2×C4#54114yes—
658C2×C4#54614yes—
659C2×C4#54814yes—
660C2×C4#55214yes—
661C2×C4#56314yes—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—44no—
666C22—48no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22C2×#53511yescharacteristic
674C22C2×#53611yescharacteristic
675C22#344911yescharacteristic
676C22C2×#53911yescharacteristic
677C22C2×#54111yescharacteristic
678C22C2×#54611yescharacteristic
679C22C2×#54811yescharacteristic
680C22C2×#55211yescharacteristic
681C22#350011yescharacteristic
682C22#350011yescharacteristic
683C22#351312yes—
684C22#351412yes—
685C22#352112yes—
686C22#352412yes—
687C22C2×#56311yescharacteristic
688C22#354212yes—
689C22#354712yes—
690C22#354712yes—
691C22#355112yes—
692C22#355412yes—
693C22#355712yes—
694C22#356512yes—
695C22#356512yes—
696C4—24no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#269018yes—
714C2—84no—
715C2#3742311yescharacteristic
716C2#3763411yescharacteristic
717C2#3808411yescharacteristic
718C2#3894811yescharacteristic
719C2#3916311yescharacteristic
720C2#3917111yescharacteristic
721C2#4256912yes—
722C2#4271812yes—
723C2#4505412yes—
724C2C2×#269011yescharacteristic
725C2#5311112yes—
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
1. C2×#2570C2semidirect
3. #17195C2semidirect
4. #16452C2semidirect
5. #16366C2semidirect
6. #15796C2semidirect
7. #15239C2semidirect
8. #15234C2semidirect
9. #15230C2semidirect
12. #15153C2semidirect
13. #14282C2semidirect
15. #14051C2semidirect
19. #13846C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25702
C2×#24522
#171952
#164522
#163662
#157962
#152392
#152342
#152302
#152242
#151532
#142822
#142602
#140512
#140022
#139792
#139062
#138462