Finite Group Databasebeta

← #29652683

Subgroups — #29652683

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455077C212yesmaximal, complemented
2#21449034C211yesmaximal, characteristic
3#21444793C212yesmaximal, complemented
4#21444776C212yesmaximal, complemented
5#21444759C212yesmaximal, complemented
6#21444742C212yesmaximal, complemented
7#21444339C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—22no—
12C2×C4×C22⋊D61—21no—
13#16839922—24no—
14#16839917C2214yes—
15#16839912C2214yes—
16#16839761—22no—
17#16839726C2212yes—
18#16839682C2211yescharacteristic
19#16839679C2211yescharacteristic
20#16839677—21no—
21#16839674C2211yescharacteristic
22#16839671C2211yescharacteristic
23#16839670—21no—
24C4×#11758198—21no—
25C4×C22⋊Dic61C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—44no—
44C22×Dic122C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49#14519808—48no—
50C2×#11758202—24no—
51C2×#11758201—24no—
52C2×#11758194—22no—
53C4×C22⋊D61—44no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×C4×Dic61—21no—
61#14519700—24no—
62#14519697—24no—
63C22⋊#92418—22no—
64C22⋊#92418C2314yes—
65#14519684—21no—
66#14519684—21no—
67#14519684C2311yescharacteristic
68#14519684C2311yescharacteristic
69C22×Q8×C61—21no—
70C22×Q8×C61D412yescomplemented
71C22×D4×C61D411yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—24no—
77#125×C61—22no—
78#125×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61D414yescomplemented
90C22⋊Q8×C61C2314yes—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—24no—
94C4×D4×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61D412yescomplemented
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61C2311yescharacteristic
107C2×C42×C61D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—44no—
115C2×Q8×C61—82no—
116C2×Q8×C61C2×D414yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144#11758222—28no—
145#11758222—44no—
146#11758222—44no—
147#11758222—44no—
148C2×D244—22no—
149C2×D244—42no—
150C2×C4×D61—24no—
151C2×C4×D61—42no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×Dic122—22no—
155C2×Dic122—42no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D414yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—22no—
172C42×C61—24no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177#11758202—48no—
178#11758201—48no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758196—42no—
183C4×Dic61—42no—
184#11758194—44no—
185C2×#92418—24no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61#12012yes—
195C23×C61#13712yescomplemented
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204Q8×C61—44no—
205Q8×C61—44no—
206Q8×C61—82no—
207Q8×C61—82no—
208D4×C61—28no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—81no—
223C2×C4×C61—82no—
224C2×C4×C61—82no—
225C2×C4×C61—82no—
226C2×C4×C61—82no—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#13712yescomplemented
230C2×C4×C61C22×D412yes—
231C22⋊D61—48no—
232C22⋊D61—48no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242D244—24no—
243D244—42no—
244C4×D61—44no—
245C4×D61—44no—
246Dic122—24no—
247Dic122—42no—
248#92418—44no—
249#18167—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—42no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—44no—
277C4×C61—41no—
278C4×C61—44no—
279C4×C61B(4,𝔽2)12yes—
280Dic61—42no—
281Dic61—42no—
282Dic61—81no—
283#3501D6111yescharacteristic, complemented
284#3007—612no—
285#2388—611no—
286#1605—612no—
287#1602—612no—
288#1599—612no—
289#1596—612no—
290#1508—612no—
291#1465—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—42no—
295C2×C61—42no—
296C2×C61#300411yescharacteristic
297C2×C61#301811yescharacteristic
298C2×C61C2×B(4,𝔽2)11yescharacteristic
299D61—82no—
300C22×#41—611no—
301C2×#137—1222no—
302#554—24no—
303#554—22no—
304D4×Q8—22no—
305#544C2×D6112yes—
306#543—21no—
307#543C2×D6112yes—
308#533—21no—
309C2×C22⋊Q8—21no—
310C2×C22⋊Q8C2×D6112yes—
311#518—21no—
312C2×C4×D4—1221no—
313C2×C4×D4C2×D6111yescharacteristic
314#452—1224no—
315#451—614no—
316#450—614no—
317#417—1222no—
318#407—612no—
319#395—611no—
320#394—611no—
321#394—1221no—
322#393—611no—
323#392—611no—
324#392—1221no—
325C4×C22⋊C4—611no—
326C4×C22⋊C4—1221no—
327#328—614no—
328#328—614no—
329#324—612no—
330#323—612no—
331C61#1816711yescharacteristic, complemented, Sylow
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—2444no—
335C22×Q8—21no—
336C22×Q8—611no—
337C22×Q8C22⋊D6112yescomplemented
338C22×D4—611no—
339C22×D4—1221no—
340C22×D4—1221no—
341C22×D4C22⋊D6111yescharacteristic
342C23×C4—1221no—
343C23×C4C22⋊D6111yescharacteristic
344C23×C4C22×D6111yescharacteristic
345#137—2448no—
346C2×Q16—1224no—
347C2×SD16—1224no—
348C2×M16—1222no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—24no—
352#125—22no—
353#125—24no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—24no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—24no—
364C22⋊Q8C22⋊D6114yescomplemented
365C22⋊Q8C22×D6114yes—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—2444no—
373#117—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22⋊D6112yescomplemented
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4C22×D6111yescharacteristic
389C2×C42—1221no—
390C2×C42C22⋊D6111yescharacteristic
391#103—1224no—
392#102—1224no—
393C22⋊C8—614no—
394C22⋊C8—1222no—
395#95—611no—
396#95—611no—
397#95—1221no—
398#95—1221no—
399C24—1221no—
400C24C2×C22⋊D6111yescharacteristic
401C24C2×C22⋊D6111yescharacteristic
402#41—1228no—
403#41—2444no—
404#41—2444no—
405#41—2444no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—1222no—
412C2×Q8—2442no—
413C2×Q8C2×C22⋊D6114yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—1222no—
418C2×D4—2442no—
419C2×D4—2442no—
420C2×D4—2442no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2442no—
424C2×D4—2442no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1224no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—2442no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6112yes—
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6112yes—
446C22×C4C23×D6111yescharacteristic
447Q16—2448no—
448SD16—2448no—
449M16—2444no—
450C2×C8—1224no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—2442no—
462C4⋊C4C2×C22⋊D6114yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—2442no—
469C22⋊C4—2442no—
470C22⋊C4—2442no—
471C22⋊C4—2442no—
472C22⋊C4—2442no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—2442no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23#1451974612yes—
493C23#1451980812yescomplemented
494C23C22×C22⋊D6111yescharacteristic
495Q8—44no—
496Q8—44no—
497Q8—82no—
498Q8—82no—
499Q8—1224no—
500Q8—2442no—
501D4—28no—
502D4—1224no—
503D4—2442no—
504D4—2448no—
505D4—2448no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2444no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2444no—
535C2×C4#1451974611yescharacteristic
536C2×C4#1451974611yescharacteristic
537C2×C4#1451980812yescomplemented
538C2×C4C22×C22⋊D6112yes—
539C8—2444no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—4881no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22C2×#1451974611yescharacteristic
560C22C2×#1451980811yescharacteristic
561C22C2×#1451980811yescharacteristic
562C4—22no—
563C4—21no—
564C4—22no—
565C4—41no—
566C4—44no—
567C4—41no—
568C4—44no—
569C4—4881no—
570C4—2442no—
571C4—2442no—
572C4#1683994912yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—42no—
577C2—4882no—
578C2#2145505311yescharacteristic
579C2#2145516811yescharacteristic
580C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21449034C2
hypercenter#3501D61
hypocenterC61#18167
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455077C2semidirect
3. #21444793C2semidirect
4. #21444776C2semidirect
5. #21444759C2semidirect
6. #21444742C2semidirect
7. #21444339C2semidirect
8. #21444157C2semidirect
9. C61×#3501C2semidirect
70. C22×Q8×C61D4semidirect
89. C22⋊Q8×C61D4semidirect
101. C2×C4⋊C4×C61D4semidirect
195. C23×C61#137semidirect
229. C2×C4×C61#137semidirect
283. #3501D61semidirect
331. C61#18167semidirect
337. C22×Q8C22⋊D61semidirect
364. C22⋊Q8C22⋊D61semidirect
378. C2×C4⋊C4C22⋊D61semidirect
493. C23#14519808semidirect
537. C2×C4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214490342
#214447932
#214447762
#214447592
#214447422
#214443392
#214441572
C61×#35012
#1816761