Finite Group Databasebeta

← #29656330

Subgroups — #29656330

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445047C212yesmaximal, complemented
5#21445040C212yesmaximal, complemented
6#21445019C212yesmaximal, complemented
7#21445012C212yesmaximal, complemented
8#21444554C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444258C211yesmaximal, characteristic, complemented
11C61×#3483C211yesmaximal, characteristic
12C2×#14519807—21no—
13C2×#14519777C2211yescharacteristic
14#16840121—22no—
15#16840112C2212yes—
16#16840108C2212yes—
17#16840057—21no—
18#16840052—21no—
19#16840047C2212yes—
20#16840042C2212yes—
21#16839972—21no—
22#16839972C2212yes—
23#16839967—21no—
24#16839967C2212yes—
25#16839805—21no—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839770—21no—
29#16839763—21no—
30#16839763—22no—
31C4×#11758194C2211yescharacteristic
32#16839511C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38Q82×C61C2211yescharacteristic
39C61×#555—22no—
40C61×#555—21no—
41C61×#555—21no—
42C61×#555C2211yescharacteristic
43C61×#550—21no—
44C61×#550—21no—
45C61×#550C2211yescharacteristic
46C61×#550C2211yescharacteristic
47C61×#532—22no—
48C61×#532—21no—
49C61×#532—21no—
50C61×#532C2211yescharacteristic
51C4×#41×C61—21no—
52C4×#41×C61—21no—
53C4×#41×C61C2211yescharacteristic
54C4×#41×C61C2211yescharacteristic
55C22×Dic122—21no—
56#14519807—44no—
57C2×#11758202—22no—
58C2×#11758200—22no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777—21no—
65#14519777C2312yes—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519710—22no—
69#14519708—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519701—24no—
76#14519700—22no—
77#14519700—22no—
78#14519698—22no—
79#14519698—24no—
80#14519688—21no—
81C4×#92418C2312yes—
82C2×#41×C61—22no—
83C2×#41×C61—21no—
84C2×#41×C61—21no—
85C2×#41×C61D411yescharacteristic
86C2×#41×C61D411yescharacteristic
87C2×#41×C61D412yes—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61C2311yescharacteristic
92#127×C61—22no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61C2311yescharacteristic
96#124×C61—22no—
97#124×C61—22no—
98#124×C61—22no—
99#124×C61—22no—
100#124×C61D412yes—
101#121×C61—22no—
102#121×C61—22no—
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—21no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61—21no—
112#121×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61D411yescharacteristic
116C4×Q8×C61D411yescharacteristic
117C4×Q8×C61C2311yescharacteristic
118C4×Q8×C61C2311yescharacteristic
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—22no—
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61D411yescharacteristic
132C4×D4×C61D411yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#117×C61—22no—
136#117×C61—22no—
137#117×C61—21no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61—21no—
141#117×C61D411yescharacteristic
142#117×C61D411yescharacteristic
143C2×C42×C61—22no—
144C2×C42×C61—21no—
145C2×C42×C61—21no—
146C2×C42×C61C2311yescharacteristic
147#41×C61—22no—
148#41×C61—22no—
149#41×C61—41no—
150#41×C61—41no—
151#41×C61—44no—
152#41×C61—42no—
153#41×C61—42no—
154#41×C61—42no—
155#41×C61—42no—
156#41×C61—84no—
157C2×Q8×C61—22no—
158C2×Q8×C61C2×D411yescharacteristic
159C2×Q8×C61C2×D411yescharacteristic
160C2×Q8×C61C2×D412yes—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—22no—
166C2×D4×C61—22no—
167C2×D4×C61—22no—
168C2×D4×C61—22no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61C2×D411yescharacteristic
172C2×D4×C61C2×D411yescharacteristic
173C2×D4×C61C2×D412yes—
174C22×C4×C61—22no—
175C22×C4×C61—22no—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—22no—
179C22×C4×C61—21no—
180C22×C4×C61—21no—
181C22×C4×C61—21no—
182C22×C4×C61—21no—
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D411yescharacteristic
185C22×C4×C61C2×D411yescharacteristic
186C22×Dic61—41no—
187C2×Dic122—22no—
188C2×Dic122—42no—
189C2×Dic122—42no—
190C2×Dic122—41no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C42×C61—22no—
216C42×C61—22no—
217C42×C61—21no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61—21no—
222C42×C61—21no—
223C42×C61C2×D411yescharacteristic
224C42×C61C2×D411yescharacteristic
225C42×C61C2×D411yescharacteristic
226C42×C61C2411yescharacteristic
227#11758202—44no—
228#11758200—44no—
229C4⋊Dic61—22no—
230C4⋊Dic61—22no—
231C4⋊Dic61—41no—
232C4⋊Dic61—41no—
233#11758194—42no—
234#11758194—42no—
235#11758194—42no—
236C2×#92418—22no—
237C2×#92418—22no—
238C2×#92418—22no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—22no—
242C23×C61#12011yescharacteristic
243Q8×C61—21no—
244Q8×C61—21no—
245Q8×C61—42no—
246Q8×C61—82no—
247Q8×C61AGL(1,ℤ/8)12yes—
248Q8×C61#13712yescomplemented
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—21no—
252D4×C61—21no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—42no—
256D4×C61—44no—
257D4×C61—44no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—41no—
262D4×C61—82no—
263D4×C61—82no—
264D4×C61—82no—
265D4×C61—82no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—22no—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—22no—
276C2×C4×C61—21no—
277C2×C4×C61—21no—
278C2×C4×C61—42no—
279C2×C4×C61—42no—
280C2×C4×C61—44no—
281C2×C4×C61—44no—
282C2×C4×C61—41no—
283C2×C4×C61—41no—
284C2×C4×C61—41no—
285C2×C4×C61—41no—
286C2×C4×C61—44no—
287C2×C4×C61—42no—
288C2×C4×C61—42no—
289C2×C4×C61#12011yescharacteristic
290C2×C4×C61#12011yescharacteristic
291C2×C4×C61#12011yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C2×Dic61—42no—
296C2×Dic61—82no—
297Dic122—42no—
298Dic122—42no—
299Dic122—82no—
300#92418—42no—
301#92418—42no—
302#92418—42no—
303#18553—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—41no—
308C22×C61—42no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—42no—
312C22×C61—41no—
313C22×C61—42no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61C2×#12011yescharacteristic
317C4×C61—21no—
318C4×C61—22no—
319C4×C61—21no—
320C4×C61—22no—
321C4×C61—21no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61#45512yes—
328C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
329C4×C61C2×#13711yescharacteristic
330C4×C61#57511yescharacteristic
331C4×C61#57511yescharacteristic
332Dic61—82no—
333#3483D6111yescharacteristic
334#3141—611no—
335#3079—611no—
336#3039—611no—
337#1646—612no—
338#1645—612no—
339#1642—612no—
340#1641—612no—
341#1560—611no—
342#1489—611no—
343#1489—611no—
344C2×C61—41no—
345C2×C61—81no—
346C2×C61—81no—
347C2×C61—82no—
348C2×C61#300611yescharacteristic
349C2×C61#300711yescharacteristic
350C2×C61C2×#45511yescharacteristic
351C2×#137—1221no—
352Q82C2×D6111yescharacteristic
353#555—22no—
354#555—21no—
355#555—21no—
356#555C2×D6111yescharacteristic
357#550—21no—
358#550—21no—
359#550C2×D6111yescharacteristic
360#550C2×D6111yescharacteristic
361#532—22no—
362#532—21no—
363#532—21no—
364#532C2×D6111yescharacteristic
365C2×C4⋊Q8—611no—
366C4×#41—21no—
367C4×#41—21no—
368C4×#41C2×D6111yescharacteristic
369C4×#41C2×D6111yescharacteristic
370#489—1222no—
371#487—612no—
372#486—612no—
373#477—1221no—
374#476—1221no—
375#475—612no—
376#474—612no—
377#462—612no—
378#462—1221no—
379#461—612no—
380#461—1221no—
381#428—1221no—
382#423—611no—
383#423—611no—
384#419—1221no—
385#417—1221no—
386#417—1222no—
387C4×M16—611no—
388#333—612no—
389#331—612no—
390Q8⋊C8—612no—
391Q8⋊C8—612no—
392D4⋊C8—612no—
393D4⋊C8—612no—
394C61#1855311yescharacteristic, complemented, Sylow
395C2×#41—22no—
396C2×#41—21no—
397C2×#41—21no—
398C2×#41C22⋊D6111yescharacteristic
399C2×#41C22⋊D6111yescharacteristic
400C2×#41C22⋊D6112yes—
401C22×Q8—1221no—
402#137—2444no—
403C2×Q16—1222no—
404C2×SD16—1222no—
405C2×M16—1221no—
406C2×M16—1221no—
407C2×M16—1221no—
408C4⋊Q8—22no—
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411C4⋊Q8—612no—
412C4⋊Q8—1221no—
413C4⋊Q8—1221no—
414C4⋊Q8—1221no—
415C4⋊Q8C22×D6111yescharacteristic
416#127—22no—
417#127—21no—
418#127—21no—
419#127C22×D6111yescharacteristic
420#124—22no—
421#124—22no—
422#124—22no—
423#124—22no—
424#124C22⋊D6112yes—
425#121—22no—
426#121—22no—
427#121—22no—
428#121—22no—
429#121—21no—
430#121—21no—
431#121—22no—
432#121—22no—
433#121—21no—
434#121—21no—
435#121—21no—
436#121—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8C22⋊D6111yescharacteristic
440C4×Q8C22⋊D6111yescharacteristic
441C4×Q8C22×D6111yescharacteristic
442C4×Q8C22×D6111yescharacteristic
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—22no—
446C4×D4—22no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4C22⋊D6111yescharacteristic
456C4×D4C22⋊D6111yescharacteristic
457C4×D4C22×D6111yescharacteristic
458C4×D4C22×D6111yescharacteristic
459#117—22no—
460#117—22no—
461#117—21no—
462#117—21no—
463#117—21no—
464#117—21no—
465#117C22⋊D6111yescharacteristic
466#117C22⋊D6111yescharacteristic
467C2×C4⋊C4—1221no—
468C2×C4⋊C4—1221no—
469C2×C42—22no—
470C2×C42—21no—
471C2×C42—21no—
472C2×C42C22×D6111yescharacteristic
473#107—1222no—
474#106—1222no—
475C4⋊C8—612no—
476C4⋊C8—612no—
477C4⋊C8—1221no—
478C4⋊C8—1221no—
479#103—1222no—
480#103—1224no—
481#103—1222no—
482#103—1222no—
483#102—1222no—
484#102—1224no—
485#97—1221no—
486C4×C8—612no—
487#41—22no—
488#41—22no—
489#41—41no—
490#41—41no—
491#41—44no—
492#41—42no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—84no—
497C2×Q8—22no—
498C2×Q8—1222no—
499C2×Q8—2442no—
500C2×Q8—2442no—
501C2×Q8—2441no—
502C2×Q8C2×C22⋊D6111yescharacteristic
503C2×Q8C2×C22⋊D6111yescharacteristic
504C2×Q8C2×C22⋊D6112yes—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4C2×C22⋊D6111yescharacteristic
516C2×D4C2×C22⋊D6111yescharacteristic
517C2×D4C2×C22⋊D6112yes—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—2441no—
528C22×C4C2×C22⋊D6111yescharacteristic
529C22×C4C2×C22⋊D6111yescharacteristic
530C22×C4C2×C22⋊D6111yescharacteristic
531Q16—2444no—
532SD16—2444no—
533M16—2442no—
534M16—2442no—
535M16—2442no—
536C2×C8—1222no—
537C2×C8—1222no—
538C2×C8—1222no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—2441no—
550C4⋊C4—2441no—
551C4⋊C4C2×C22⋊D6111yescharacteristic
552C4⋊C4C2×C22⋊D6111yescharacteristic
553C4⋊C4C2×C22⋊D6111yescharacteristic
554C4⋊C4C2×C22⋊D6111yescharacteristic
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—21no—
566C22⋊C4—21no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42C2×C22⋊D6111yescharacteristic
576C42C2×C22⋊D6111yescharacteristic
577C42C2×C22⋊D6111yescharacteristic
578C42C23×D6111yescharacteristic
579C23—22no—
580C23—21no—
581C23—21no—
582C23#1451974611yescharacteristic
583Q8—21no—
584Q8—21no—
585Q8—42no—
586Q8—82no—
587Q8—4882no—
588Q8—2442no—
589Q8—2442no—
590Q8#1451980212yes—
591Q8#1451980712yescomplemented
592D4—21no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—44no—
600D4—44no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—82no—
606D4—82no—
607D4—82no—
608D4—82no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—4882no—
633C2×C4—2442no—
634C2×C4#1451974611yescharacteristic
635C2×C4#1451974611yescharacteristic
636C2×C4#1451974611yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C22×C22⋊D6111yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C8—2442no—
641C8—2442no—
642C8—2442no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22C2×#1451974611yescharacteristic
656C4—21no—
657C4—21no—
658C4—22no—
659C4—21no—
660C4—22no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—4882no—
667C4#1683994012yes—
668C4C2×#1451980211yescharacteristic
669C4C2×#1451980711yescharacteristic
670C4#1684056011yescharacteristic
671C4#1684056011yescharacteristic
672C2—41no—
673C2—82no—
674C2—81no—
675C2—81no—
676C2#2145507111yescharacteristic
677C2#2145507911yescharacteristic
678C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3483C2
socleC22×C61C2×#120
hypercenter#3483D61
hypocenterC61#18553
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3483

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
2. #21455634C2semidirect
3. #21455306C2semidirect
4. #21445047C2semidirect
5. #21445040C2semidirect
6. #21445019C2semidirect
7. #21445012C2semidirect
8. #21444554C2semidirect
9. #21444258C2semidirect
10. #21444258C2semidirect
248. Q8×C61#137semidirect
394. C61#18553semidirect
591. Q8#14519807semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214556342
#214553062
#214450472
#214450402
#214450192
#214450122
#214445542
#214442582
C61×#34832
#1855361