Finite Group Databasebeta

← #30548546

Subgroups — #30548546

Order 15616.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2#21456114C214yesmaximal, complemented
3#21456066C214yesmaximal, complemented
4#21456027C218yesmaximal, complemented
5#21455763C212yesmaximal, complemented
6#21455637C212yesmaximal, complemented
7#21455302C212yesmaximal, complemented
8#21455077C212yesmaximal
9C4×#14519808C214yesmaximal, complemented
10#21454457C211yesmaximal, characteristic, complemented
11C2×C61×#541C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519808C2214yes—
14#16840243—22no—
15#16840242C2214yes—
16#16840239C2212yes—
17#16840236C2212yes—
18#16840235—22no—
19C4×#11758222C2212yes—
20C4×#11758222C2212yes—
21#16840230—22no—
22C2×#14519736C2211yescharacteristic
23#117×D61—22no—
24#16840100C2218yes—
25#16840095C2214yescomplemented
26#16840090C2214yescomplemented
27#16840066C2218yescomplemented
28#16840059C2218yescomplemented
29#16839969C2218yescomplemented
30#16839964C2218yescomplemented
31#16839922C2218yes—
32#16839917C2214yes—
33#16839912C2214yes—
34#16839867C2218yescomplemented
35#16839860C2218yescomplemented
36C4×#11758202C2218yescomplemented
37C4×#11758201C2218yescomplemented
38#16839823C2214yes—
39#16839823C2214yescomplemented
40#16839802C2214yescomplemented
41#16839786C2212yescomplemented
42#16839771C2214yescomplemented
43#16839761C2214yes—
44#16839726C2212yes—
45C4×#11758194C2212yescomplemented
46C61×#541—22no—
47C61×#541—22no—
48C61×#541C2214yes—
49C61×#541C2214yescomplemented
50C2×C4⋊Q8×C61C2211yescharacteristic, complemented
51C2×#126×C61—22no—
52C2×#125×C61—22no—
53C2×#125×C61C2211yescharacteristic, complemented
54C2×C22⋊Q8×C61C2212yes—
55C2×C4×Q8×C61C2212yescomplemented
56C2×#117×C61C2211yescharacteristic
57C2×#11758222—21no—
58C2×#11758222—21no—
59C2×#11758222—28no—
60C2×#11758222C2312yes—
61C2×#11758222C2312yes—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C4×D61—21no—
66#14519808—216no—
67C2×#11758202C2318yes—
68C2×#11758201C2318yes—
69C2×#11758194C2314yes—
70C4×Dic122C2314yes—
71C4×Dic122C2314yes—
72#14519742—24no—
73C4×D244C2314yes—
74C4×D244C2314yes—
75#14519739—24no—
76C4×C22⋊D61—24no—
77C4×C22⋊D61—24no—
78#14519737—24no—
79#14519737—24no—
80#14519736—21no—
81#14519736—21no—
82#14519736C2312yes—
83#14519736C2312yes—
84#14519735—24no—
85#14519734—22no—
86#14519734—22no—
87#14519733—24no—
88C2×C4⋊Dic61C2312yes—
89C4⋊C4×D61—24no—
90C2×C22⋊Dic61—21no—
91C22⋊C4×D61—24no—
92C2×C4×Dic61—21no—
93C42×D61—22no—
94C42×D61—22no—
95#14519710C2318yes—
96#14519708C2318yes—
97C4⋊#92418C2318yes—
98#14519701C2318yes—
99#14519701C2318yes—
100#14519700C2318yes—
101#14519697C2318yes—
102C22⋊#92418C2314yes—
103C22⋊#92418C2314yes—
104#14519688C2314yes—
105C4×#92418C2314yes—
106C22×Q8×C61C2312yes—
107C23×C4×C61C2311yescharacteristic
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61C2314yes—
110#126×C61—28no—
111#126×C61—28no—
112#125×C61—28no—
113#125×C61—28no—
114#125×C61—22no—
115#125×C61C2314yes—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61C2314yes—
119C22⋊Q8×C61C2314yes—
120C4×Q8×C61—24no—
121C4×Q8×C61C2318yes—
122#117×C61D412yes—
123#117×C61D412yes—
124#117×C61C2312yes—
125#117×C61C2312yes—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61—22no—
129C2×C4⋊C4×C61—22no—
130C2×C4⋊C4×C61D412yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C22⋊C4×C61—22no—
133C2×C22⋊C4×C61D412yes—
134C2×C42×C61—22no—
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—28no—
141C2×Q8×C61C2414yes—
142C22×C4×C61—24no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—28no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158#11758222—28no—
159#11758222—28no—
160#11758222—24no—
161#11758222—24no—
162#11758222—24no—
163#11758222#4118yes—
164C2×D244—21no—
165C2×D244—24no—
166C2×D244C2412yes—
167C2×C4×D61—28no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122C2412yes—
175C4⋊C4×C61—28no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—28no—
179C4⋊C4×C61—28no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C2×D414yes—
182C4⋊C4×C61C2×D414yes—
183C4⋊C4×C61C2414yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61C2×D414yes—
187C22⋊C4×C61C2×D414yes—
188C42×C61—28no—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191C42×C61C2412yes—
192C42×C61C2412yes—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758202—216no—
196#11758201—216no—
197#11758198—24no—
198#11758198—24no—
199C4⋊Dic61—22no—
200C4⋊Dic61C2414yes—
201#11758196—24no—
202#11758196—24no—
203C4×Dic61—22no—
204C4×Dic61—24no—
205C4×Dic61—24no—
206C4×Dic61—22no—
207#11758194—28no—
208C2×#92418C2418yes—
209C23×C61—22no—
210C23×C61—42no—
211C23×C61C22×D411yescharacteristic
212C23×C61C22×D411yescharacteristic
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—82no—
217Q8×C61—28no—
218Q8×C61—44no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D414yes—
232C2×C4×C61C22×D414yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4114yes—
236C2×C4×C61C2511yescharacteristic
237C22⋊D61—28no—
238C22⋊D61—28no—
239C2×Dic61—24no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—82no—
243D244—24no—
244D244—22no—
245D244C2×#4114yes—
246C4×D61—28no—
247C4×D61—28no—
248C4×D61—84no—
249C4×D61—84no—
250Dic122—24no—
251Dic122—22no—
252Dic122C2×#4114yes—
253#92418—28no—
254#60569—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—82no—
261C22×C61#52512yes—
262C22×C61C23×D411yescharacteristic
263C2×D61—24no—
264C2×D61—44no—
265C2×D61—84no—
266C4×C61—24no—
267C4×C61—24no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—44no—
271C4×C61—44no—
272C4×C61#52512yes—
273C4×C61C22×#4112yes—
274Dic61—24no—
275Dic61—44no—
276C2×#541D6111yescharacteristic, complemented
277#3428—611no—
278#3127—614no—
279#3123—614no—
280#3120—618no—
281#3095—612no—
282#3080—612no—
283#3039—612no—
284#3007—612no—
285C4×#137—614no—
286#2921—611no—
287C2×C61—22no—
288C2×C61—41no—
289C2×C61—41no—
290C2×C61C2×#52511yescharacteristic
291C2×C61#358612yes—
292D61—44no—
293C22×#41—611no—
294C2×#137—614no—
295#541—22no—
296#541—22no—
297#541C2×D6114yes—
298#541C2×D6114yescomplemented
299C2×C4⋊Q8C2×D6111yescharacteristic, complemented
300C2×#126—22no—
301C2×#125—22no—
302C2×#125C2×D6111yescharacteristic, complemented
303C2×C22⋊Q8C2×D6112yes—
304#520—614no—
305#520—1222no—
306#519—612no—
307#518—612no—
308#518—1222no—
309C4×#41—612no—
310C4×#41—612no—
311C4×#41—1222no—
312C2×C4×Q8C2×D6112yescomplemented
313C2×#117—611no—
314C2×#117—1222no—
315C2×#117C2×D6111yescharacteristic
316#485—618no—
317#484—614no—
318#483—614no—
319#479—618no—
320#478—618no—
321#462—618no—
322#461—618no—
323#452—618no—
324#451—614no—
325#450—614no—
326#441—618no—
327#440—618no—
328C4×Q16—618no—
329C4×SD16—618no—
330#433—614no—
331#433—614no—
332#428—614no—
333#423—612no—
334#419—614no—
335#417—614no—
336#407—612no—
337C4×M16—612no—
338C61#6056911yescharacteristic, complemented, Sylow
339C2×#41—612no—
340C2×#41—612no—
341C2×#41—1228no—
342C2×#41—1221no—
343C2×#41—1221no—
344C22×Q8—611no—
345C22×Q8C22×D6112yes—
346C22×D4—611no—
347C22×D4—1221no—
348C23×C4—1221no—
349C23×C4C22×D6111yescharacteristic
350#137—12216no—
351C2×Q16—618no—
352C2×SD16—618no—
353C2×M16—614no—
354C4⋊Q8—22no—
355C4⋊Q8C22×D6114yes—
356#126—28no—
357#126—28no—
358#125—28no—
359#125—28no—
360#125—22no—
361#125C22×D6114yes—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8C22×D6114yes—
365C22⋊Q8C22×D6114yes—
366C4×Q8—24no—
367C4×Q8—614no—
368C4×Q8—614no—
369C4×Q8—1224no—
370C4×Q8C22×D6118yes—
371C4×D4—614no—
372C4×D4—614no—
373C4×D4—1224no—
374C4×D4—1224no—
375C4×D4—1224no—
376C4×D4—1224no—
377C4×D4—1224no—
378#117—612no—
379#117—612no—
380#117—1221no—
381#117—1221no—
382#117—1224no—
383#117—1224no—
384#117—1222no—
385#117—1222no—
386#117C22⋊D6112yes—
387#117C22⋊D6112yes—
388#117C22×D6112yes—
389#117C22×D6112yes—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4C22⋊D6112yes—
397C2×C4⋊C4C22×D6112yes—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4C22⋊D6112yes—
402C2×C42—22no—
403C2×C42—1221no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#107—618no—
409#106—618no—
410C4⋊C8—618no—
411#103—618no—
412#103—618no—
413#103—618no—
414#102—618no—
415C22⋊C8—614no—
416C22⋊C8—614no—
417#97—614no—
418C4×C8—614no—
419C24—1221no—
420C24C2×C22⋊D6111yescharacteristic
421#41—618no—
422#41—1224no—
423#41—1224no—
424#41—1228no—
425#41—1228no—
426#41—1224no—
427C2×Q8—22no—
428C2×Q8—28no—
429C2×Q8—612no—
430C2×Q8—1224no—
431C2×Q8—1221no—
432C2×Q8C23×D6114yes—
433C2×D4—612no—
434C2×D4—1224no—
435C2×D4—1228no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1221no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—1228no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6112yes—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C23×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457Q16—12216no—
458SD16—12216no—
459M16—1228no—
460C2×C8—618no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—614no—
468C4⋊C4—1222no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4C2×C22⋊D6114yes—
472C4⋊C4C2×C22⋊D6114yes—
473C4⋊C4C23×D6114yes—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C22⋊C4C2×C22⋊D6114yes—
481C22⋊C4C2×C22⋊D6114yes—
482C42—28no—
483C42—1222no—
484C42—1222no—
485C42—1224no—
486C42—1224no—
487C42C2×C22⋊D6112yes—
488C42C2×C22⋊D6112yes—
489C42C23×D6112yes—
490C42C23×D6112yes—
491C23—22no—
492C23—42no—
493C23—1224no—
494C23—1222no—
495C23—4882no—
496C23C22×C22⋊D6111yescharacteristic
497C23C22×C22⋊D6111yescharacteristic
498C23C22×C22⋊D6111yescharacteristic
499Q8—28no—
500Q8—44no—
501Q8—614no—
502Q8—1224no—
503Q8—1222no—
504D4—614no—
505D4—1224no—
506D4—1228no—
507D4—1228no—
508D4—1222no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—1228no—
521C2×C4—1224no—
522C2×C4—1222no—
523C2×C4—1228no—
524C2×C4—1224no—
525C2×C4—4884no—
526C2×C4—4884no—
527C2×C4—4882no—
528C2×C4C22×C22⋊D6114yes—
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C22×C22⋊D6114yes—
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4C22×C22⋊D6111yescharacteristic
533C2×C4#41×D6114yes—
534C2×C4C24×D6111yescharacteristic
535C8—1228no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—82no—
542C22—1224no—
543C22—4884no—
544C22—2444no—
545C22#1684026912yes—
546C22C23×C22⋊D6111yescharacteristic
547C4—21no—
548C4—21no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—1224no—
554C4—2444no—
555C4#1684026912yes—
556C4C2×#41×D6112yes—
557C2—22no—
558C2—41no—
559C2—41no—
560C2—2444no—
561C2C2×#1684026911yescharacteristic
562C2#2146060612yes—
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#541C2
socleC22×C61C23×D4
prime radicalC22×#11758222C22
hypercenterC2×#541D61
hypocenterC61#60569
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
2. #21456114C2semidirect
3. #21456066C2semidirect
4. #21456027C2semidirect
5. #21455763C2semidirect
6. #21455637C2semidirect
7. #21455302C2semidirect
9. C4×#14519808C2semidirect
10. #21454457C2semidirect
11. C2×C61×#541C2semidirect
25. #16840095C22semidirect
26. #16840090C22semidirect
27. #16840066C22semidirect
28. #16840059C22semidirect
29. #16839969C22semidirect
30. #16839964C22semidirect
34. #16839867C22semidirect
35. #16839860C22semidirect
36. C4×#11758202C22semidirect
37. C4×#11758201C22semidirect
39. #16839823C22semidirect
40. #16839802C22semidirect
41. #16839786C22semidirect
42. #16839771C22semidirect
45. C4×#11758194C22semidirect
49. C61×#541C22semidirect
50. C2×C4⋊Q8×C61C22semidirect
53. C2×#125×C61C22semidirect
55. C2×C4×Q8×C61C22semidirect
276. C2×#541D61semidirect
298. #541C2×D61semidirect
299. C2×C4⋊Q8C2×D61semidirect
302. C2×#125C2×D61semidirect
312. C2×C4×Q8C2×D61semidirect
338. C61#60569semidirect

Maximal subgroups

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

subgroupindex
#214590202
#214561142
#214560662
#214560272
#214557632
#214556372
#214553022
#214550772
C4×#145198082
#214544572
C2×C61×#5412
#6056961