Finite Group Databasebeta

← #612649

Subgroups — #612649

Order 512.

4686 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C22×#380C211yesmaximal, characteristic
4C22×C4×C22⋊C4C211yesmaximal, characteristic
5#15195C212yesmaximal, complemented
6#15182C212yesmaximal, complemented
7#14840C214yesmaximal, complemented
8#14766C214yesmaximal, complemented
9#14702C212yesmaximal, complemented
10#14686C212yesmaximal, complemented
11#14139C211yesmaximal, characteristic, complemented
12#14044C214yesmaximal, complemented
13#13924C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13877C214yesmaximal
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C42C2212yes—
25#2848—28no—
26#2502—28no—
27C2×#398C2214yescomplemented
28C2×#391C2212yescomplemented
29C2×#391C2212yescomplemented
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#384C2212yescomplemented
36C2×#384C2212yescomplemented
37C2×#380—28no—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2218yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4—28no—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54C22×#95C2212yes—
55#1445C2214yescomplemented
56#1444C2214yescomplemented
57#1442C2218yescomplemented
58#1441C2214yescomplemented
59#1441C2218yescomplemented
60#1439C2218yes—
61#1439C2218yes—
62#1436C2218yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C4⋊#95C2212yescomplemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2218yescomplemented
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22⋊#95C2218yes—
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2318yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2318yes—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—48no—
115C22×C22⋊C4Q8116yescomplemented
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42—28no—
128C22×C42—28no—
129C22×C42—44no—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133#398—216no—
134#391—28no—
135#391—44no—
136#385—28no—
137#385—28no—
138#385—28no—
139#385—216no—
140#385—216no—
141#384—28no—
142#384—44no—
143#380—24no—
144#380—24no—
145#380—24no—
146#380—28no—
147#380—28no—
148#380—28no—
149#380—28no—
150#380—416no—
151#379—28no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4D418yes—
157C2×#95—28no—
158C2×#95—28no—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2312yes—
173C25—22no—
174C25—24no—
175C25—22no—
176C25—24no—
177C25—24no—
178C25—42no—
179C25—42no—
180C25C2×D414yes—
181C25C2×Q818yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—232no—
201C23×C4—232no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—28no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—416no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—432no—
226C23×C4—432no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×Q8116yes—
239C23×C4#4114yes—
240C23×C4#4118yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C23×C4C2414yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C22⋊Q8—416no—
254C22⋊Q8—416no—
255C22⋊Q8—416no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—416no—
284C2×C4⋊C4—416no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—264no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—216no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—432no—
312C2×C22⋊C4C2×D418yes—
313C2×C22⋊C4C2×Q818yes—
314C2×C22⋊C4C2×Q8116yes—
315C2×C22⋊C4C2×Q818yes—
316C2×C22⋊C4#4118yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—432no—
325C2×C42—44no—
326C2×C42C2×D418yes—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—216no—
335#95—216no—
336#95—48no—
337#95—48no—
338#95—48no—
339#95—48no—
340#95C2×D418yes—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—28no—
351C24—216no—
352C24—28no—
353C24—28no—
354C24—216no—
355C24—28no—
356C24—232no—
357C24—21no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—44no—
365C24—44no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—416no—
370C24—48no—
371C24—48no—
372C24—416no—
373C24C4⋊Q8116yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C22×Q814yes—
379C24C22×Q814yes—
380C24C22×Q814yes—
381C24C2×#4112yes—
382C24C2×#4114yes—
383C24C2×#4114yes—
384C24H(5,𝔽2)14yes—
385C24C2511yescharacteristic
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—232no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—232no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—4128no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—416no—
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—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—816no—
483C22×C4#121116yes—
484C22×C4#123116yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D418yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×Q818yes—
493C22×C4C22×Q818yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4118yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)14yes—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—816no—
511C4⋊C4—816no—
512C22⋊C4—216no—
513C22⋊C4—232no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—416no—
522C22⋊C4C22⋊Q8132yescomplemented
523C42—216no—
524C42—416no—
525C42—48no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—232no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—28no—
562C23—232no—
563C23—416no—
564C23—416no—
565C23—416no—
566C23—464no—
567C23—464no—
568C23—416no—
569C23—416no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23C2×#12114yes—
577C23C2×C22⋊Q818yes—
578C23C2×#12314yes—
579C23C2×C4⋊Q814yes—
580C23C2×C4⋊Q814yes—
581C23C2×C4⋊Q814yes—
582C23#53214yes—
583C23#53318yes—
584C23#53514yes—
585C23#554116yes—
586C23C23×D411yescharacteristic
587C23C23×D411yescharacteristic
588C23C23×Q812yes—
589C23C22×#4111yescharacteristic
590C23C22×#4112yes—
591C23C22×#4112yes—
592C23C2×H(5,𝔽2)12yes—
593C23C2×H(5,𝔽2)12yes—
594C23#58512yes—
595Q8—88no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—232no—
611C2×C4—232no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—416no—
619C2×C4—464no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4C2×#12114yes—
660C2×C4C2×#12114yes—
661C2×C4C2×C22⋊Q8116yes—
662C2×C4C2×#12314yes—
663C2×C4C2×#12314yes—
664C2×C4#53414yes—
665C2×C4#54618yes—
666C2×C4#55118yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—216no—
671C22—216no—
672C22—216no—
673C22—216no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—464no—
687C22—42no—
688C22—48no—
689C22—48no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22#250218yescomplemented
694C22#285718yes—
695C22C22×#12111yescharacteristic
696C22C22×C22⋊Q812yes—
697C22C22×#12311yescharacteristic
698C22C22×C4⋊Q811yescharacteristic
699C22C2×#53211yescharacteristic
700C22C2×#53312yes—
701C22C2×#53411yescharacteristic
702C22C2×#53511yescharacteristic
703C22#344911yescharacteristic
704C22C2×#54612yes—
705C22C2×#55112yes—
706C22C2×#55414yes—
707C22#347514yes—
708C22#348812yes—
709C22#349412yes—
710C22#350114yes—
711C22#351914yes—
712C4—216no—
713C4—24no—
714C4—24no—
715C4—48no—
716C4—42no—
717C4—82no—
718C4—84no—
719C4—82no—
720C4—82no—
721C4—84no—
722C4—82no—
723C4#241218yes—
724C2—28no—
725C2—28no—
726C2—44no—
727C2C2×#241211yescharacteristic
728C2C2×#250211yescharacteristic
729C2#3481011yescharacteristic
730C2#3501111yescharacteristic
731C2#3507811yescharacteristic
732C2#3525211yescharacteristic
733C2#3615414yes—
734C2#3619014yes—
735C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2502C2semidirect
5. #15195C2semidirect
6. #15182C2semidirect
7. #14840C2semidirect
8. #14766C2semidirect
9. #14702C2semidirect
10. #14686C2semidirect
11. #14139C2semidirect
12. #14044C2semidirect
13. #13924C2semidirect
14. #13880C2semidirect
27. C2×#398C22semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
35. C2×#384C22semidirect
36. C2×#384C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
62. #1436C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
115. C22×C22⋊C4Q8semidirect
522. C22⋊C4C22⋊Q8semidirect
693. C22#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#25022
C22×#3802
C22×C4×C22⋊C42
#151952
#151822
#148402
#147662
#147022
#146862
#141392
#140442
#139242
#138802
#138772