Finite Group Databasebeta

← #29655377

Subgroups — #29655377

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456629C211yesmaximal, characteristic, complemented
2#21456572C211yesmaximal, characteristic, complemented
3#21453841C211yesmaximal, characteristic, complemented
4#21444833C212yesmaximal, complemented
5#21444810C212yesmaximal, complemented
6#21444765C212yesmaximal, complemented
7#21444742C212yesmaximal, complemented
8#21444375C211yesmaximal, characteristic, complemented
9#21444352C211yesmaximal, characteristic, complemented
10#21444179C211yesmaximal, characteristic, complemented
11C61×#3536C211yesmaximal, characteristic, complemented
12C2×#14519751C2211yescharacteristic
13#16840090C2212yes—
14#16840090C2211yescharacteristic
15#16840086—21no—
16#16840080—21no—
17#16840073C2211yescharacteristic
18#16840013—21no—
19#16840010—21no—
20#16839993—22no—
21#16839912C2211yescharacteristic
22#16839912C2212yes—
23#16839907—21no—
24#16839901—21no—
25#16839895C2211yescharacteristic
26#16839726C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28#16839688—21no—
29#16839687C2212yes—
30#16839671C2212yes—
31#16839670—22no—
32#16839670—21no—
33#16839625C2211yescharacteristic
34#16839499C2212yes—
35#16839499C2212yes—
36#16839499C2212yes—
37#16839496C2212yes—
38#16839486C2212yes—
39#16839486C2212yes—
40C61×#562—22no—
41C61×#554—21no—
42C61×#548—21no—
43C61×#547—21no—
44C61×#546—21no—
45C61×#546—22no—
46C61×#543C2212yes—
47C61×#542—21no—
48C61×#539—21no—
49C61×#538C2212yes—
50C61×#537—21no—
51C61×#537—21no—
52C61×#525C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C22×C22⋊D61—22no—
58C2×#11758201—22no—
59C2×#11758201—21no—
60C2×#11758199—21no—
61C2×#11758194—21no—
62C22×#92418—21no—
63#14519751—21no—
64#14519751—21no—
65#14519751—21no—
66#14519751—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—21no—
70C2×#11758198—21no—
71C2×#11758198—22no—
72#14519710—21no—
73#14519708—22no—
74#14519708—21no—
75#14519701—22no—
76#14519701—21no—
77#14519698—21no—
78#14519697—22no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C22⋊#92418C2312yes—
84C22⋊#92418C2311yescharacteristic
85C22⋊#92418C2311yescharacteristic
86#14519684—22no—
87#14519684—21no—
88#14519684C2312yes—
89C2×#41×C61—22no—
90C2×#41×C61—21no—
91C2×#41×C61D411yescharacteristic, complemented
92C22×Q8×C61D411yescharacteristic, complemented
93C22×D4×C61D412yes—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—21no—
97#126×C61—22no—
98#125×C61—22no—
99#124×C61—21no—
100#124×C61—22no—
101#124×C61—22no—
102#123×C61—22no—
103#123×C61—22no—
104#123×C61—22no—
105#123×C61—21no—
106#123×C61—22no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—22no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61D411yescharacteristic, complemented
117C22⋊Q8×C61D412yescomplemented
118C22⋊Q8×C61C2312yes—
119C22⋊Q8×C61C2311yescharacteristic
120#121×C61—21no—
121#121×C61—22no—
122#121×C61—22no—
123#121×C61—22no—
124#121×C61—22no—
125#121×C61—22no—
126#121×C61—21no—
127#121×C61—21no—
128#121×C61—21no—
129#121×C61—21no—
130#121×C61D411yescharacteristic, complemented
131#121×C61C2311yescharacteristic
132#120×C61—22no—
133#120×C61—21no—
134C4×Q8×C61—21no—
135C4×D4×C61—22no—
136C4×D4×C61—21no—
137C4×D4×C61—22no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140#117×C61D411yescharacteristic, complemented
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C4⋊C4×C61D411yescharacteristic, complemented
145C2×C22⋊C4×C61—22no—
146C2×C22⋊C4×C61—21no—
147C2×C22⋊C4×C61C2312yes—
148C24×C61C2×D412yes—
149C24×C61C2×D411yescharacteristic
150C23×D61—21no—
151#41×C61—81no—
152#41×C61—84no—
153#41×C61—82no—
154C2×Q8×C61—22no—
155C2×Q8×C61—21no—
156C2×Q8×C61—81no—
157C2×Q8×C61C2×D411yescharacteristic
158C2×Q8×C61C2×D412yes—
159C2×D4×C61—21no—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—22no—
163C2×D4×C61—22no—
164C2×D4×C61—21no—
165C2×D4×C61—22no—
166C2×D4×C61—21no—
167C2×D4×C61—21no—
168C2×D4×C61—81no—
169C2×D4×C61—82no—
170C2×D4×C61C2×D411yescharacteristic
171C22×C4×C61—21no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—81no—
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2411yescharacteristic
185C2×C22⋊D61—41no—
186C2×C22⋊D61—41no—
187C2×C22⋊D61—82no—
188C2×C22⋊D61—82no—
189C2×C22⋊D61—82no—
190C2×C22⋊D61—82no—
191C22×Dic61—22no—
192C22×Dic61—21no—
193C2×D244—21no—
194C2×D244—21no—
195C2×D244—81no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61C2×D412yes—
208C4⋊C4×C61C2×D411yescharacteristic
209C4⋊C4×C61C2×D411yescharacteristic
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—21no—
218C22⋊C4×C61—22no—
219C22⋊C4×C61—21no—
220C22⋊C4×C61—21no—
221C22⋊C4×C61—22no—
222C22⋊C4×C61—22no—
223C22⋊C4×C61—21no—
224C42×C61—22no—
225C42×C61—21no—
226C22⋊Dic61—82no—
227C22⋊Dic61—81no—
228#11758201—82no—
229#11758201—81no—
230#11758199—81no—
231#11758198—41no—
232#11758198—41no—
233#11758198—82no—
234C4⋊Dic61—21no—
235C4⋊Dic61—21no—
236#11758194—81no—
237C2×#92418—21no—
238C2×#92418—22no—
239C2×#92418—21no—
240C2×#92418—81no—
241C23×C61—22no—
242C23×C61—22no—
243C23×C61—21no—
244C23×C61—22no—
245C23×C61—22no—
246C23×C61—41no—
247C23×C61—41no—
248C23×C61—42no—
249C23×C61—42no—
250C23×C61#12011yescharacteristic
251C23×C61#12011yescharacteristic
252C23×C61C22×D411yescharacteristic
253C22×D61—41no—
254C22×D61—42no—
255C22×D61—41no—
256C22×D61—41no—
257C22×D61—42no—
258Q8×C61—42no—
259Q8×C61—42no—
260Q8×C61—41no—
261Q8×C61—82no—
262D4×C61—41no—
263D4×C61—42no—
264D4×C61—44no—
265D4×C61—42no—
266D4×C61—42no—
267D4×C61—81no—
268D4×C61—82no—
269D4×C61—81no—
270D4×C61—82no—
271D4×C61—82no—
272C2×C4×C61—21no—
273C2×C4×C61—22no—
274C2×C4×C61—21no—
275C2×C4×C61—22no—
276C2×C4×C61—21no—
277C2×C4×C61—22no—
278C2×C4×C61—41no—
279C2×C4×C61—41no—
280C2×C4×C61—41no—
281C2×C4×C61—41no—
282C2×C4×C61—41no—
283C2×C4×C61—41no—
284C2×C4×C61—42no—
285C2×C4×C61—42no—
286C2×C4×C61—42no—
287C2×C4×C61—81no—
288C2×C4×C61—82no—
289C2×C4×C61—82no—
290C2×C4×C61—82no—
291C2×C4×C61#12011yescharacteristic
292C2×C4×C61#12011yescharacteristic
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C22⋊D61—82no—
296C22⋊D61—82no—
297C22⋊D61—88no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—41no—
301C2×Dic61—42no—
302C2×Dic61—42no—
303D244—41no—
304D244—41no—
305D244—42no—
306#92418—42no—
307#92418—41no—
308#92418—41no—
309#18452—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—21no—
318C22×C61—41no—
319C22×C61—42no—
320C22×C61—41no—
321C22×C61—42no—
322C22×C61—41no—
323C22×C61—42no—
324C22×C61—82no—
325C22×C61—82no—
326C22×C61—82no—
327C22×C61C2×#12011yescharacteristic
328C22×C61#57712yes—
329C2×D61—82no—
330C2×D61—82no—
331C2×D61—81no—
332C2×D61—81no—
333C2×D61—81no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—21no—
337C4×C61—21no—
338C4×C61—41no—
339C4×C61—42no—
340C4×C61—42no—
341C4×C61—41no—
342C4×C61—41no—
343C4×C61—42no—
344Dic61—82no—
345Dic61—81no—
346#3536D6111yescharacteristic, complemented
347#3194—611no—
348#3187—611no—
349#2815—611no—
350#1612—612no—
351#1608—612no—
352#1600—612no—
353#1596—612no—
354#1516—611no—
355#1511—611no—
356#1471—611no—
357C2×C61—22no—
358C2×C61—41no—
359C2×C61—41no—
360C2×C61—81no—
361C2×C61—82no—
362C2×C61#301011yescharacteristic
363C2×C61#301911yescharacteristic
364C2×C61#302611yescharacteristic
365D61—161no—
366#562—22no—
367#554—21no—
368#548—21no—
369#547—21no—
370#546—22no—
371#546—21no—
372#543C2×D6112yes—
373#542—21no—
374#539—21no—
375#538C2×D6112yes—
376#537—21no—
377#537—21no—
378#525C2×D6111yescharacteristic
379C2×C22⋊Q8C2×D6111yescharacteristic
380C2×#121—611no—
381C2×#121C2×D6111yescharacteristic
382#483—611no—
383#483—612no—
384#482—1221no—
385#481—1221no—
386#480—611no—
387#469—1221no—
388#468—1221no—
389#465—1222no—
390#450—611no—
391#450—612no—
392#449—1221no—
393#448—1221no—
394#447—611no—
395#407—611no—
396C2×C22⋊C8—611no—
397#397—612no—
398#397—1221no—
399#392—612no—
400#392—1221no—
401#392—1222no—
402#381—611no—
403#328—612no—
404#328—612no—
405#328—612no—
406#327—612no—
407#323—612no—
408#323—612no—
409C61#1845211yescharacteristic, complemented, Sylow
410C2×#41—22no—
411C2×#41—21no—
412C2×#41C22⋊D6111yescharacteristic, complemented
413C22×Q8C22⋊D6111yescharacteristic, complemented
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4—1221no—
417C22×D4C22⋊D6112yes—
418C22×D4C22⋊D6111yescharacteristic
419C23×C4C22×D6111yescharacteristic
420C2×SD16—1222no—
421C2×SD16—1221no—
422C2×D8—1221no—
423C2×M16—1221no—
424C22×C8—1221no—
425C4⋊Q8—21no—
426#126—22no—
427#125—22no—
428#124—21no—
429#124—22no—
430#124—22no—
431#123—22no—
432#123—22no—
433#123—22no—
434#123—21no—
435#123—22no—
436C22⋊Q8—21no—
437C22⋊Q8—22no—
438C22⋊Q8—22no—
439C22⋊Q8—22no—
440C22⋊Q8—21no—
441C22⋊Q8—21no—
442C22⋊Q8—22no—
443C22⋊Q8—22no—
444C22⋊Q8—22no—
445C22⋊Q8C22⋊D6112yescomplemented
446C22⋊Q8C22⋊D6111yescharacteristic, complemented
447C22⋊Q8C22×D6112yes—
448C22⋊Q8C22×D6111yescharacteristic
449#121—21no—
450#121—22no—
451#121—22no—
452#121—22no—
453#121—22no—
454#121—22no—
455#121—21no—
456#121—21no—
457#121—21no—
458#121—21no—
459#121—1221no—
460#121—1221no—
461#121—1221no—
462#121—1221no—
463#121C22⋊D6111yescharacteristic, complemented
464#121C22×D6111yescharacteristic
465#120—21no—
466#120—22no—
467C4×Q8—21no—
468C4×D4—22no—
469C4×D4—21no—
470C4×D4—21no—
471C4×D4—21no—
472C4×D4—22no—
473#117C22⋊D6111yescharacteristic, complemented
474C2×C4⋊C4—21no—
475C2×C4⋊C4—21no—
476C2×C4⋊C4—611no—
477C2×C4⋊C4C22⋊D6111yescharacteristic
478C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
479C2×C22⋊C4—21no—
480C2×C22⋊C4—22no—
481C2×C22⋊C4—1221no—
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1222no—
484C2×C22⋊C4—1221no—
485C2×C22⋊C4C22×D6112yes—
486#107—1221no—
487#106—1222no—
488#106—1221no—
489#103—1222no—
490#103—1221no—
491#102—1221no—
492#102—1221no—
493#102—1222no—
494#102—1221no—
495C22⋊C8—612no—
496C22⋊C8—611no—
497C22⋊C8—611no—
498C22⋊C8—1221no—
499C22⋊C8—1221no—
500#95—612no—
501#95—1221no—
502#95—1222no—
503C24—1221no—
504C24C2×C22⋊D6112yes—
505C24C2×C22⋊D6111yescharacteristic
506#41—81no—
507#41—84no—
508#41—82no—
509C2×Q8—22no—
510C2×Q8—21no—
511C2×Q8—81no—
512C2×Q8C2×C22⋊D6111yescharacteristic
513C2×Q8C2×C22⋊D6112yes—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—21no—
520C2×D4—22no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—81no—
524C2×D4—82no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—4881no—
528C2×D4—4882no—
529C2×D4—4882no—
530C2×D4—4882no—
531C2×D4—4882no—
532C2×D4—2441no—
533C2×D4—2441no—
534C2×D4C2×C22⋊D6111yescharacteristic
535C22×C4—21no—
536C22×C4—22no—
537C22×C4—22no—
538C22×C4—22no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—81no—
542C22×C4—1222no—
543C22×C4—1221no—
544C22×C4C2×C22⋊D6112yes—
545C22×C4C2×C22⋊D6111yescharacteristic
546C22×C4C2×C22⋊D6111yescharacteristic
547C22×C4C2×C22⋊D6111yescharacteristic
548C22×C4C2×C22⋊D6111yescharacteristic
549C22×C4C2×C22⋊D6111yescharacteristic
550C22×C4C23×D6111yescharacteristic
551SD16—4882no—
552SD16—4881no—
553D8—4881no—
554M16—4881no—
555C2×C8—1221no—
556C2×C8—1222no—
557C2×C8—1221no—
558C2×C8—4881no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C4⋊C4C2×C22⋊D6112yes—
573C4⋊C4C2×C22⋊D6111yescharacteristic
574C4⋊C4C2×C22⋊D6111yescharacteristic
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—21no—
582C22⋊C4—22no—
583C22⋊C4—21no—
584C22⋊C4—21no—
585C22⋊C4—22no—
586C22⋊C4—22no—
587C22⋊C4—21no—
588C22⋊C4—21no—
589C22⋊C4—4882no—
590C22⋊C4—4882no—
591C22⋊C4—4881no—
592C22⋊C4—2441no—
593C22⋊C4—2441no—
594C42—22no—
595C42—21no—
596C23—22no—
597C23—22no—
598C23—21no—
599C23—22no—
600C23—22no—
601C23—41no—
602C23—41no—
603C23—42no—
604C23—42no—
605C23—2441no—
606C23—2441no—
607C23—2442no—
608C23—2441no—
609C23—2442no—
610C23#1451974611yescharacteristic
611C23#1451974611yescharacteristic
612C23C22×C22⋊D6111yescharacteristic
613Q8—42no—
614Q8—41no—
615Q8—42no—
616Q8—82no—
617D4—41no—
618D4—42no—
619D4—44no—
620D4—42no—
621D4—42no—
622D4—81no—
623D4—82no—
624D4—81no—
625D4—82no—
626D4—82no—
627D4—4882no—
628D4—4882no—
629D4—4888no—
630D4—2441no—
631D4—2441no—
632D4—2442no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—81no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—2442no—
653C2×C4—2442no—
654C2×C4—2441no—
655C2×C4—2442no—
656C2×C4—2442no—
657C2×C4#1451974611yescharacteristic
658C2×C4#1451974611yescharacteristic
659C2×C4C22×C22⋊D6111yescharacteristic
660C2×C4C22×C22⋊D6111yescharacteristic
661C8—2442no—
662C8—2441no—
663C8—2441no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—42no—
676C22—41no—
677C22—42no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—4882no—
682C22—4882no—
683C22—4881no—
684C22—4881no—
685C22—4881no—
686C22C2×#1451974611yescharacteristic
687C22#1684057012yes—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—41no—
693C4—42no—
694C4—42no—
695C4—41no—
696C4—41no—
697C4—42no—
698C4—4882no—
699C4—4881no—
700C2—22no—
701C2—41no—
702C2—41no—
703C2—82no—
704C2—81no—
705C2—9761no—
706C2#2145510511yescharacteristic
707C2#2145517611yescharacteristic
708C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3536C2
socleC22×C61C2×#120
hypercenter#3536D61
hypocenterC61#18452
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3536

Complements

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

NH ≅ G/Ntype
1. #21456629C2semidirect
2. #21456572C2semidirect
3. #21453841C2semidirect
4. #21444833C2semidirect
5. #21444810C2semidirect
6. #21444765C2semidirect
7. #21444742C2semidirect
8. #21444375C2semidirect
9. #21444352C2semidirect
10. #21444179C2semidirect
11. C61×#3536C2semidirect
91. C2×#41×C61D4semidirect
92. C22×Q8×C61D4semidirect
116. C22⋊Q8×C61D4semidirect
117. C22⋊Q8×C61D4semidirect
130. #121×C61D4semidirect
140. #117×C61D4semidirect
144. C2×C4⋊C4×C61D4semidirect
346. #3536D61semidirect
409. C61#18452semidirect
412. C2×#41C22⋊D61semidirect
413. C22×Q8C22⋊D61semidirect
445. C22⋊Q8C22⋊D61semidirect
446. C22⋊Q8C22⋊D61semidirect
463. #121C22⋊D61semidirect
473. #117C22⋊D61semidirect
478. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214566292
#214565722
#214538412
#214448332
#214448102
#214447652
#214447422
#214443752
#214443522
#214441792
C61×#35362
#1845261