Finite Group Databasebeta

← #768868

Subgroups — #768868

Order 512.

3366 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2537C211yesmaximal, characteristic
3C2×#2491C212yesmaximal, complemented
4#15777C214yesmaximal, complemented
5#15770C214yesmaximal, complemented
6#15761C214yesmaximal, complemented
7#15673C214yesmaximal, complemented
8#15126C212yesmaximal, complemented
9#15125C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13941C211yesmaximal, characteristic, complemented
12#13905C211yesmaximal, characteristic, complemented
13#13847C211yesmaximal, characteristic, complemented
14#13785C211yesmaximal, characteristic, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2212yes—
18C22×#124C2214yes—
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2551—28no—
24#2537—28no—
25#2491—216no—
26C2×#395C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#393C2214yes—
31C2×#393C2214yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50#1446C2214yes—
51#1445C2212yes—
52#1445C2212yes—
53#1443C2214yes—
54#1442C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2314yes—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2314yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8—48no—
89C2×#124—216no—
90C2×#124—48no—
91C2×#124—48no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121—28no—
95C2×#121—48no—
96C2×#121—44no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#395—216no—
136#394—216no—
137#394—216no—
138#394—48no—
139#393—216no—
140#393—48no—
141#390—28no—
142#390—24no—
143#390D418yescomplemented
144#389—28no—
145#389—28no—
146#388—216no—
147#384D418yes—
148#382—24no—
149#380—28no—
150#380—28no—
151#380—28no—
152#380—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—44no—
158#376—24no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2412yes—
181C25C2412yes—
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4C2×D414yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8—432no—
234C4⋊Q8—432no—
235#124—464no—
236#121—432no—
237#121—432no—
238C4×D4—416no—
239C4×D4—416no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D4116yes—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330C2×C42#4114yes—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—44no—
345#95—44no—
346#95—42no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24C22×D414yes—
377C24H(5,𝔽2)14yes—
378C24C2511yescharacteristic
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—416no—
382C2×Q8—88no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×Q8—88no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
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—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+414yes—
497C22×C42−1+412yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—416no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C22⋊C4—216no—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—28no—
525C42—416no—
526C42—48no—
527C42—48no—
528C42—44no—
529C42#124116yescomplemented
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23—44no—
587C23—42no—
588C23C23×D411yescharacteristic
589C23C23×D411yescharacteristic
590C23C22×#4111yescharacteristic
591C23C22×#4111yescharacteristic
592C23C22×#4111yescharacteristic
593C23C2×H(5,𝔽2)11yescharacteristic
594C23C2×H(5,𝔽2)12yes—
595C23C2×H(5,𝔽2)12yes—
596C23C2×2−1+411yescharacteristic
597C23C2×2−1+412yes—
598C23#58512yes—
599Q8—832no—
600D4—416no—
601D4—416no—
602D4—832no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—816no—
676C2×C4—816no—
677C2×C4C2×#12414yes—
678C2×C4C2×#12414yes—
679C2×C4C2×#12414yes—
680C2×C4#53214yes—
681C2×C4#54214yes—
682C2×C4D4214yes—
683C2×C4#54714yes—
684C2×C4#54814yes—
685C2×C4#55514yes—
686C22—44no—
687C22—48no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—48no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—84no—
712C22C22×#12411yescharacteristic
713C22C2×#53211yescharacteristic
714C22C2×#54211yescharacteristic
715C22C2×D4211yescharacteristic
716C22C2×#54711yescharacteristic
717C22C2×#54811yescharacteristic
718C22C2×#55511yescharacteristic
719C22#348411yescharacteristic
720C22#348411yescharacteristic
721C22#348412yes—
722C22#348611yescharacteristic
723C22#348711yescharacteristic
724C22#348912yes—
725C22#349412yes—
726C22#349412yes—
727C22#351112yes—
728C22#351112yes—
729C22#351412yes—
730C22#352212yes—
731C22#353712yes—
732C22#354112yes—
733C22#354412yes—
734C22#354712yes—
735C4—24no—
736C4—24no—
737C4—48no—
738C4—44no—
739C4—44no—
740C4—44no—
741C4—44no—
742C4—44no—
743C4—82no—
744C4—84no—
745C4—82no—
746C4—82no—
747C4#249118yes—
748C2—84no—
749C2—84no—
750C2C2×#249111yescharacteristic
751C2#3558411yescharacteristic
752C2#3559211yescharacteristic
753C2#3560712yes—
754C2#3734012yes—
755C2#4492812yes—
756C2#4493711yescharacteristic
757C2#4493811yescharacteristic
758C2#4520812yes—
759C2#4526711yescharacteristic
760C2#4535511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2551C2semidirect
3. C2×#2491C2semidirect
4. #15777C2semidirect
5. #15770C2semidirect
6. #15761C2semidirect
7. #15673C2semidirect
8. #15126C2semidirect
9. #15125C2semidirect
10. #13993C2semidirect
11. #13941C2semidirect
12. #13905C2semidirect
13. #13847C2semidirect
14. #13785C2semidirect
15. #13779C2semidirect
143. #390D4semidirect
529. C42#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#25372
C2×#24912
#157772
#157702
#157612
#156732
#151262
#151252
#139932
#139412
#139052
#138472
#137852
#137792