Finite Group Databasebeta

← #29654355

Subgroups — #29654355

Order 15616.

1034 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C212yesmaximal, complemented
2#21450772C211yesmaximal, characteristic, complemented
3#21444867C214yesmaximal, complemented
4#21444844C212yesmaximal, complemented
5#21444839C212yesmaximal, complemented
6#21444344C212yesmaximal, complemented
7#21444160C211yesmaximal, characteristic, complemented
8C61×#3459C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic, complemented
10C2×#14519756—21no—
11C2×#14519756—21no—
12#16840172—22no—
13#16840121—22no—
14#16840095C2214yescomplemented
15#16840078C2214yes—
16#16840047—22no—
17#16840042—22no—
18#16839819C2212yescomplemented
19#16839698C2212yes—
20#16839698C2212yescomplemented
21#16839665—21no—
22#16839631—21no—
23C4×C4⋊Dic61C2211yescharacteristic, complemented
24#16839499C2214yescomplemented
25#16839496C2214yes—
26#16839486C2212yes—
27#16839486C2212yescomplemented
28C61×#543C2211yescharacteristic
29C61×#539—22no—
30C61×#539—21no—
31C61×#538—22no—
32C61×#538C2212yes—
33C61×#534—21no—
34C61×#534C2211yescharacteristic
35C61×#529—22no—
36C61×#529C2212yes—
37C61×#525—21no—
38C61×#525—22no—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122—21no—
42#14519777—22no—
43#14519777—22no—
44#14519756—44no—
45#14519756—44no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61C2312yes—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×C22⋊Dic61—22no—
52C2×C22⋊Dic61—22no—
53C2×C4×Dic61—21no—
54#14519710—24no—
55#14519708—24no—
56C4⋊#92418—22no—
57#14519701—24no—
58#14519698—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61#14519684C2312yes—
62C2×#41×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61D411yescharacteristic
65C23×C4×C61—21no—
66C23×C4×C61D411yescharacteristic
67C23×C4×C61D411yescharacteristic
68#126×C61—22no—
69#126×C61—22no—
70#126×C61—22no—
71#125×C61—22no—
72#125×C61D412yes—
73#124×C61—22no—
74#124×C61D412yes—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80#123×C61—22no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61C2312yes—
87#121×C61—22no—
88#121×C61—22no—
89#121×C61—21no—
90#121×C61—22no—
91#121×C61—21no—
92#121×C61C2312yes—
93#120×C61—22no—
94#120×C61—22no—
95#120×C61—22no—
96C4×Q8×C61D412yes—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61D412yes—
105#117×C61—21no—
106#117×C61—22no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117C24×C61C2×D411yescharacteristic
118#41×C61—82no—
119C2×Q8×C61—21no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—41no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—82no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—22no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—41no—
154C2×Dic122—41no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182C22⋊Dic61—44no—
183C22⋊Dic61—44no—
184C4⋊Dic61—24no—
185C4⋊Dic61—24no—
186C4⋊Dic61—42no—
187#11758196—44no—
188#11758196—44no—
189C4×Dic61—42no—
190C2×#92418—24no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—81no—
203C23×C61—81no—
204C23×C61#13712yes—
205C23×C61C22×D411yescharacteristic
206Q8×C61—42no—
207Q8×C61—81no—
208D4×C61—44no—
209D4×C61—42no—
210D4×C61—44no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—82no—
214D4×C61—81no—
215D4×C61—82no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—44no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—81no—
232C2×C4×C61—81no—
233C2×C4×C61—81no—
234C2×C4×C61—82no—
235C2×C4×C61—81no—
236C2×C4×C61#12012yes—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61C22×D412yes—
240C2×Dic61—24no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246Dic122—82no—
247Dic122—82no—
248#92418—44no—
249#18343—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—84no—
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C22×C61#57511yescharacteristic
269C4×C61—22no—
270C4×C61—21no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#45312yes—
278Dic61—44no—
279Dic61—82no—
280#3459D6111yescharacteristic
281#3239—612no—
282#2542—611no—
283#1618—614no—
284#1614—612no—
285#1613—612no—
286#1509—612no—
287#1466—611no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—42no—
291C2×C61—81no—
292C2×C61—82no—
293C2×C61#300711yescharacteristic
294C2×C61C2×#45311yescharacteristic
295C2×C61#302811yescharacteristic
296#543C2×D6111yescharacteristic
297#539—22no—
298#539—21no—
299#538—22no—
300#538C2×D6112yes—
301#534—21no—
302#534C2×D6111yescharacteristic
303C2×C4⋊Q8—611no—
304#529—22no—
305#529C2×D6112yes—
306#525—21no—
307#525—22no—
308C2×C22⋊Q8—1221no—
309C2×C22⋊Q8—1221no—
310C2×C4×D4—21no—
311C2×C4×D4C2×D6111yescharacteristic
312#501—1222no—
313#489—1222no—
314#484—614no—
315#481—614no—
316#475—1222no—
317#474—1222no—
318#432—612no—
319#399—612no—
320#399—612no—
321#390—1221no—
322#384—1221no—
323C4×C4⋊C4—611no—
324#328—614no—
325#327—614no—
326#323—612no—
327#323—612no—
328C61#1834311yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C22×Q8—1221no—
331C22×D4—21no—
332C22×D4C22⋊D6111yescharacteristic
333C23×C4—21no—
334C23×C4C22⋊D6111yescharacteristic
335C23×C4C22⋊D6111yescharacteristic
336C4⋊Q8—1222no—
337C4⋊Q8—1222no—
338#126—22no—
339#126—22no—
340#126—22no—
341#125—22no—
342#125C22⋊D6112yes—
343#124—22no—
344#124C22⋊D6112yes—
345#123—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8—2444no—
357C22⋊Q8—2444no—
358C22⋊Q8C22×D6112yes—
359#121—22no—
360#121—21no—
361#121—22no—
362#121—22no—
363#121—21no—
364#121C22×D6112yes—
365#120—22no—
366#120—22no—
367#120—22no—
368C4×Q8C22⋊D6112yes—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4C22⋊D6112yes—
377#117—21no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4C22×D6111yescharacteristic
392C2×C42—21no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—1224no—
396#106—1224no—
397C4⋊C8—1222no—
398#103—1224no—
399#102—1224no—
400C22⋊C8—614no—
401#97—1222no—
402#95—612no—
403C24—21no—
404C24C2×C22⋊D6111yescharacteristic
405C24C2×C22⋊D6111yescharacteristic
406#41—82no—
407C2×Q8—21no—
408C2×Q8—2442no—
409C2×Q8—2442no—
410C2×Q8—2441no—
411C2×Q8—2441no—
412C2×Q8C2×C22⋊D6112yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×C22⋊D6112yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443C2×C8—1224no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—2444no—
456C4⋊C4—2444no—
457C4⋊C4—2442no—
458C4⋊C4C2×C22⋊D6112yes—
459C4⋊C4C2×C22⋊D6112yes—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—2444no—
473C22⋊C4—2444no—
474C42—22no—
475C42—22no—
476C42—2442no—
477C42C2×C22⋊D6112yes—
478C42C2×C22⋊D6112yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23#1451980712yes—
493C23C22×C22⋊D6111yescharacteristic
494Q8—42no—
495Q8—81no—
496Q8—4882no—
497Q8—4882no—
498D4—44no—
499D4—42no—
500D4—44no—
501D4—42no—
502D4—42no—
503D4—82no—
504D4—81no—
505D4—82no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—1224no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4#1451974612yes—
533C2×C4#1451974611yescharacteristic
534C2×C4#1451974611yescharacteristic
535C2×C4C22×C22⋊D6112yes—
536C8—2444no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—84no—
553C22C2×#1451974611yescharacteristic
554C22C2×#1451980711yescharacteristic
555C22#1684056011yescharacteristic
556C4—21no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—4882no—
565C4—2444no—
566C4#1683993012yes—
567C2—21no—
568C2—21no—
569C2—42no—
570C2—81no—
571C2—82no—
572C2#2145507911yescharacteristic
573C2C2×#1683993011yescharacteristic
574C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3459C2
socleC22×C61C2×#120
hypercenter#3459D61
hypocenterC61#18343
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3459

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21450772C2semidirect
3. #21444867C2semidirect
4. #21444844C2semidirect
5. #21444839C2semidirect
6. #21444344C2semidirect
7. #21444160C2semidirect
9. C2×#14519777C22semidirect
14. #16840095C22semidirect
18. #16839819C22semidirect
20. #16839698C22semidirect
23. C4×C4⋊Dic61C22semidirect
24. #16839499C22semidirect
27. #16839486C22semidirect
328. C61#18343semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214507722
#214448672
#214448442
#214448392
#214443442
#214441602
C61×#34592
#1834361