Finite Group Databasebeta

← #29653665

Subgroups — #29653665

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C211yesmaximal, characteristic
2C2×#16840090C211yesmaximal, characteristic, complemented
3#21450298C211yesmaximal, characteristic, complemented
4#21444827C214yesmaximal, complemented
5#21444816C214yesmaximal, complemented
6#21444339C211yesmaximal, characteristic, complemented
7#21444310C211yesmaximal, characteristic
8#21444183C211yesmaximal, characteristic, complemented
9C61×#3538C211yesmaximal, characteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16840095—21no—
12#16840090—22no—
13#16840090C2214yes—
14#16840090C2212yes—
15#16840078—21no—
16#16840073C2212yes—
17#16839805—21no—
18C2×#14519708—21no—
19#16839763—21no—
20C2×#14519701—21no—
21#16839726C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839692C2214yes—
24#16839669—21no—
25#16839634—22no—
26#16839631—21no—
27C4×C22⋊Dic61C2211yescharacteristic
28#16839499C2214yes—
29#16839499C2212yes—
30#16839496C2212yes—
31#16839488C2214yes—
32C61×#563—22no—
33C61×#552—21no—
34C61×#552—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37D4×Q8×C61—21no—
38C61×#544—21no—
39C61×#544C2212yes—
40C61×#541—21no—
41C61×#539—21no—
42C61×#539C2212yes—
43C61×#536—21no—
44C61×#525C2211yescharacteristic
45C2×C22⋊Q8×C61C2211yescharacteristic
46C2×C22⋊Q8×C61C2211yescharacteristic
47C23×Dic61—21no—
48C2×#11758194—21no—
49C22×#92418—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—24no—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61—21no—
59#14519710—22no—
60#14519708—24no—
61#14519708—22no—
62#14519701—24no—
63#14519701—22no—
64#14519698—22no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67C22⋊#92418C2312yes—
68C22⋊#92418C2312yes—
69#14519684—24no—
70#14519684C2312yes—
71C2×#41×C61D411yescharacteristic
72C22×Q8×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×Q8×C61D411yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—22no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61—22no—
80#126×C61—22no—
81#126×C61—22no—
82#126×C61—22no—
83#126×C61—22no—
84#125×C61—21no—
85#125×C61—22no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61—22no—
89#124×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—22no—
94#123×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61D411yescharacteristic
108C22⋊Q8×C61D412yes—
109C22⋊Q8×C61C2312yes—
110C22⋊Q8×C61C2311yescharacteristic
111#121×C61D411yescharacteristic
112#121×C61C2311yescharacteristic
113#120×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120#117×C61—21no—
121#117×C61—22no—
122#117×C61—22no—
123#117×C61D411yescharacteristic
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61C2312yes—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61D412yes—
130C24×C61C2×D411yescharacteristic
131#41×C61—42no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61—42no—
138C2×Q8×C61—82no—
139C2×Q8×C61C2×D412yes—
140C2×Q8×C61C2×D411yescharacteristic
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—41no—
161C22×Dic61—41no—
162C22×Dic61—82no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C42×C61—22no—
196C42×C61—22no—
197C42×C61—22no—
198C42×C61—21no—
199C22⋊Dic61—24no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204#11758196—84no—
205C4×Dic61—42no—
206#11758194—42no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—42no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—41no—
214C23×C61#12011yescharacteristic
215C23×C61#12011yescharacteristic
216C23×C61C22×D411yescharacteristic
217Q8×C61—42no—
218Q8×C61—41no—
219Q8×C61—42no—
220Q8×C61—42no—
221Q8×C61—81no—
222Q8×C61—82no—
223D4×C61—41no—
224D4×C61—42no—
225D4×C61—42no—
226D4×C61—81no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—81no—
245C2×C4×C61—82no—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×Dic61—24no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—44no—
255C2×Dic61—82no—
256C2×Dic61—82no—
257#92418—42no—
258#92418—42no—
259#18268—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—44no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#45612yes—
272C22×C61C2×#12011yescharacteristic
273C22×C61#57712yes—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#45812yes—
285Dic61—44no—
286Dic61—82no—
287#3538D6111yescharacteristic
288#3092—611no—
289C2×#483—611no—
290#2502—611no—
291#1611—614no—
292#1609—614no—
293#1508—611no—
294#1501—611no—
295#1472—611no—
296C2×C61—22no—
297C2×C61—22no—
298C2×C61—41no—
299C2×C61—81no—
300C2×C61#301011yescharacteristic
301C2×C61C2×#45611yescharacteristic
302C2×C61C2×#45811yescharacteristic
303#563—22no—
304#552—21no—
305#552—21no—
306#551—22no—
307D4×Q8—21no—
308D4×Q8—21no—
309#544—21no—
310#544C2×D6112yes—
311#541—21no—
312#539—21no—
313#539C2×D6112yes—
314#536—21no—
315#525C2×D6111yescharacteristic
316C2×C22⋊Q8C2×D6111yescharacteristic
317C2×C22⋊Q8C2×D6111yescharacteristic
318C22×C4⋊C4—611no—
319#484—1221no—
320#483—614no—
321#483—612no—
322#483—1222no—
323#481—1221no—
324#480—612no—
325#428—1221no—
326C2×#106—1221no—
327#417—1221no—
328C2×#103—1221no—
329#407—611no—
330C2×C22⋊C8—611no—
331#398—614no—
332#391—1221no—
333#385—1222no—
334#384—1221no—
335C4×C22⋊C4—611no—
336#328—614no—
337#328—612no—
338#327—612no—
339#324—614no—
340C61#1826811yescharacteristic, complemented, Sylow
341C2×#41C22⋊D6111yescharacteristic
342C22×Q8—21no—
343C22×Q8C22⋊D6111yescharacteristic
344C22×Q8C22⋊D6111yescharacteristic
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C2×M16—1221no—
348C22×C8—1221no—
349C4⋊Q8—21no—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#126—22no—
357#125—21no—
358#125—22no—
359#125—22no—
360#125—22no—
361#125—22no—
362#124—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—21no—
367#123—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8C22⋊D6112yes—
381C22⋊Q8C22⋊D6111yescharacteristic
382C22⋊Q8C22×D6112yes—
383C22⋊Q8C22×D6111yescharacteristic
384#121C22⋊D6111yescharacteristic
385#121C22×D6111yescharacteristic
386#120—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—22no—
393#117—21no—
394#117—22no—
395#117—22no—
396#117C22⋊D6111yescharacteristic
397C2×C4⋊C4—21no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22⋊D6111yescharacteristic
406C2×C4⋊C4C22⋊D6111yescharacteristic
407C2×C4⋊C4C22×D6112yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4C22⋊D6112yes—
411C2×C42—1221no—
412#107—1222no—
413#106—1224no—
414#106—1222no—
415#103—1224no—
416#103—1222no—
417#102—1222no—
418C22⋊C8—612no—
419C22⋊C8—612no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422#95—612no—
423#95—1224no—
424C24C2×C22⋊D6111yescharacteristic
425#41—42no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—82no—
433C2×Q8C2×C22⋊D6112yes—
434C2×Q8C2×C22⋊D6111yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×C22⋊D6111yescharacteristic
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—4882no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4C2×C22⋊D6112yes—
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457M16—2442no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—2442no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—4884no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4C2×C22⋊D6112yes—
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—1224no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—2442no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—41no—
508C23#1451974611yescharacteristic
509C23#1451974611yescharacteristic
510C23C22×C22⋊D6111yescharacteristic
511Q8—42no—
512Q8—41no—
513Q8—42no—
514Q8—42no—
515Q8—81no—
516Q8—82no—
517D4—41no—
518D4—42no—
519D4—42no—
520D4—81no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—1224no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2442no—
544C2×C4—2444no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974611yescharacteristic
548C2×C4#1451974611yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C8—2442no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—44no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22#1683994512yes—
565C22C2×#1451974611yescharacteristic
566C22#1684057012yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—4882no—
578C4—2444no—
579C4#1683995212yes—
580C2—22no—
581C2—22no—
582C2—41no—
583C2—81no—
584C2#2145510511yescharacteristic
585C2C2×#1683994511yescharacteristic
586C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3538C2
socleC22×C61C2×#120
prime radicalC61×#525C22
hypercenter#3538D61
hypocenterC61#18268
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
2. C2×#16840090C2semidirect
3. #21450298C2semidirect
4. #21444827C2semidirect
5. #21444816C2semidirect
6. #21444339C2semidirect
8. #21444183C2semidirect
340. C61#18268semidirect

Maximal subgroups

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

subgroupindex
#214557392
C2×#168400902
#214502982
#214448272
#214448162
#214443392
#214443102
#214441832
C61×#35382
#1826861