Finite Group Databasebeta

← #29655118

Subgroups — #29655118

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456764C211yesmaximal, characteristic, complemented
3#21451759C211yesmaximal, characteristic, complemented
4#21444799C214yesmaximal, complemented
5#21444776C214yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444171C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840157—21no—
12#16840141—21no—
13#16840132—21no—
14#16840114—21no—
15#16839972—22no—
16#16839972—21no—
17#16839967—21no—
18#16839917C2214yes—
19#16839917C2212yes—
20#16839900C2212yes—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839687—22no—
24#16839674C2214yes—
25#16839669—21no—
26#16839665—21no—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839499C2214yes—
30#16839496C2212yes—
31#16839486C2214yes—
32C61×#563—21no—
33C61×#562—21no—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543C2212yes—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538C2212yes—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C2×#11758202—24no—
49C2×#11758202—22no—
50C2×#11758200—22no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C22⋊Dic61—24no—
58C2×C4×Dic61—21no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519700—22no—
62#14519700—24no—
63#14519700—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—24no—
70#14519684C2312yes—
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×C61—21no—
84#125×C61D411yescharacteristic
85#124×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61—21no—
91#124×C61—21no—
92#124×C61D411yescharacteristic
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—22no—
96#123×C61—21no—
97#123×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2312yes—
108C22⋊Q8×C61C2311yescharacteristic
109#121×C61—21no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61C2311yescharacteristic
113#120×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61D412yes—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—21no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61D411yescharacteristic
123#117×C61—22no—
124#117×C61—21no—
125#117×C61—21no—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61D412yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C24×C61C2×D412yes—
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—21no—
145C2×D4×C61—42no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C22×Dic61—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—82no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—21no—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C22⋊Dic61—84no—
199#11758202—82no—
200#11758202—84no—
201#11758200—82no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206C4×Dic61—42no—
207C2×#92418—22no—
208C2×#92418—22no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—24no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61C22×D411yescharacteristic
215Q8×C61—41no—
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—42no—
219Q8×C61—81no—
220Q8×C61—82no—
221D4×C61—41no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—81no—
225D4×C61—82no—
226D4×C61—82no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61—82no—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—44no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252Dic122—42no—
253Dic122—42no—
254Dic122—44no—
255#92418—42no—
256#92418—42no—
257#18429—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—41no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—41no—
266C22×C61—84no—
267C22×C61C2×#12011yescharacteristic
268C22×C61#57812yes—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61#45512yes—
281Dic61—82no—
282Dic61—82no—
283#3561D6111yescharacteristic
284#3218—611no—
285#3209—611no—
286#2640—611no—
287#1606—614no—
288#1602—614no—
289#1506—611no—
290#1500—611no—
291#1469—611no—
292C2×C61—22no—
293C2×C61—81no—
294C2×C61—82no—
295C2×C61#301111yescharacteristic
296C2×C61C2×#45511yescharacteristic
297C2×C61#302711yescharacteristic
298#563—21no—
299#562—21no—
300#556—21no—
301#551—22no—
302D4×Q8—21no—
303#548—21no—
304#546—21no—
305#543C2×D6112yes—
306#542—21no—
307#542—22no—
308#538C2×D6112yes—
309#536—21no—
310#533C2×D6111yescharacteristic
311C2×C4⋊Q8—611no—
312#529C2×D6111yescharacteristic
313C2×#124C2×D6111yescharacteristic
314#497—1221no—
315#494—1221no—
316#491—1221no—
317#487—1221no—
318#462—1222no—
319#462—1221no—
320#461—1221no—
321#451—612no—
322#451—614no—
323#448—612no—
324#432—611no—
325#431—611no—
326#397—1222no—
327#393—614no—
328#391—1221no—
329#390—1221no—
330#376—611no—
331#328—612no—
332#328—614no—
333#327—612no—
334#323—614no—
335C61#1842911yescharacteristic, complemented, Sylow
336C2×#41—21no—
337C22×Q8—21no—
338C22×Q8—612no—
339C22×Q8C22⋊D6111yescharacteristic
340C22×D4C22⋊D6111yescharacteristic
341C2×Q16—1224no—
342C2×Q16—1222no—
343C2×SD16—1222no—
344C4⋊Q8—21no—
345C4⋊Q8—22no—
346C4⋊Q8—1221no—
347C4⋊Q8—1221no—
348C4⋊Q8—1221no—
349C4⋊Q8—1221no—
350C4⋊Q8C22⋊D6111yescharacteristic
351C4⋊Q8C22⋊D6111yescharacteristic
352#126—22no—
353#126—22no—
354#126—22no—
355#126—21no—
356#125—21no—
357#125C22⋊D6111yescharacteristic
358#124—21no—
359#124—22no—
360#124—22no—
361#124—21no—
362#124—21no—
363#124—21no—
364#124—21no—
365#124C22⋊D6111yescharacteristic
366#123—22no—
367#123—22no—
368#123—22no—
369#123—21no—
370#123—22no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8C22×D6111yescharacteristic
381C22⋊Q8C22×D6112yes—
382#121—21no—
383#121—22no—
384#121—22no—
385#121C22×D6111yescharacteristic
386#120—22no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8C22⋊D6111yescharacteristic
390C4×Q8C22⋊D6112yes—
391C4×D4—21no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—22no—
395C4×D4C22⋊D6111yescharacteristic
396#117—22no—
397#117—21no—
398#117—21no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4C22⋊D6112yes—
404C2×C22⋊C4C22×D6112yes—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409#103—1222no—
410#103—1222no—
411#103—1224no—
412C22⋊C8—612no—
413C22⋊C8—612no—
414#97—1221no—
415C4×C8—1221no—
416#95—612no—
417#95—1221no—
418#95—1224no—
419C24C2×C22⋊D6112yes—
420#41—82no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—4882no—
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×Q8C2×C22⋊D6112yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4C2×C22⋊D6111yescharacteristic
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6112yes—
448C22×C4C23×D6111yescharacteristic
449Q16—4882no—
450Q16—4884no—
451SD16—4882no—
452C2×C8—1222no—
453C2×C8—1222no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C4⋊C4C2×C22⋊D6112yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—4884no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—2442no—
494C42C2×C22⋊D6111yescharacteristic
495C42C2×C22⋊D6111yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C23—22no—
499C23—21no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23C22×C22⋊D6111yescharacteristic
504Q8—41no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—81no—
509Q8—82no—
510Q8—2442no—
511Q8—2442no—
512Q8—2444no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—81no—
517D4—82no—
518D4—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4#1451974611yescharacteristic
539C2×C4#1451974611yescharacteristic
540C2×C4#1451974611yescharacteristic
541C2×C4#1451974611yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C8—2442no—
545C8—2442no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—84no—
555C22C2×#1451974611yescharacteristic
556C22#1684057712yes—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—4882no—
569C4—4882no—
570C4#1683994012yes—
571C2—22no—
572C2—82no—
573C2—81no—
574C2#2145511411yescharacteristic
575C2C2×#1683994011yescharacteristic
576C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#18429
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21456764C2semidirect
3. #21451759C2semidirect
4. #21444799C2semidirect
5. #21444776C2semidirect
6. #21444331C2semidirect
7. #21444305C2semidirect
8. #21444171C2semidirect
335. C61#18429semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214567642
#214517592
#214447992
#214447762
#214443312
#214443052
#214441712
C61×#35612
#1842961