Finite Group Databasebeta

← #764385

Subgroups — #764385

Order 512.

4190 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2413C212yesmaximal, complemented
3C22×#398C211yesmaximal, characteristic, complemented
4#15912C214yesmaximal, complemented
5#15848C214yesmaximal, complemented
6#15614C214yesmaximal, complemented
7#15612C214yesmaximal, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14926C211yesmaximal, characteristic
10#14720C211yesmaximal, characteristic, complemented
11#14719C211yesmaximal, characteristic
12#14135C212yesmaximal, complemented
13#14015C212yesmaximal, complemented
14#13968C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C2×#539—216no—
17C2×#539—28no—
18C22×#126C2212yes—
19C22×#125C2212yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2212yes—
25C22×#121C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32#2413—216no—
33C2×#398—28no—
34C2×#398C2214yes—
35C2×#398C2214yes—
36C2×#398C2214yes—
37C2×#396C2214yes—
38C2×#395C2212yes—
39C2×#394C2212yes—
40C2×#389C2214yes—
41C2×#388C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#385C2214yes—
47C2×#381C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1446C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90#539—2128no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#126—28no—
94C2×#125—28no—
95C2×#125—28no—
96C2×#125—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—44no—
105C2×#123D4116yes—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—28no—
111C2×#121—28no—
112C2×#121—24no—
113C2×C4×D4—28no—
114C2×C4×D4—28no—
115C2×C4×D4—28no—
116C2×#117—28no—
117C2×#117—28no—
118C2×#117—44no—
119C22×C4⋊C4—28no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—28no—
124C22×C4⋊C4—44no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D418yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2312yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2314yes—
153C22×C42C2312yes—
154C22×C42C2314yes—
155#398—216no—
156#398—216no—
157#398—216no—
158#396—216no—
159#395—44no—
160#394—44no—
161#389—216no—
162#388—216no—
163#386—28no—
164#386—28no—
165#386—44no—
166#385—28no—
167#385—48no—
168#381—28no—
169C2×#95—24no—
170C2×#95—24no—
171C2×#95—24no—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4C2×D418yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D418yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258#126—264no—
259#125—264no—
260#123—264no—
261#123—264no—
262#123—432no—
263C22⋊Q8—264no—
264C22⋊Q8—232no—
265#121—232no—
266C4×D4—264no—
267#117—432no—
268C2×C4⋊C4—232no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—416no—
309C2×C4⋊C4—416no—
310C2×C4⋊C4—416no—
311C2×C4⋊C4C2×D418yes—
312C2×C4⋊C4C2×D418yes—
313C2×C4⋊C4C2×D418yes—
314C2×C4⋊C4C2×D418yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4C2×D418yes—
345C2×C22⋊C4C2×D418yes—
346C2×C22⋊C4C2×D418yes—
347C2×C22⋊C4#4118yes—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—28no—
352C2×C42—28no—
353C2×C42—44no—
354#95—28no—
355#95—28no—
356#95—24no—
357#95—24no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—42no—
366#95—44no—
367#95—42no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—42no—
372#95—42no—
373#95—44no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—24no—
388C24C22×D414yes—
389C24C22×D412yes—
390C24C22×D414yes—
391C24C2×#4114yes—
392C24H(5,𝔽2)12yes—
393C242−1+412yes—
394C24C2511yescharacteristic
395C2×Q8—28no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—88no—
399C2×D4—216no—
400C2×D4—216no—
401C2×D4—216no—
402C2×D4—28no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—88no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—416no—
485C22×C4—416no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—84no—
500C22×C4—84no—
501C22×C4—88no—
502C22×C4—88no—
503C22×C4—88no—
504C22×C4—88no—
505C22×C4#120116yes—
506C22×C4C22×D414yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D414yes—
511C22×C4C22×D412yes—
512C22×C4C22×D414yes—
513C22×C4C22×D414yes—
514C22×C4C22×D414yes—
515C22×C4C2×#4118yes—
516C22×C4H(5,𝔽2)14yes—
517C22×C42−1+414yes—
518C22×C42−1+414yes—
519C4⋊C4—216no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—28no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—416no—
528C4⋊C4—416no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—416no—
532C4⋊C4—416no—
533C4⋊C4—416no—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—416no—
541C22⋊C4—416no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—88no—
545C42—216no—
546C42—416no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—28no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23C2×#12014yes—
583C23#53214yes—
584C23#53714yes—
585C23#54618yes—
586C23#54718yes—
587C23C23×D411yescharacteristic
588C23C23×D412yes—
589C23C22×#4112yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)12yes—
592C23C2×2−1+411yescharacteristic
593C23C2×2−1+412yes—
594C23C2×2−1+412yes—
595C23#58512yes—
596Q8—88no—
597D4—432no—
598D4—88no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—232no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—416no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—88no—
676C2×C4—88no—
677C2×C4—816no—
678C2×C4—88no—
679C2×C4C2×#12018yes—
680C22—216no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—216no—
688C22—48no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—48no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—84no—
699C22#243018yes—
700C22C22×#12011yescharacteristic
701C22C2×#53211yescharacteristic
702C22C2×#53711yescharacteristic
703C22C2×#54612yes—
704C22C2×#54712yes—
705C22#348411yescharacteristic
706C22#348511yescharacteristic
707C22#348611yescharacteristic
708C22#348611yescharacteristic
709C22#348814yes—
710C22#349914yes—
711C22#351514yes—
712C22#352014yes—
713C22#353612yes—
714C22#353812yes—
715C22#355912yes—
716C22#356312yes—
717C4—28no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—44no—
723C4—44no—
724C4—42no—
725C4—42no—
726C4—44no—
727C4—48no—
728C4—82no—
729C4—84no—
730C2—28no—
731C2—84no—
732C2C2×#243011yescharacteristic
733C2#3539711yescharacteristic
734C2#3546811yescharacteristic
735C2#3551211yescharacteristic
736C2#3566611yescharacteristic
737C2#3671911yescharacteristic
738C2#3696811yescharacteristic
739C2#4397414yes—
740C2#4404214yes—
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
2. C2×#2413C2semidirect
3. C22×#398C2semidirect
4. #15912C2semidirect
5. #15848C2semidirect
6. #15614C2semidirect
7. #15612C2semidirect
8. #14934C2semidirect
10. #14720C2semidirect
12. #14135C2semidirect
13. #14015C2semidirect
14. #13968C2semidirect
15. #13795C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24132
C22×#3982
#159122
#158482
#156142
#156122
#149342
#149262
#147202
#147192
#141352
#140152
#139682
#137952