Finite Group Databasebeta

← #763382

Subgroups — #763382

Order 512.

3720 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2C2×#2311C212yesmaximal
3#15961C212yesmaximal, complemented
4#15951C212yesmaximal, complemented
5#15511C212yesmaximal, complemented
6#15510C212yesmaximal, complemented
7#15330C214yesmaximal, complemented
8#15325C214yesmaximal, complemented
9#15135C211yesmaximal, characteristic, complemented
10#15131C212yesmaximal, complemented
11#15125C211yesmaximal, characteristic, complemented
12#14129C212yesmaximal, complemented
13#13951C212yesmaximal, complemented
14#13880C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2214yes—
19C22×#117C2212yes—
20C23×C4⋊C4C2212yes—
21#2440—216no—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#395C2214yes—
26C2×#393C2212yes—
27#2311—216no—
28C2×#391C2214yes—
29C2×#390C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#1446C2212yes—
49#1446C2211yescharacteristic
50#1446C2211yescharacteristic
51#1445C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2211yescharacteristic
66#1438C2211yescharacteristic
67#1438C2212yes—
68#1438C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2214yes—
76#1434C2214yes—
77C22×#41—28no—
78C22×#41—28no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41Q818yescomplemented
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—42no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—48no—
93C2×C22⋊Q8—48no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129#399—44no—
130#398—216no—
131#396—48no—
132#395—216no—
133#393—44no—
134#391—216no—
135#390—216no—
136#389—28no—
137#389—28no—
138#389—44no—
139#386—216no—
140#386—216no—
141#385—216no—
142#385—48no—
143#385—48no—
144#384—28no—
145#384—42no—
146#380—28no—
147#380—28no—
148#380—28no—
149#380D4116yes—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—42no—
153C4×C22⋊C4—42no—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C25C2411yescharacteristic
178C25C2412yes—
179C2×#41—264no—
180C2×#41—416no—
181C2×#41—416no—
182C2×#41—432no—
183C2×#41—416no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C2×Q814yes—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4C2×Q814yes—
205C22×D4C2×Q818yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D418yes—
222C23×C4C2×Q814yes—
223C23×C4C2×Q818yes—
224C23×C4#4118yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C4⋊Q8—416no—
239C22⋊Q8—464no—
240#117—432no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
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—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D4116yes—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—44no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—41no—
339#95—41no—
340#95—41no—
341#95—41no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95C2×D4116yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24C22×D414yes—
371C24C22×Q812yes—
372C24C22×Q814yes—
373C24C2×#4114yes—
374C24C2511yescharacteristic
375#41—4128no—
376C2×Q8—216no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—416no—
382C2×Q8—48no—
383C2×Q8—88no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—44no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
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—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—416no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4C22⋊Q8132yescomplemented
479C22×C4C22×D418yes—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×Q812yes—
484C22×C4C22×Q814yes—
485C22×C4C22×Q812yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C22×C42−1+414yes—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—416no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—416no—
522C42—44no—
523C42—48no—
524C42—44no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
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—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×C22⋊Q818yes—
583C23C23×D412yes—
584C23C23×Q811yescharacteristic
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×2−1+411yescharacteristic
592C23C2×2−1+412yes—
593C23C2×2−1+411yescharacteristic
594Q8—416no—
595Q8—816no—
596D4—432no—
597D4—416no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—264no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—416no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—832no—
674C2×C4C2×C22⋊Q818yes—
675C2×C4C2×C22⋊Q818yes—
676C2×C4#54314yes—
677C2×C4D4214yes—
678C2×C4#54618yes—
679C2×C4#55114yes—
680C22—28no—
681C22—28no—
682C22—28no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—81no—
701C22—81no—
702C22—82no—
703C22—81no—
704C22—81no—
705C22—82no—
706C22C22×C22⋊Q812yes—
707C22C2×#54311yescharacteristic
708C22C2×D4211yescharacteristic
709C22C2×#54612yes—
710C22C2×#55111yescharacteristic
711C22#348412yes—
712C22#348612yes—
713C22#349212yes—
714C22#349212yes—
715C22#349812yes—
716C22#349912yes—
717C22#350312yes—
718C22#350311yescharacteristic
719C22#350311yescharacteristic
720C22#351112yes—
721C22#351112yes—
722C22#353612yes—
723C22#353711yescharacteristic
724C22#353811yescharacteristic
725C22#354811yescharacteristic
726C22#354811yescharacteristic
727C22#354911yescharacteristic
728C22#355011yescharacteristic
729C4—28no—
730C4—42no—
731C4—42no—
732C4—48no—
733C4—44no—
734C4—48no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—44no—
739C4—84no—
740C4—84no—
741C4#243718yes—
742C2—48no—
743C2—82no—
744C2C2×#243711yescharacteristic
745C2#3553212yes—
746C2#3557412yes—
747C2#3730611yescharacteristic
748C2#3730711yescharacteristic
749C2#4486912yes—
750C2#4487011yescharacteristic
751C2#4491411yescharacteristic
752C2#4491512yes—
753C2#4532111yescharacteristic
754C2#4533011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2440C2semidirect
3. #15961C2semidirect
4. #15951C2semidirect
5. #15511C2semidirect
6. #15510C2semidirect
7. #15330C2semidirect
8. #15325C2semidirect
9. #15135C2semidirect
10. #15131C2semidirect
11. #15125C2semidirect
12. #14129C2semidirect
13. #13951C2semidirect
14. #13880C2semidirect
81. C22×#41Q8semidirect
478. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#23112
#159612
#159512
#155112
#155102
#153302
#153252
#151352
#151312
#151252
#141292
#139512
#138802
#138512