Finite Group Databasebeta

← #29834495

Subgroups — #29834495

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×M32×D61C211yesmaximal, characteristic, complemented
2#21455396C211yesmaximal, characteristic
3C2×#16840000C211yesmaximal, characteristic, complemented
4#21447929C212yesmaximal, complemented
5#21447924C212yesmaximal, complemented
6#21447912C212yesmaximal, complemented
7#21447907C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447539C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C2×C61×#510C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14C2×#14519753—21no—
15C2×#14519717C2211yescharacteristic
16C2×#14519716C2212yes—
17M32×D61—24no—
18C2×C16×D61C2212yes—
19#16840000—21no—
20#16840000—21no—
21#16840000C2212yes—
22#16840000C2212yes—
23#16840000C2211yescharacteristic
24#16840000C2211yescharacteristic
25#16839995C2211yescharacteristic
26#16839995C2211yescharacteristic
27#16839988—21no—
28#16839988—21no—
29#16839800C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839758—21no—
32C2×#14519701—21no—
33#16839590C2212yescomplemented
34#16839588C2212yescomplemented
35#16839566C2212yescomplemented
36#16839566C2212yescomplemented
37#16839566C2212yescomplemented
38#16839563C2212yescomplemented
39C2×#135×C61C2211yescharacteristic
40C22×Q16×C61C2211yescharacteristic
41C61×#510—24no—
42C2×Q32×C61C2212yescomplemented
43C2×SD32×C61C2212yescomplemented
44C2×M32×C61C2211yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×#92418D411yescharacteristic
47C2×C8×D61—22no—
48C2×C8×D61—22no—
49C2×C8×D61D411yescharacteristic
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52C2×C8×D61C2311yescharacteristic
53#14519758—41no—
54#14519753—21no—
55#14519753—21no—
56#14519753—22no—
57#14519753—22no—
58#14519753—41no—
59#14519753—41no—
60#14519748—21no—
61#14519748—21no—
62#14519745—41no—
63C4×C22⋊D61—41no—
64C4×C22⋊D61—41no—
65#14519736—21no—
66C2×C4⋊Dic61—21no—
67C2×#11758196—41no—
68C2×#11758198—41no—
69#14519717—22no—
70#14519716—24no—
71C2×#11758185C2312yes—
72C16×D61—24no—
73#14519711—21no—
74#14519711C2311yescharacteristic
75#14519711C2311yescharacteristic
76#14519711C2312yes—
77#14519709—21no—
78#14519701—21no—
79#14519701—21no—
80#14519701—22no—
81#14519701—22no—
82#14519698—21no—
83#14519698—21no—
84C2×#41×C61—21no—
85C22×Q8×C61—21no—
86#135×C61—22no—
87#135×C61—22no—
88C2×Q16×C61—22no—
89C2×Q16×C61—21no—
90C2×Q16×C61—22no—
91C2×Q16×C61C2312yes—
92C2×Q16×C61C2311yescharacteristic
93C2×SD16×C61—21no—
94C2×D8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96Q32×C61—24no—
97SD32×C61—24no—
98M32×C61D414yescomplemented
99C2×C16×C61C2312yes—
100C23×D61D811yescharacteristic
101#41×C61—42no—
102#41×C61—42no—
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—41no—
106C2×Q8×C61—42no—
107C2×Q8×C61—42no—
108C2×D4×C61—21no—
109C2×D4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C22×Dic61—41no—
116C22×Dic61D811yescharacteristic
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61D811yescharacteristic
120C2×C4×D61D811yescharacteristic
121C2×C4×D61C2×D411yescharacteristic
122C2×C4×D61C2×D411yescharacteristic
123Q16×C61—22no—
124Q16×C61—21no—
125Q16×C61—22no—
126Q16×C61—21no—
127Q16×C61—42no—
128SD16×C61—42no—
129D8×C61—21no—
130D8×C61—21no—
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D412yes—
133C2×C8×C61#4112yes—
134C2×C8×C61C2411yescharacteristic
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—41no—
138#11758198—42no—
139#11758198—42no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—22no—
145C4⋊Dic61—41no—
146#11758196—42no—
147#11758196—42no—
148#11758196—41no—
149#11758196—41no—
150C4×Dic61—41no—
151C2×#92418—22no—
152C2×#92418C2×D411yescharacteristic
153C2×#92418C2×D411yescharacteristic
154C8×D61—22no—
155C8×D61—22no—
156C8×D61—21no—
157C8×D61—21no—
158C8×D61—21no—
159C8×D61—21no—
160C16×C61C2×D414yes—
161#11758185—22no—
162C23×C61—41no—
163C23×C61C2×D811yescharacteristic
164C22×D61—22no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61C2×D811yescharacteristic
168C22×D61C2×D811yescharacteristic
169Q8×C61—41no—
170Q8×C61—42no—
171Q8×C61—42no—
172Q8×C61—41no—
173Q8×C61—82no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—82no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—82no—
180C2×C4×C61—82no—
181C2×C4×C61#12112yes—
182C2×C4×C61C2×D811yescharacteristic
183C2×C4×C61#13512yes—
184C2×C4×C61C22×D411yescharacteristic
185C22⋊D61—82no—
186C22⋊D61—82no—
187C2×Dic61—22no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—82no—
192C2×Dic61C2×D811yescharacteristic
193C2×Dic61C2×D811yescharacteristic
194C4×D61—21no—
195C4×D61—21no—
196C4×D61—21no—
197C4×D61—21no—
198C4×D61—22no—
199C4×D61—22no—
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—81no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61#46612yes—
212C22×C61C22×D811yescharacteristic
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—44no—
219C2×D61#51012yescomplemented
220C4×C61—81no—
221C4×C61—82no—
222C4×C61#46612yes—
223C4×C61C2×#12111yescharacteristic
224C4×C61C2×#13511yescharacteristic
225Dic61—21no—
226Dic61—21no—
227Dic61—81no—
228Dic61—82no—
229Dic61—81no—
230C2×#510D6111yescharacteristic, complemented
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#2141—611no—
240#2139—611no—
241C2×C61—22no—
242C2×C61—161no—
243C2×C61C2×#46611yescharacteristic
244C2×C61C2×#51011yescharacteristic
245C2×C61#341411yescharacteristic
246D61—22no—
247D61—41no—
248C2×#135C2×D6111yescharacteristic
249C22×Q16C2×D6111yescharacteristic
250C23×C8—611no—
251#525—1221no—
252C2×C22⋊Q8—1221no—
253#510—24no—
254C2×Q32C2×D6112yescomplemented
255C2×SD32C2×D6112yescomplemented
256C2×M32—611no—
257C2×M32—612no—
258C2×M32—1224no—
259C2×M32C2×D6111yescharacteristic
260C22×C16—612no—
261#467—611no—
262#467—612no—
263#467—611no—
264#467—612no—
265#467—1221no—
266#467—1221no—
267#466—611no—
268#466—611no—
269#465—1221no—
270#465—1221no—
271#427—611no—
272C2×#107—611no—
273#416—1221no—
274C2×#103—1221no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358—612no—
279#358—612no—
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×Q8—21no—
284C23×C4—611no—
285#135—22no—
286#135—22no—
287C2×Q16—21no—
288C2×Q16—22no—
289C2×Q16—22no—
290C2×Q16C22×D6112yes—
291C2×Q16C22×D6111yescharacteristic
292C2×SD16—21no—
293C2×D8C22×D6111yescharacteristic
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—1222no—
304C22⋊Q8—1222no—
305C22⋊Q8—1221no—
306C22⋊Q8—1221no—
307C22⋊Q8—2441no—
308C22⋊Q8—2441no—
309#121—1221no—
310#121—1221no—
311#120—2441no—
312C4×D4—2441no—
313C4×D4—2441no—
314#117—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—24no—
319SD32—24no—
320M32—1222no—
321M32—1224no—
322M32C22⋊D6114yescomplemented
323C2×C16—612no—
324C2×C16—1224no—
325C2×C16C22×D6112yes—
326#107—612no—
327#107—611no—
328#107—611no—
329#107—1221no—
330#106—1221no—
331#103—1222no—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336#102—1221no—
337C24—611no—
338#41—42no—
339#41—42no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—2441no—
359C22×C4D4×D6111yescharacteristic
360Q16—22no—
361Q16—21no—
362Q16—22no—
363Q16—21no—
364Q16—42no—
365SD16—42no—
366D8—21no—
367D8—21no—
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×C8D4×D6111yescharacteristic
378C2×C8#1175822412yes—
379C2×C8C2×C22⋊D6112yes—
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—2442no—
385C4⋊C4—2442no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C42—2441no—
397C16—1222no—
398C16C2×C22⋊D6114yes—
399C23—41no—
400C23—611no—
401C23—611no—
402C23—1222no—
403C23—1222no—
404C23—2441no—
405C23D8×D6111yescharacteristic
406Q8—41no—
407Q8—42no—
408Q8—42no—
409Q8—41no—
410Q8—82no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—4882no—
415D4—4882no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—82no—
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—2441no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4#1451974812yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451979612yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451974812yes—
440C8C22×C22⋊D6111yescharacteristic
441C8C2×#1175822411yescharacteristic
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—612no—
448C22—1221no—
449C22—1221no—
450C22—1221no—
451C22—1221no—
452C22—1221no—
453C22—2444no—
454C22#1683999512yes—
455C22C2×D8×D6111yescharacteristic
456C4—81no—
457C4—82no—
458C4—1221no—
459C4—1221no—
460C4—4881no—
461C4—4882no—
462C4—4881no—
463C4#1683999512yes—
464C4C2×#1451974811yescharacteristic
465C4C2×#1451979611yescharacteristic
466C2—22no—
467C2—161no—
468C2—1222no—
469C2—2441no—
470C2C2×#1683999511yescharacteristic
471C2#510×D6111yescharacteristic
472C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC22×C61C22×D8
prime radical#21455396C2
hypercenterC2×#510D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C2×M32×D61C2semidirect
3. C2×#16840000C2semidirect
4. #21447929C2semidirect
5. #21447924C2semidirect
6. #21447912C2semidirect
7. #21447907C2semidirect
8. #21447607C2semidirect
9. #21447539C2semidirect
10. #21447531C2semidirect
11. C2×C61×#510C2semidirect
33. #16839590C22semidirect
34. #16839588C22semidirect
35. #16839566C22semidirect
36. #16839566C22semidirect
37. #16839566C22semidirect
38. #16839563C22semidirect
42. C2×Q32×C61C22semidirect
43. C2×SD32×C61C22semidirect
98. M32×C61D4semidirect
219. C2×D61#510semidirect
230. C2×#510D61semidirect
254. C2×Q32C2×D61semidirect
255. C2×SD32C2×D61semidirect
281. C61#33606semidirect
322. M32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D612
#214553962
C2×#168400002
#214479292
#214479242
#214479122
#214479072
#214476072
#214475392
#214475312
C2×C61×#5102
#3360661