Finite Group Databasebeta

← #29653154

Subgroups — #29653154

Order 15616.

1078 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#21455823C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic, complemented
4#21444833C214yesmaximal, complemented
5#21444765C214yesmaximal, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444379C211yesmaximal, characteristic, complemented
8#21444140C211yesmaximal, characteristic, complemented
9C61×#3506C211yesmaximal, characteristic, complemented
10C2×#14519748—21no—
11C2×C4×D244C2211yescharacteristic
12#16840090C2212yes—
13#16840073C2212yes—
14#16840049—21no—
15#16840039—21no—
16#16839964—21no—
17#16839954—21no—
18#16839912C2212yes—
19#16839895C2212yes—
20C4×#11758201—22no—
21C4×#11758199—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839687C2212yes—
25#16839671C2212yes—
26#16839641—22no—
27#16839628C2211yescharacteristic
28#16839619—21no—
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—
50#14519748—44no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244—21no—
54C4×D244—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C22⋊Dic61—22no—
58C2×#11758198—21no—
59C2×#11758198—22no—
60#14519710—22no—
61#14519708—22no—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64#14519701—22no—
65#14519698—22no—
66#14519697—22no—
67#14519697—22no—
68C22⋊#92418C2312yes—
69C22⋊#92418C2312yes—
70C4×#92418—21no—
71C4×#92418—21no—
72#14519684—22no—
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—42no—
123#41×C61—42no—
124#41×C61—84no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—22no—
129C2×D4×C61—24no—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61C2×D412yes—
138C22×C4×C61—24no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—82no—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—44no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—82no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—42no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C42×C61—24no—
179C42×C61—22no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C22⋊Dic61—44no—
185#11758201—44no—
186#11758199—44no—
187#11758198—41no—
188#11758198—41no—
189#11758198—82no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C22×D411yescharacteristic
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209Q8×C61—42no—
210Q8×C61—42no—
211Q8×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—84no—
220D4×C61—84no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61—82no—
231C2×C4×C61—84no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#13512yes—
237C2×C4×C61AGL(1,ℤ/8)12yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—88no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—82no—
252C4×D61—82no—
253#92418—42no—
254#92418—42no—
255#18222—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—44no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—82no—
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#13511yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#44912yes—
285C4×C61#44912yes—
286Dic61—82no—
287Dic61—81no—
288#3506D6111yescharacteristic, complemented
289#3101—611no—
290#3100—611no—
291#2535—611no—
292#1612—614no—
293#1600—614no—
294#1518—611no—
295#1517—611no—
296#1460—611no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—82no—
300C2×C61—82no—
301C2×C61C2×#44911yescharacteristic
302C2×C61#301911yescharacteristic
303C2×C61#302511yescharacteristic
304D61—82no—
305#559—22no—
306#556—21no—
307#555—21no—
308#553—22no—
309#550—21no—
310#547—21no—
311#538C2×D6114yes—
312#535—21no—
313#533C2×D6111yescharacteristic
314#532C2×D6111yescharacteristic
315#525—22no—
316C2×#121—1221no—
317#520—21no—
318C2×C4×D4—611no—
319C2×C4×D4C2×D6111yescharacteristic
320#483—612no—
321#480—612no—
322#476—1221no—
323#474—1221no—
324#461—1221no—
325#459—1221no—
326#450—612no—
327#447—612no—
328C4×SD16—1222no—
329C4×D8—1222no—
330#431—611no—
331#431—611no—
332#397—612no—
333#392—612no—
334#386—1222no—
335#382—611no—
336#380—1221no—
337#328—614no—
338#327—614no—
339#323—614no—
340C61#1822211yescharacteristic, complemented, Sylow
341C2×#41—22no—
342C2×#41—21no—
343C22×D4—611no—
344C22×D4—1222no—
345C22×D4C22⋊D6111yescharacteristic
346C23×C4—1221no—
347C23×C4C22⋊D6112yes—
348C2×SD16—1222no—
349C2×D8—1222no—
350C4⋊Q8C22⋊D6111yescharacteristic, complemented
351C4⋊Q8C22⋊D6111yescharacteristic, complemented
352C4⋊Q8C22⋊D6111yescharacteristic, complemented
353#127—22no—
354#127C22⋊D6111yescharacteristic, complemented
355#126—24no—
356#125—21no—
357#124—22no—
358#124—24no—
359#124—21no—
360#123—24no—
361#123—24no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8C22×D6112yes—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—24no—
369#121—21no—
370#121—2444no—
371#121C22×D6112yes—
372#120—24no—
373C4×Q8—22no—
374C4×Q8—21no—
375C4×Q8C22⋊D6112yescomplemented
376C4×D4—22no—
377C4×D4—24no—
378C4×D4—24no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4—1221no—
387C4×D4—1221no—
388C4×D4C22⋊D6112yescomplemented
389#117—22no—
390#117—22no—
391#117—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4C22⋊D6111yescharacteristic
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4C22×D6112yes—
400C2×C42C22×D6111yescharacteristic
401#107—1222no—
402#106—1222no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405#103—1222no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409C22⋊C8—612no—
410C22⋊C8—612no—
411C4×C8—1221no—
412C4×C8—1221no—
413#95—612no—
414#95—1222no—
415#95—1221no—
416C24—1221no—
417C24C2×C22⋊D6112yes—
418#41—42no—
419#41—42no—
420#41—84no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C2×C22⋊D6112yes—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—4882no—
436C2×D4—2442no—
437C2×D4—2444no—
438C2×D4—2444no—
439C2×D4C2×C22⋊D6112yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—82no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455SD16—2444no—
456D8—2444no—
457C2×C8—1222no—
458C2×C8—1222no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2442no—
471C4⋊C4C2×C22⋊D6112yes—
472C4⋊C4C2×C22⋊D6112yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—4882no—
482C22⋊C4—2444no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C42—24no—
486C42—22no—
487C42C2×C22⋊D6111yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C2×C22⋊D6111yescharacteristic
490C42C2×C22⋊D6111yescharacteristic
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—1222no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23C22×C22⋊D6111yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—84no—
516D4—84no—
517D4—1222no—
518D4—1222no—
519D4—4888no—
520D4—2441no—
521D4—2441no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2442no—
541C2×C4#1451974611yescharacteristic
542C2×C4#1451974611yescharacteristic
543C2×C4#1451974611yescharacteristic
544C2×C4#1451974611yescharacteristic
545C2×C4#1451979712yes—
546C2×C4#1451980212yes—
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C22×C22⋊D6111yescharacteristic
549C8—2442no—
550C8—2442no—
551C22—21no—
552C22—21no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22C2×#1451974611yescharacteristic
569C22C2×#1451979711yescharacteristic
570C22C2×#1451980211yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—4882no—
580C4—4881no—
581C4#1683990712yes—
582C4#1683990712yes—
583C2—21no—
584C2—21no—
585C2—82no—
586C2—82no—
587C2—4882no—
588C2C2×#1683990711yescharacteristic
589C2#2145517611yescharacteristic
590C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18222
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. #21455823C2semidirect
2. #21455815C2semidirect
3. #21450716C2semidirect
4. #21444833C2semidirect
5. #21444765C2semidirect
6. #21444383C2semidirect
7. #21444379C2semidirect
8. #21444140C2semidirect
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
288. #3506D61semidirect
340. C61#18222semidirect
350. C4⋊Q8C22⋊D61semidirect
351. C4⋊Q8C22⋊D61semidirect
352. C4⋊Q8C22⋊D61semidirect
354. #127C22⋊D61semidirect
375. C4×Q8C22⋊D61semidirect
388. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214558152
#214507162
#214448332
#214447652
#214443832
#214443792
#214441402
C61×#35062
#1822261