Finite Group Databasebeta

← #29669766

Subgroups — #29669766

Order 15616.

598 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
1D8×Dic122C211yesmaximal, characteristic
2#21458096C211yesmaximal, characteristic, complemented
3#21457689C211yesmaximal, characteristic, complemented
4#21445468C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445244C211yesmaximal, characteristic, complemented
8#21445213C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic, complemented
12#21444983C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3360C211yesmaximal, characteristic
16D4×Dic122C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840166—21no—
20#16840158C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840103—21no—
25#16840097—21no—
26#16840054C2211yescharacteristic
27#16840044C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840040—21no—
31#16840040—21no—
32#16840018—21no—
33#16840001—21no—
34#16839972C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839917—21no—
37#16839917—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758202C2211yescharacteristic
40D8×Dic61—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839770—21no—
44C2×#14519700—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#488—21no—
59C61×#487C2211yescharacteristic
60C61×#479—21no—
61C61×#476C2211yescharacteristic
62C61×#463—21no—
63C61×#461C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C22×Dic122—21no—
68C22×Dic122—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202D411yescharacteristic
72C2×#11758194—21no—
73C22×#92418—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519756—21no—
79#14519756—21no—
80#14519754—41no—
81#14519754—41no—
82C4×Dic122C2311yescharacteristic
83D4×Dic61—21no—
84D4×Dic61—21no—
85#14519711—21no—
86#14519711—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—41no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—41no—
98#14519701D411yescharacteristic
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×SD16×C61—21no—
116C2×D8×C61D411yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#125×C61—21no—
121#124×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131C2×C4⋊C4×C61—21no—
132#107×C61D411yescharacteristic
133#106×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×Dic122—41no—
159C2×Dic122—41no—
160C2×Dic122—81no—
161C2×Dic122—81no—
162C2×Dic122C2×D411yescharacteristic
163SD16×C61—42no—
164D8×C61—22no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188C22⋊Dic61—41no—
189#11758202—22no—
190#11758202—81no—
191#11758202—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200C4×Dic61—21no—
201#11758194—81no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61#13712yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—161no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—82no—
239C2×Dic61—82no—
240Dic122—21no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—82no—
246Dic122—82no—
247Dic122#13512yes—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19782—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61D4211yescharacteristic
265C4×C61—21no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61C2×#13511yescharacteristic
273C4×C61C2×#13711yescharacteristic
274C4×C61#57611yescharacteristic
275C4×C61#57811yescharacteristic
276Dic61—22no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Q8×D8—611no—
281#3360D6111yescharacteristic
282#3328—611no—
283#3299—611no—
284#1718—611no—
285#1703—611no—
286#1684—611no—
287#1679—611no—
288#1674—611no—
289#1652—611no—
290#1651—611no—
291#1644—611no—
292#1637—611no—
293#1581—611no—
294#1579—611no—
295C2×C61—42no—
296C2×C61—81no—
297C2×C61#327011yescharacteristic
298C2×C61#328011yescharacteristic
299C2×C61#328911yescharacteristic
300#556C2×D6111yescharacteristic
301#554C2×D6111yescharacteristic
302D4×Q8—611no—
303D4×Q8—611no—
304#500—611no—
305#500—1221no—
306#498—611no—
307#488—21no—
308#487—611no—
309#487—611no—
310#487—611no—
311#487C2×D6111yescharacteristic
312#485—1221no—
313#484—1221no—
314#479—21no—
315#477—611no—
316#476C2×D6111yescharacteristic
317#475—611no—
318#474—611no—
319#474—611no—
320#474—1221no—
321#474—1221no—
322#470—1221no—
323#467—1221no—
324#463—21no—
325#462—611no—
326#462—611no—
327#461C2×D6111yescharacteristic
328#451—1221no—
329#451—1221no—
330C8×Q8—611no—
331C8×Q8C2×D6111yescharacteristic
332C4×Q16—611no—
333C4×SD16—21no—
334C4×D8—1221no—
335C4×D8C2×D6111yescharacteristic
336#436—611no—
337C8×D4—611no—
338#419—1221no—
339C2×#103—1221no—
340#335—611no—
341#334—611no—
342#333—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350#322—611no—
351C61#1978211yescharacteristic, complemented, Sylow
352C22×Q8—1221no—
353C22×Q8—1221no—
354C2×Q16—611no—
355C2×Q16—1221no—
356C2×Q16—1221no—
357C2×SD16—21no—
358C2×D8C22⋊D6111yescharacteristic
359C2×M16—1221no—
360C22×C8—1221no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#126—41no—
368#125—21no—
369#124—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—2441no—
376C22⋊Q8—2441no—
377C4×Q8—21no—
378C4×Q8—611no—
379C4×Q8C22×D6111yescharacteristic
380C4×D4—21no—
381C4×D4—1221no—
382C4×D4—1221no—
383C4×D4C22×D6111yescharacteristic
384C4×D4C22×D6111yescharacteristic
385#117—21no—
386C2×C4⋊C4—21no—
387#107—611no—
388#107—1221no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107C22⋊D6111yescharacteristic
394#106—21no—
395#106—1221no—
396#106—2441no—
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—21no—
404#103—21no—
405#103—21no—
406#103—611no—
407#103—611no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#102—21no—
418#102—1221no—
419#102—1221no—
420#102C22⋊D6111yescharacteristic
421#102C22⋊D6111yescharacteristic
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C4×C8—21no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8C22×D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—611no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—4881no—
436C2×Q8—2441no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×Q8D4×D6111yescharacteristic
441C2×D4C2×C22⋊D6111yescharacteristic
442C2×D4C2×C22⋊D6111yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—2441no—
447C22×C4—2441no—
448Q16—1222no—
449Q16—4881no—
450Q16—4881no—
451SD16—42no—
452D8—22no—
453M16—4881no—
454C2×C8—21no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—4881no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—611no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4D4×D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—21no—
491C42—41no—
492C42—1221no—
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—612no—
502Q8—1221no—
503Q8—4882no—
504Q8—4882no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8—2441no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4#1451980712yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—161no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4882no—
527C2×C4—4882no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#1451982211yescharacteristic
533C8—21no—
534C8—21no—
535C8—41no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22D4×C22⋊D6111yescharacteristic
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—1222no—
557C4—4881no—
558C4—4881no—
559C4—2441no—
560C4#135×D6111yescharacteristic
561C4C2×#1451980711yescharacteristic
562C4#1684056411yescharacteristic
563C4#1684057711yescharacteristic
564C2—42no—
565C2—81no—
566C2#2145738511yescharacteristic
567C2#2145749811yescharacteristic
568C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3360C2
socleC22×C61D42
prime radicalC2×D8×C61D4
hypercenter#3360D61
hypocenterC61#19782
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
2. #21458096C2semidirect
3. #21457689C2semidirect
4. #21445468C2semidirect
5. #21445396C2semidirect
6. #21445274C2semidirect
7. #21445244C2semidirect
8. #21445213C2semidirect
9. #21445089C2semidirect
10. #21445080C2semidirect
11. #21445033C2semidirect
12. #21444983C2semidirect
14. #21444636C2semidirect
351. C61#19782semidirect

Maximal subgroups

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

subgroupindex
D8×Dic1222
#214580962
#214576892
#214454682
#214453962
#214452742
#214452442
#214452132
#214450892
#214450802
#214450332
#214449832
#214446492
#214446362
C61×#33602
#1978261