Finite Group Databasebeta

← #30610989

Subgroups — #30610989

Order 15616.

1166 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462679C211yesmaximal, characteristic, complemented
2#21462258C211yesmaximal, characteristic, complemented
3#21461221C211yesmaximal, characteristic, complemented
4#21461215C212yesmaximal, complemented
5#21461208C211yesmaximal, characteristic, complemented
6#21461204C212yesmaximal, complemented
7#21461199C211yesmaximal, characteristic, complemented
8#21461195C212yesmaximal, complemented
9#21461190C211yesmaximal, characteristic, complemented
10C2×#16840613C212yesmaximal, complemented
11#21459842C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13#16840796—21no—
14#16840791—22no—
15D4×#92420—21no—
16#16840784C2211yescharacteristic, complemented
17#16840781C2211yescharacteristic, complemented
18#16840733C2212yescomplemented
19C2×#14519834C2212yescomplemented
20C4⋊C4×#6923—21no—
21C4×C4⋊#6923C2211yescharacteristic, complemented
22C22⋊C4×#6923—22no—
23C4×C22⋊#6923C2212yescomplemented
24#16840698C2211yescharacteristic, complemented
25C2×(C2×#6923)⋊C4C2212yescomplemented
26#16840615C2211yescharacteristic, complemented
27#16840615C2211yescharacteristic, complemented
28#16840615C2212yescomplemented
29#16840613C2211yescharacteristic
30#16840613C2212yes—
31#16840613C2211yescharacteristic
32#16840613C2212yescomplemented
33#16840613C2212yescomplemented
34#16840613C2212yescomplemented
35#16840613C2212yescomplemented
36#16840477C411yescharacteristic, complemented
37#16840469C411yescharacteristic, complemented
38#16840464C412yescomplemented
39#16840459—21no—
40#16840432C411yescharacteristic, complemented
41#16840406C412yescomplemented
42#16840399—22no—
43D4×C22⋊D61—21no—
44#16840365—21no—
45#16840347—22no—
46#16840310C2211yescharacteristic
47#16840294C2211yescharacteristic
48#16840272C2212yes—
49C2×#14519748C2212yes—
50#16840237—21no—
51C4×D4×D61C2211yescharacteristic
52D42×C61C411yescharacteristic, complemented
53C23×#6923—22no—
54C2×#11758213—22no—
55C22×#92420—22no—
56D4×#6923—28no—
57C2×C4⋊#6923—21no—
58C2×C22⋊#6923—22no—
59C2×C4×#6923—21no—
60C2×C4×#6923—22no—
61C2×C4×#6923—21no—
62C4⋊#92420—21no—
63C4⋊#92420—21no—
64#14519834—22no—
65#14519834C2311yescharacteristic
66#14519834C2312yes—
67#14519834C2311yescharacteristic
68C22⋊#92420—22no—
69#14519832—21no—
70C4×#92420—21no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—21no—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2312yes—
75(C2×#6923)⋊C4C2311yescharacteristic
76C2×#11758224—21no—
77C2×#11758224C2×C412yescomplemented
78C22×C22⋊D61—22no—
79C2×D4×D61D411yescharacteristic
80C22×C4×D61C2312yes—
81#14519777C2×C411yescharacteristic, complemented
82#14519773—21no—
83#14519770—22no—
84#14519770—21no—
85#14519768—21no—
86#14519768C2×C411yescharacteristic, complemented
87#14519764—21no—
88#14519764C2×C411yescharacteristic, complemented
89#14519763—22no—
90#14519761—22no—
91#14519759—22no—
92#14519759—21no—
93#14519758—22no—
94#14519757C2311yescharacteristic
95#14519756C2×C412yescomplemented
96#14519754—22no—
97#14519751—21no—
98#14519749—22no—
99#14519749—22no—
100#14519749—22no—
101#14519748—22no—
102#14519748—21no—
103#14519748—22no—
104#14519748C2×C412yescomplemented
105#14519748C2×C412yescomplemented
106#14519748C2311yescharacteristic
107#14519748C2312yes—
108#14519746—22no—
109#14519745—22no—
110C4×Dic122C2×C411yescharacteristic, complemented
111C4×D244C2×C411yescharacteristic
112D4×Dic61—21no—
113D4×Dic61—22no—
114D4×Dic61—22no—
115D4×Dic61—21no—
116D4×Dic61—21no—
117D4×Dic61C2×C412yescomplemented
118D4×Dic61D411yescharacteristic
119#14519739D411yescharacteristic
120C4×C22⋊D61—21no—
121C4×C22⋊D61C2×C412yes—
122#14519737—22no—
123#14519737D412yes—
124#14519736—21no—
125#14519736—21no—
126#14519735C2×C412yescomplemented
127C2×C4⋊Dic61C2×C412yescomplemented
128C4⋊C4×D61D411yescharacteristic
129C2×C22⋊Dic61—22no—
130C2×C22⋊Dic61—22no—
131C22⋊C4×D61D412yes—
132C42×D61C2311yescharacteristic
133C22×D4×C61—21no—
134C22×D4×C61C2×C412yescomplemented
135#127×C61C2×C411yescharacteristic, complemented
136#121×C61—21no—
137#121×C61C2×C412yescomplemented
138#120×C61—22no—
139C4×D4×C61C2×C411yescharacteristic
140C4×D4×C61C2×C411yescharacteristic, complemented
141C24×C61—22no—
142C23×D61C2×D412yes—
143C22×#6923—21no—
144C22×#6923—21no—
145C22×#6923—22no—
146C22×#6923—28no—
147C22×#6923—82no—
148C22×#6923—82no—
149C2×D4×C61—22no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—22no—
153C2×D4×C61—44no—
154C2×D4×C61—82no—
155C2×D4×C61C22⋊C411yescharacteristic
156C2×D4×C61C22×C412yes—
157C2×D4×C61C22×C411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61C22×C412yes—
160C2×C22⋊D61—21no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C2×C22⋊D61—82no—
166C2×C22⋊D61—82no—
167C2×C22⋊D61C22⋊C412yes—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61C2×D412yes—
172#11758224—82no—
173#11758224—81no—
174#11758224—81no—
175D4×D61—24no—
176C2×D244C22⋊C411yescharacteristic
177C2×C4×D61—82no—
178C2×C4×D61C22×C412yes—
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D412yes—
183C2×C4×D61C2411yescharacteristic
184C2×Dic122—21no—
185C2×Dic122C22×C411yescharacteristic
186C22⋊#6923—28no—
187C2×#92420—21no—
188C2×#92420—21no—
189C2×#92420—22no—
190C2×#92420—82no—
191C4⋊#6923—24no—
192C4×#6923—24no—
193C4×#6923—81no—
194C4×#6923—82no—
195#11758213—82no—
196C4⋊C4×C61C22⋊C411yescharacteristic
197C4⋊C4×C61C22×C411yescharacteristic
198C22⋊C4×C61—21no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61C22⋊C412yes—
201C42×C61C22×C411yescharacteristic
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204C22⋊Dic61—21no—
205C22⋊Dic61—22no—
206C22⋊Dic61—21no—
207C22⋊Dic61—22no—
208C22⋊Dic61—21no—
209C22⋊Dic61—22no—
210C22⋊Dic61C22⋊C412yes—
211#11758198—22no—
212#11758198—21no—
213#11758198C22⋊C412yes—
214#11758198C22⋊C412yes—
215C4⋊Dic61—22no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61C22⋊C411yescharacteristic
220C4⋊Dic61C22×C411yescharacteristic
221C4⋊Dic61C22×C412yes—
222#11758196—22no—
223#11758196—21no—
224#11758196—21no—
225#11758196C22⋊C412yes—
226C4×Dic61—21no—
227C4×Dic61—22no—
228C4×Dic61C22×C411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C23×C61—21no—
232C23×C61—22no—
233C23×C61—44no—
234C23×C61—82no—
235C23×C61—82no—
236C23×C61C2×C22⋊C412yes—
237C22×D61—24no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—42no—
241C22×D61C2×C22⋊C412yes—
242C22×D61C22×D411yescharacteristic
243C2×#6923—22no—
244C2×#6923—22no—
245C2×#6923—24no—
246C2×#6923—41no—
247C2×#6923—42no—
248C2×#6923—41no—
249C2×#6923—42no—
250C2×#6923—41no—
251C2×#6923—41no—
252C2×#6923—88no—
253D4×C61—42no—
254D4×C61—41no—
255D4×C61—44no—
256D4×C61—42no—
257D4×C61—81no—
258D4×C61—82no—
259D4×C61#10414yes—
260C2×C4×C61—21no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C412yes—
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C23×C411yescharacteristic
267C22⋊D61—24no—
268C22⋊D61—42no—
269C22⋊D61—41no—
270C22⋊D61—42no—
271C22⋊D61—41no—
272C22⋊D61—82no—
273C22⋊D61—82no—
274C22⋊D61—82no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—42no—
279C2×Dic61—82no—
280C2×Dic61—82no—
281C2×Dic61—82no—
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C22×D411yescharacteristic
286D244—22no—
287C4×D61—21no—
288C4×D61—21no—
289C4×D61—22no—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—41no—
293C4×D61—42no—
294Dic122—41no—
295Dic122—81no—
296#92420—41no—
297#92420—42no—
298#92420—41no—
299#17547—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—42no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—42no—
308C22×C61—164no—
309C22×C61C2×#10414yes—
310C22×C61C22×C22⋊C411yescharacteristic
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—44no—
317C2×D61—44no—
318C2×D61#57611yescharacteristic
319C2×D61#57711yescharacteristic
320#6923—42no—
321#6923—44no—
322#6923—42no—
323C4×C61—21no—
324C4×C61—22no—
325C4×C61—21no—
326C4×C61—41no—
327C4×C61C2×#10412yes—
328Dic61—21no—
329Dic61—22no—
330Dic61—21no—
331Dic61—21no—
332Dic61—41no—
333Dic61—42no—
334Dic61—41no—
335#3519—611no—
336#2974—611no—
337#2275—611no—
338#1509—611no—
339#1507—612no—
340#1505—611no—
341#1504—612no—
342#1502—611no—
343#1501—612no—
344#1500—611no—
345C2×#327—612no—
346C2×C61—24no—
347C2×C61—82no—
348C2×C61—81no—
349C2×C61#288311yescharacteristic
350C2×C61#289711yescharacteristic
351C2×C61C22×#10411yescharacteristic
352D61—21no—
353D61—21no—
354D61—42no—
355#555—611no—
356#554—611no—
357#553—612no—
358#553—1221no—
359#550—611no—
360#546—612no—
361#546—1222no—
362D42—1221no—
363D42#692311yescharacteristic, complemented
364#540—1221no—
365#538—1222no—
366#532—611no—
367#529—611no—
368#525—612no—
369C2×#121—612no—
370#518—1221no—
371C2×C4×D4—611no—
372C2×C4×D4—1221no—
373#436—1221no—
374#435—1222no—
375C8×D4—1221no—
376#432—611no—
377#431—611no—
378#407—612no—
379C2×C22⋊C8—612no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C22⋊C4—612no—
383C4×C22⋊C4—1222no—
384#376—611no—
385C2×#95—612no—
386#328—611no—
387#328—611no—
388#328—612no—
389#327—611no—
390#327—612no—
391#327—611no—
392#327—612no—
393#327—612no—
394#327—612no—
395#327—612no—
396C61#1754711yescharacteristic, complemented, Sylow
397C2×#41—612no—
398C2×#41—1221no—
399C22×D4—21no—
400C22×D4—611no—
401C22×D4—1222no—
402C22×D4C2×#692312yescomplemented
403C23×C4—612no—
404C23×C4—1222no—
405C2×M16—1222no—
406C22×C8—1222no—
407C4⋊Q8—611no—
408#127—1221no—
409#127C2×#692311yescharacteristic, complemented
410#126—1221no—
411#126—1222no—
412#125—611no—
413#125—1221no—
414#124—611no—
415#124—1221no—
416#124—1222no—
417#123—1222no—
418#123—1222no—
419#123—1221no—
420#123—1222no—
421C22⋊Q8—612no—
422C22⋊Q8—611no—
423C22⋊Q8—1222no—
424#121—21no—
425#121—611no—
426#121—612no—
427#121—612no—
428#121—612no—
429#121—1222no—
430#121—1222no—
431#121—1221no—
432#121—1222no—
433#121—1222no—
434#121—1221no—
435#121—1222no—
436#121C2×#692312yescomplemented
437#120—22no—
438#120—1222no—
439#120—1222no—
440C4×Q8—611no—
441C4×D4—611no—
442C4×D4—612no—
443C4×D4—611no—
444C4×D4—612no—
445C4×D4—612no—
446C4×D4—611no—
447C4×D4—1221no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1221no—
451C4×D4—1221no—
452C4×D4—1222no—
453C4×D4—1228no—
454C4×D4—1221no—
455C4×D4C2×#692311yescharacteristic
456C4×D4C2×#692311yescharacteristic, complemented
457#117—612no—
458#117—1221no—
459#117—1221no—
460C2×C4⋊C4—611no—
461C2×C4⋊C4—612no—
462C2×C4⋊C4—1221no—
463C2×C22⋊C4—612no—
464C2×C22⋊C4—1222no—
465C2×C22⋊C4—1222no—
466C2×C22⋊C4—1222no—
467C2×C42—611no—
468C2×C42—1221no—
469C2×C42—1222no—
470C2×C42—1221no—
471C4⋊C8—1221no—
472C4⋊C8—1221no—
473C22⋊C8—611no—
474C22⋊C8—612no—
475C22⋊C8—611no—
476C22⋊C8—1222no—
477C22⋊C8—1222no—
478#97—1221no—
479C4×C8—1221no—
480#95—611no—
481#95—612no—
482#95—611no—
483#95—1222no—
484#95—1221no—
485C24—22no—
486C24—612no—
487#41—4881no—
488#41—4881no—
489#41—4882no—
490C2×Q8—611no—
491C2×Q8—1221no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—44no—
497C2×D4—82no—
498C2×D4—612no—
499C2×D4—611no—
500C2×D4—1224no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1221no—
506C2×D4—4882no—
507C2×D4—4882no—
508C2×D4C22⋊#692311yescharacteristic
509C2×D4C22×#692312yes—
510C2×D4C22×#692311yescharacteristic
511C22×C4—21no—
512C22×C4—612no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—612no—
516C22×C4—612no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1228no—
522C22×C4—1222no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1222no—
526C22×C4—4882no—
527C22×C4—4882no—
528C22×C4—4882no—
529C22×C4C22×#692312yes—
530M16—4882no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1222no—
534C2×C8—4882no—
535C4⋊C4—612no—
536C4⋊C4—611no—
537C4⋊C4—611no—
538C4⋊C4—612no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1224no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4C22⋊#692311yescharacteristic
548C4⋊C4C22×#692311yescharacteristic
549C22⋊C4—22no—
550C22⋊C4—21no—
551C22⋊C4—612no—
552C22⋊C4—612no—
553C22⋊C4—612no—
554C22⋊C4—1222no—
555C22⋊C4—1221no—
556C22⋊C4—1221no—
557C22⋊C4—1222no—
558C22⋊C4—1221no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C22⋊C4—1221no—
563C22⋊C4—1228no—
564C22⋊C4—1222no—
565C22⋊C4C22⋊#692312yes—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—1222no—
570C42—1224no—
571C42—1221no—
572C42—4881no—
573C42—4882no—
574C42C22×#692311yescharacteristic
575C23—21no—
576C23—22no—
577C23—44no—
578C23—82no—
579C23—82no—
580C23—611no—
581C23—612no—
582C23—1222no—
583C23—1222no—
584C23—1224no—
585C23—2442no—
586C23C2×C22⋊#692312yes—
587Q8—4881no—
588Q8—2441no—
589D4—41no—
590D4—42no—
591D4—44no—
592D4—42no—
593D4—82no—
594D4—81no—
595D4—1224no—
596D4—1222no—
597D4—4882no—
598D4—4882no—
599D4—4882no—
600D4—2441no—
601D4—2441no—
602D4—2442no—
603D4—2442no—
604D4#1451984114yes—
605C2×C4—21no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—612no—
611C2×C4—611no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1224no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—4882no—
622C2×C4—4882no—
623C2×C4—4882no—
624C2×C4—4888no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2441no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2442no—
635C2×C4—2441no—
636C2×C4C2×C22⋊#692311yescharacteristic
637C2×C4C2×C22⋊#692312yes—
638C2×C4C2×C22⋊#692311yescharacteristic
639C2×C4C23×#692311yescharacteristic
640C8—2441no—
641C8—2442no—
642C8—2441no—
643C22—22no—
644C22—22no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—164no—
652C22—611no—
653C22—611no—
654C22—1221no—
655C22—1221no—
656C22—1222no—
657C22—1222no—
658C22—1222no—
659C22—2444no—
660C22—2444no—
661C22C2×#1451984114yes—
662C22C22×C22⋊#692311yescharacteristic
663C4—21no—
664C4—22no—
665C4—21no—
666C4—41no—
667C4—1221no—
668C4—1222no—
669C4—1221no—
670C4—1221no—
671C4—2442no—
672C4—2444no—
673C4—2442no—
674C4—2441no—
675C4—2441no—
676C4—2442no—
677C4C2×#1451984112yes—
678C2—24no—
679C2—81no—
680C2—82no—
681C2—1221no—
682C2—1221no—
683C2—2442no—
684C2#2146241811yescharacteristic
685C2#2146244011yescharacteristic
686C2C22×#1451984111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingD42×C61C4
socleC22×C61C22×C22⋊C4
prime radical#21459842C2
cosocleC2×C4×D61C24
hypercenterD42#6923
hypocenterC61#17547
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #21462679C2semidirect
2. #21462258C2semidirect
3. #21461221C2semidirect
4. #21461215C2semidirect
5. #21461208C2semidirect
6. #21461204C2semidirect
7. #21461199C2semidirect
8. #21461195C2semidirect
9. #21461190C2semidirect
10. C2×#16840613C2semidirect
16. #16840784C22semidirect
17. #16840781C22semidirect
18. #16840733C22semidirect
19. C2×#14519834C22semidirect
21. C4×C4⋊#6923C22semidirect
23. C4×C22⋊#6923C22semidirect
24. #16840698C22semidirect
25. C2×(C2×#6923)⋊C4C22semidirect
26. #16840615C22semidirect
27. #16840615C22semidirect
28. #16840615C22semidirect
32. #16840613C22semidirect
33. #16840613C22semidirect
34. #16840613C22semidirect
35. #16840613C22semidirect
36. #16840477C4semidirect
37. #16840469C4semidirect
38. #16840464C4semidirect
40. #16840432C4semidirect
41. #16840406C4semidirect
52. D42×C61C4semidirect
77. C2×#11758224C2×C4semidirect
81. #14519777C2×C4semidirect
86. #14519768C2×C4semidirect
88. #14519764C2×C4semidirect
95. #14519756C2×C4semidirect
104. #14519748C2×C4semidirect
105. #14519748C2×C4semidirect
110. C4×Dic122C2×C4semidirect
117. D4×Dic61C2×C4semidirect
126. #14519735C2×C4semidirect
127. C2×C4⋊Dic61C2×C4semidirect
134. C22×D4×C61C2×C4semidirect
135. #127×C61C2×C4semidirect
137. #121×C61C2×C4semidirect
140. C4×D4×C61C2×C4semidirect
363. D42#6923semidirect
396. C61#17547semidirect
402. C22×D4C2×#6923semidirect
409. #127C2×#6923semidirect
436. #121C2×#6923semidirect
456. C4×D4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626792
#214622582
#214612212
#214612152
#214612082
#214612042
#214611992
#214611952
#214611902
C2×#168406132
#214598422
#1754761