Finite Group Databasebeta

← #647162

Subgroups — #647162

Order 512.

2898 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17050C214yesmaximal, complemented
4#16395C212yesmaximal, complemented
5#16263C212yesmaximal, complemented
6#15202C211yesmaximal, characteristic, complemented
7#15194C211yesmaximal, characteristic, complemented
8#15183C212yesmaximal, complemented
9#15157C211yesmaximal, characteristic, complemented
10#15147C211yesmaximal, characteristic, complemented
11#15126C212yesmaximal
12#15050C212yesmaximal, complemented
13#14969C212yesmaximal, complemented
14#14685C211yesmaximal, characteristic, complemented
15#14300C214yesmaximal, complemented
16#14042C212yesmaximal, complemented
17#14018C212yesmaximal, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C42C2211yescharacteristic
24C2×#397C2212yes—
25C2×#394C2214yes—
26C2×#393C2212yes—
27C4×C22⋊Q8—28no—
28C4×#121—28no—
29C2×#391C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—44no—
102C2×C4×D4—48no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42—44no—
130C22×C42D418yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#397—44no—
139#394—216no—
140#393—44no—
141#391—24no—
142#390—28no—
143#390—24no—
144#390—44no—
145#389—28no—
146#389—24no—
147#389—24no—
148#389—44no—
149#388—28no—
150#387—28no—
151#387—48no—
152#386—28no—
153#385—28no—
154#385—48no—
155#381—44no—
156#380—24no—
157#380—24no—
158#380—44no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C22⋊C4—28no—
163C4×C22⋊C4—28no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—42no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—44no—
205C22×D4#4114yes—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—41no—
217C23×C4—41no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C22⋊Q8—232no—
242#121—232no—
243C4×Q8—416no—
244C4×D4—416no—
245C4×D4—432no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4#4114yes—
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—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
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⋊C4—44no—
315C2×C22⋊C4#4118yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—416no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—42no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24C22×D412yes—
378C24C2×#4114yes—
379C24C2×#4114yes—
380C24C2511yescharacteristic
381C2×Q8—28no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×D4—216no—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—48no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)14yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C42−1+412yes—
502C22×C42−1+414yes—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—416no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—88no—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C42—28no—
530C42—28no—
531C42—44no—
532C42—44no—
533C42—48no—
534C42—48no—
535C42—48no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—28no—
550C23—28no—
551C23—24no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23#52514yes—
582C23C23×D411yescharacteristic
583C23C22×#4111yescharacteristic
584C23C22×#4111yescharacteristic
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)12yes—
587C23C2×H(5,𝔽2)12yes—
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×2−1+412yes—
590C23C2×2−1+411yescharacteristic
591C23#58511yescharacteristic
592C23#58511yescharacteristic
593Q8—416no—
594D4—432no—
595D4—416no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—416no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—816no—
680C2×C4#52514yes—
681C2×C4#53414yes—
682C2×C4#53514yes—
683C2×C4#53614yes—
684C2×C4#53714yes—
685C2×C4#53918yes—
686C2×C4#54218yes—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—48no—
702C22—48no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22C2×#52511yescharacteristic
710C22C2×#52511yescharacteristic
711C22C2×#53411yescharacteristic
712C22C2×#53511yescharacteristic
713C22C2×#53611yescharacteristic
714C22C2×#53711yescharacteristic
715C22#344911yescharacteristic
716C22C2×#53912yes—
717C22C2×#54212yes—
718C22#348112yes—
719C22#348112yes—
720C22#350712yes—
721C22#350712yes—
722C22#351411yescharacteristic
723C22#351511yescharacteristic
724C22#351711yescharacteristic
725C22#351811yescharacteristic
726C22#351912yes—
727C22#352012yes—
728C22#354212yes—
729C22#354212yes—
730C22#354812yes—
731C22#355012yes—
732C4—24no—
733C4—24no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—44no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—84no—
745C4—84no—
746C4#263818yes—
747C2—44no—
748C2—84no—
749C2C2×#263811yescharacteristic
750C2#3496311yescharacteristic
751C2#3507711yescharacteristic
752C2#3511111yescharacteristic
753C2#3518911yescharacteristic
754C2#3743711yescharacteristic
755C2#3764011yescharacteristic
756C2#4014912yes—
757C2#4015112yes—
758C2#4017212yes—
759C2#4017412yes—
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
1. C2×C4×C22⋊Q8C2semidirect
3. #17050C2semidirect
4. #16395C2semidirect
5. #16263C2semidirect
6. #15202C2semidirect
7. #15194C2semidirect
8. #15183C2semidirect
9. #15157C2semidirect
10. #15147C2semidirect
12. #15050C2semidirect
13. #14969C2semidirect
14. #14685C2semidirect
15. #14300C2semidirect
16. #14042C2semidirect
17. #14018C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×C4×#1212
#170502
#163952
#162632
#152022
#151942
#151832
#151572
#151472
#151262
#150502
#149692
#146852
#143002
#140422
#140182