Finite Group Databasebeta

← #29656747

Subgroups — #29656747

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16839972C211yesmaximal, characteristic, complemented
4#21445096C212yesmaximal, complemented
5#21445089C212yesmaximal, complemented
6#21445068C212yesmaximal, complemented
7#21445061C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic
11C61×#3523C211yesmaximal, characteristic
12C22×#11758202—21no—
13C2×#14519777C2211yescharacteristic
14#16840172—21no—
15#16840172—22no—
16#16840165C2212yes—
17#16840158C2212yes—
18#16840121—21no—
19#16840095—21no—
20#16840078—21no—
21#16840047C2212yes—
22#16840042C2212yes—
23#16839972—21no—
24#16839972—21no—
25#16839972C2212yes—
26#16839972C2212yes—
27#16839819C2211yescharacteristic
28#16839805—22no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519700—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#557—21no—
39C61×#557C2211yescharacteristic
40C61×#557C2211yescharacteristic
41C61×#556—21no—
42C61×#556C2211yescharacteristic
43C61×#552—22no—
44C61×#552—21no—
45D4×Q8×C61—21no—
46D4×Q8×C61C2211yescharacteristic
47C61×#541—22no—
48C61×#541—21no—
49C61×#533—21no—
50C61×#533C2211yescharacteristic
51C61×#529—21no—
52C61×#529C2211yescharacteristic
53C61×#519—21no—
54C2×C4×Q8×C61C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758202—22no—
57C2×#11758202—22no—
58C2×#11758202—44no—
59C2×#11758194—21no—
60C22×#92418—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67#14519710—22no—
68#14519710—24no—
69#14519708—22no—
70#14519708—24no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519698—22no—
78C22⋊#92418—21no—
79#14519688—21no—
80#14519688—21no—
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C22×Q8×C61D411yescharacteristic
84C4⋊Q8×C61—22no—
85C4⋊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#126×C61—22no—
94#126×C61—21no—
95#125×C61—22no—
96#125×C61—22no—
97#125×C61—22no—
98#125×C61—22no—
99#125×C61—22no—
100#125×C61—21no—
101#125×C61—21no—
102#125×C61D412yes—
103#125×C61D411yescharacteristic
104#124×C61—21no—
105#124×C61D411yescharacteristic
106#123×C61—22no—
107#123×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61D411yescharacteristic
116#121×C61D411yescharacteristic
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61D411yescharacteristic
129C4×Q8×C61D411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—22no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#117×C61—22no—
138#117×C61—22no—
139#117×C61—22no—
140#117×C61—21no—
141#117×C61—21no—
142#117×C61D412yes—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61D411yescharacteristic
145C2×C42×C61—21no—
146C2×C42×C61C2311yescharacteristic
147C2×Q8×C61—22no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—22no—
152C2×Q8×C61—41no—
153C2×Q8×C61—42no—
154C2×Q8×C61—42no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×Q8×C61C2×D411yescharacteristic
157C2×Q8×C61C2×D411yescharacteristic
158C2×D4×C61—21no—
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—22no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×Dic61—41no—
168C2×Dic122—22no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—41no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D412yes—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C42×C61—22no—
209C42×C61—22no—
210C42×C61—22no—
211C42×C61—21no—
212C42×C61—21no—
213C42×C61—22no—
214C42×C61—21no—
215C42×C61—21no—
216C42×C61—21no—
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221#11758202—44no—
222#11758202—44no—
223C4⋊Dic61—22no—
224C4⋊Dic61—22no—
225C4⋊Dic61—41no—
226C4⋊Dic61—41no—
227#11758194—24no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—42no—
232C23×C61—21no—
233C23×C61#12011yescharacteristic
234Q8×C61—21no—
235Q8×C61—21no—
236Q8×C61—21no—
237Q8×C61—22no—
238Q8×C61—42no—
239Q8×C61—44no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—42no—
243Q8×C61AGL(1,ℤ/8)12yes—
244D4×C61—22no—
245D4×C61—42no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—42no—
260C2×C4×C61—44no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61#12011yescharacteristic
265C2×C4×C61#12011yescharacteristic
266C2×C4×C61#12011yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×Dic61—42no—
271C2×Dic61—82no—
272Dic122—42no—
273Dic122—42no—
274Dic122—82no—
275#92418—24no—
276#92418—42no—
277#92418—42no—
278#18588—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61#45612yes—
285C22×C61C2×#12011yescharacteristic
286C4×C61—21no—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C61#57511yescharacteristic
301C4×C61#57812yes—
302Dic61—82no—
303#3523D6111yescharacteristic
304#3239—611no—
305#3165—611no—
306C2×#462—611no—
307#1653—612no—
308#1652—612no—
309#1649—612no—
310#1648—612no—
311#1552—611no—
312#1491—611no—
313#1483—611no—
314C2×C61—22no—
315C2×C61—81no—
316C2×C61#300611yescharacteristic
317C2×C61#301111yescharacteristic
318C2×C61C2×#45611yescharacteristic
319C22×Q16—1221no—
320#557—21no—
321#557C2×D6111yescharacteristic
322#557C2×D6111yescharacteristic
323#556—21no—
324#556C2×D6111yescharacteristic
325#552—22no—
326#552—21no—
327D4×Q8—21no—
328D4×Q8C2×D6111yescharacteristic
329#541—22no—
330#541—21no—
331#533—21no—
332#533C2×D6111yescharacteristic
333C2×C4⋊Q8—611no—
334#529—21no—
335#529C2×D6111yescharacteristic
336#519—21no—
337C2×C4×Q8C2×D6111yescharacteristic
338#501—1222no—
339#501—1221no—
340#500—612no—
341#498—612no—
342#489—1221no—
343#484—1221no—
344#481—1221no—
345#475—612no—
346#474—612no—
347#462—612no—
348#462—612no—
349#462—1221no—
350#462—1221no—
351#432—611no—
352#428—1222no—
353C2×C4⋊C8—611no—
354C2×#103—1221no—
355C4×M16—611no—
356#332—612no—
357#332—612no—
358Q8⋊C8—612no—
359Q8⋊C8—612no—
360Q8⋊C8—612no—
361D4⋊C8—612no—
362C61#1858811yescharacteristic, complemented, Sylow
363C22×Q8—21no—
364C22×Q8—1221no—
365C22×Q8C22⋊D6111yescharacteristic
366C2×Q16—1222no—
367C2×Q16—1222no—
368C2×Q16—2444no—
369C2×M16—1221no—
370C22×C8—1221no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—612no—
376C4⋊Q8—1221no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#126—21no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—22no—
391#125—22no—
392#125—21no—
393#125C22⋊D6111yescharacteristic
394#125C22⋊D6112yes—
395#124—21no—
396#124C22⋊D6111yescharacteristic
397#123—22no—
398#123—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C22⋊D6111yescharacteristic
407#121C22⋊D6111yescharacteristic
408C4×Q8—22no—
409C4×Q8—22no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8C22⋊D6111yescharacteristic
420C4×Q8C22⋊D6111yescharacteristic
421C4×Q8C22×D6111yescharacteristic
422C4×Q8C22×D6111yescharacteristic
423C4×Q8C22×D6111yescharacteristic
424C4×D4—22no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4C22×D6111yescharacteristic
428#117—22no—
429#117—22no—
430#117—22no—
431#117—21no—
432#117—21no—
433#117C22⋊D6112yes—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22⋊D6111yescharacteristic
438C2×C42—21no—
439C2×C42C22×D6111yescharacteristic
440#107—1224no—
441#107—1222no—
442#106—1224no—
443#106—1222no—
444C4⋊C8—612no—
445C4⋊C8—612no—
446C4⋊C8—1221no—
447#103—1222no—
448#103—1222no—
449#103—1222no—
450#102—1222no—
451C22⋊C8—1221no—
452#97—1221no—
453#97—1221no—
454C4×C8—612no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—41no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—1222no—
464C2×Q8—2442no—
465C2×Q8—2442no—
466C2×Q8—2441no—
467C2×Q8C2×C22⋊D6111yescharacteristic
468C2×Q8C2×C22⋊D6111yescharacteristic
469C2×Q8C2×C22⋊D6111yescharacteristic
470C2×D4—21no—
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—2441no—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480Q16—2444no—
481Q16—2444no—
482M16—1224no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—2442no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4C2×C22⋊D6111yescharacteristic
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C4⋊C4C2×C22⋊D6112yes—
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C4⋊C4C2×C22⋊D6112yes—
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42C2×C22⋊D6111yescharacteristic
537C42C2×C22⋊D6111yescharacteristic
538C42C2×C22⋊D6111yescharacteristic
539C42C23×D6111yescharacteristic
540C23—21no—
541C23#1451974611yescharacteristic
542Q8—21no—
543Q8—21no—
544Q8—21no—
545Q8—22no—
546Q8—42no—
547Q8—44no—
548Q8—41no—
549Q8—41no—
550Q8—42no—
551Q8—4882no—
552Q8—2442no—
553Q8—2442no—
554Q8#1451980212yes—
555D4—22no—
556D4—42no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—4882no—
576C2×C4—2442no—
577C2×C4#1451974611yescharacteristic
578C2×C4#1451974611yescharacteristic
579C2×C4#1451974611yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C8—1224no—
584C8—2442no—
585C8—2442no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22#1683994512yes—
592C22C2×#1451974611yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—22no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—4882no—
607C4C2×#1451980211yescharacteristic
608C4#1684056011yescharacteristic
609C4#1684057712yes—
610C2—22no—
611C2—81no—
612C2#2145507111yescharacteristic
613C2#2145511411yescharacteristic
614C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3523C2
socleC22×C61C2×#120
prime radical#121×C61D4
hypercenter#3523D61
hypocenterC61#18588
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3523

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. C2×#16839972C2semidirect
4. #21445096C2semidirect
5. #21445089C2semidirect
6. #21445068C2semidirect
7. #21445061C2semidirect
8. #21444519C2semidirect
9. #21444266C2semidirect
362. C61#18588semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214563662
C2×#168399722
#214450962
#214450892
#214450682
#214450612
#214445192
#214442662
#214442322
C61×#35232
#1858861