Finite Group Databasebeta

← #26371018

Subgroups — #26371018

Order 12032.

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#20253900C211yesmaximal, characteristic, complemented
2C2×#15689240C211yesmaximal, characteristic
3#20245562C214yesmaximal, complemented
4#20245557C214yesmaximal, complemented
5#20245161C211yesmaximal, characteristic, complemented
6#20245160C211yesmaximal, characteristic, complemented
7C2×#15688943C211yesmaximal, characteristic, complemented
8#20245124C211yesmaximal, characteristic
9C47×#2145C211yesmaximal, characteristic
10C2×#13094031—21no—
11#15689676—21no—
12C2×#13093964C2211yescharacteristic
13#15689527—21no—
14#15689524—21no—
15#15689523C2212yescomplemented
16#15689520C2212yescomplemented
17#15689240—21no—
18#15689240—21no—
19#15689240C2212yes—
20#15689240C2212yes—
21C2×#13093960C2211yescharacteristic
22#15689161C2211yescharacteristic
23#15689159—21no—
24C2×#13093951—21no—
25C2×C8×Dic47C2211yescharacteristic
26#15688946C2214yescomplemented
27#15688945C2214yescomplemented
28#15688943C2212yescomplemented
29#15688943C2212yescomplemented
30#15688933C2214yes—
31C2×#135×C47C2211yescharacteristic
32C2×M32×C47C2211yescharacteristic
33C47×#427C2211yescharacteristic
34C47×#360C2214yes—
35#13094031—22no—
36#13094031—22no—
37#13094031—41no—
38#13094031—41no—
39C2×#10625329—21no—
40C2×#10625329—41no—
41C22×C47⋊C8—41no—
42C22×C47⋊C8D411yescharacteristic
43#13094023—22no—
44#13094015—22no—
45#13094001—42no—
46C4×Dic94—42no—
47#13093983—21no—
48C2×C4×Dic47D411yescharacteristic
49#13093964—21no—
50#13093964C2312yes—
51C2×C47⋊C16—21no—
52#13093960—21no—
53#13093960C2312yes—
54#13093958—21no—
55#13093957—22no—
56#13093956C2312yes—
57#13093955—22no—
58#13093951—22no—
59#13093951—22no—
60C8×Dic47D411yescharacteristic
61C8×Dic47D411yescharacteristic
62C8×Dic47C2311yescharacteristic
63C8×Dic47C2311yescharacteristic
64C2×#41×C47—21no—
65#135×C47—21no—
66#135×C47—21no—
67#135×C47Q812yescomplemented
68#135×C47C2312yes—
69C2×Q16×C47Q811yescharacteristic, complemented
70C2×SD16×C47—21no—
71C2×D8×C47Q811yescharacteristic, complemented
72C22×C8×C47C2311yescharacteristic
73#117×C47—21no—
74M32×C47—21no—
75M32×C47C2312yes—
76C2×C16×C47—21no—
77#107×C47—21no—
78#106×C47C2312yes—
79#41×C47—22no—
80#41×C47—22no—
81#41×C47—41no—
82#41×C47—41no—
83C2×Q8×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C22×Dic47—21no—
89C2×Dic94—42no—
90Q16×C47—21no—
91Q16×C47C2×Q812yes—
92SD16×C47—22no—
93SD16×C47—22no—
94D8×C47—21no—
95D8×C47C2×Q812yes—
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2×Q811yescharacteristic
98C2×C8×C47C2×Q811yescharacteristic
99C2×C8×C47#4111yescharacteristic
100C2×C8×C47#4111yescharacteristic
101C2×C8×C47C2411yescharacteristic
102C4⋊C4×C47—22no—
103C22⋊C4×C47—41no—
104C42×C47—41no—
105C22⋊Dic47—41no—
106C4⋊Dic47—22no—
107C4⋊Dic47—42no—
108Dic47⋊C4—42no—
109Dic47⋊C4—42no—
110C4×Dic47—21no—
111C4×Dic47—41no—
112C4×Dic47C2×D411yescharacteristic
113C4×Dic47C2×D411yescharacteristic
114#10625329—22no—
115#10625329—41no—
116#10625329—42no—
117#10625329—42no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—42no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8SD1611yescharacteristic
122C2×C47⋊C8SD1611yescharacteristic
123C2×C47⋊C8SD1611yescharacteristic
124C2×C47⋊C8SD1611yescharacteristic
125C2×C47⋊C8C2×D411yescharacteristic
126C2×C47⋊C8C2×D411yescharacteristic
127C16×C47—22no—
128C47⋊C16—22no—
129C23×C47—41no—
130C23×C47AGL(1,ℤ/8)11yescharacteristic
131Q8×C47—22no—
132Q8×C47—41no—
133D4×C47—22no—
134D4×C47—42no—
135D4×C47—42no—
136D4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47C22⋊Q811yescharacteristic
142C2×C4×C47C22⋊Q811yescharacteristic
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47C2×SD1611yescharacteristic
145C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C47C22×D411yescharacteristic
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151Dic94—84no—
152C8×C47C22⋊Q811yescharacteristic
153C8×C47C22⋊Q811yescharacteristic
154C8×C47C22×Q811yescharacteristic
155C8×C47C2×#4111yescharacteristic
156C47⋊C8—42no—
157C47⋊C8C2×SD1611yescharacteristic
158C47⋊C8C2×SD1611yescharacteristic
159C47⋊C8C2×SD1611yescharacteristic
160C47⋊C8C2×SD1611yescharacteristic
161#33637—471nomaximal, Sylow
162C22×C47—41no—
163C22×C47—42no—
164C22×C47—82no—
165C22×C47#47611yescharacteristic
166C22×C47#47611yescharacteristic
167C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
168C4×C47—42no—
169C4×C47—82no—
170C4×C47#47611yescharacteristic
171C4×C47#47611yescharacteristic
172C4×C47C2×C22⋊Q811yescharacteristic
173C4×C47C22×SD1611yescharacteristic
174Dic47—22no—
175Dic47—41no—
176Dic47—82no—
177#3160—471no—
178C2×#443—471no—
179#2231—474no—
180#2230—474no—
181#2155—471no—
182#2155—471no—
183C2×#364—471no—
184#2145—471no—
185#2145D4711yescharacteristic
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—82no—
190C2×C47C2×#47611yescharacteristic
191C2×#135C2×D4711yescharacteristic
192C2×#131—941no—
193#533—941no—
194C2×M32—471no—
195C2×M32C2×D4711yescharacteristic
196#500—941no—
197#499—472no—
198#499—941no—
199#498—472no—
200#443—472no—
201#443—472no—
202#443—941no—
203#443—941no—
204C2×#108—471no—
205#427—471no—
206#427—941no—
207#427C2×D4711yescharacteristic
208C2×#104—941no—
209C2×C4×C8—471no—
210#365—474no—
211#365—474no—
212#364—472no—
213#364—472no—
214#360—474no—
215#360C2×D4714yes—
216C47#3363711yescharacteristic, complemented, Sylow
217C2×#41—21no—
218#135—21no—
219#135—21no—
220#135Dic9412yescomplemented
221#135C22×D4712yes—
222C2×Q16Dic9411yescharacteristic, complemented
223C2×SD16—21no—
224C2×D8Dic9411yescharacteristic, complemented
225#131—942no—
226#131—942no—
227#131—1881no—
228#131—1881no—
229C2×M16—941no—
230C2×M16—1881no—
231C22×C8—471no—
232C22×C8—1881no—
233C22×C8C22×D4711yescharacteristic
234C4⋊Q8—942no—
235#125—942no—
236C22⋊Q8—1882no—
237C4×Q8—1882no—
238#117—21no—
239#117—941no—
240C2×C42—471no—
241M32—21no—
242M32—472no—
243M32—941no—
244M32C22×D4712yes—
245C2×C16—21no—
246C2×C16—941no—
247#108—472no—
248#108—941no—
249#107—21no—
250#107—942no—
251#107—941no—
252#106—472no—
253#106—942no—
254#106C22×D4712yes—
255#104—942no—
256#104—942no—
257C4×C8—471no—
258C4×C8—471no—
259C4×C8—471no—
260C4×C8—471no—
261#41—22no—
262#41—22no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—1882no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—941no—
271C22×C4D4×D4711yescharacteristic
272Q16—21no—
273Q16C2×Dic9412yes—
274SD16—22no—
275SD16—22no—
276D8—21no—
277D8C2×Dic9412yes—
278M16—942no—
279M16—1882no—
280M16—1882no—
281M16—1881no—
282C2×C8—471no—
283C2×C8—471no—
284C2×C8—471no—
285C2×C8—471no—
286C2×C8—471no—
287C2×C8—471no—
288C2×C8—1882no—
289C2×C8—1882no—
290C2×C8—1881no—
291C2×C8C2×Dic9411yescharacteristic
292C2×C8C2×Dic9411yescharacteristic
293C2×C8D4×D4711yescharacteristic
294C2×C8#1062535211yescharacteristic
295C2×C8#1062535211yescharacteristic
296C2×C8C23×D4711yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—942no—
299C4⋊C4—1882no—
300C4⋊C4—1882no—
301C4⋊C4—1882no—
302C22⋊C4—41no—
303C22⋊C4—1881no—
304C42—41no—
305C42—471no—
306C42—471no—
307C42—941no—
308C42—1881no—
309C16—22no—
310C16—942no—
311C23—41no—
312C23#1309404811yescharacteristic
313Q8—22no—
314Q8—41no—
315Q8—3764no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—941no—
325C2×C4—941no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4#1309400111yescharacteristic
329C2×C4#1309400111yescharacteristic
330C2×C4SD16×D4711yescharacteristic
331C2×C4SD16×D4711yescharacteristic
332C2×C4#1309404811yescharacteristic
333C2×C4C2×D4×D4711yescharacteristic
334C8—471no—
335C8—471no—
336C8—471no—
337C8—471no—
338C8—1882no—
339C8#1309400111yescharacteristic
340C8#1309400111yescharacteristic
341C8C22×Dic9411yescharacteristic
342C8C2×#1062535211yescharacteristic
343C22—41no—
344C22—42no—
345C22—82no—
346C22#1568941111yescharacteristic
347C22#1568941111yescharacteristic
348C22C2×#1309404811yescharacteristic
349C4—42no—
350C4—82no—
351C4—942no—
352C4—3762no—
353C4—1881no—
354C4#1568941111yescharacteristic
355C4#1568941111yescharacteristic
356C4C2×#1309400111yescharacteristic
357C4C2×SD16×D4711yescharacteristic
358C2—21no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2C2×#1568941111yescharacteristic
subgroupG/H
centerC4#15689411
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2145C2
socleC2×C47C2×#476
prime radicalC2×#135×C47C22
hypercenter#2145D47
hypocenterC47#33637
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
1. #20253900C2semidirect
3. #20245562C2semidirect
4. #20245557C2semidirect
5. #20245161C2semidirect
6. #20245160C2semidirect
7. C2×#15688943C2semidirect
15. #15689523C22semidirect
16. #15689520C22semidirect
26. #15688946C22semidirect
27. #15688945C22semidirect
28. #15688943C22semidirect
29. #15688943C22semidirect
67. #135×C47Q8semidirect
69. C2×Q16×C47Q8semidirect
71. C2×D8×C47Q8semidirect
216. C47#33637semidirect
220. #135Dic94semidirect
222. C2×Q16Dic94semidirect
224. C2×D8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539002
C2×#156892402
#202455622
#202455572
#202451612
#202451602
C2×#156889432
#202451242
C47×#21452
#3363747