Finite Group Databasebeta

← #645123

Subgroups — #645123

Order 512.

2852 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2663C212yesmaximal, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4#17078C211yesmaximal, characteristic, complemented
5#17077C211yesmaximal, characteristic, complemented
6#17074C211yesmaximal, characteristic, complemented
7#17071C211yesmaximal, characteristic, complemented
8#16513C214yesmaximal, complemented
9#15176C212yesmaximal, complemented
10#15175C212yesmaximal, complemented
11#15061C212yesmaximal, complemented
12#15012C212yesmaximal, complemented
13#14996C212yesmaximal
14#14982C212yesmaximal, complemented
15#14287C212yesmaximal, complemented
16#14200C214yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2854—28no—
24#2663—216no—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#397C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#394C2214yes—
31C2×#393C2212yes—
32C4×C22⋊Q8—28no—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2214yes—
39C2×#389C2211yescharacteristic
40C2×#388C2212yes—
41C2×#388C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1438C2214yes—
74#1438C2212yes—
75#1438C2212yes—
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82#1434C2212yes—
83#1434C2214yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—28no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8D418yescomplemented
98C2×#121—24no—
99C2×#121—44no—
100C2×C4×Q8—24no—
101C2×C4×Q8—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
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⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C42—44no—
131C22×C42D418yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#400—28no—
140#399—28no—
141#397—216no—
142#396—216no—
143#395—28no—
144#394—216no—
145#393—28no—
146#391—24no—
147#390—24no—
148#390—24no—
149#389—216no—
150#389—24no—
151#389—24no—
152#389—24no—
153#388—216no—
154#388—28no—
155#388—28no—
156#388—48no—
157#387—28no—
158#386—28no—
159#385—28no—
160#380—24no—
161#380—24no—
162#380—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—44no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—28no—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8C2×D414yes—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—44no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#123—432no—
240C22⋊Q8—232no—
241#121—416no—
242C4×Q8—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4#4118yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D418yes—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—416no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C2×#4112yes—
381C24C2511yescharacteristic
382C2×Q8—28no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—88no—
396C2×D4—88no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
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—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4#121116yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4118yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—28no—
522C42—28no—
523C42—44no—
524C42—44no—
525C42—48no—
526C42—48no—
527C42—88no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
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—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23C2×#12114yes—
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+411yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23#58511yescharacteristic
582C23#58511yescharacteristic
583C23#58512yes—
584C23#58512yes—
585Q8—416no—
586D4—88no—
587D4—816no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
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—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
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—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4C2×#12114yes—
668C2×C4C2×#12114yes—
669C2×C4#53214yes—
670C2×C4#53414yes—
671C2×C4#53614yes—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—82no—
684C22—84no—
685C22—84no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—84no—
692C22—82no—
693C22—82no—
694C22C22×#12111yescharacteristic
695C22C2×#53211yescharacteristic
696C22C2×#53411yescharacteristic
697C22C2×#53611yescharacteristic
698C22#344911yescharacteristic
699C22#344911yescharacteristic
700C22#344911yescharacteristic
701C22#345912yes—
702C22#346012yes—
703C22D4×#4112yes—
704C22#348012yes—
705C22#349612yes—
706C22#349711yescharacteristic
707C22#349711yescharacteristic
708C22#350512yes—
709C22#350711yescharacteristic
710C22#350711yescharacteristic
711C22#351512yes—
712C22#351812yes—
713C22#353512yes—
714C22#353512yes—
715C22#356712yes—
716C22#356812yes—
717C4—24no—
718C4—24no—
719C4—24no—
720C4—24no—
721C4—24no—
722C4—42no—
723C4—48no—
724C4—44no—
725C4—44no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C2—44no—
733C2—162no—
734C2#3457511yescharacteristic
735C2#3493711yescharacteristic
736C2#3500011yescharacteristic
737C2#3517811yescharacteristic
738C2#3520111yescharacteristic
739C2#3744111yescharacteristic
740C2#3754111yescharacteristic
741C2#3945312yes—
742C2#3945512yes—
743C2#3983712yes—
744C2#3983912yes—
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×#2663C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
4. #17078C2semidirect
5. #17077C2semidirect
6. #17074C2semidirect
7. #17071C2semidirect
8. #16513C2semidirect
9. #15176C2semidirect
10. #15175C2semidirect
11. #15061C2semidirect
12. #15012C2semidirect
14. #14982C2semidirect
15. #14287C2semidirect
16. #14200C2semidirect
17. C4×#380C2semidirect
97. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#26632
C2×C4×C22⋊Q82
#170782
#170772
#170742
#170712
#165132
#151762
#151752
#150612
#150122
#149962
#149822
#142872
#142002
C4×#3802