Finite Group Databasebeta

← #29572305

Subgroups — #29572305

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446805C211yesmaximal, characteristic, complemented
2#21446288C211yesmaximal, characteristic, complemented
3#21446253C211yesmaximal, characteristic, complemented
4#21446122C211yesmaximal, characteristic
5#21444362C211yesmaximal, characteristic
6#21444339C211yesmaximal, characteristic, complemented
7C61×#3538C211yesmaximal, characteristic
8C22×#11758194C2211yescharacteristic
9C2×#14519704—21no—
10C2×#14519701—21no—
11C2×#14519696—21no—
12#16839726C2211yescharacteristic
13#16839622C2211yescharacteristic
14C4×C22⋊Dic61C2211yescharacteristic
15#16839531—21no—
16#16839529—22no—
17#16839524—21no—
18#16839521—21no—
19#16839499—21no—
20#16839499—22no—
21#16839496—21no—
22C61×#563—22no—
23C61×#552—21no—
24C61×#552—21no—
25C61×#551—22no—
26D4×Q8×C61—21no—
27D4×Q8×C61—21no—
28C61×#544—21no—
29C61×#544C412yescomplemented
30C61×#541—21no—
31C61×#539—21no—
32C61×#539C412yescomplemented
33C61×#536—21no—
34C61×#525C2211yescharacteristic
35C2×C22⋊Q8×C61C2211yescharacteristic
36C2×C22⋊Q8×C61C2211yescharacteristic
37C2×#11758194—21no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194—42no—
41C22×#92418—21no—
42C2×C4⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C4×Dic61—21no—
46#14519704—42no—
47#14519701—42no—
48#14519696—42no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—21no—
52#14519684—21no—
53C2×#41×C61D411yescharacteristic
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×Q8×C61D411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61—22no—
62#126×C61—22no—
63#126×C61—22no—
64#126×C61—22no—
65#126×C61—22no—
66#125×C61—21no—
67#125×C61—22no—
68#125×C61—22no—
69#125×C61—22no—
70#125×C61—22no—
71#124×C61—22no—
72#123×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—21no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61C2×C412yes—
90C22⋊Q8×C61C2×C411yescharacteristic
91C22⋊Q8×C61D411yescharacteristic
92C22⋊Q8×C61D412yes—
93#121×C61C2×C411yescharacteristic
94#121×C61D411yescharacteristic
95#120×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102#117×C61—21no—
103#117×C61—22no—
104#117×C61—22no—
105#117×C61D411yescharacteristic
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2×C412yes—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D412yes—
112C24×C61C2×D411yescharacteristic
113#41×C61—41no—
114#41×C61—41no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—41no—
120C2×Q8×C61—41no—
121C2×Q8×C61—41no—
122C2×Q8×C61—41no—
123C2×Q8×C61—81no—
124C2×Q8×C61—81no—
125C2×Q8×C61C22⋊C412yes—
126C2×Q8×C61C22⋊C411yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C22⋊C411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—41no—
136C22×C4×C61—41no—
137C22×C4×C61C22⋊C412yes—
138C22×C4×C61C22⋊C412yes—
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×Dic61—41no—
146C22×Dic61—41no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C22⋊C412yes—
165C4⋊C4×C61C22⋊C411yescharacteristic
166C4⋊C4×C61C22⋊C411yescharacteristic
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61—21no—
183C22⋊Dic61—22no—
184C22⋊Dic61—41no—
185C22⋊Dic61—42no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188C4×Dic61—41no—
189C4×Dic61—41no—
190#11758194—22no—
191#11758194—22no—
192#11758194—41no—
193#11758194—41no—
194#11758194—41no—
195#11758194—41no—
196C2×#92418—22no—
197C2×#92418—41no—
198C2×#92418—41no—
199C2×#92418—41no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61C2×C22⋊C411yescharacteristic
206C23×C61#12011yescharacteristic
207C23×C61#12011yescharacteristic
208Q8×C61—42no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215Q8×C61—82no—
216D4×C61—41no—
217D4×C61—42no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—81no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—81no—
244C2×C4×C61—82no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×Dic61—42no—
250C2×Dic61—81no—
251C2×Dic61—81no—
252C2×Dic61—81no—
253C2×Dic61—81no—
254C2×Dic61—81no—
255#92418—42no—
256#92418—81no—
257#8482—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61#37911yescharacteristic
272C22×C61#45411yescharacteristic
273C22×C61#45611yescharacteristic
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×C61B(4,𝔽2)11yescharacteristic
285C4×C61#45811yescharacteristic
286Dic61—82no—
287Dic61—161no—
288#3538D6111yescharacteristic
289#1996—611no—
290#1901—611no—
291#1894—611no—
292#1871—611no—
293#1513—611no—
294#1508—611no—
295C2×C61—21no—
296C2×C61—21no—
297C2×C61—41no—
298C2×C61—41no—
299C2×C61—81no—
300C2×C61#179011yescharacteristic
301C2×C61#179311yescharacteristic
302C2×C61#179611yescharacteristic
303C22×M16—611no—
304#563—22no—
305#552—21no—
306#552—21no—
307#551—22no—
308D4×Q8—21no—
309D4×Q8—21no—
310#544—21no—
311#544Dic6112yescomplemented
312#541—21no—
313#539—21no—
314#539Dic6112yescomplemented
315#536—21no—
316#525C2×D6111yescharacteristic
317C2×C22⋊Q8C2×D6111yescharacteristic
318C2×C22⋊Q8C2×D6111yescharacteristic
319C2×#104—1221no—
320C2×#103—1221no—
321C2×#101—1221no—
322#407—611no—
323#380—611no—
324C4×C22⋊C4—611no—
325#343—1221no—
326#342—1222no—
327#340—1221no—
328#339—1221no—
329#328—1222no—
330#328—1221no—
331#327—1221no—
332C61#848211yescharacteristic, complemented, Sylow
333C2×#41C22⋊D6111yescharacteristic
334C22×Q8—21no—
335C22×Q8C22⋊D6111yescharacteristic
336C22×Q8C22⋊D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C2×M16—1221no—
339C2×M16—1221no—
340C2×M16—1221no—
341C2×M16—2442no—
342C22×C8—1221no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347#126—22no—
348#126—22no—
349#126—22no—
350#126—22no—
351#125—21no—
352#125—22no—
353#125—22no—
354#125—22no—
355#125—22no—
356#124—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8C2×Dic6112yes—
375C22⋊Q8C2×Dic6111yescharacteristic
376C22⋊Q8C22⋊D6112yes—
377C22⋊Q8C22⋊D6111yescharacteristic
378#121C2×Dic6111yescharacteristic
379#121C22⋊D6111yescharacteristic
380#120—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—21no—
386C4×D4—21no—
387#117—21no—
388#117—22no—
389#117—22no—
390#117C22⋊D6111yescharacteristic
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C2×Dic6112yes—
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C4⋊C4C22⋊D6111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4C22⋊D6112yes—
400C2×C42—1221no—
401#104—2442no—
402#103—2442no—
403#101—2442no—
404C22⋊C8—1221no—
405C22⋊C8—1221no—
406#95—1221no—
407#95—1221no—
408C24C2×C22⋊D6111yescharacteristic
409#41—41no—
410#41—41no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8C22⋊Dic6112yes—
422C2×Q8C22⋊Dic6111yescharacteristic
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C22⋊Dic6111yescharacteristic
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4C22⋊Dic6112yes—
436C22×C4C22⋊Dic6112yes—
437C22×C4C22×Dic6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443M16—1222no—
444M16—1222no—
445M16—2441no—
446M16—2441no—
447M16—2441no—
448M16—2441no—
449C2×C8—1222no—
450C2×C8—2441no—
451C2×C8—2441no—
452C2×C8—2441no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4C22⋊Dic6112yes—
473C4⋊C4C22⋊Dic6111yescharacteristic
474C4⋊C4C22⋊Dic6111yescharacteristic
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—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—1222no—
488C22⋊C4—2442no—
489C22⋊C4—2441no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—2441no—
495C42—2441no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23C2×C22⋊Dic6111yescharacteristic
502C23#1451974611yescharacteristic
503C23#1451974611yescharacteristic
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—81no—
511Q8—82no—
512D4—41no—
513D4—42no—
514D4—41no—
515D4—41no—
516D4—81no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—4881no—
542C2×C4—4881no—
543C2×C4—4881no—
544C2×C4—4881no—
545C2×C4—4881no—
546C2×C4—2442no—
547C2×C4C2×C22⋊Dic6111yescharacteristic
548C2×C4C2×C22⋊Dic6111yescharacteristic
549C2×C4#1451974611yescharacteristic
550C2×C4#1451974611yescharacteristic
551C8—4881no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22#1683961811yescharacteristic
567C22#1683993511yescharacteristic
568C22#1683994511yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—9761no—
580C4—4882no—
581C4#1683994911yescharacteristic
582C4#1683995211yescharacteristic
583C2—21no—
584C2—21no—
585C2—41no—
586C2—41no—
587C2—81no—
588C2#2144576711yescharacteristic
589C2#2144577811yescharacteristic
590C2#2144579411yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#3538C2
socleC22×C61#379
prime radicalC61×#525C22
cosocleC23×C4×C61C23
hypercenter#3538D61
hypocenterC61#8482
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
1. #21446805C2semidirect
2. #21446288C2semidirect
3. #21446253C2semidirect
6. #21444339C2semidirect
29. C61×#544C4semidirect
32. C61×#539C4semidirect
311. #544Dic61semidirect
314. #539Dic61semidirect
332. C61#8482semidirect

Maximal subgroups

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

subgroupindex
#214468052
#214462882
#214462532
#214461222
#214443622
#214443392
C61×#35382
#848261