Finite Group Databasebeta

← #769920

Subgroups — #769920

Order 512.

3982 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C212yesmaximal, complemented
2C2×#2409C212yesmaximal, complemented
3C22×#395C211yesmaximal, characteristic, complemented
4C2×#2325C211yesmaximal, characteristic
5#15856C214yesmaximal, complemented
6#15825C214yesmaximal, complemented
7#15430C214yesmaximal, complemented
8#15325C214yesmaximal, complemented
9#14946C212yesmaximal, complemented
10#14777C212yesmaximal, complemented
11#14152C211yesmaximal, characteristic
12#14137C211yesmaximal, characteristic
13#13965C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13778C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2579—216no—
24#2409—216no—
25C2×#399C2214yes—
26C2×#398C2214yes—
27C2×#396C2214yes—
28C2×#395—28no—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#393C2214yes—
33#2325—28no—
34C2×#391C2214yes—
35C2×#390C2212yes—
36C2×#388C2212yes—
37C2×#387C2214yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#385C2214yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#384C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2211yescharacteristic
58C22×#95C2211yescharacteristic
59#1445C2214yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2214yes—
63#1444C2214yes—
64#1444C2212yes—
65#1444C2214yes—
66#1444C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—28no—
81C23×Q8—44no—
82C23×Q8—44no—
83C23×Q8C2314yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8—48no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#121Q818yescomplemented
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#399—48no—
137#398—216no—
138#396—216no—
139#395—216no—
140#395—216no—
141#395—216no—
142#393—216no—
143#391—216no—
144#390—44no—
145#388—28no—
146#387—216no—
147#387—28no—
148#386—28no—
149#386—28no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#385—28no—
155#385—28no—
156#385—28no—
157#385—28no—
158#385—216no—
159#385—28no—
160#384—44no—
161#380—24no—
162#380D4116yescomplemented
163C4×C22⋊C4—24no—
164C4×C22⋊C4—24no—
165C2×#95—24no—
166C2×#95—24no—
167C2×#95D418yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—416no—
199C22×Q8—416no—
200C22×Q8—464no—
201C22×Q8—416no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4C2×Q818yes—
213C22×D4C2×Q814yescomplemented
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D418yes—
246C23×C4C2×Q814yes—
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2414yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C22⋊Q8—464no—
262#121—232no—
263C4×D4—432no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—416no—
296C2×C4⋊C4—416no—
297C2×C4⋊C4—416no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D4116yes—
303C2×C4⋊C4C2×Q814yescomplemented
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4C2×D418yes—
334C2×C22⋊C4C2×Q818yescomplemented
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—24no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364#95C2×D4116yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24C22×D412yes—
380C24C22×D414yes—
381C24C22×Q814yes—
382C24C22×Q812yes—
383C24C2×#4112yes—
384C242−1+414yes—
385C24C2511yescharacteristic
386C2×Q8—232no—
387C2×Q8—416no—
388C2×Q8—416no—
389C2×Q8—416no—
390C2×Q8—416no—
391C2×Q8—416no—
392C2×Q8—4128no—
393C2×Q8—416no—
394C2×D4—216no—
395C2×D4—216no—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—84no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
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—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—416no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—84no—
489C22×C4—88no—
490C22×C4—84no—
491C22×C4—88no—
492C22×C4C22⋊Q8116yescomplemented
493C22×C4#127116yes—
494C22×C4C22×D414yes—
495C22×C4C22×D418yes—
496C22×C4C22×D418yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C22×Q814yes—
503C22×C4C22×Q812yes—
504C22×C4C22×Q812yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C4H(5,𝔽2)14yes—
509C22×C42−1+414yes—
510C4⋊C4—28no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4—416no—
518C4⋊C4—416no—
519C4⋊C4—48no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—88no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—88no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—48no—
535C42—48no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23C2×C22⋊Q814yes—
573C23C2×#12714yes—
574C23#53614yes—
575C23#54718yes—
576C23D4×Q818yes—
577C23C23×D411yescharacteristic
578C23C23×D412yes—
579C23C23×Q811yescharacteristic
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)12yes—
583C23C2×2−1+412yes—
584C23C2×2−1+412yes—
585C23#58512yes—
586Q8—464no—
587D4—432no—
588D4—88no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—48no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—816no—
663C2×C4—88no—
664C2×C4C2×C22⋊Q814yes—
665C2×C4C2×C22⋊Q814yes—
666C2×C4C2×#12718yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—216no—
671C22—216no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—48no—
677C22—48no—
678C22—44no—
679C22—44no—
680C22—82no—
681C22—82no—
682C22—84no—
683C22—84no—
684C22—82no—
685C22—82no—
686C22#254218yes—
687C22C22×C22⋊Q811yescharacteristic
688C22C22×#12711yescharacteristic
689C22C2×#53611yescharacteristic
690C22C2×#54712yes—
691C22C2×D4×Q812yes—
692C22#348411yescharacteristic
693C22#348414yes—
694C22#348411yescharacteristic
695C22#348511yescharacteristic
696C22#348611yescharacteristic
697C22#348914yes—
698C22#350112yes—
699C22#350312yes—
700C22#350512yes—
701C22#351112yes—
702C22#354114yes—
703C22#355214yes—
704C4—24no—
705C4—24no—
706C4—28no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—48no—
711C4—48no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—82no—
717C2—28no—
718C2—84no—
719C2C2×#254211yescharacteristic
720C2#3534911yescharacteristic
721C2#3539011yescharacteristic
722C2#3557411yescharacteristic
723C2#3563411yescharacteristic
724C2#3671611yescharacteristic
725C2#3723311yescharacteristic
726C2#4309714yes—
727C2#4310014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2579C2semidirect
2. C2×#2409C2semidirect
3. C22×#395C2semidirect
5. #15856C2semidirect
6. #15825C2semidirect
7. #15430C2semidirect
8. #15325C2semidirect
9. #14946C2semidirect
10. #14777C2semidirect
13. #13965C2semidirect
14. #13880C2semidirect
15. #13778C2semidirect
99. C2×#121Q8semidirect
162. #380D4semidirect
213. C22×D4C2×Q8semidirect
303. C2×C4⋊C4C2×Q8semidirect
334. C2×C22⋊C4C2×Q8semidirect
492. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
C2×#24092
C22×#3952
C2×#23252
#158562
#158252
#154302
#153252
#149462
#147772
#141522
#141372
#139652
#138802
#137782