Finite Group Databasebeta

← #29662817

Subgroups — #29662817

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456366C211yesmaximal, characteristic, complemented
2#21455823C211yesmaximal, characteristic
3C2×#16840053C211yesmaximal, characteristic
4#21454554C211yesmaximal, characteristic
5#21445341C212yesmaximal, complemented
6#21445317C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444501C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840172—22no—
16#16840165C2212yescomplemented
17#16840158C2212yescomplemented
18#16840090—21no—
19#16840053—21no—
20#16840053—21no—
21#16840053C2212yes—
22#16840053C2212yes—
23#16840049C2211yescharacteristic
24#16840049C2211yescharacteristic
25#16839964C2211yescharacteristic
26#16839964C2211yescharacteristic
27#16839912—21no—
28C4×#11758201—21no—
29C4×#11758201—21no—
30#16839839C2212yes—
31#16839832—22no—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34#16839805—21no—
35#16839805—21no—
36C2×#14519709—21no—
37C2×#14519699—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839510C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×Dic122—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758201—21no—
53C2×#11758201—21no—
54C2×#11758194D411yescharacteristic
55C2×#11758194D411yescharacteristic
56C2×#11758189—22no—
57M16×D61—44no—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2312yes—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D411yescharacteristic
66C4×D244D411yescharacteristic
67C4×D244C2311yescharacteristic
68C4×D244C2311yescharacteristic
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73#14519710—22no—
74#14519710—22no—
75#14519709—22no—
76#14519709—22no—
77#14519708—21no—
78#14519708—21no—
79#14519708—22no—
80#14519708—22no—
81C4⋊#92418C2311yescharacteristic
82C4⋊#92418C2311yescharacteristic
83#14519701—21no—
84#14519701—21no—
85#14519699—22no—
86#14519699—22no—
87#14519697—21no—
88#14519697—21no—
89C22⋊#92418—21no—
90#14519689—22no—
91#14519689—22no—
92#14519688D412yes—
93C4×#92418—21no—
94C4×#92418C2312yes—
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—41no—
118C2×D244—22no—
119C2×D244—21no—
120C2×D244—41no—
121C2×D244C2×D411yescharacteristic
122C2×D244C2×D411yescharacteristic
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×Dic122—22no—
130C2×Dic122—41no—
131C2×Dic122—82no—
132M16×C61—42no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—42no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C42×C61—41no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148#11758201—42no—
149#11758201—42no—
150#11758198—21no—
151#11758198—21no—
152#11758198—21no—
153#11758198—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—41no—
158C4⋊Dic61—41no—
159C4⋊Dic61C2×D411yescharacteristic
160C4⋊Dic61C2×D411yescharacteristic
161#11758194—22no—
162#11758194SD1614yescomplemented
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418C2×D412yes—
166C2×#92418C2×D412yes—
167#11758189—44no—
168C8×D61—44no—
169C23×C61#12111yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—81no—
175Q8×C61—41no—
176Q8×C61—41no—
177Q8×C61—81no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61C2×SD1612yes—
187C2×C4×C61#13712yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196D244—21no—
197D244—21no—
198D244—21no—
199D244—42no—
200D244#13712yes—
201C4×D61—41no—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61—41no—
205C4×D61—42no—
206C4×D61—42no—
207Dic122—82no—
208Dic122—82no—
209C8×C61—42no—
210C8×C61—42no—
211#92418—22no—
212#92418—41no—
213#92418—41no—
214#92418C2×SD1614yes—
215#19165—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61#46112yes—
219C22×C61C2×#12111yescharacteristic
220C2×D61—22no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—84no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#46112yes—
233C4×C61C22×SD1611yescharacteristic
234C4×C61C2×#13711yescharacteristic
235C4×C61C2×#13711yescharacteristic
236C4×C61#57511yescharacteristic
237Dic61—41no—
238Dic61—41no—
239Dic61—82no—
240#3165—611no—
241#3101—611no—
242C2×#476—611no—
243D4×M16—611no—
244#1695—612no—
245#1691—612no—
246#1675—612no—
247#1671—612no—
248#1565—611no—
249#1550D6111yescharacteristic, complemented
250#1548—611no—
251C2×C61—22no—
252C2×C61C2×#46111yescharacteristic
253C2×C61#305611yescharacteristic
254C2×C61#306811yescharacteristic
255D61—42no—
256D61—81no—
257C22×M16—1221no—
258#533C2×D6111yescharacteristic
259C2×C4⋊Q8—611no—
260C2×C4×D4—611no—
261#501—1222no—
262#500—612no—
263#498—612no—
264#483—1221no—
265#476—611no—
266#476—611no—
267#476—612no—
268#476—612no—
269#476—1221no—
270#476—1221no—
271#461—611no—
272#461—611no—
273#450—1221no—
274C4×SD16—1221no—
275C4×SD16—1221no—
276#436—612no—
277#435—1222no—
278C8×D4—612no—
279#431—611no—
280#428—1221no—
281#428—1221no—
282C2×#106—1221no—
283#423C2×D6111yescharacteristic
284C2×C4⋊C8C2×D6111yescharacteristic
285C2×#102—1221no—
286#407—1221no—
287C4×M16—611no—
288#335—612no—
289#334—612no—
290#333—612no—
291#333—612no—
292#333C2×D6112yescomplemented
293#332C2×D6112yescomplemented
294C61#1916511yescharacteristic, complemented, Sylow
295C22×Q8—1221no—
296C22×D4—611no—
297C23×C4—1221no—
298C2×SD16—1221no—
299C2×SD16—1221no—
300C2×M16—21no—
301C2×M16—611no—
302C2×M16—611no—
303C2×M16—1222no—
304C2×M16—2444no—
305C22×C8—21no—
306C22×C8—1222no—
307C4⋊Q8—612no—
308C4⋊Q8—1221no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8C22×D6111yescharacteristic
312C4⋊Q8C22×D6111yescharacteristic
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—1221no—
323C4×D4—1221no—
324#117—21no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C42C22×D6111yescharacteristic
330#107—1222no—
331#107—1222no—
332#106—1221no—
333#106—1221no—
334#106—1222no—
335#106—1222no—
336#106—1222no—
337#106—1222no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8C22×D6112yes—
343C4⋊C8C22×D6112yes—
344#103—1221no—
345#103—1221no—
346#102—1221no—
347#102—1221no—
348#102—1222no—
349#102—1222no—
350C22⋊C8—1222no—
351C22⋊C8—1222no—
352C22⋊C8—1221no—
353#97—612no—
354C4×C8—612no—
355C4×C8—1221no—
356C24—1221no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—1222no—
360C2×Q8—4882no—
361C2×Q8—2441no—
362C2×D4—611no—
363C2×D4—611no—
364C2×D4—1222no—
365C2×D4—1221no—
366C2×D4—2441no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—2441no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4#1175822211yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378SD16—2442no—
379SD16—2442no—
380M16—42no—
381M16—614no—
382M16—1222no—
383M16—2444no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—42no—
387C2×C8—612no—
388C2×C8—612no—
389C2×C8—1221no—
390C2×C8—1221no—
391C2×C8—2444no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—611no—
399C4⋊C4—611no—
400C4⋊C4—1221no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C22⋊C4—41no—
406C22⋊C4—1221no—
407C22⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C42—41no—
411C42#1175822211yescharacteristic
412C42D4×D6111yescharacteristic
413C42D4×D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—1221no—
416C23—1221no—
417C23—4881no—
418C23—2442no—
419C23—2442no—
420C23#1451975011yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—81no—
424Q8—4882no—
425Q8—4882no—
426D4—612no—
427D4—1221no—
428D4—1221no—
429D4—1221no—
430D4—2442no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—4882no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4#1451975011yescharacteristic
448C2×C4#1451975011yescharacteristic
449C2×C4#1451975011yescharacteristic
450C2×C4SD16×D6112yes—
451C2×C4#1451980912yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×#1175822211yescharacteristic
455C8—42no—
456C8—42no—
457C8—614no—
458C8—1222no—
459C8—2441no—
460C8—2441no—
461C22—21no—
462C22—21no—
463C22—1222no—
464C22—4884no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22—2441no—
469C22—2441no—
470C22#1683996612yes—
471C22C2×#1451975011yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—81no—
476C4—81no—
477C4—4882no—
478C4—2441no—
479C4—2441no—
480C4#1683996612yes—
481C4C2×SD16×D6111yescharacteristic
482C4#137×D6111yescharacteristic
483C4C2×#1451980911yescharacteristic
484C4#1684055811yescharacteristic
485C2—22no—
486C2—4881no—
487C2—2442no—
488C2C2×#1683996611yescharacteristic
489C2#2145545211yescharacteristic
490C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1550D61
hypocenterC61#19165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
5. #21445341C2semidirect
6. #21445317C2semidirect
7. #21445217C2semidirect
8. #21445193C2semidirect
9. #21444569C2semidirect
10. #21444501C2semidirect
11. C61×#1550C2semidirect
16. #16840165C22semidirect
17. #16840158C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839510C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
162. #11758194SD16semidirect
249. #1550D61semidirect
292. #333C2×D61semidirect
293. #332C2×D61semidirect
294. C61#19165semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214558232
C2×#168400532
#214545542
#214453412
#214453172
#214452172
#214451932
#214445692
#214445012
C61×#15502
#1916561