Finite Group Databasebeta

← #29834502

Subgroups — #29834502

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×#16840205C211yesmaximal, characteristic, complemented
2C2×M32×D61C211yesmaximal, characteristic, complemented
3#21455401C211yesmaximal, characteristic
4C2×#16840004C211yesmaximal, characteristic, complemented
5#21447933C212yesmaximal, complemented
6#21447928C212yesmaximal, complemented
7#21447916C212yesmaximal, complemented
8#21447911C212yesmaximal, complemented
9#21447538C211yesmaximal, characteristic, complemented
10#21447529C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×Dic244C2211yescharacteristic
14C22×C8×D61C2211yescharacteristic
15#16840274—21no—
16C2×#14519757—21no—
17#16840205—24no—
18C2×Dic488C2212yescomplemented
19C2×#14519722C2212yescomplemented
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yes—
22M32×D61—24no—
23C2×C16×D61C2212yes—
24#16840004—21no—
25#16840004—21no—
26#16840004C2212yes—
27#16840004C2212yes—
28#16840004C2211yescharacteristic
29#16840004C2211yescharacteristic
30#16839999C2211yescharacteristic
31#16839999C2211yescharacteristic
32#16839992—21no—
33#16839992—21no—
34#16839756—21no—
35C2×#14519700—21no—
36#16839565C2212yescomplemented
37#16839565C2212yescomplemented
38#16839565C2212yescomplemented
39#16839562C2212yescomplemented
40C2×M32×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C2×#107×C61C2211yescharacteristic
43C61×#367C2212yescomplemented
44C61×#366C2212yescomplemented
45C2×#11758222—21no—
46C22×Dic122—21no—
47C22×C4×D61D411yescharacteristic
48#14519799—22no—
49#14519799—22no—
50C2×Dic244—22no—
51C2×Dic244—22no—
52C2×Dic244—21no—
53C2×Dic244C2312yes—
54C2×Dic244C2311yescharacteristic
55C2×#11758190—21no—
56C2×D488C2311yescharacteristic
57C22×#92418D411yescharacteristic
58C2×C8×D61—22no—
59C2×C8×D61—22no—
60C2×C8×D61D411yescharacteristic
61C2×C8×D61D411yescharacteristic
62C2×C8×D61C2311yescharacteristic
63C2×C8×D61C2311yescharacteristic
64#14519761—41no—
65#14519757—21no—
66#14519757—21no—
67#14519757—22no—
68#14519757—22no—
69#14519757—41no—
70#14519757—41no—
71#14519752—21no—
72#14519752—21no—
73#14519747—41no—
74C4×D244—41no—
75C4×D244—41no—
76C2×C4⋊Dic61—41no—
77C2×#11758198—41no—
78Dic488—24no—
79#14519722—24no—
80#14519717—22no—
81#14519716—24no—
82C2×#11758185C2312yes—
83C16×D61—24no—
84#14519700—21no—
85#14519700—21no—
86#14519700—22no—
87#14519700—22no—
88#14519697—21no—
89#14519697—21no—
90C22×C8×C61C2311yescharacteristic
91#117×C61—21no—
92C2×C4⋊C4×C61—21no—
93M32×C61D414yescomplemented
94C2×C16×C61C2312yes—
95#107×C61—21no—
96#107×C61C2311yescharacteristic
97#107×C61C2311yescharacteristic
98#107×C61C2312yes—
99#106×C61—21no—
100C23×D61D811yescharacteristic
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C22×Dic61—41no—
105C22×Dic61D811yescharacteristic
106#11758222—42no—
107#11758222—42no—
108C2×D244—21no—
109C2×D244—41no—
110C2×D244—41no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—41no—
114C2×C4×D61D811yescharacteristic
115C2×C4×D61D811yescharacteristic
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—41no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D412yes—
125C2×C8×C61#4112yes—
126C2×C8×C61C2411yescharacteristic
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C42×C61—41no—
133#11758198—42no—
134#11758198—42no—
135#11758198—41no—
136#11758198—41no—
137#11758198—41no—
138#11758198—41no—
139C4⋊Dic61—42no—
140C4⋊Dic61—42no—
141C4⋊Dic61—41no—
142C4⋊Dic61—41no—
143C2×#92418—22no—
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146Dic244—22no—
147Dic244—21no—
148Dic244—22no—
149Dic244—21no—
150Dic244—42no—
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×C61C2×D414yes—
161#11758185—22no—
162C23×C61C2×D811yescharacteristic
163C22×D61—22no—
164C22×D61—22no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61C2×D811yescharacteristic
168C22×D61C2×D811yescharacteristic
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—82no—
173C2×C4×C61#12112yes—
174C2×C4×C61C2×D811yescharacteristic
175C2×C4×C61#13512yes—
176C2×C4×C61C22×D411yescharacteristic
177C22⋊D61—82no—
178C2×Dic61—22no—
179C2×Dic61—42no—
180C2×Dic61—41no—
181C2×Dic61—82no—
182C2×Dic61C2×D811yescharacteristic
183C2×Dic61C2×D811yescharacteristic
184D244—41no—
185D244—41no—
186D244—82no—
187D244—82no—
188C4×D61—21no—
189C4×D61—21no—
190C4×D61—21no—
191C4×D61—21no—
192C4×D61—22no—
193C4×D61—22no—
194C4×D61—82no—
195Dic122—41no—
196Dic122—42no—
197Dic122—42no—
198Dic122—41no—
199Dic122—82no—
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#2159D6111yescharacteristic, complemented
239#2141—611no—
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×Q16—611no—
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—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#427C2×D6111yescharacteristic
272C2×#107C2×D6111yescharacteristic
273#416—1221no—
274C2×#103—1221no—
275#367C2×D6112yescomplemented
276#366C2×D6112yescomplemented
277#358—612no—
278#358—612no—
279#358—612no—
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—1221no—
283C22×Q8—1221no—
284C23×C4—611no—
285#135—1222no—
286#135—1222no—
287C2×Q16—612no—
288C2×Q16—611no—
289C2×Q16—1222no—
290C2×Q16—1221no—
291C2×Q16—1222no—
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—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—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—1224no—
319SD32—1224no—
320M32—1222no—
321M32—1224no—
322M32D24414yescomplemented
323C2×C16—612no—
324C2×C16—1224no—
325C2×C16C22×D6112yes—
326#107—21no—
327#107C22×D6112yes—
328#107C22×D6111yescharacteristic
329#107C22×D6111yescharacteristic
330#106—21no—
331#103—1222no—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336#102—1221no—
337C24—611no—
338#41—2442no—
339#41—2442no—
340C2×Q8—1221no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×Q8—2441no—
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—1222no—
361Q16—1221no—
362Q16—1222no—
363Q16—1221no—
364Q16—2442no—
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×C8C2×D24412yes—
378C2×C8D4×D6111yescharacteristic
379C2×C8#1175822612yes—
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
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—1222no—
398C16C2×D24414yes—
399C23—611no—
400C23—611no—
401C23—1222no—
402C23—1222no—
403C23—2441no—
404C23—2441no—
405C23D8×D6111yescharacteristic
406Q8—4882no—
407Q8—2441no—
408Q8—2442no—
409Q8—2442no—
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—2441no—
432C2×C4—2442no—
433C2×C4#1451975212yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451980012yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451975212yes—
440C8C22×D24411yescharacteristic
441C8C2×#1175822611yescharacteristic
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#1683999912yes—
455C22C2×D8×D6111yescharacteristic
456C4—81no—
457C4—82no—
458C4—81no—
459C4—1221no—
460C4—1221no—
461C4—4881no—
462C4—4882no—
463C4#1683999912yes—
464C4C2×#1451975211yescharacteristic
465C4C2×#1451980011yescharacteristic
466C2—22no—
467C2—1222no—
468C2—9761no—
469C2—2441no—
470C2C2×#1683999911yescharacteristic
471C2#510×D6111yescharacteristic
472C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radical#21455401C2
hypercenter#2159D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
2. C2×M32×D61C2semidirect
4. C2×#16840004C2semidirect
5. #21447933C2semidirect
6. #21447928C2semidirect
7. #21447916C2semidirect
8. #21447911C2semidirect
9. #21447538C2semidirect
10. #21447529C2semidirect
11. C61×#2159C2semidirect
18. C2×Dic488C22semidirect
19. C2×#14519722C22semidirect
36. #16839565C22semidirect
37. #16839565C22semidirect
38. #16839565C22semidirect
39. #16839562C22semidirect
43. C61×#367C22semidirect
44. C61×#366C22semidirect
93. M32×C61D4semidirect
219. C2×D61#510semidirect
238. #2159D61semidirect
275. #367C2×D61semidirect
276. #366C2×D61semidirect
281. C61#33606semidirect
322. M32D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
C2×M32×D612
#214554012
C2×#168400042
#214479332
#214479282
#214479162
#214479112
#214475382
#214475292
C61×#21592
#3360661