Finite Group Databasebeta

← #4416309

Subgroups — #4416309

Order 512.

2484 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#17063C211yesmaximal, characteristic, complemented
4#17062C211yesmaximal, characteristic, complemented
5#16632C212yesmaximal
6#16606C212yesmaximal, complemented
7#16560C211yesmaximal, characteristic, complemented
8#16511C211yesmaximal, characteristic, complemented
9#16394C212yesmaximal, complemented
10#16392C212yesmaximal
11#15121C212yesmaximal, complemented
12#15097C212yesmaximal, complemented
13#15063C212yesmaximal, complemented
14#15050C212yesmaximal, complemented
15#14893C212yesmaximal, complemented
16#14825C212yesmaximal
17#14684C211yesmaximal, characteristic, complemented
18#14629C212yesmaximal, complemented
19#14628C212yesmaximal, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2857—28no—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#398C2212yes—
31C2×#397C2212yes—
32C2×#395C2212yes—
33C2×#393C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37#2290—28no—
38C2×#388C2211yescharacteristic
39C2×#388C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#386C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#385C2211yescharacteristic
45C2×#385C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#381C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1446C2212yes—
55#1446C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1444C2212yes—
63#1444C2214yes—
64#1444C2212yes—
65#1443C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2212yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2212yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2212yes—
81#1438C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1434C2212yes—
85#1434C2214yes—
86#1434C2212yes—
87#1434C2211yescharacteristic
88#1434C2211yescharacteristic
89#1434C2212yes—
90C22⋊#95C2214yes—
91C22⋊#95C2212yes—
92C22⋊#95C2212yes—
93C23×Q8C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—44no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×#117—24no—
108C2×#117—44no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2314yes—
134C22×C42—44no—
135C22×C42D418yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#400—28no—
144#400—44no—
145#399—28no—
146#399—44no—
147#398—44no—
148#397—44no—
149#395—44no—
150#393—44no—
151#391—28no—
152#389—28no—
153#389—28no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#386—24no—
159#386—24no—
160#385—24no—
161#385—24no—
162#384—24no—
163#381—44no—
164#380—28no—
165#380—28no—
166C4×C4⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169#376—24no—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C25C2411yescharacteristic
204C25C2411yescharacteristic
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—42no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C4⋊Q8—416no—
251C22⋊Q8—432no—
252C4×D4—232no—
253#117—416no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
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—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—44no—
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—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4#4114yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
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—44no—
316C2×C22⋊C4#4114yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—416no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24C22×D412yes—
382C24C2×#4112yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—88no—
389C2×Q8—88no—
390C2×D4—28no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—44no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
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—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—44no—
482C22×C4—416no—
483C22×C4—44no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C42−1+414yes—
498C4⋊C4—28no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—416no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—28no—
524C42—28no—
525C42—28no—
526C42—44no—
527C42—44no—
528C42—44no—
529C42—44no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—28no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23#53614yes—
564C23C23×D411yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×2−1+412yes—
570C23C2×2−1+412yes—
571C23#58512yes—
572C23#58512yes—
573Q8—816no—
574D4—416no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
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—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
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—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4#52514yes—
661C2×C4#53414yes—
662C2×C4#53514yes—
663C2×C4#53614yes—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—48no—
670C22—48no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—48no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—84no—
680C22—81no—
681C22—81no—
682C22C2×#52511yescharacteristic
683C22C2×#53411yescharacteristic
684C22C2×#53511yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#53611yescharacteristic
687C22#344911yescharacteristic
688C22#344911yescharacteristic
689C22#349512yes—
690C22#349512yes—
691C22#351412yes—
692C22#351712yes—
693C22#352012yes—
694C22#352012yes—
695C22#353114yes—
696C22#353514yes—
697C22#354214yes—
698C22#355214yes—
699C4—24no—
700C4—24no—
701C4—24no—
702C4—24no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—81no—
709C4—81no—
710C4—84no—
711C4—84no—
712C4—84no—
713C4—81no—
714C4—84no—
715C4—81no—
716C2—44no—
717C2—82no—
718C2#3508411yescharacteristic
719C2#3519011yescharacteristic
720C2#3532511yescharacteristic
721C2#3669911yescharacteristic
722C2#3744711yescharacteristic
723C2#3744811yescharacteristic
724C2#3761711yescharacteristic
725C2#4616514yes—
726C2#4616714yes—
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×#2857C2semidirect
3. #17063C2semidirect
4. #17062C2semidirect
6. #16606C2semidirect
7. #16560C2semidirect
8. #16511C2semidirect
9. #16394C2semidirect
11. #15121C2semidirect
12. #15097C2semidirect
13. #15063C2semidirect
14. #15050C2semidirect
15. #14893C2semidirect
17. #14684C2semidirect
18. #14629C2semidirect
19. #14628C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#22902
#170632
#170622
#166322
#166062
#165602
#165112
#163942
#163922
#151212
#150972
#150632
#150502
#148932
#148252
#146842
#146292
#146282