Finite Group Databasebeta

← #30611407

Subgroups — #30611407

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462682C211yesmaximal, characteristic, complemented
2#21462332C211yesmaximal, characteristic, complemented
3#21461251C211yesmaximal, characteristic, complemented
4#21461249C211yesmaximal, characteristic, complemented
5#21461236C212yesmaximal, complemented
6#21461235C212yesmaximal, complemented
7#21461207C212yesmaximal, complemented
8#21461204C212yesmaximal, complemented
9#21461202C211yesmaximal, characteristic, complemented
10#21461198C211yesmaximal, characteristic, complemented
11#21459661C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13#16840797—21no—
14#16840796—21no—
15#16840795—22no—
16#16840781C2211yescharacteristic
17#16840780C2211yescharacteristic
18#16840734C2212yes—
19#16840733C2212yes—
20#16840713—21no—
21#16840709—22no—
22#16840706C2211yescharacteristic
23#16840704C2212yes—
24C4×C4⋊#6923C2211yescharacteristic
25C2×(C2×#6923)⋊C4C2212yes—
26#16840615C2212yes—
27#16840615C2212yes—
28#16840615C2212yes—
29#16840615C2212yes—
30#16840614C2212yes—
31#16840614C2211yescharacteristic
32#16840614C2211yescharacteristic
33#16840613C2212yes—
34#16840613C2211yescharacteristic
35#16840613C2211yescharacteristic
36#16840494—22no—
37#16840480C412yescomplemented
38#16840478C411yescharacteristic, complemented
39#16840464—21no—
40#16840456—21no—
41#16840430C412yescomplemented
42#16840417C411yescharacteristic, complemented
43#16840416C411yescharacteristic, complemented
44#16840346—22no—
45#16840328—21no—
46#16840317C2211yescharacteristic
47#16840309C2211yescharacteristic
48#16840274C2212yes—
49#16840272C2212yes—
50#16840242—21no—
51C4×D4×D61C2211yescharacteristic
52C61×#555C411yescharacteristic, complemented
53C23×#6923—22no—
54C2×#11758213—22no—
55C2×#11758217—22no—
56D4×#6923—44no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C4×#6923—21no—
62#14519844—21no—
63C4⋊#92420—21no—
64#14519834—22no—
65#14519834C2311yescharacteristic
66#14519834C2312yes—
67#14519834C2311yescharacteristic
68C22⋊#92420—22no—
69C4×#92420—21no—
70C4×#92420—21no—
71(C2×#6923)⋊C4—21no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2312yes—
75(C2×#6923)⋊C4C2311yescharacteristic
76C2×#11758222—21no—
77C2×#11758222C2×C412yescomplemented
78C2×#11758224C2×C412yescomplemented
79C2×D4×D61D411yescharacteristic
80C22×C4×D61C2312yes—
81#14519777C2×C411yescharacteristic, complemented
82#14519776C2×C411yescharacteristic, complemented
83#14519773—22no—
84#14519773C2×C411yescharacteristic, complemented
85#14519771—22no—
86#14519770—22no—
87#14519768—21no—
88#14519766—22no—
89#14519763—22no—
90#14519763—21no—
91#14519761—22no—
92#14519761—22no—
93#14519759—22no—
94#14519759—22no—
95#14519757C2×C412yescomplemented
96#14519757C2312yes—
97#14519756C2×C412yescomplemented
98#14519753—21no—
99#14519752—21no—
100#14519752C2×C412yescomplemented
101#14519752C2311yescharacteristic
102#14519751—21no—
103#14519750—21no—
104#14519750—22no—
105#14519749—22no—
106#14519748—21no—
107#14519748C2311yescharacteristic
108#14519747—22no—
109#14519745—22no—
110Q8×Dic61—21no—
111C4×Dic122—22no—
112#14519742C2×C412yescomplemented
113C4×D244—22no—
114C4×D244—22no—
115C4×D244C2×C411yescharacteristic
116C4×D244C2×C411yescharacteristic, complemented
117D4×Dic61—22no—
118D4×Dic61D411yescharacteristic
119#14519739D411yescharacteristic
120C4×C22⋊D61—21no—
121C4×C22⋊D61C2×C412yes—
122#14519737—22no—
123#14519737—22no—
124#14519737D412yes—
125#14519735—21no—
126#14519735—21no—
127#14519735C2×C412yescomplemented
128C2×C4⋊Dic61—21no—
129C4⋊C4×D61—21no—
130C4⋊C4×D61D411yescharacteristic
131C22⋊C4×D61D412yes—
132C42×D61C2311yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic, complemented
134#124×C61—22no—
135#121×C61—21no—
136#121×C61C2×C412yescomplemented
137C4×D4×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic, complemented
139#117×C61—21no—
140#117×C61C2×C412yescomplemented
141C23×D61C2×D412yes—
142C22×#6923—21no—
143C22×#6923—22no—
144C22×#6923—21no—
145C22×#6923—44no—
146C22×#6923—44no—
147C22×#6923—82no—
148C2×Q8×C61—21no—
149C2×D4×C61—22no—
150C2×D4×C61C22⋊C411yescharacteristic
151C2×D4×C61C22×C411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61C22×C412yes—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—21no—
158C2×C22⋊D61—21no—
159C2×C22⋊D61C22⋊C412yes—
160C22×Dic61—22no—
161C22×Dic61C2×D412yes—
162#11758224—82no—
163D4×D61—22no—
164D4×D61—22no—
165#11758222—42no—
166#11758222—42no—
167#11758222—82no—
168C2×D244—21no—
169C2×D244—22no—
170C2×D244C22⋊C411yescharacteristic
171C2×D244C22×C411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—22no—
174C2×C4×D61—82no—
175C2×C4×D61C22×C412yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D412yes—
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2411yescharacteristic
181C2×Dic122—21no—
182C2×Dic122C22×C412yes—
183C22⋊#6923—44no—
184#11758217—82no—
185C2×#92420—21no—
186C2×#92420—22no—
187C2×#92420—21no—
188C4⋊#6923—42no—
189C4⋊#6923—42no—
190C4⋊#6923—82no—
191C4×#6923—42no—
192#11758213—82no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C22⋊C411yescharacteristic
195C4⋊C4×C61C22×C411yescharacteristic
196C4⋊C4×C61C22×C412yes—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61C22⋊C412yes—
201C42×C61—22no—
202C42×C61C22×C411yescharacteristic
203C22⋊Dic61—22no—
204C22⋊Dic61—21no—
205C22⋊Dic61C22⋊C412yes—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—21no—
210#11758198—22no—
211#11758198—21no—
212#11758198C22⋊C412yes—
213#11758198C22⋊C412yes—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—22no—
217C4⋊Dic61—22no—
218C4⋊Dic61C22⋊C411yescharacteristic
219C4⋊Dic61C22×C411yescharacteristic
220#11758196—22no—
221#11758196—21no—
222#11758196—22no—
223#11758196C22⋊C412yes—
224C4×Dic61—22no—
225C4×Dic61—22no—
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C23×C61—21no—
230C23×C61C2×C22⋊C412yes—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—21no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61C2×C22⋊C412yes—
237C22×D61C22×D411yescharacteristic
238C2×#6923—22no—
239C2×#6923—41no—
240C2×#6923—41no—
241C2×#6923—41no—
242C2×#6923—41no—
243C2×#6923—42no—
244C2×#6923—42no—
245C2×#6923—41no—
246C2×#6923—42no—
247C2×#6923—88no—
248Q8×C61—81no—
249D4×C61—22no—
250D4×C61—41no—
251D4×C61—82no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C22⋊D61—24no—
261C22⋊D61—41no—
262C22⋊D61—41no—
263C22⋊D61—42no—
264C22⋊D61—81no—
265C22⋊D61—82no—
266C22⋊D61—82no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—42no—
270C2×Dic61—82no—
271C2×Dic61C2×C22⋊C412yes—
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C22×D411yescharacteristic
275D244—22no—
276D244—41no—
277D244—41no—
278D244—41no—
279D244—82no—
280C4×D61—42no—
281C4×D61—41no—
282C4×D61—42no—
283C4×D61—41no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—82no—
287C4×D61AGL(1,ℤ/8)12yes—
288C4×D61#13712yes—
289Dic122—42no—
290Dic122—42no—
291#92420—41no—
292#92420—42no—
293#92420—41no—
294#17630—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61C22×C22⋊C411yescharacteristic
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—44no—
308C2×D61—44no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
313C2×D61C2×#13711yescharacteristic
314#6923—42no—
315#6923—81no—
316#6923—82no—
317C4×C61—21no—
318C4×C61—22no—
319C4×C61—21no—
320C4×C61—41no—
321C4×C61—42no—
322C4×C61#41712yes—
323Dic61—21no—
324Dic61—22no—
325Dic61—21no—
326Dic61—42no—
327Dic61—41no—
328Dic61#41712yes—
329#3506—611no—
330#2975—611no—
331#2349—611no—
332#1518—611no—
333#1517—611no—
334#1513—612no—
335#1513—612no—
336#1504—612no—
337#1504—612no—
338#1503—611no—
339#1502—611no—
340C2×C61—42no—
341C2×C61—81no—
342C2×C61#288311yescharacteristic
343C2×C61C2×#41711yescharacteristic
344C2×C61#290111yescharacteristic
345D61—21no—
346D61—21no—
347D61—42no—
348D61—81no—
349#559—1222no—
350#556—612no—
351#555—611no—
352#555#692311yescharacteristic, complemented
353#553—1221no—
354#553—1221no—
355#550—612no—
356#547—611no—
357#547—611no—
358#538—1222no—
359#535—1221no—
360#533—611no—
361#532—611no—
362#525—612no—
363#525—612no—
364#520—1221no—
365C2×C4×D4—611no—
366C2×C4×D4—1221no—
367#436—1222no—
368#436—1221no—
369#436—1221no—
370#431—611no—
371#431—611no—
372#407—612no—
373#407—612no—
374#389—1221no—
375#385—1222no—
376#382—611no—
377#380—612no—
378C4×C4⋊C4—611no—
379C2×#95—612no—
380#328—612no—
381#328—612no—
382#328—612no—
383#328—612no—
384#327—611no—
385#327—611no—
386#327—612no—
387#327—611no—
388#327—611no—
389#327—612no—
390C61#1763011yescharacteristic, complemented, Sylow
391C2×#41—612no—
392C2×#41—612no—
393C2×#41—1221no—
394C22×D4—611no—
395C23×C4—612no—
396C23×C4—1222no—
397C2×M16—1222no—
398C2×M16—1222no—
399C4⋊Q8—611no—
400C4⋊Q8—611no—
401C4⋊Q8C2×#692311yescharacteristic, complemented
402#127—611no—
403#127—1222no—
404#126—1222no—
405#126—1222no—
406#125—1221no—
407#124—22no—
408#124—1221no—
409#124—1222no—
410#124—1222no—
411#123—1222no—
412#123—1222no—
413#123—1222no—
414#123—1222no—
415C22⋊Q8—612no—
416C22⋊Q8—612no—
417C22⋊Q8—612no—
418C22⋊Q8—1221no—
419#121—21no—
420#121—611no—
421#121—611no—
422#121—612no—
423#121—1222no—
424#121—1221no—
425#121—1222no—
426#121—1221no—
427#121—1221no—
428#121—1221no—
429#121C2×#692312yescomplemented
430#120—1222no—
431#120—1222no—
432C4×Q8—612no—
433C4×Q8—1222no—
434C4×Q8—1221no—
435C4×D4—611no—
436C4×D4—612no—
437C4×D4—611no—
438C4×D4—612no—
439C4×D4—611no—
440C4×D4—611no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4—1222no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—2444no—
448C4×D4C2×#692311yescharacteristic
449C4×D4C2×#692311yescharacteristic, complemented
450#117—21no—
451#117—612no—
452#117—1221no—
453#117—1221no—
454#117C2×#692312yescomplemented
455C2×C4⋊C4—611no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1222no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C22⋊C4—612no—
462C2×C22⋊C4—1222no—
463C2×C42—611no—
464C2×C42—1221no—
465C4⋊C8—1221no—
466C4⋊C8—1221no—
467C22⋊C8—611no—
468C22⋊C8—612no—
469C22⋊C8—611no—
470C22⋊C8—1222no—
471C22⋊C8—1222no—
472C4×C8—1221no—
473C4×C8—1221no—
474#95—611no—
475#95—612no—
476#95—611no—
477#95—1222no—
478#95—1221no—
479C24—612no—
480#41—4882no—
481#41—4882no—
482#41—2442no—
483#41—2442no—
484C2×Q8—21no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×D4—22no—
488C2×D4—612no—
489C2×D4—611no—
490C2×D4—611no—
491C2×D4—1222no—
492C2×D4—1221no—
493C2×D4—1222no—
494C2×D4—1222no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4C22⋊#692311yescharacteristic
501C2×D4C22×#692311yescharacteristic
502C22×C4—21no—
503C22×C4—611no—
504C22×C4—612no—
505C22×C4—611no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—1222no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—1221no—
514C22×C4—1222no—
515C22×C4—1221no—
516C22×C4—4882no—
517C22×C4—4882no—
518C22×C4—2444no—
519C22×C4—2444no—
520C22×C4C22×#692312yes—
521M16—4882no—
522M16—4882no—
523C2×C8—1221no—
524C2×C8—1222no—
525C2×C8—1221no—
526C4⋊C4—21no—
527C4⋊C4—611no—
528C4⋊C4—612no—
529C4⋊C4—611no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4—4882no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4C22⋊#692311yescharacteristic
541C4⋊C4C22×#692312yes—
542C4⋊C4C22×#692311yescharacteristic
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—612no—
547C22⋊C4—612no—
548C22⋊C4—612no—
549C22⋊C4—1222no—
550C22⋊C4—1221no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1221no—
555C22⋊C4—1222no—
556C22⋊C4—1221no—
557C22⋊C4—2444no—
558C22⋊C4C22⋊#692312yes—
559C42—22no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—1222no—
564C42—1222no—
565C42—2442no—
566C42C22×#692311yescharacteristic
567C23—21no—
568C23—611no—
569C23—612no—
570C23—1222no—
571C23—1222no—
572C23—1221no—
573C23—2442no—
574C23—2442no—
575C23C2×C22⋊#692312yes—
576Q8—81no—
577Q8—2442no—
578Q8—2442no—
579D4—22no—
580D4—41no—
581D4—82no—
582D4—1224no—
583D4—1222no—
584D4—4882no—
585D4—4881no—
586D4—4882no—
587D4—4882no—
588D4—2441no—
589D4—2441no—
590D4—2441no—
591D4—2442no—
592D4—2441no—
593D4—2441no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—611no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—611no—
602C2×C4—612no—
603C2×C4—611no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—4882no—
608C2×C4—4882no—
609C2×C4—4888no—
610C2×C4—2442no—
611C2×C4—2441no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2441no—
619C2×C4—2442no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2442no—
625C2×C4C2×C22⋊#692312yes—
626C2×C4C2×C22⋊#692311yescharacteristic
627C2×C4C2×C22⋊#692311yescharacteristic
628C2×C4C23×#692311yescharacteristic
629C8—2441no—
630C8—2442no—
631C8—2441no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—611no—
639C22—611no—
640C22—1221no—
641C22—1221no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—2444no—
646C22—2444no—
647C22—2441no—
648C22—2441no—
649C22—2441no—
650C22C22×C22⋊#692311yescharacteristic
651C4—21no—
652C4—22no—
653C4—21no—
654C4—41no—
655C4—42no—
656C4—612no—
657C4—1221no—
658C4—1222no—
659C4—1221no—
660C4—4882no—
661C4—4881no—
662C4—2442no—
663C4—2442no—
664C4—2441no—
665C4#1684075312yes—
666C2—42no—
667C2—81no—
668C2—1221no—
669C2—1221no—
670C2—4881no—
671C2—2442no—
672C2#2146241911yescharacteristic
673C2C2×#1684075311yescharacteristic
674C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#555C4
socleC22×C61C22×C22⋊C4
prime radical#21459661C2
cosocleC2×C4×D61C24
hypercenter#555#6923
hypocenterC61#17630
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #21462682C2semidirect
2. #21462332C2semidirect
3. #21461251C2semidirect
4. #21461249C2semidirect
5. #21461236C2semidirect
6. #21461235C2semidirect
7. #21461207C2semidirect
8. #21461204C2semidirect
9. #21461202C2semidirect
10. #21461198C2semidirect
37. #16840480C4semidirect
38. #16840478C4semidirect
41. #16840430C4semidirect
42. #16840417C4semidirect
43. #16840416C4semidirect
52. C61×#555C4semidirect
77. C2×#11758222C2×C4semidirect
78. C2×#11758224C2×C4semidirect
81. #14519777C2×C4semidirect
82. #14519776C2×C4semidirect
84. #14519773C2×C4semidirect
95. #14519757C2×C4semidirect
97. #14519756C2×C4semidirect
100. #14519752C2×C4semidirect
112. #14519742C2×C4semidirect
116. C4×D244C2×C4semidirect
127. #14519735C2×C4semidirect
133. C4⋊Q8×C61C2×C4semidirect
136. #121×C61C2×C4semidirect
138. C4×D4×C61C2×C4semidirect
140. #117×C61C2×C4semidirect
352. #555#6923semidirect
390. C61#17630semidirect
401. C4⋊Q8C2×#6923semidirect
429. #121C2×#6923semidirect
449. C4×D4C2×#6923semidirect
454. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626822
#214623322
#214612512
#214612492
#214612362
#214612352
#214612072
#214612042
#214612022
#214611982
#214596612
#1763061