Finite Group Databasebeta

← #29656248

Subgroups — #29656248

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456366C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445082C212yesmaximal, complemented
6#21445061C212yesmaximal, complemented
7#21445054C212yesmaximal, complemented
8#21444557C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444258C211yesmaximal, characteristic, complemented
11C61×#3482C211yesmaximal, characteristic
12C2×#14519807—21no—
13C2×#14519777C2211yescharacteristic
14#16840172—22no—
15#16840165C2212yes—
16#16840158C2212yes—
17#16840057—21no—
18#16840052—21no—
19#16840047C2212yes—
20#16840042C2212yes—
21#16839972—21no—
22#16839972C2212yes—
23#16839967—21no—
24#16839967C2212yes—
25#16839805—21no—
26#16839805—22no—
27#16839786C2211yescharacteristic
28#16839786C2211yescharacteristic
29#16839770—21no—
30#16839763—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#557C2211yescharacteristic
39C61×#556—21no—
40C61×#556C2211yescharacteristic
41C61×#552—22no—
42C61×#552—21no—
43D4×Q8×C61C2211yescharacteristic
44C61×#548—21no—
45C61×#547—21no—
46C61×#547C2211yescharacteristic
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C61×#529—21no—
50C61×#529—22no—
51C4×#41×C61—21no—
52C4×#41×C61C2211yescharacteristic
53C4×#41×C61C2211yescharacteristic
54C2×C4×Q8×C61—21no—
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#14519710—24no—
70#14519708—22no—
71#14519708—24no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2312yes—
75C4⋊#92418C2312yes—
76#14519701—22no—
77#14519700—22no—
78#14519700—22no—
79#14519698—22no—
80#14519688—21no—
81C4×#92418C2312yes—
82C2×#41×C61—21no—
83C2×#41×C61D411yescharacteristic
84C2×#41×C61D411yescharacteristic
85C22×Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#126×C61—22no—
91#126×C61—22no—
92#126×C61—22no—
93#125×C61—22no—
94#125×C61—22no—
95#125×C61—21no—
96#125×C61D412yes—
97#124×C61—22no—
98#124×C61—21no—
99#123×C61—22no—
100#123×C61—22no—
101#123×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109#121×C61—22no—
110#121×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61D411yescharacteristic
119C4×Q8×C61D411yescharacteristic
120C4×Q8×C61C2311yescharacteristic
121C4×Q8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61D411yescharacteristic
130C4×D4×C61D411yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#117×C61—22no—
134#117×C61—22no—
135#117×C61—21no—
136#117×C61—21no—
137#117×C61—21no—
138#117×C61D412yes—
139#117×C61D411yescharacteristic
140#117×C61D411yescharacteristic
141C2×C4⋊C4×C61—22no—
142C2×C4⋊C4×C61—21no—
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—42no—
152#41×C61—42no—
153C2×Q8×C61—22no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—42no—
157C2×Q8×C61—42no—
158C2×Q8×C61C2×D411yescharacteristic
159C2×Q8×C61C2×D411yescharacteristic
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×Dic61—41no—
177C2×Dic122—22no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—41no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C4⋊C4×C61C2×D412yes—
200C4⋊C4×C61C2×D412yes—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—21no—
212C42×C61—22no—
213C42×C61—22no—
214C42×C61—21no—
215C42×C61—21no—
216C42×C61—21no—
217C42×C61—22no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225#11758202—44no—
226#11758200—44no—
227C4⋊Dic61—22no—
228C4⋊Dic61—22no—
229C4⋊Dic61—41no—
230C4⋊Dic61—41no—
231#11758194—24no—
232#11758194—42no—
233#11758194—42no—
234C2×#92418—22no—
235C2×#92418—22no—
236C2×#92418—22no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61#12011yescharacteristic
240Q8×C61—21no—
241Q8×C61—21no—
242Q8×C61—21no—
243Q8×C61—44no—
244Q8×C61—41no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248Q8×C61#13712yescomplemented
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—21no—
252D4×C61—42no—
253D4×C61—42no—
254D4×C61—44no—
255D4×C61AGL(1,ℤ/8)12yes—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—22no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—44no—
272C2×C4×C61—44no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—42no—
278C2×C4×C61—42no—
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61#12011yescharacteristic
281C2×C4×C61#12011yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D411yescharacteristic
285C2×Dic61—42no—
286C2×Dic61—82no—
287Dic122—42no—
288Dic122—42no—
289Dic122—82no—
290#92418—24no—
291#92418—42no—
292#92418—42no—
293#18543—611nomaximal, Sylow
294C22×C61—21no—
295C22×C61—22no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61C2×#12011yescharacteristic
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—22no—
307C4×C61—21no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61#45612yes—
316C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C4×C61C2×#13711yescharacteristic
318C4×C61#57511yescharacteristic
319C4×C61#57511yescharacteristic
320Dic61—82no—
321#3482D6111yescharacteristic
322#3165—611no—
323#3079—611no—
324#3039—611no—
325#1652—612no—
326#1651—612no—
327#1648—612no—
328#1647—612no—
329#1561—611no—
330#1489—611no—
331#1489—611no—
332C2×C61—41no—
333C2×C61—42no—
334C2×C61—81no—
335C2×C61#300611yescharacteristic
336C2×C61#300711yescharacteristic
337C2×C61C2×#45611yescharacteristic
338C2×#137—1221no—
339#557C2×D6111yescharacteristic
340#556—21no—
341#556C2×D6111yescharacteristic
342#552—22no—
343#552—21no—
344D4×Q8C2×D6111yescharacteristic
345#548—21no—
346#547—21no—
347#547C2×D6111yescharacteristic
348#533—21no—
349#533C2×D6111yescharacteristic
350C2×C4⋊Q8—611no—
351#529—21no—
352#529—22no—
353C4×#41—21no—
354C4×#41C2×D6111yescharacteristic
355C4×#41C2×D6111yescharacteristic
356C2×C4×Q8—21no—
357#501—1222no—
358#500—612no—
359#498—612no—
360#477—1221no—
361#476—1221no—
362#475—612no—
363#474—612no—
364#462—612no—
365#462—1221no—
366#461—612no—
367#461—1221no—
368#428—1221no—
369#428—1222no—
370#423—611no—
371#423—611no—
372#419—1221no—
373#417—1221no—
374C4×M16—611no—
375#332—612no—
376#332—612no—
377Q8⋊C8—612no—
378Q8⋊C8—612no—
379D4⋊C8—612no—
380D4⋊C8—612no—
381C61#1854311yescharacteristic, complemented, Sylow
382C2×#41—21no—
383C2×#41C22⋊D6111yescharacteristic
384C2×#41C22⋊D6111yescharacteristic
385C22×Q8—21no—
386C22×Q8—1221no—
387#137—2444no—
388C2×Q16—1222no—
389C2×SD16—1222no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—612no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400C4⋊Q8C22×D6111yescharacteristic
401#126—22no—
402#126—22no—
403#126—22no—
404#125—22no—
405#125—22no—
406#125—21no—
407#125C22⋊D6112yes—
408#124—22no—
409#124—21no—
410#123—22no—
411#123—22no—
412#123—22no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8—21no—
420#121—22no—
421#121—21no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8C22⋊D6111yescharacteristic
430C4×Q8C22⋊D6111yescharacteristic
431C4×Q8C22×D6111yescharacteristic
432C4×Q8C22×D6111yescharacteristic
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4C22⋊D6111yescharacteristic
441C4×D4C22⋊D6111yescharacteristic
442C4×D4C22×D6111yescharacteristic
443C4×D4C22×D6111yescharacteristic
444#117—22no—
445#117—22no—
446#117—21no—
447#117—21no—
448#117—21no—
449#117C22⋊D6111yescharacteristic
450#117C22⋊D6111yescharacteristic
451#117C22⋊D6112yes—
452C2×C4⋊C4—22no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C42—22no—
457C2×C42—21no—
458C2×C42—21no—
459C2×C42C22×D6111yescharacteristic
460#107—1222no—
461#107—1224no—
462#106—1222no—
463#106—1224no—
464C4⋊C8—612no—
465C4⋊C8—612no—
466C4⋊C8—1221no—
467C4⋊C8—1221no—
468#103—1222no—
469#103—1222no—
470#103—1222no—
471#102—1222no—
472#97—1221no—
473C4×C8—612no—
474#41—22no—
475#41—22no—
476#41—41no—
477#41—41no—
478#41—42no—
479#41—42no—
480C2×Q8—22no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—1222no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2441no—
489C2×Q8C2×C22⋊D6111yescharacteristic
490C2×Q8C2×C22⋊D6111yescharacteristic
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×C22⋊D6111yescharacteristic
495C2×D4C2×C22⋊D6111yescharacteristic
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—2441no—
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508Q16—2444no—
509SD16—2444no—
510M16—1224no—
511M16—2442no—
512M16—2442no—
513C2×C8—1222no—
514C2×C8—1222no—
515C2×C8—1222no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—2441no—
533C4⋊C4—2441no—
534C4⋊C4C2×C22⋊D6111yescharacteristic
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6111yescharacteristic
537C4⋊C4C2×C22⋊D6112yes—
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C4⋊C4C2×C22⋊D6112yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42C2×C22⋊D6111yescharacteristic
561C42C2×C22⋊D6111yescharacteristic
562C42C2×C22⋊D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—21no—
565C23—21no—
566C23#1451974611yescharacteristic
567Q8—21no—
568Q8—21no—
569Q8—21no—
570Q8—44no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—4882no—
576Q8—2442no—
577Q8—2442no—
578Q8#1451980712yescomplemented
579D4—21no—
580D4—21no—
581D4—21no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4#1451980212yes—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—4882no—
610C2×C4—2442no—
611C2×C4#1451974611yescharacteristic
612C2×C4#1451974611yescharacteristic
613C2×C4#1451974611yescharacteristic
614C2×C4C22×C22⋊D6111yescharacteristic
615C2×C4C22×C22⋊D6111yescharacteristic
616C2×C4C22×C22⋊D6111yescharacteristic
617C8—1224no—
618C8—2442no—
619C8—2442no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—21no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22C2×#1451974611yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—4882no—
642C4#1683994512yes—
643C4C2×#1451980211yescharacteristic
644C4C2×#1451980711yescharacteristic
645C4#1684056011yescharacteristic
646C4#1684056011yescharacteristic
647C2—41no—
648C2—42no—
649C2—81no—
650C2#2145507111yescharacteristic
651C2#2145507911yescharacteristic
652C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3482C2
socleC22×C61C2×#120
hypercenter#3482D61
hypocenterC61#18543
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3482

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
2. #21455634C2semidirect
3. #21455306C2semidirect
4. #21445089C2semidirect
5. #21445082C2semidirect
6. #21445061C2semidirect
7. #21445054C2semidirect
8. #21444557C2semidirect
9. #21444258C2semidirect
10. #21444258C2semidirect
248. Q8×C61#137semidirect
381. C61#18543semidirect
578. Q8#14519807semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214556342
#214553062
#214450892
#214450822
#214450612
#214450542
#214445572
#214442582
C61×#34822
#1854361