Finite Group Databasebeta

← #29663543

Subgroups — #29663543

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic
2#21456195C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21445382C212yesmaximal, complemented
5#21445354C212yesmaximal, complemented
6#21445258C212yesmaximal, complemented
7#21445230C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#3246C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13C2×#14519774C2211yescharacteristic
14#16840138—21no—
15#16840128—22no—
16#16840110—21no—
17#16840110C2212yes—
18#16840110C2212yes—
19#16839957C2212yes—
20#16839957C2212yes—
21#16839895—22no—
22#16839840C2212yescomplemented
23#16839834—22no—
24D4×#92418C2212yescomplemented
25#16839815C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839766—21no—
28#16839766—21no—
29#16839729—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839511C2211yescharacteristic, complemented
34#16839511C2211yescharacteristic, complemented
35#16839508C2211yescharacteristic, complemented
36#16839508C2211yescharacteristic, complemented
37#16839505C2212yes—
38#16839505C2212yes—
39C61×#533C2211yescharacteristic
40C61×#532C2211yescharacteristic
41C61×#476C2212yescomplemented
42C61×#475—21no—
43C61×#465—22no—
44C61×#461C2212yescomplemented
45C61×#460—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52#14519784—44no—
53C2×#11758194—22no—
54C2×#11758194D411yescharacteristic
55C2×#11758194D411yescharacteristic
56C22×#92418—21no—
57C22×#92418—22no—
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774C2312yes—
62C4⋊#92418—21no—
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67#14519697—22no—
68#14519697—22no—
69#14519697—22no—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73C22⋊#92418—21no—
74C22⋊#92418—22no—
75C22⋊#92418—22no—
76#14519688D412yes—
77C4×#92418—21no—
78C4×#92418C2312yes—
79C2×#41×C61—21no—
80C2×SD16×C61—22no—
81C22×C8×C61—21no—
82C4⋊Q8×C61D411yescharacteristic
83C4⋊Q8×C61C2311yescharacteristic
84C4⋊Q8×C61C2311yescharacteristic
85#127×C61D411yescharacteristic
86#125×C61—21no—
87#124×C61—21no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61—21no—
90#121×C61—21no—
91#121×C61D412yes—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61C2312yes—
96#117×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#106×C61—22no—
99C4⋊C8×C61—21no—
100C4⋊C8×C61C2312yes—
101#103×C61—22no—
102#102×C61—22no—
103C23×D61—41no—
104#41×C61—42no—
105#41×C61—42no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—81no—
126C2×D244—81no—
127C2×D244—82no—
128SD16×C61—82no—
129C2×C8×C61—22no—
130C2×C8×C61—81no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61C2×D412yes—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—41no—
142C42×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147#11758199—82no—
148#11758199—82no—
149#11758194—22no—
150#11758194—22no—
151#11758194—44no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—44no—
157C2×#92418—42no—
158C2×#92418C2×D412yes—
159C2×#92418C2×D412yes—
160C23×C61—22no—
161C23×C61#12111yescharacteristic
162C22×D61—42no—
163C22×D61—82no—
164C22×D61—82no—
165C22×D61—82no—
166Q8×C61—42no—
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—81no—
170D4×C61—22no—
171D4×C61—22no—
172D4×C61—44no—
173D4×C61—42no—
174D4×C61—81no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#13512yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191D244—42no—
192D244—42no—
193D244—42no—
194D244—42no—
195D244—82no—
196D244—82no—
197D244—82no—
198D244—82no—
199C8×C61—42no—
200#92418—22no—
201#92418—22no—
202#92418—41no—
203#92418—41no—
204#92418—41no—
205#92418—41no—
206#19234—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—42no—
212C22×C61#46312yes—
213C22×C61C2×#12111yescharacteristic
214C2×D61—82no—
215C2×D61—82no—
216C2×D61—82no—
217C2×D61—164no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61#46312yes—
225C4×C61C2×#13511yescharacteristic
226C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C61#57611yescharacteristic
228C4×C61#57711yescharacteristic
229#3246D6111yescharacteristic, complemented
230#3205—611no—
231#3138—611no—
232#2935—611no—
233#1701—612no—
234#1697—612no—
235#1681—612no—
236#1677—612no—
237#1565—611no—
238#1542—611no—
239C2×#331—611no—
240C2×C61—22no—
241C2×C61—82no—
242C2×C61C2×#46311yescharacteristic
243C2×C61#306211yescharacteristic
244C2×C61#306811yescharacteristic
245D61—162no—
246C2×#131—1221no—
247#533C2×D6111yescharacteristic
248#532C2×D6111yescharacteristic
249C2×#127—611no—
250#493—1221no—
251#490—1222no—
252#486—612no—
253#486—612no—
254#486—1221no—
255#476C2×D6112yescomplemented
256#475—21no—
257#465—22no—
258#461C2×D6112yescomplemented
259#460—21no—
260#459—612no—
261#459—612no—
262#447—1222no—
263#436—612no—
264#435—1222no—
265C8×D4—612no—
266#431—611no—
267C2×C4⋊C8—611no—
268C2×C4⋊C8C2×D6111yescharacteristic
269#418—1221no—
270#418—1221no—
271#408—1221no—
272C4×M16—611no—
273#335—612no—
274#334—612no—
275#333—611no—
276#333—611no—
277#332—611no—
278#332—611no—
279#331—612no—
280#331—612no—
281C61#1923411yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×D4—1221no—
284C22×D4—1221no—
285C22×D4—1221no—
286C2×SD16—22no—
287C2×D8—1222no—
288C2×D8—1222no—
289#131—2444no—
290C2×M16—611no—
291C2×M16—611no—
292C2×M16—1222no—
293C22×C8—21no—
294C22×C8—1222no—
295C22×C8—1221no—
296C4⋊Q8C22⋊D6111yescharacteristic
297C4⋊Q8C22×D6111yescharacteristic
298C4⋊Q8C22×D6111yescharacteristic
299#127—612no—
300#127—1221no—
301#127—1221no—
302#127—1221no—
303#127C22⋊D6111yescharacteristic
304#125—21no—
305#124—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308#121—21no—
309#121C22⋊D6112yes—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4C22×D6112yes—
314#117—21no—
315C2×C42C22×D6111yescharacteristic
316#106—22no—
317C4⋊C8—21no—
318C4⋊C8—611no—
319C4⋊C8—611no—
320C4⋊C8—611no—
321C4⋊C8—611no—
322C4⋊C8—1221no—
323C4⋊C8C22×D6112yes—
324#103—22no—
325#102—22no—
326#102—1222no—
327#102—1222no—
328#102—1222no—
329#102—1222no—
330#102—1222no—
331#102—1222no—
332C22⋊C8—1222no—
333C22⋊C8—1222no—
334C22⋊C8—1221no—
335#97—612no—
336C4×C8—612no—
337C4×C8—1221no—
338C24—2441no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—1222no—
348C2×D4—1222no—
349C2×D4—1222no—
350C2×D4—4881no—
351C2×D4—4881no—
352C2×D4—4882no—
353C2×D4—2441no—
354C2×D4—2441no—
355C2×D4—2441no—
356C2×D4—2442no—
357C2×D4—2442no—
358C2×D4C2×C22⋊D6112yes—
359C22×C4—22no—
360C22×C4D4×D6111yescharacteristic
361C22×C4#1175822411yescharacteristic
362C22×C4C2×C22⋊D6111yescharacteristic
363SD16—82no—
364D8—4882no—
365D8—4882no—
366M16—1222no—
367M16—1222no—
368M16—2444no—
369C2×C8—22no—
370C2×C8—81no—
371C2×C8—612no—
372C2×C8—612no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1221no—
376C2×C8—1221no—
377C2×C8—2444no—
378C2×C8—2442no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4C2×C22⋊D6112yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—41no—
390C42—41no—
391C42D4×D6111yescharacteristic
392C42#1175822411yescharacteristic
393C42C2×C22⋊D6111yescharacteristic
394C42C23×D6111yescharacteristic
395C23—22no—
396C23—4882no—
397C23—4882no—
398C23—4882no—
399C23—2442no—
400C23#1451974911yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—41no—
404Q8—81no—
405D4—22no—
406D4—22no—
407D4—44no—
408D4—42no—
409D4—81no—
410D4—4882no—
411D4—4882no—
412D4—4882no—
413D4—4882no—
414D4—2442no—
415D4—2442no—
416D4—2442no—
417D4—2442no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4#1451974911yescharacteristic
427C2×C4#1451974911yescharacteristic
428C2×C4#1451974911yescharacteristic
429C2×C4#1451979812yes—
430C2×C4#1451980412yes—
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C22×C22⋊D6111yescharacteristic
433C2×C4C2×#1175822411yescharacteristic
434C8—42no—
435C8—1222no—
436C8—1222no—
437C8—2441no—
438C8—2441no—
439C8—2441no—
440C8—2441no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—9764no—
447C22—4882no—
448C22—4882no—
449C22—4882no—
450C22#1683997912yes—
451C22C2×#1451974911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4#1683997912yes—
459C4C2×#1451979811yescharacteristic
460C4C2×#1451980411yescharacteristic
461C4#1684056511yescharacteristic
462C4#1684057011yescharacteristic
463C2—22no—
464C2—82no—
465C2—9762no—
466C2C2×#1683997911yescharacteristic
467C2#2145550111yescharacteristic
468C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3246C2
socleC22×C61C2×#121
prime radical#21456734C2
hypercenter#3246D61
hypocenterC61#19234
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
2. #21456195C2semidirect
3. #21454563C2semidirect
4. #21445382C2semidirect
5. #21445354C2semidirect
6. #21445258C2semidirect
7. #21445230C2semidirect
8. #21444569C2semidirect
9. #21444476C2semidirect
10. C2×#16839505C2semidirect
11. C61×#3246C2semidirect
22. #16839840C22semidirect
24. D4×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839511C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
36. #16839508C22semidirect
41. C61×#476C22semidirect
44. C61×#461C22semidirect
229. #3246D61semidirect
255. #476C2×D61semidirect
258. #461C2×D61semidirect
281. C61#19234semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214561952
#214545632
#214453822
#214453542
#214452582
#214452302
#214445692
#214444762
C2×#168395052
C61×#32462
#1923461