Finite Group Databasebeta

← #641727

Subgroups — #641727

Order 512.

3308 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic, complemented
3C2×#2290C211yesmaximal, characteristic
4C22×#384C211yesmaximal, characteristic, complemented
5#16563C214yesmaximal, complemented
6#16515C214yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14821C214yesmaximal, complemented
10#14729C214yesmaximal
11#14340C211yesmaximal, characteristic, complemented
12#14283C211yesmaximal, characteristic, complemented
13#14276C211yesmaximal, characteristic, complemented
14#13907C212yesmaximal, complemented
15#13872C212yesmaximal
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2214yes—
21C23×C42C2211yescharacteristic
22#2851—28no—
23#2629—28no—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31#2290—28no—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#384—28no—
40C2×#384C2211yescharacteristic
41C2×#384C2212yes—
42C2×#384C2214yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C2×#125—24no—
81C2×#125—24no—
82C2×#125—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4Q818yes—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—216no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—48no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42D418yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#398—216no—
129#398—48no—
130#396—216no—
131#396—48no—
132#389—28no—
133#389—28no—
134#389—44no—
135#388—28no—
136#387—216no—
137#387—28no—
138#386—28no—
139#386—48no—
140#385—216no—
141#385—28no—
142#384—28no—
143#384—216no—
144#384—24no—
145#384D418yescomplemented
146#382—24no—
147#380—28no—
148#380—28no—
149#380—48no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—24no—
154#376—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4C2×Q814yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D418yes—
206C23×C4C2×D414yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q814yes—
209C23×C4#4114yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#125—416no—
224C4×D4—232no—
225#117—416no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—464no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D4116yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
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—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×Q814yes—
299C2×C22⋊C4C2×Q814yes—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—232no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42C2×D414yes—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×Q814yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24C22×D412yes—
356C24C22×Q812yes—
357C24C22×Q812yes—
358C24C2×#4112yes—
359C24H(5,𝔽2)14yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—44no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C22⋊Q8116yes—
462C22×C4C22×D418yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+414yes—
481C22×C42−1+412yes—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—216no—
507C42—28no—
508C42—28no—
509C42—44no—
510C42—44no—
511C42#121116yes—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C2×C22⋊Q814yes—
554C23#53718yes—
555C23#54014yes—
556C23#54114yes—
557C23#54418yes—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C23×Q811yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23C2×2−1+411yescharacteristic
568C23C2×2−1+412yes—
569D4—416no—
570C2×C4—264no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—232no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—832no—
651C2×C4C2×#12118yes—
652C2×C4C2×#12114yes—
653C2×C4C2×C22⋊Q814yes—
654C2×C4C2×C22⋊Q814yes—
655C2×C4#53718yes—
656C2×C4#53714yes—
657C2×C4#54614yes—
658C2×C4D4×Q814yes—
659C2×C4#55214yes—
660C2×C4#55414yes—
661C22—28no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—24no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—24no—
671C22—42no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—81no—
677C22—81no—
678C22—82no—
679C22—82no—
680C22—81no—
681C22—81no—
682C22#262918yescomplemented
683C22C22×#12111yescharacteristic
684C22C22×C22⋊Q811yescharacteristic
685C22C2×#53712yes—
686C22C2×#53711yescharacteristic
687C22C2×#53712yes—
688C22C2×#54011yescharacteristic
689C22C2×#54111yescharacteristic
690C22C2×#54412yes—
691C22C2×#54611yescharacteristic
692C22C2×D4×Q811yescharacteristic
693C22C2×#55211yescharacteristic
694C22C2×#55411yescharacteristic
695C22#348412yes—
696C22#348612yes—
697C22#349212yes—
698C22#349212yes—
699C22#350312yes—
700C22#350811yescharacteristic
701C22#350911yescharacteristic
702C22#354312yes—
703C22#354412yes—
704C22#355212yes—
705C22#355212yes—
706C4—24no—
707C4—28no—
708C4—28no—
709C4—42no—
710C4—48no—
711C4—42no—
712C4—42no—
713C4—84no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—84no—
718C4#241718yes—
719C2—28no—
720C2—82no—
721C2C2×#262911yescharacteristic
722C2C2×#241711yescharacteristic
723C2#3522312yes—
724C2#3528011yescharacteristic
725C2#3762012yes—
726C2#3768612yes—
727C2#3769411yescharacteristic
728C2#3769511yescharacteristic
729C2#3787412yes—
730C2#3788211yescharacteristic
731C2#3788311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2851C2semidirect
2. C2×#2629C2semidirect
4. C22×#384C2semidirect
5. #16563C2semidirect
6. #16515C2semidirect
7. #15133C2semidirect
8. #15131C2semidirect
9. #14821C2semidirect
11. #14340C2semidirect
12. #14283C2semidirect
13. #14276C2semidirect
14. #13907C2semidirect
145. #384D4semidirect
682. C22#2629semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#26292
C2×#22902
C22×#3842
#165632
#165152
#151332
#151312
#148212
#147292
#143402
#142832
#142762
#139072
#138722