Finite Group Databasebeta

← #29652882

Subgroups — #29652882

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455914C211yesmaximal, characteristic, complemented
2#21455906C211yesmaximal, characteristic, complemented
3#21449098C211yesmaximal, characteristic, complemented
4#21444867C214yesmaximal, complemented
5#21444765C214yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444327C211yesmaximal, characteristic, complemented
8#21444160C211yesmaximal, characteristic, complemented
9C61×#3506C211yesmaximal, characteristic, complemented
10C2×C4×D244C2211yescharacteristic
11C2×C4×C22⋊D61—22no—
12#16840095C2212yes—
13#16840078C2212yes—
14#16840049—21no—
15#16840039—21no—
16#16839964—21no—
17#16839954—21no—
18#16839912C2212yes—
19#16839895C2212yes—
20#16839872—22no—
21#16839860—22no—
22#16839819C2211yescharacteristic
23#16839819C2211yescharacteristic
24#16839698C2212yes—
25#16839697—21no—
26#16839671C2212yes—
27#16839670—21no—
28#16839608C2211yescharacteristic
29#16839499C2214yes—
30#16839496C2214yes—
31#16839486C2214yes—
32C61×#559—22no—
33C61×#556—21no—
34C61×#555—21no—
35C61×#553—22no—
36C61×#550—21no—
37C61×#547—21no—
38C61×#538C2214yes—
39C61×#535—21no—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#525—22no—
43C61×#520—21no—
44C2×C4×D4×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758199—22no—
50C4×D244—21no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244—21no—
54C4×C22⋊D61—48no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—22no—
57C2×C22⋊Dic61—22no—
58C2×#11758198—22no—
59C2×#11758198—21no—
60C2×C4×Dic61—21no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65#14519701—22no—
66#14519698—22no—
67#14519697—22no—
68#14519697—22no—
69C22⋊#92418C2312yes—
70C22⋊#92418C2312yes—
71#14519688—21no—
72#14519688—21no—
73#14519684—21no—
74#14519684C2312yes—
75C2×#41×C61—22no—
76C2×#41×C61—21no—
77C22×D4×C61D411yescharacteristic
78C23×C4×C61D412yes—
79C4⋊Q8×C61D411yescharacteristic, complemented
80C4⋊Q8×C61D411yescharacteristic, complemented
81C4⋊Q8×C61D411yescharacteristic, complemented
82#127×C61—22no—
83#127×C61D411yescharacteristic, complemented
84#126×C61—24no—
85#125×C61—21no—
86#124×C61—22no—
87#124×C61—24no—
88#124×C61—21no—
89#123×C61—24no—
90#123×C61—24no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61C2312yes—
94#121×C61—22no—
95#121×C61—22no—
96#121×C61—22no—
97#121×C61—24no—
98#121×C61—21no—
99#121×C61C2312yes—
100#120×C61—24no—
101C4×Q8×C61—22no—
102C4×Q8×C61—21no—
103C4×Q8×C61D412yescomplemented
104C4×D4×C61—22no—
105C4×D4×C61—24no—
106C4×D4×C61—24no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C4×D4×C61—21no—
112C4×D4×C61D412yescomplemented
113#117×C61—22no—
114#117×C61—22no—
115#117×C61—21no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61C2312yes—
119C2×C42×C61C2311yescharacteristic
120C24×C61C2×D412yes—
121C23×D61—21no—
122#41×C61—44no—
123#41×C61—84no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61C2×D412yes—
127C2×D4×C61—22no—
128C2×D4×C61—24no—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—42no—
135C2×D4×C61C2×D412yes—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—44no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—44no—
146C2×C22⋊D61—44no—
147C2×C22⋊D61—44no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—42no—
153C2×C4×D61—44no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61—24no—
176C42×C61—22no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C22⋊Dic61—44no—
182#11758201—44no—
183#11758199—44no—
184#11758198—44no—
185#11758198—41no—
186#11758198—41no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189#11758196—44no—
190C4×Dic61—24no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—42no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61—42no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—44no—
213D4×C61—42no—
214D4×C61—84no—
215D4×C61—84no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61—82no—
227C2×C4×C61—84no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C22⋊D61—416no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240D244—22no—
241D244—22no—
242D244—41no—
243D244—41no—
244C4×D61—82no—
245C4×D61—82no—
246#92418—42no—
247#92418—42no—
248#18190—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—44no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—82no—
259C22×C61C2×#12011yescharacteristic
260C22×C61#57512yes—
261C2×D61—44no—
262C2×D61—42no—
263C2×D61—82no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61#45312yes—
276Dic61—44no—
277Dic61—81no—
278#3506D6111yescharacteristic, complemented
279#3110—611no—
280#3109—611no—
281#2397—611no—
282#1618—614no—
283#1600—614no—
284#1506—611no—
285#1505—611no—
286#1466—611no—
287C2×C61—22no—
288C2×C61—82no—
289C2×C61—82no—
290C2×C61#300511yescharacteristic
291C2×C61C2×#45311yescharacteristic
292C2×C61#302511yescharacteristic
293D61—82no—
294#559—22no—
295#556—21no—
296#555—21no—
297#553—22no—
298#550—21no—
299#547—21no—
300#538C2×D6114yes—
301#535—21no—
302#533C2×D6111yescharacteristic
303#532C2×D6111yescharacteristic
304#525—22no—
305#520—21no—
306C2×C4×D4—611no—
307C2×C4×D4—1222no—
308C2×C4×D4C2×D6111yescharacteristic
309#484—612no—
310#481—612no—
311#476—1221no—
312#474—1221no—
313#461—1221no—
314#459—1221no—
315#450—612no—
316#447—612no—
317#442—1222no—
318#440—1222no—
319#432—611no—
320#432—611no—
321#399—612no—
322#399—1221no—
323#392—612no—
324#392—1221no—
325#376—611no—
326#328—614no—
327#327—614no—
328#323—614no—
329C61#1819011yescharacteristic, complemented, Sylow
330C2×#41—22no—
331C2×#41—21no—
332C22×D4—611no—
333C22×D4—1222no—
334C22×D4C22⋊D6111yescharacteristic
335C23×C4—1221no—
336C23×C4C22⋊D6112yes—
337C2×SD16—1222no—
338C2×D8—1222no—
339C4⋊Q8C22⋊D6111yescharacteristic, complemented
340C4⋊Q8C22⋊D6111yescharacteristic, complemented
341C4⋊Q8C22⋊D6111yescharacteristic, complemented
342#127—22no—
343#127C22⋊D6111yescharacteristic, complemented
344#126—24no—
345#125—21no—
346#124—22no—
347#124—24no—
348#124—21no—
349#123—24no—
350#123—24no—
351C22⋊Q8—22no—
352C22⋊Q8—21no—
353C22⋊Q8C22×D6112yes—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—24no—
358#121—21no—
359#121C22×D6112yes—
360#120—24no—
361C4×Q8—22no—
362C4×Q8—21no—
363C4×Q8C22⋊D6112yescomplemented
364C4×D4—22no—
365C4×D4—24no—
366C4×D4—24no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—1221no—
373C4×D4—1221no—
374C4×D4—1221no—
375C4×D4—1221no—
376C4×D4—2448no—
377C4×D4C22⋊D6112yescomplemented
378#117—22no—
379#117—22no—
380#117—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4C22⋊D6111yescharacteristic
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4C22×D6112yes—
389C2×C42—1221no—
390C2×C42C22×D6111yescharacteristic
391#107—1222no—
392#106—1222no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395#103—1222no—
396#102—1222no—
397#102—1222no—
398#102—1222no—
399C22⋊C8—612no—
400C22⋊C8—612no—
401#97—1221no—
402#97—1221no—
403#95—612no—
404#95—1221no—
405C24—1221no—
406C24C2×C22⋊D6112yes—
407#41—44no—
408#41—84no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×C22⋊D6112yes—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—2442no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4C2×C22⋊D6112yes—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—2444no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6112yes—
440C22×C4C23×D6111yescharacteristic
441SD16—2444no—
442D8—2444no—
443C2×C8—1222no—
444C2×C8—1222no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—2444no—
457C4⋊C4C2×C22⋊D6112yes—
458C4⋊C4C2×C22⋊D6112yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—2444no—
468C22⋊C4—2444no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—24no—
472C42—22no—
473C42—1224no—
474C42C2×C22⋊D6111yescharacteristic
475C42C2×C22⋊D6111yescharacteristic
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—1222no—
485C23—2442no—
486C23—2441no—
487C23—2441no—
488C23—2442no—
489C23C22×C22⋊D6111yescharacteristic
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—84no—
500D4—84no—
501D4—1222no—
502D4—1222no—
503D4—2441no—
504D4—2441no—
505D4—24416no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—1224no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4#1451974611yescharacteristic
526C2×C4#1451974611yescharacteristic
527C2×C4#1451974611yescharacteristic
528C2×C4#1451974611yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C8—2442no—
532C8—2442no—
533C22—21no—
534C22—21no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—4882no—
544C22—4881no—
545C22—4881no—
546C22—2444no—
547C22—2442no—
548C22C2×#1451974611yescharacteristic
549C22#1684056012yes—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—4881no—
560C4—2444no—
561C4#1683993012yes—
562C2—22no—
563C2—82no—
564C2—82no—
565C2—4882no—
566C2#2145506111yescharacteristic
567C2C2×#1683993011yescharacteristic
568C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18190
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21455914C2semidirect
2. #21455906C2semidirect
3. #21449098C2semidirect
4. #21444867C2semidirect
5. #21444765C2semidirect
6. #21444331C2semidirect
7. #21444327C2semidirect
8. #21444160C2semidirect
9. C61×#3506C2semidirect
79. C4⋊Q8×C61D4semidirect
80. C4⋊Q8×C61D4semidirect
81. C4⋊Q8×C61D4semidirect
83. #127×C61D4semidirect
103. C4×Q8×C61D4semidirect
112. C4×D4×C61D4semidirect
278. #3506D61semidirect
329. C61#18190semidirect
339. C4⋊Q8C22⋊D61semidirect
340. C4⋊Q8C22⋊D61semidirect
341. C4⋊Q8C22⋊D61semidirect
343. #127C22⋊D61semidirect
363. C4×Q8C22⋊D61semidirect
377. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214559062
#214490982
#214448672
#214447652
#214443312
#214443272
#214441602
C61×#35062
#1819061