Finite Group Databasebeta

← #29661558

Subgroups — #29661558

Order 15616.

884 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#21457037C211yesmaximal, characteristic, complemented
2#21456201C211yesmaximal, characteristic, complemented
3C2×#16840121C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445142C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172C2212yescomplemented
12#16840132C2214yescomplemented
13#16840121—22no—
14#16840121C2214yes—
15#16840121C2212yescomplemented
16#16840114—21no—
17#16840114—21no—
18#16840090—21no—
19#16840073—21no—
20#16840057—21no—
21#16840052—21no—
22#16839819C2211yescharacteristic
23#16839770—22no—
24C2×#14519701—21no—
25C2×#14519698—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2214yescomplemented
29#16839504C2212yescomplemented
30#16839502C2214yes—
31#16839502C2212yescomplemented
32C61×#563C2212yes—
33C61×#562C2212yes—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543—21no—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538—21no—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59#14519701—24no—
60#14519701—22no—
61#14519700—24no—
62#14519700—24no—
63#14519698—24no—
64#14519698—22no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C2×#41×C61—21no—
72C22×Q8×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61D411yescharacteristic
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79#126×C61—22no—
80#126×C61—22no—
81#126×C61—22no—
82#126×C61—21no—
83#125×C61C2312yes—
84#125×C61C2311yescharacteristic
85#124×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61D412yes—
91#124×C61C2311yescharacteristic
92#124×C61C2312yes—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—21no—
96#123×C61—22no—
97#123×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61D411yescharacteristic
109#121×C61—21no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61D411yescharacteristic
113#120×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61D411yescharacteristic
123#117×C61—22no—
124#117×C61—21no—
125#117×C61D412yes—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61—21no—
131#41×C61—82no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61—82no—
138C2×Q8×C61C2×D411yescharacteristic
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—42no—
145C2×D4×C61C2×D411yescharacteristic
146C2×D4×C61C2×D412yes—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×Dic61—41no—
156C2×Dic122—22no—
157C2×Dic122—44no—
158C2×Dic122—41no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—21no—
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201#11758194—42no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—42no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—42no—
209C23×C61—81no—
210C23×C61—81no—
211C23×C61#12011yescharacteristic
212Q8×C61—42no—
213Q8×C61—41no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—81no—
217Q8×C61—82no—
218D4×C61—42no—
219D4×C61—41no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—82no—
223D4×C61—82no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—81no—
235C2×C4×C61—82no—
236C2×C4×C61—82no—
237C2×C4×C61—82no—
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×Dic61—42no—
245C2×Dic61—82no—
246Dic122—44no—
247Dic122—82no—
248#92418—42no—
249#92418—42no—
250#92418—42no—
251#19027—611nomaximal, Sylow
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—41no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—162no—
261C22×C61#45412yes—
262C22×C61C2×#12011yescharacteristic
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61#45512yes—
275Dic61—82no—
276#3561D6111yescharacteristic
277#3238—611no—
278#3139—611no—
279C2×#489—611no—
280#1664—614no—
281#1662—614no—
282#1531—611no—
283#1527—611no—
284#1524—611no—
285C2×C61—22no—
286C2×C61—81no—
287C2×C61—82no—
288C2×C61C2×#45511yescharacteristic
289C2×C61C2×#45411yescharacteristic
290C2×C61#302211yescharacteristic
291#563C2×D6112yes—
292#562C2×D6112yes—
293#556—21no—
294#551—22no—
295D4×Q8—21no—
296#548—21no—
297#546—21no—
298#543—21no—
299#542—21no—
300#542—22no—
301#538—21no—
302#536—21no—
303#533C2×D6111yescharacteristic
304C2×C4⋊Q8—611no—
305#529C2×D6111yescharacteristic
306C2×#124C2×D6111yescharacteristic
307#501—612no—
308#491—614no—
309#489—614no—
310#489—612no—
311#489—1222no—
312#487—1221no—
313#487—1221no—
314#483—1221no—
315#480—1221no—
316#477—1221no—
317#476—1221no—
318#432—611no—
319#419—1222no—
320C2×#103—1221no—
321C2×#102—1221no—
322C4×M16—611no—
323C2×#97—611no—
324#330—614no—
325#330—612no—
326#329—614no—
327#329—612no—
328C61#1902711yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C22×Q8—21no—
331C22×Q8—1221no—
332C22×Q8C22⋊D6111yescharacteristic
333C22×D4C22⋊D6111yescharacteristic
334C2×M16—1221no—
335C22×C8—1221no—
336C4⋊Q8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—612no—
339C4⋊Q8—1221no—
340C4⋊Q8—1221no—
341C4⋊Q8—1221no—
342C4⋊Q8C22⋊D6111yescharacteristic
343C4⋊Q8C22⋊D6111yescharacteristic
344#126—22no—
345#126—22no—
346#126—22no—
347#126—21no—
348#125C22×D6111yescharacteristic
349#125C22×D6112yes—
350#124—21no—
351#124—22no—
352#124—22no—
353#124—21no—
354#124—21no—
355#124C22⋊D6112yes—
356#124C22×D6112yes—
357#124C22×D6111yescharacteristic
358#123—22no—
359#123—22no—
360#123—21no—
361#123—22no—
362#123—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8C22⋊D6111yescharacteristic
374#121—21no—
375#121—22no—
376#121—22no—
377#121C22⋊D6111yescharacteristic
378#120—22no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C22⋊D6111yescharacteristic
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4C22⋊D6111yescharacteristic
388#117—22no—
389#117—21no—
390#117C22⋊D6112yes—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42C22×D6111yescharacteristic
397#107—1222no—
398#106—1222no—
399C4⋊C8—1221no—
400#103—1224no—
401#103—1222no—
402#103—1224no—
403#103—1224no—
404#102—1224no—
405#102—1222no—
406C22⋊C8—1221no—
407#97—612no—
408#97—612no—
409#97—612no—
410#97—1221no—
411C4×C8—1221no—
412C24—21no—
413#41—82no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8—1222no—
421C2×Q8—2444no—
422C2×Q8—2441no—
423C2×Q8C2×C22⋊D6111yescharacteristic
424C2×Q8C2×C22⋊D6112yes—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—42no—
430C2×D4C2×C22⋊D6111yescharacteristic
431C2×D4C2×C22⋊D6112yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—2441no—
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441M16—2442no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—2442no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42C2×C22⋊D6111yescharacteristic
485C42C2×C22⋊D6111yescharacteristic
486C42C2×C22⋊D6111yescharacteristic
487C42C23×D6111yescharacteristic
488C23—22no—
489C23—21no—
490C23—42no—
491C23—81no—
492C23—81no—
493C23#1451974611yescharacteristic
494Q8—42no—
495Q8—41no—
496Q8—42no—
497Q8—42no—
498Q8—81no—
499Q8—82no—
500Q8—4882no—
501Q8—2444no—
502D4—42no—
503D4—41no—
504D4—42no—
505D4—42no—
506D4—82no—
507D4—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—4882no—
523C2×C4—2442no—
524C2×C4#1451974611yescharacteristic
525C2×C4#1451974611yescharacteristic
526C2×C4#1451974611yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C8—2442no—
531C8—2442no—
532C8—2442no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—162no—
542C22#1683993512yes—
543C22C2×#1451974611yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—4882no—
556C4#1683994012yes—
557C2—22no—
558C2—82no—
559C2—81no—
560C2C2×#1683994011yescharacteristic
561C2C2×#1683993511yescharacteristic
562C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#19027
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456201C2semidirect
3. C2×#16840121C2semidirect
4. #21445153C2semidirect
5. #21445142C2semidirect
6. #21444434C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
11. #16840172C22semidirect
12. #16840132C22semidirect
15. #16840121C22semidirect
28. #16839504C22semidirect
29. #16839504C22semidirect
31. #16839502C22semidirect
328. C61#19027semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562012
C2×#168401212
#214451532
#214451422
#214444342
#214444172
#214444072
C61×#35612
#1902761