Finite Group Databasebeta

← #29834753

Subgroups — #29834753

Order 15616.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456331C211yesmaximal, characteristic, complemented
2C2×#16839879C211yesmaximal, characteristic
3#21447993C214yesmaximal, complemented
4#21447988C214yesmaximal, complemented
5#21447592C211yesmaximal, characteristic, complemented
6#21447591C211yesmaximal, characteristic, complemented
7C2×#16839582C211yesmaximal, characteristic, complemented
8#21447555C211yesmaximal, characteristic
9C61×#2145C211yesmaximal, characteristic
10C2×#14519784—21no—
11#16840315—21no—
12C2×#14519717C2211yescharacteristic
13#16840166—21no—
14#16840163—21no—
15#16840162C2212yescomplemented
16#16840159C2212yescomplemented
17#16839879—21no—
18#16839879—21no—
19#16839879C2212yes—
20#16839879C2212yes—
21C2×#14519713C2211yescharacteristic
22#16839800C2211yescharacteristic
23#16839798—21no—
24C2×#14519704—21no—
25C2×C8×Dic61C2211yescharacteristic
26#16839585C2214yescomplemented
27#16839584C2214yescomplemented
28#16839582C2212yescomplemented
29#16839582C2212yescomplemented
30#16839572C2214yes—
31C2×#135×C61C2211yescharacteristic
32C2×M32×C61C2211yescharacteristic
33C61×#427C2211yescharacteristic
34C61×#360C2214yes—
35#14519784—22no—
36#14519784—22no—
37#14519784—41no—
38#14519784—41no—
39C2×#11758194—21no—
40C2×#11758194—41no—
41C22×#92418—41no—
42C22×#92418D411yescharacteristic
43#14519776—22no—
44#14519768—22no—
45#14519754—42no—
46C4×Dic122—42no—
47#14519736—21no—
48C2×C4×Dic61D411yescharacteristic
49#14519717—21no—
50#14519717C2312yes—
51C2×#11758185—21no—
52#14519713—21no—
53#14519713C2312yes—
54#14519711—21no—
55#14519710—22no—
56#14519709C2312yes—
57#14519708—22no—
58#14519704—22no—
59#14519704—22no—
60C8×Dic61D411yescharacteristic
61C8×Dic61D411yescharacteristic
62C8×Dic61C2311yescharacteristic
63C8×Dic61C2311yescharacteristic
64C2×#41×C61—21no—
65#135×C61—21no—
66#135×C61—21no—
67#135×C61Q812yescomplemented
68#135×C61C2312yes—
69C2×Q16×C61Q811yescharacteristic, complemented
70C2×SD16×C61—21no—
71C2×D8×C61Q811yescharacteristic, complemented
72C22×C8×C61C2311yescharacteristic
73#117×C61—21no—
74M32×C61—21no—
75M32×C61C2312yes—
76C2×C16×C61—21no—
77#107×C61—21no—
78#106×C61C2312yes—
79#41×C61—22no—
80#41×C61—22no—
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C22×Dic61—21no—
89C2×Dic122—42no—
90Q16×C61—21no—
91Q16×C61C2×Q812yes—
92SD16×C61—22no—
93SD16×C61—22no—
94D8×C61—21no—
95D8×C61C2×Q812yes—
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61C2×Q811yescharacteristic
98C2×C8×C61C2×Q811yescharacteristic
99C2×C8×C61#4111yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C2×C8×C61C2411yescharacteristic
102C4⋊C4×C61—22no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C22⋊Dic61—41no—
106C4⋊Dic61—22no—
107C4⋊Dic61—42no—
108#11758196—42no—
109#11758196—42no—
110C4×Dic61—21no—
111C4×Dic61—41no—
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61C2×D411yescharacteristic
114#11758194—22no—
115#11758194—41no—
116#11758194—42no—
117#11758194—42no—
118C2×#92418—41no—
119C2×#92418—42no—
120C2×#92418—42no—
121C2×#92418SD1611yescharacteristic
122C2×#92418SD1611yescharacteristic
123C2×#92418SD1611yescharacteristic
124C2×#92418SD1611yescharacteristic
125C2×#92418C2×D411yescharacteristic
126C2×#92418C2×D411yescharacteristic
127C16×C61—22no—
128#11758185—22no—
129C23×C61—41no—
130C23×C61AGL(1,ℤ/8)11yescharacteristic
131Q8×C61—22no—
132Q8×C61—41no—
133D4×C61—22no—
134D4×C61—42no—
135D4×C61—42no—
136D4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61C22⋊Q811yescharacteristic
142C2×C4×C61C22⋊Q811yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61C2×SD1611yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61C22×D411yescharacteristic
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151Dic122—84no—
152C8×C61C22⋊Q811yescharacteristic
153C8×C61C22⋊Q811yescharacteristic
154C8×C61C22×Q811yescharacteristic
155C8×C61C2×#4111yescharacteristic
156#92418—42no—
157#92418C2×SD1611yescharacteristic
158#92418C2×SD1611yescharacteristic
159#92418C2×SD1611yescharacteristic
160#92418C2×SD1611yescharacteristic
161#33637—611nomaximal, Sylow
162C22×C61—41no—
163C22×C61—42no—
164C22×C61—82no—
165C22×C61#47611yescharacteristic
166C22×C61#47611yescharacteristic
167C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
168C4×C61—42no—
169C4×C61—82no—
170C4×C61#47611yescharacteristic
171C4×C61#47611yescharacteristic
172C4×C61C2×C22⋊Q811yescharacteristic
173C4×C61C22×SD1611yescharacteristic
174Dic61—22no—
175Dic61—41no—
176Dic61—82no—
177#3160—611no—
178C2×#443—611no—
179#2231—614no—
180#2230—614no—
181#2155—611no—
182#2155—611no—
183C2×#364—611no—
184#2145—611no—
185#2145D6111yescharacteristic
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—82no—
190C2×C61C2×#47611yescharacteristic
191C2×#135C2×D6111yescharacteristic
192C2×#131—1221no—
193#533—1221no—
194C2×M32—611no—
195C2×M32C2×D6111yescharacteristic
196#500—1221no—
197#499—612no—
198#499—1221no—
199#498—612no—
200#443—612no—
201#443—612no—
202#443—1221no—
203#443—1221no—
204C2×#108—611no—
205#427—611no—
206#427—1221no—
207#427C2×D6111yescharacteristic
208C2×#104—1221no—
209C2×C4×C8—611no—
210#365—614no—
211#365—614no—
212#364—612no—
213#364—612no—
214#360—614no—
215#360C2×D6114yes—
216C61#3363711yescharacteristic, complemented, Sylow
217C2×#41—21no—
218#135—21no—
219#135—21no—
220#135Dic12212yescomplemented
221#135C22×D6112yes—
222C2×Q16Dic12211yescharacteristic, complemented
223C2×SD16—21no—
224C2×D8Dic12211yescharacteristic, complemented
225#131—1222no—
226#131—1222no—
227#131—2441no—
228#131—2441no—
229C2×M16—1221no—
230C2×M16—2441no—
231C22×C8—611no—
232C22×C8—2441no—
233C22×C8C22×D6111yescharacteristic
234C4⋊Q8—1222no—
235#125—1222no—
236C22⋊Q8—2442no—
237C4×Q8—2442no—
238#117—21no—
239#117—1221no—
240C2×C42—611no—
241M32—21no—
242M32—612no—
243M32—1221no—
244M32C22×D6112yes—
245C2×C16—21no—
246C2×C16—1221no—
247#108—612no—
248#108—1221no—
249#107—21no—
250#107—1222no—
251#107—1221no—
252#106—612no—
253#106—1222no—
254#106C22×D6112yes—
255#104—1222no—
256#104—1222no—
257C4×C8—611no—
258C4×C8—611no—
259C4×C8—611no—
260C4×C8—611no—
261#41—22no—
262#41—22no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—2442no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—1221no—
271C22×C4D4×D6111yescharacteristic
272Q16—21no—
273Q16C2×Dic12212yes—
274SD16—22no—
275SD16—22no—
276D8—21no—
277D8C2×Dic12212yes—
278M16—1222no—
279M16—2442no—
280M16—2442no—
281M16—2441no—
282C2×C8—611no—
283C2×C8—611no—
284C2×C8—611no—
285C2×C8—611no—
286C2×C8—611no—
287C2×C8—611no—
288C2×C8—2442no—
289C2×C8—2442no—
290C2×C8—2441no—
291C2×C8C2×Dic12211yescharacteristic
292C2×C8C2×Dic12211yescharacteristic
293C2×C8D4×D6111yescharacteristic
294C2×C8#1175822411yescharacteristic
295C2×C8#1175822411yescharacteristic
296C2×C8C23×D6111yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—1222no—
299C4⋊C4—2442no—
300C4⋊C4—2442no—
301C4⋊C4—2442no—
302C22⋊C4—41no—
303C22⋊C4—2441no—
304C42—41no—
305C42—611no—
306C42—611no—
307C42—1221no—
308C42—2441no—
309C16—22no—
310C16—1222no—
311C23—41no—
312C23#1451980111yescharacteristic
313Q8—22no—
314Q8—41no—
315Q8—4884no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4#1451975411yescharacteristic
329C2×C4#1451975411yescharacteristic
330C2×C4SD16×D6111yescharacteristic
331C2×C4SD16×D6111yescharacteristic
332C2×C4#1451980111yescharacteristic
333C2×C4C2×D4×D6111yescharacteristic
334C8—611no—
335C8—611no—
336C8—611no—
337C8—611no—
338C8—2442no—
339C8#1451975411yescharacteristic
340C8#1451975411yescharacteristic
341C8C22×Dic12211yescharacteristic
342C8C2×#1175822411yescharacteristic
343C22—41no—
344C22—42no—
345C22—82no—
346C22#1684005011yescharacteristic
347C22#1684005011yescharacteristic
348C22C2×#1451980111yescharacteristic
349C4—42no—
350C4—82no—
351C4—1222no—
352C4—4882no—
353C4—2441no—
354C4#1684005011yescharacteristic
355C4#1684005011yescharacteristic
356C4C2×#1451975411yescharacteristic
357C4C2×SD16×D6111yescharacteristic
358C2—21no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2C2×#1684005011yescharacteristic
subgroupG/H
centerC4#16840050
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2145C2
socleC2×C61C2×#476
prime radicalC2×#135×C61C22
hypercenter#2145D61
hypocenterC61#33637
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
1. #21456331C2semidirect
3. #21447993C2semidirect
4. #21447988C2semidirect
5. #21447592C2semidirect
6. #21447591C2semidirect
7. C2×#16839582C2semidirect
15. #16840162C22semidirect
16. #16840159C22semidirect
26. #16839585C22semidirect
27. #16839584C22semidirect
28. #16839582C22semidirect
29. #16839582C22semidirect
67. #135×C61Q8semidirect
69. C2×Q16×C61Q8semidirect
71. C2×D8×C61Q8semidirect
216. C61#33637semidirect
220. #135Dic122semidirect
222. C2×Q16Dic122semidirect
224. C2×D8Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563312
C2×#168398792
#214479932
#214479882
#214475922
#214475912
C2×#168395822
#214475552
C61×#21452
#3363761