Finite Group Databasebeta

← #29834522

Subgroups — #29834522

Order 15616.

682 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
1C2×SD32×D61C211yesmaximal, characteristic, complemented
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21455469C211yesmaximal, characteristic, complemented
4#21455462C211yesmaximal, characteristic
5#21447933C211yesmaximal, characteristic, complemented
6#21447932C211yesmaximal, characteristic, complemented
7#21447929C211yesmaximal, characteristic, complemented
8#21447918C211yesmaximal, characteristic, complemented
9#21447917C211yesmaximal, characteristic, complemented
10#21447914C211yesmaximal, characteristic, complemented
11#21447913C211yesmaximal, characteristic, complemented
12#21447600C211yesmaximal, characteristic, complemented
13#21447537C211yesmaximal, characteristic, complemented
14#21447533C211yesmaximal, characteristic, complemented
15C61×#2220C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840335—21no—
20#16840328—21no—
21C2×#14519722C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C2×#14519720C2211yescharacteristic
24SD32×D61—24no—
25#16840180—24no—
26C2×#14519716C2211yescharacteristic
27C2×#14519716C2211yescharacteristic, complemented
28C22×#11758185C2211yescharacteristic, complemented
29C2×C16×D61C2211yescharacteristic
30C2×C16×D61C2211yescharacteristic, complemented
31#16840019—21no—
32#16840018—21no—
33#16840014—21no—
34#16840013—21no—
35#16840004C2211yescharacteristic, complemented
36#16840003C2211yescharacteristic, complemented
37#16840000C2211yescharacteristic, complemented
38#16839999C2211yescharacteristic
39#16839998C2211yescharacteristic
40#16839995C2211yescharacteristic
41#16839799C2211yescharacteristic
42#16839799C2211yescharacteristic
43#16839769—21no—
44#16839765—21no—
45#16839589C2211yescharacteristic, complemented
46#16839589C2211yescharacteristic, complemented
47#16839588C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic
52#16839563C2211yescharacteristic
53#16839562C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C2×M32×C61C2211yescharacteristic, complemented
56C61×#467C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic
58C61×#366C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic
61C2×#11758222D411yescharacteristic
62C2×Q8×D61—21no—
63C2×D4×D61—21no—
64C2×Dic244C2311yescharacteristic
65C2×#11758202—21no—
66Q16×D61—22no—
67Q16×D61—22no—
68C2×D488C2311yescharacteristic
69C2×#11758199—21no—
70D8×D61—22no—
71D8×D61—22no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194D411yescharacteristic
75C2×#11758189D411yescharacteristic
76C2×#11758189D411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519776—41no—
80#14519773—41no—
81#14519763—41no—
82#14519763—41no—
83#14519762—41no—
84#14519759—41no—
85#14519757—21no—
86#14519756—21no—
87#14519753—21no—
88#14519752—21no—
89#14519751—21no—
90#14519748—21no—
91#14519747—41no—
92#14519745—41no—
93#14519735—21no—
94#14519735—21no—
95#14519722—22no—
96#14519721—22no—
97#14519720—22no—
98#14519716—22no—
99#14519716—22no—
100C2×#11758185—22no—
101C2×#11758185C2311yescharacteristic
102C2×#11758185C2311yescharacteristic
103C16×D61—22no—
104C16×D61D414yescomplemented
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519708—21no—
108#14519708—21no—
109#14519702—21no—
110#14519701—21no—
111#14519700—21no—
112#14519699—21no—
113#14519698—21no—
114#14519697—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121M32×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C23×D61—41no—
129C2×Q8×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61D811yescharacteristic
137C2×C22⋊D61D811yescharacteristic
138Q8×D61—42no—
139Q8×D61—42no—
140D4×D61—42no—
141D4×D61—42no—
142#11758222—22no—
143#11758222—22no—
144C2×D244—21no—
145C2×D244—41no—
146C2×D244D811yescharacteristic
147C2×C4×D61—41no—
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—41no—
152C2×Dic122D811yescharacteristic
153Q16×C61—21no—
154Q16×C61—21no—
155D8×C61—21no—
156D8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C22⋊Dic61—41no—
167C22⋊Dic61—41no—
168#11758202—42no—
169#11758199—42no—
170#11758198—41no—
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—41no—
179#11758196—41no—
180#11758196—41no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183#11758194—22no—
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186Dic244—21no—
187Dic244—21no—
188D488—21no—
189D488—21no—
190#11758189—22no—
191#11758189—22no—
192C8×D61—21no—
193C8×D61—21no—
194C8×D61C2×D412yes—
195C8×D61#4112yes—
196C16×C61—21no—
197C16×C61C2×D412yes—
198#11758185—21no—
199#11758185C2×D412yes—
200C23×C61—41no—
201C23×C61C2×D811yescharacteristic
202C22×D61—41no—
203C22×D61—82no—
204C22×D61—161no—
205C22×D61—161no—
206C22×D61C2×D811yescharacteristic
207C22×D61C2×D811yescharacteristic
208Q8×C61—41no—
209Q8×C61—41no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—161no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C2×D811yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C22⋊D61—22no—
221C22⋊D61—22no—
222C22⋊D61—82no—
223C22⋊D61—161no—
224C22⋊D61—161no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61C2×D811yescharacteristic
229C2×Dic61C2×D811yescharacteristic
230D244—22no—
231D244—41no—
232D244—41no—
233D244—161no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—82no—
237C4×D61#12112yes—
238C4×D61#13512yes—
239Dic122—22no—
240Dic122—41no—
241Dic122—41no—
242Dic122—82no—
243C8×C61—21no—
244C8×C61C22×D411yescharacteristic
245C8×C61C2×#4111yescharacteristic
246#92418—21no—
247#92418#12112yes—
248#33609—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61C22×D811yescharacteristic
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—164no—
265C2×D61#46612yes—
266C4×C61—21no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×#13511yescharacteristic
272Dic61—21no—
273Dic61—81no—
274Dic61—81no—
275Dic61—81no—
276Dic61#46612yes—
277C22×SD32—611no—
278C2×#504—611no—
279#3059—611no—
280#3058—611no—
281#2220—611no—
282#2220—611no—
283#2220—611no—
284#2220D6111yescharacteristic, complemented
285#2217—611no—
286#2217—611no—
287#2217—611no—
288#2217—611no—
289#2158—611no—
290#2141—611no—
291#2140—611no—
292C2×C61—41no—
293C2×C61—161no—
294C2×C61C2×#46611yescharacteristic
295C2×C61#341611yescharacteristic
296C2×C61#341611yescharacteristic
297D61—22no—
298D61—41no—
299D61—161no—
300C22×Q16—611no—
301C22×D8—611no—
302C2×#131—611no—
303#536—1221no—
304#535—1221no—
305C2×SD32—611no—
306C2×SD32—611no—
307C2×SD32—611no—
308C2×SD32—1224no—
309C2×SD32C2×D6111yescharacteristic
310#504—1224no—
311C2×M32—611no—
312C2×M32—611no—
313C2×M32C2×D6111yescharacteristic, complemented
314C22×C16—611no—
315C22×C16—611no—
316C22×C16—611no—
317#470—1221no—
318#470—1221no—
319#469—1221no—
320#469—1221no—
321#467—611no—
322#467—611no—
323#467—611no—
324#467C2×D6111yescharacteristic, complemented
325#466—611no—
326#466—611no—
327#466—611no—
328#466C2×D6111yescharacteristic
329#427—611no—
330#427—611no—
331#419—1221no—
332#418—1221no—
333#367—611no—
334#367—611no—
335#366—611no—
336#366C2×D6111yescharacteristic, complemented
337#358—611no—
338#358—611no—
339#358—611no—
340#358C2×D6111yescharacteristic, complemented
341#357—611no—
342#357—611no—
343#357—611no—
344#357C2×D6111yescharacteristic
345C61#3360911yescharacteristic, complemented, Sylow
346C2×#41—611no—
347C22×Q8—1221no—
348C22×D4—1221no—
349C2×Q16—611no—
350C2×Q16—1222no—
351C2×Q16—1222no—
352C2×Q16—1221no—
353C2×Q16C22×D6111yescharacteristic
354C2×D8—611no—
355C2×D8—1222no—
356C2×D8—1222no—
357C2×D8—1221no—
358C2×D8C22×D6111yescharacteristic
359#131—1222no—
360#131—1222no—
361C2×M16—611no—
362C2×M16—611no—
363C2×M16—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8C22×D6111yescharacteristic
367C4⋊Q8—2441no—
368#127—2441no—
369#124—2441no—
370#124—2441no—
371#123—2441no—
372#123—2441no—
373C22⋊Q8—21no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377#121—21no—
378#121—1221no—
379#121—1221no—
380#121—1221no—
381#120—2441no—
382#120—2441no—
383#117—1221no—
384#117—1221no—
385SD32—22no—
386SD32—1222no—
387SD32—1222no—
388SD32—1222no—
389M32—22no—
390M32—1222no—
391M32—1222no—
392C2×C16—611no—
393C2×C16—614no—
394C2×C16—611no—
395C2×C16—1222no—
396C2×C16—1222no—
397C2×C16C22×D6111yescharacteristic
398C2×C16C22×D6111yescharacteristic
399#107—611no—
400#107—611no—
401#107C22×D6111yescharacteristic
402#107C22×D6111yescharacteristic
403#106—1221no—
404#106—1221no—
405#103—21no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#102—21no—
410#102—1221no—
411#102—1221no—
412#102—1221no—
413C24—2441no—
414#41—1222no—
415#41—1222no—
416C2×Q8—21no—
417C2×Q8—611no—
418C2×Q8—1221no—
419C2×Q8—2441no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×D4—21no—
423C2×D4—41no—
424C2×D4—611no—
425C2×D4—611no—
426C2×D4—611no—
427C2×D4—1221no—
428C2×D4—2441no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2441no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—2441no—
437C22×C4D4×D6111yescharacteristic
438Q16—21no—
439Q16—21no—
440Q16—1221no—
441Q16—1221no—
442Q16—2442no—
443D8—21no—
444D8—21no—
445D8—1221no—
446D8—1221no—
447D8—2442no—
448M16—1222no—
449M16—1222no—
450M16—1222no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—611no—
454C2×C8—612no—
455C2×C8—612no—
456C2×C8—611no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8D4×D6111yescharacteristic
460C2×C8C23×D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C42—2441no—
480C42—2441no—
481C16—21no—
482C16—612no—
483C16—1221no—
484C16D4×D6112yes—
485C23—41no—
486C23—611no—
487C23—611no—
488C23—9761no—
489C23—9761no—
490C23—4882no—
491C23—2441no—
492C23D8×D6111yescharacteristic
493Q8—41no—
494Q8—41no—
495Q8—1222no—
496Q8—4882no—
497Q8—2441no—
498Q8—2441no—
499D4—41no—
500D4—41no—
501D4—161no—
502D4—1222no—
503D4—1222no—
504D4—1222no—
505D4—9761no—
506D4—9761no—
507D4—9761no—
508D4—4882no—
509D4—2441no—
510D4—2441no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—4882no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C8—21no—
529C8—612no—
530C8—1221no—
531C8C2×D4×D6111yescharacteristic
532C8#41×D6111yescharacteristic
533C22—21no—
534C22—21no—
535C22—21no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—612no—
540C22—1221no—
541C22—1221no—
542C22—1221no—
543C22—1221no—
544C22—1221no—
545C22—9764no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22C2×D8×D6111yescharacteristic
550C4—21no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—612no—
555C4—1221no—
556C4—4881no—
557C4—4881no—
558C4—4881no—
559C4#121×D6111yescharacteristic
560C4#135×D6111yescharacteristic
561C2—41no—
562C2—161no—
563C2—1222no—
564C2—9761no—
565C2—2441no—
566C2#466×D6111yescharacteristic
567C2#2145896411yescharacteristic
568C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2220C2
socleC22×C61C22×D8
prime radical#21455462C2
hypercenter#2220D61
hypocenterC61#33609
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. C2×SD32×D61C2semidirect
2. C2×#16840180C2semidirect
3. #21455469C2semidirect
5. #21447933C2semidirect
6. #21447932C2semidirect
7. #21447929C2semidirect
8. #21447918C2semidirect
9. #21447917C2semidirect
10. #21447914C2semidirect
11. #21447913C2semidirect
12. #21447600C2semidirect
13. #21447537C2semidirect
14. #21447533C2semidirect
15. C61×#2220C2semidirect
27. C2×#14519716C22semidirect
28. C22×#11758185C22semidirect
30. C2×C16×D61C22semidirect
35. #16840004C22semidirect
36. #16840003C22semidirect
37. #16840000C22semidirect
45. #16839589C22semidirect
46. #16839589C22semidirect
47. #16839588C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839565C22semidirect
55. C2×M32×C61C22semidirect
56. C61×#467C22semidirect
58. C61×#366C22semidirect
59. C61×#358C22semidirect
104. C16×D61D4semidirect
284. #2220D61semidirect
313. C2×M32C2×D61semidirect
324. #467C2×D61semidirect
336. #366C2×D61semidirect
340. #358C2×D61semidirect
345. C61#33609semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D612
C2×#168401802
#214554692
#214554622
#214479332
#214479322
#214479292
#214479182
#214479172
#214479142
#214479132
#214476002
#214475372
#214475332
C61×#22202
#3360961