Finite Group Databasebeta

← #732284

Subgroups — #732284

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic
4C2×#2354C211yesmaximal, characteristic
5#17069C212yesmaximal
6#17011C212yesmaximal
7#15751C214yesmaximal, complemented
8#15632C214yesmaximal, complemented
9#15183C212yesmaximal, complemented
10#15156C212yesmaximal, complemented
11#15132C212yesmaximal, complemented
12#15090C212yesmaximal, complemented
13#14283C211yesmaximal, characteristic, complemented
14#13951C212yesmaximal, complemented
15#13951C212yesmaximal, complemented
16#13926C211yesmaximal, characteristic, complemented
17#13783C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2584—28no—
23#2562—28no—
24#2398—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#400C2211yescharacteristic
28C2×#398C2212yes—
29C2×#397C2212yes—
30C2×#397C2214yes—
31C2×#395C2212yes—
32C2×#394C2212yes—
33C2×#394C2211yescharacteristic
34C2×#394C2211yescharacteristic
35C2×#393C2212yes—
36#2354—28no—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#388C2214yes—
47C2×#388C2214yes—
48C2×#384C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2211yescharacteristic
58#1445C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2214yes—
85#1434C2212yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×#125—24no—
91C2×#125—24no—
92C2×#125—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4D418yes—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—28no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#400—28no—
138#400—24no—
139#400—24no—
140#398—28no—
141#397—28no—
142#397—48no—
143#395—28no—
144#394—28no—
145#394—24no—
146#394—24no—
147#393—28no—
148#391—44no—
149#390—28no—
150#389—24no—
151#389—24no—
152#389—24no—
153#389—24no—
154#389—28no—
155#389—28no—
156#389—28no—
157#388—216no—
158#388—216no—
159#384—24no—
160#380—28no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4D418yes—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—28no—
168#376—24no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C25C2412yes—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—41no—
203C22×Q8—41no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4C2×D414yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4C2×D418yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#125—416no—
233C4×Q8—416no—
234C4×D4—232no—
235#117—432no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4#4118yes—
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—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D418yes—
315C2×C42—28no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C2×D418yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C2×D418yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24C22×D414yes—
377C24C2511yescharacteristic
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+414yes—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—88no—
512C4⋊C4—84no—
513C4⋊C4#124116yes—
514C22⋊C4—216no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—48no—
524C42—44no—
525C42—44no—
526C42—416no—
527C42—48no—
528C42—84no—
529C42#121116yes—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
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—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C23×D411yescharacteristic
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4112yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×2−1+412yes—
571C23C2×2−1+411yescharacteristic
572C23#58512yes—
573C23#58512yes—
574C23#58511yescharacteristic
575C23#58511yescharacteristic
576Q8—416no—
577D4—416no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—432no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4C2×#12114yes—
670C2×C4C2×#12114yes—
671C2×C4C2×#12114yes—
672C2×C4C2×#12418yes—
673C2×C4C2×#12414yes—
674C2×C4#53614yes—
675C2×C4#54718yes—
676C2×C4#55618yes—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22C22×#12111yescharacteristic
693C22C22×#12411yescharacteristic
694C22C2×#53611yescharacteristic
695C22#344912yes—
696C22C2×#54712yes—
697C22C2×#55612yes—
698C22#348511yescharacteristic
699C22#348711yescharacteristic
700C22#348811yescharacteristic
701C22#348811yescharacteristic
702C22#348911yescharacteristic
703C22#348911yescharacteristic
704C22#349211yescharacteristic
705C22#349312yes—
706C22#349311yescharacteristic
707C22#351512yes—
708C22#351712yes—
709C22#352112yes—
710C22#352412yes—
711C22#354712yes—
712C22#355512yes—
713C22#356512yes—
714C22#356512yes—
715C4—24no—
716C4—24no—
717C4—28no—
718C4—44no—
719C4—44no—
720C4—42no—
721C4—42no—
722C4—81no—
723C4—81no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—84no—
730C4#254718yes—
731C2—84no—
732C2C2×#254711yescharacteristic
733C2#3515011yescharacteristic
734C2#3551811yescharacteristic
735C2#3604211yescharacteristic
736C2#3604911yescharacteristic
737C2#3804812yes—
738C2#3804912yes—
739C2#4268111yescharacteristic
740C2#4289111yescharacteristic
741C2#4498312yes—
742C2#4516612yes—
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
2. C2×#2584C2semidirect
7. #15751C2semidirect
8. #15632C2semidirect
9. #15183C2semidirect
10. #15156C2semidirect
11. #15132C2semidirect
12. #15090C2semidirect
13. #14283C2semidirect
14. #13951C2semidirect
15. #13951C2semidirect
16. #13926C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#25842
C2×#23982
C2×#23542
#170692
#170112
#157512
#156322
#151832
#151562
#151322
#150902
#142832
#139512
#139262
#137832