Finite Group Databasebeta

← #29834499

Subgroups — #29834499

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C2×M32×D61C211yesmaximal, characteristic, complemented
3#21455401C211yesmaximal, characteristic
4C2×#16840000C211yesmaximal, characteristic, complemented
5#21447935C212yesmaximal, complemented
6#21447926C212yesmaximal, complemented
7#21447918C212yesmaximal, complemented
8#21447909C212yesmaximal, complemented
9#21447608C211yesmaximal, characteristic, complemented
10#21447532C211yesmaximal, characteristic, complemented
11C61×#2141C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840274—21no—
15C2×#14519753—21no—
16#16840203—24no—
17C2×#14519723C2212yescomplemented
18C2×#14519720C2212yescomplemented
19C2×#14519717C2211yescharacteristic
20C2×#14519716C2212yes—
21M32×D61—24no—
22C2×C16×D61C2212yes—
23#16840004C2211yescharacteristic
24#16840004C2211yescharacteristic
25#16840000—21no—
26#16840000—21no—
27#16840000C2212yes—
28#16840000C2212yes—
29#16839999C2211yescharacteristic
30#16839999C2211yescharacteristic
31#16839992—21no—
32#16839992—21no—
33C2×#14519711C2211yescharacteristic
34#16839756—21no—
35C2×#14519701—21no—
36#16839589C2212yescomplemented
37#16839587C2212yescomplemented
38#16839567C2212yescomplemented
39#16839564C2212yescomplemented
40C22×Q16×C61C2211yescharacteristic
41C2×M32×C61C2211yescharacteristic
42C61×#427C2211yescharacteristic
43C61×#358C2212yescomplemented
44C61×#358C2212yescomplemented
45C2×#11758222—21no—
46C22×C4×D61D411yescharacteristic
47#14519799—22no—
48#14519799—22no—
49C2×Dic244C2311yescharacteristic
50C2×#11758190—21no—
51C2×D488C2311yescharacteristic
52C22×#92418D411yescharacteristic
53C2×C8×D61—22no—
54C2×C8×D61—22no—
55C2×C8×D61D411yescharacteristic
56C2×C8×D61D411yescharacteristic
57C2×C8×D61C2311yescharacteristic
58C2×C8×D61C2311yescharacteristic
59#14519761—41no—
60#14519757—21no—
61#14519757—21no—
62#14519753—22no—
63#14519753—22no—
64#14519753—41no—
65#14519753—41no—
66#14519752—21no—
67#14519752—21no—
68#14519747—41no—
69C4×D244—41no—
70C4×D244—41no—
71C2×C4⋊Dic61—21no—
72C2×#11758196—41no—
73C2×#11758198—41no—
74#14519723—24no—
75#14519720—24no—
76#14519717D414yescomplemented
77#14519716—24no—
78C2×#11758185C2312yes—
79C16×D61—24no—
80#14519711—21no—
81#14519711C2312yes—
82#14519701—22no—
83#14519701—22no—
84#14519700—21no—
85#14519700—21no—
86#14519697—21no—
87#14519697—21no—
88C22×Q8×C61—21no—
89C2×Q16×C61—22no—
90C2×Q16×C61—21no—
91C2×Q16×C61—22no—
92C2×Q16×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94#117×C61—21no—
95M32×C61—22no—
96C2×C16×C61C2312yes—
97#107×C61C2311yescharacteristic
98#107×C61C2311yescharacteristic
99#106×C61—21no—
100C23×D61D811yescharacteristic
101C2×Q8×C61—22no—
102C2×Q8×C61—41no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C22×C4×C61—41no—
106C22×C4×C61C2×D411yescharacteristic
107C2×C22⋊D61—41no—
108C22×Dic61—41no—
109C22×Dic61D811yescharacteristic
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244—41no—
114C2×D244—41no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—41no—
118C2×C4×D61D811yescharacteristic
119C2×C4×D61D811yescharacteristic
120C2×C4×D61C2×D411yescharacteristic
121C2×C4×D61C2×D411yescharacteristic
122C2×Dic122—21no—
123Q16×C61—22no—
124Q16×C61—22no—
125Q16×C61—42no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D412yes—
128C2×C8×C61#4112yes—
129C2×C8×C61C2411yescharacteristic
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C22⋊C4×C61—41no—
133C42×C61—41no—
134#11758198—42no—
135#11758198—42no—
136#11758198—41no—
137#11758198—41no—
138#11758198—41no—
139#11758198—41no—
140C4⋊Dic61—22no—
141C4⋊Dic61—41no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144#11758196—42no—
145#11758196—42no—
146C2×#92418—22no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149Dic244—21no—
150Dic244—21no—
151D488—21no—
152D488—21no—
153#11758190—42no—
154C8×D61—22no—
155C8×D61—22no—
156C8×D61—21no—
157C8×D61—21no—
158C8×D61—21no—
159C8×D61—21no—
160C16×C61—22no—
161#11758185C2×D414yes—
162C23×C61C2×D811yescharacteristic
163C22×D61—22no—
164C22×D61—22no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61C2×D811yescharacteristic
168C22×D61C2×D811yescharacteristic
169Q8×C61—42no—
170Q8×C61—42no—
171Q8×C61—82no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—82no—
176C2×C4×C61#12112yes—
177C2×C4×C61C2×D811yescharacteristic
178C2×C4×C61#13512yes—
179C2×C4×C61C22×D411yescharacteristic
180C22⋊D61—82no—
181C2×Dic61—22no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—82no—
185C2×Dic61C2×D811yescharacteristic
186C2×Dic61C2×D811yescharacteristic
187D244—41no—
188D244—41no—
189D244—82no—
190D244—82no—
191C4×D61—21no—
192C4×D61—21no—
193C4×D61—21no—
194C4×D61—21no—
195C4×D61—22no—
196C4×D61—22no—
197C4×D61—82no—
198Dic122—41no—
199Dic122—41no—
200C8×C61#12112yes—
201C8×C61C22×D411yescharacteristic
202C8×C61C2×#4111yescharacteristic
203#92418—21no—
204#92418—21no—
205#33606—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61#46612yes—
209C22×C61C22×D811yescharacteristic
210C2×D61—21no—
211C2×D61—21no—
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—44no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61#51012yescomplemented
220C4×C61—81no—
221C4×C61—81no—
222C4×C61—82no—
223C4×C61#46612yes—
224C4×C61C2×#12111yescharacteristic
225C4×C61C2×#13511yescharacteristic
226Dic61—21no—
227Dic61—21no—
228Dic61—82no—
229Dic61—81no—
230C2×#510—611no—
231C22×M32—611no—
232#3050—611no—
233C2×#467—611no—
234#2220—612no—
235#2219—612no—
236#2217—612no—
237#2216—612no—
238#2159—611no—
239#2141D6111yescharacteristic, complemented
240#2139—611no—
241C2×C61—22no—
242C2×C61C2×#46611yescharacteristic
243C2×C61C2×#51011yescharacteristic
244C2×C61#341411yescharacteristic
245D61—22no—
246D61—41no—
247D61—161no—
248C2×#135—611no—
249C22×Q16C2×D6111yescharacteristic
250C23×C8—611no—
251#525—1221no—
252C2×C22⋊Q8—1221no—
253#510—1224no—
254C2×Q32—612no—
255C2×SD32—612no—
256C2×M32—611no—
257C2×M32—612no—
258C2×M32—1224no—
259C2×M32C2×D6111yescharacteristic
260C22×C16—612no—
261#467—612no—
262#467—612no—
263#467—611no—
264#467—611no—
265#467—1221no—
266#467—1221no—
267#466—611no—
268#466—611no—
269#465—1221no—
270#465—1221no—
271#427C2×D6111yescharacteristic
272C2×#107—611no—
273#416—1221no—
274C2×#103—1221no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358C2×D6112yescomplemented
279#358C2×D6112yescomplemented
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—1221no—
283C22×Q8—21no—
284C23×C4—611no—
285#135—1222no—
286#135—1222no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×Q16—611no—
291C2×Q16C22×D6112yes—
292C2×SD16—1221no—
293C2×D8—611no—
294C22×C8—611no—
295C22×C8—611no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—611no—
299C22×C8—1222no—
300C22×C8—1222no—
301C22×C8C22×D6111yescharacteristic
302#123—2441no—
303C22⋊Q8—1221no—
304C22⋊Q8—1221no—
305C22⋊Q8—1222no—
306C22⋊Q8—1222no—
307C22⋊Q8—2441no—
308C22⋊Q8—2441no—
309#121—1221no—
310#121—1221no—
311#120—2441no—
312C4×D4—2441no—
313C4×D4—2441no—
314#117—21no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—1224no—
319SD32—1224no—
320M32—22no—
321M32—614no—
322M32—1224no—
323C2×C16—612no—
324C2×C16—1224no—
325C2×C16C22×D6112yes—
326#107—612no—
327#107—1221no—
328#107C22×D6111yescharacteristic
329#107C22×D6111yescharacteristic
330#106—21no—
331#103—1221no—
332#103—1221no—
333#103—1222no—
334#103—1222no—
335#102—1221no—
336#102—1221no—
337C24—611no—
338#41—2442no—
339#41—2442no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—1221no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—41no—
350C22×C4—611no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—2441no—
358C22×C4—2441no—
359C22×C4D4×D6111yescharacteristic
360Q16—22no—
361Q16—22no—
362Q16—42no—
363Q16—1221no—
364Q16—1221no—
365SD16—2442no—
366D8—1221no—
367D8—1221no—
368C2×C8—611no—
369C2×C8—611no—
370C2×C8—1222no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8#1175822212yes—
378C2×C8D4×D6112yes—
379C2×C8D4×D6111yescharacteristic
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—1222no—
384C4⋊C4—2442no—
385C4⋊C4—2442no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—41no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C22⋊C4—2442no—
393C22⋊C4—2442no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C42—41no—
397C16—22no—
398C16—614no—
399C23—611no—
400C23—611no—
401C23—1222no—
402C23—1222no—
403C23—2441no—
404C23—2441no—
405C23D8×D6111yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—2441no—
410Q8—2441no—
411D4—4882no—
412D4—4882no—
413D4—4882no—
414D4—2441no—
415D4—2441no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—82no—
420C2×C4—611no—
421C2×C4—611no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—4882no—
430C2×C4—4882no—
431C2×C4—2442no—
432C2×C4—2441no—
433C2×C4#1451975012yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451979812yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451975012yes—
440C8C2×D4×D6111yescharacteristic
441C8C2×#1175822211yescharacteristic
442C22—21no—
443C22—21no—
444C22—612no—
445C22—1221no—
446C22—1221no—
447C22—1221no—
448C22—1221no—
449C22—1221no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22—2444no—
454C22#1683999712yes—
455C22C2×D8×D6111yescharacteristic
456C4—81no—
457C4—81no—
458C4—82no—
459C4—1221no—
460C4—1221no—
461C4—4882no—
462C4—4881no—
463C4#1683999712yes—
464C4C2×#1451975011yescharacteristic
465C4C2×#1451979811yescharacteristic
466C2—22no—
467C2—1222no—
468C2—9761no—
469C2—2441no—
470C2C2×#1683999711yescharacteristic
471C2#510×D6111yescharacteristic
472C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2141C2
socleC22×C61C22×D8
prime radical#21455401C2
hypercenter#2141D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×M32×D61C2semidirect
4. C2×#16840000C2semidirect
5. #21447935C2semidirect
6. #21447926C2semidirect
7. #21447918C2semidirect
8. #21447909C2semidirect
9. #21447608C2semidirect
10. #21447532C2semidirect
11. C61×#2141C2semidirect
17. C2×#14519723C22semidirect
18. C2×#14519720C22semidirect
36. #16839589C22semidirect
37. #16839587C22semidirect
38. #16839567C22semidirect
39. #16839564C22semidirect
43. C61×#358C22semidirect
44. C61×#358C22semidirect
76. #14519717D4semidirect
219. C2×D61#510semidirect
239. #2141D61semidirect
278. #358C2×D61semidirect
279. #358C2×D61semidirect
281. C61#33606semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×M32×D612
#214554012
C2×#168400002
#214479352
#214479262
#214479182
#214479092
#214476082
#214475322
C61×#21412
#3360661