Finite Group Databasebeta

← #1409249

Subgroups — #1409249

Order 512.

2470 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2506C211yesmaximal, characteristic
3C2×#2660C212yesmaximal
4#16608C212yesmaximal
5#16603C212yesmaximal
6#16525C214yesmaximal
7#16523C214yesmaximal
8#15158C211yesmaximal, characteristic
9#15143C211yesmaximal, characteristic
10#15137C212yesmaximal
11#15133C212yesmaximal
12#15128C211yesmaximal, characteristic
13#14901C212yesmaximal
14#13918C212yesmaximal
15#13910C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2660—216no—
23#2546—28no—
24#2506—28no—
25C2×#400C2212yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#395C2212yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#395C2211yescharacteristic
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#387C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2214yes—
47C2×#384C2212yes—
48C2×#382C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×#376C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1444C2212yes—
57#1444C2214yes—
58#1444C2211yescharacteristic
59#1444C2212yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2211yescharacteristic
73#1441C2212yes—
74#1441C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8—44no—
93C2×#125—28no—
94C2×#125—28no—
95C2×#125—44no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130#400—28no—
131#398—216no—
132#398—216no—
133#398—28no—
134#398—216no—
135#395—28no—
136#395—28no—
137#395—28no—
138#395—44no—
139#395D418yes—
140#391—28no—
141#389—24no—
142#389—24no—
143#389—24no—
144#389—28no—
145#389—28no—
146#389—24no—
147#389—44no—
148#389—44no—
149#387—28no—
150#384—216no—
151#384—24no—
152#384—44no—
153#382—216no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—44no—
156C4×C4⋊C4D418yes—
157C4×C4⋊C4Q8116yescomplemented
158#376—28no—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C2×D414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C4⋊Q8—432no—
218C4⋊Q8—416no—
219#125—432no—
220C4×Q8—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
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—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
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—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×Q818yes—
291C2×C4⋊C4C2×Q818yes—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42C2×D414yes—
308C2×C42C2×D418yes—
309C2×C42C2×Q818yes—
310C2×C42C2×Q8116yes—
311#95—24no—
312#95—28no—
313#95—22no—
314#95—22no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D414yes—
339#95C2×D418yes—
340#95C2×D418yes—
341#95C2×Q8116yes—
342C24C2511yescharacteristic
343C2×Q8—28no—
344C2×Q8—48no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—88no—
352C2×Q8—88no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q818yes—
447C22×C4C22×Q818yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+414yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—416no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4C4⋊Q8116yes—
481C42—28no—
482C42—216no—
483C42—216no—
484C42—232no—
485C42—44no—
486C42—48no—
487C42—44no—
488C23C23×D411yescharacteristic
489C23C23×D411yescharacteristic
490C23C23×Q812yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23C2×2−1+411yescharacteristic
495C23C2×2−1+412yes—
496C23C2×2−1+411yescharacteristic
497C23#58512yes—
498C23#58511yescharacteristic
499Q8—416no—
500Q8—816no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4C2×C4⋊Q818yes—
597C2×C4C2×C4⋊Q814yes—
598C2×C4#53514yes—
599C2×C4#53614yes—
600C2×C4#54418yes—
601C2×C4#54418yes—
602C2×C4D4214yes—
603C2×C4D4×Q818yes—
604C22C22×C4⋊Q811yescharacteristic
605C22C2×#53511yescharacteristic
606C22C2×#53611yescharacteristic
607C22C2×#54412yes—
608C22C2×#54412yes—
609C22C2×D4211yescharacteristic
610C22C2×D4×Q812yes—
611C22#348711yescharacteristic
612C22#348911yescharacteristic
613C22#349612yes—
614C22#349711yescharacteristic
615C22#349911yescharacteristic
616C22#349911yescharacteristic
617C22#350212yes—
618C22#350312yes—
619C22#350412yes—
620C22#353812yes—
621C22#354212yes—
622C22#354711yescharacteristic
623C22#354712yes—
624C22#355011yescharacteristic
625C22#355512yes—
626C22#356312yes—
627C4—28no—
628C4—24no—
629C4—28no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—48no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—81no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—82no—
643C4#266018yescomplemented
644C2C2×#266011yescharacteristic
645C2#3595711yescharacteristic
646C2D4×C4⋊Q811yescharacteristic
647C2#3933911yescharacteristic
648C2#3934311yescharacteristic
649C2#3936812yes—
650C2#3937012yes—
651C2#4282012yes—
652C2#4287911yescharacteristic
653C2#4297212yes—
654C2#4299411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
157. C4×C4⋊C4Q8semidirect
643. C4#2660semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25062
C2×#26602
#166082
#166032
#165252
#165232
#151582
#151432
#151372
#151332
#151282
#149012
#139182
#139102
C4×#3952
#138102