Finite Group Databasebeta

← #26369733

Subgroups — #26369733

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256522C211yesmaximal, characteristic
2#20252305C211yesmaximal, characteristic, complemented
3#20252271C211yesmaximal, characteristic, complemented
4#20245283C214yesmaximal, complemented
5#20245276C214yesmaximal, complemented
6C2×#15688945C211yesmaximal, characteristic, complemented
7#20245138C211yesmaximal, characteristic, complemented
8#20244992C211yesmaximal, characteristic, complemented
9C2×C47×#360C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11#15689890—21no—
12#15689604—21no—
13#15689564C412yescomplemented
14#15689563C2212yes—
15#15689559C412yescomplemented
16#15689558C2212yes—
17#15689553C414yescomplemented
18C2×#13093964C2211yescharacteristic
19#15689245C2214yes—
20#15689240—22no—
21#15689234C2212yescomplemented
22#15689229C2212yescomplemented
23#15689210—22no—
24C2×#13093960C2211yescharacteristic
25#15689140—21no—
26#15689122—21no—
27#15689078C2211yescharacteristic
28#15688945C2214yescomplemented
29#15688940C2212yescomplemented
30#15688937C2212yescomplemented
31#15688900C2214yescomplemented
32C2×#135×C47C2211yescharacteristic
33C2×M32×C47C2211yescharacteristic
34C2×#106×C47C2211yescharacteristic
35C47×#360C2214yes—
36C2×#10625350—21no—
37#13094046—21no—
38#13094046—21no—
39#13094046C2×C412yes—
40#13094046C2×C412yes—
41C2×Dic188C2311yescharacteristic
42C2×#10625325—21no—
43C2×D376C2311yescharacteristic
44#13094031—24no—
45#13094031—42no—
46C2×#10625329—21no—
47C2×#10625329—42no—
48C2×#10625329D411yescharacteristic
49#13093989—22no—
50#13093986—22no—
51C4×C22⋊D47—42no—
52#13093983D411yescharacteristic
53#13093982—41no—
54C4⋊C4×D47—41no—
55#13093970C2×C414yescomplemented
56#13093968C2×C414yescomplemented
57#13093967C2×C414yescomplemented
58#13093965C2×C414yescomplemented
59#13093964C2×C412yescomplemented
60#13093964C2312yes—
61C2×C47⋊C16C2×C412yescomplemented
62#13093960—21no—
63#13093960C2312yes—
64#13093951—24no—
65#13093947—22no—
66#13093944—22no—
67#13093934C2312yes—
68C8×Dic47D412yes—
69C2×#41×C47—21no—
70#135×C47—21no—
71#135×C47—21no—
72#135×C47C2×C412yescomplemented
73#135×C47C2312yes—
74C2×Q16×C47C2×C411yescharacteristic, complemented
75C2×SD16×C47—21no—
76C2×D8×C47C2×C411yescharacteristic, complemented
77C22×C8×C47C2311yescharacteristic
78C2×C4⋊C4×C47—21no—
79M32×C47—21no—
80M32×C47C2312yes—
81C2×C16×C47—21no—
82#106×C47—21no—
83#106×C47C2312yes—
84#41×C47—22no—
85#41×C47—22no—
86#41×C47—41no—
87#41×C47—41no—
88C2×Q8×C47—21no—
89C2×D4×C47—21no—
90C2×D4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47C2×D411yescharacteristic
94C2×C22⋊D47—41no—
95#10625350—22no—
96#10625350—22no—
97#10625350—41no—
98#10625350—41no—
99C2×D188—21no—
100C2×C4×D47—41no—
101C2×C4×D47—42no—
102C2×Dic94—21no—
103Q16×C47—21no—
104Q16×C47C22×C412yes—
105SD16×C47—22no—
106SD16×C47—22no—
107D8×C47—21no—
108D8×C47C22×C412yes—
109C2×C8×C47C22×C411yescharacteristic
110C2×C8×C47C22×C411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47#4111yescharacteristic
113C2×C8×C47#4111yescharacteristic
114C2×C8×C47C2411yescharacteristic
115C4⋊C4×C47—22no—
116C4⋊C4×C47—41no—
117C22⋊Dic47—21no—
118(C2×D47)⋊C4—42no—
119C4⋊Dic47—21no—
120Dic47⋊C4—42no—
121C4×Dic47—42no—
122C4×Dic47C2×D412yes—
123#10625329—21no—
124#10625329—21no—
125#10625329—22no—
126#10625329—44no—
127#10625329—41no—
128C2×C47⋊C8—44no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8C2×D412yes—
131Dic188—21no—
132Dic188C22×C412yes—
133D376—21no—
134D376C22×C412yes—
135#10625325—22no—
136#10625325—22no—
137C16×C47—22no—
138C47⋊C16C22×C414yes—
139C23×C47—41no—
140C23×C47AGL(1,ℤ/8)11yescharacteristic
141C22×D47—41no—
142Q8×C47—22no—
143Q8×C47—41no—
144D4×C47—22no—
145D4×C47—42no—
146D4×C47—42no—
147D4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—161no—
153C2×C4×C47C4×D411yescharacteristic
154C2×C4×C47C4×D411yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47C22×D411yescharacteristic
159C22⋊D47—42no—
160C22⋊D47—42no—
161C2×Dic47—22no—
162C2×Dic47—41no—
163D188—22no—
164D188—41no—
165C4×D47—42no—
166C4×D47—42no—
167C4×D47—162no—
168Dic94—22no—
169Dic94—41no—
170C8×C47C4×D411yescharacteristic
171C8×C47C4×D411yescharacteristic
172C8×C47C23×C411yescharacteristic
173C8×C47C2×#4111yescharacteristic
174C47⋊C8—22no—
175C47⋊C8—42no—
176#33469—471nomaximal, Sylow
177C22×C47—41no—
178C22×C47—42no—
179C22×C47—161no—
180C22×C47#44211yescharacteristic
181C22×C47#44211yescharacteristic
182C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D47—42no—
184C2×D47—81no—
185C2×D47—81no—
186C4×C47—42no—
187C4×C47—82no—
188C4×C47#44211yescharacteristic
189C4×C47#44211yescharacteristic
190C4×C47C2×C4×D411yescharacteristic
191C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
192Dic47—42no—
193Dic47—42no—
194#3414—471no—
195#2957—471no—
196#2952—471no—
197AGL(1,ℤ/16)—474no—
198#2180—474no—
199C2×#365—471no—
200#2148—471no—
201C2×#360D4711yescharacteristic, complemented
202#2110—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—82no—
207C2×C47C2×#44211yescharacteristic
208D47—82no—
209C2×#135—471no—
210C2×#135C2×D4711yescharacteristic
211#568—941no—
212#520—941no—
213#510—472no—
214#510—472no—
215#509—472no—
216#509—472no—
217#508—474no—
218C2×M32—471no—
219C2×M32C2×D4711yescharacteristic
220#444—474no—
221#443—942no—
222#442—472no—
223#441—472no—
224C4×SD16—942no—
225C2×#108—471no—
226C2×#106C2×D4711yescharacteristic
227#421—941no—
228#417—941no—
229#405—471no—
230#365—474no—
231#362—472no—
232#361—472no—
233#360C2×D4714yes—
234#347—474no—
235C47#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—941no—
238#135—21no—
239#135—21no—
240#135—472no—
241#135—472no—
242#135—941no—
243#135—941no—
244#135C4×D4712yescomplemented
245#135C22×D4712yes—
246C2×Q16—471no—
247C2×Q16C4×D4711yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—941no—
250C2×D8—471no—
251C2×D8C4×D4711yescharacteristic, complemented
252#131—944no—
253#131—1882no—
254C2×M16—471no—
255C2×M16—941no—
256C2×M16—1882no—
257C22×C8C22×D4711yescharacteristic
258C4×Q8—942no—
259C4×D4—942no—
260C4×D4—1882no—
261#117—471no—
262#117—1881no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1881no—
265Q32—474no—
266SD32—474no—
267SD32—474no—
268D16—474no—
269M32—21no—
270M32—472no—
271M32—472no—
272M32C22×D4712yes—
273C2×C16—21no—
274C2×C16—472no—
275#108—472no—
276#108—941no—
277#106—21no—
278#106C22×D4712yes—
279#104—944no—
280#103—942no—
281#102—942no—
282#97—472no—
283C4×C8—472no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—942no—
289#41—942no—
290#41—1881no—
291#41—1881no—
292C2×Q8—21no—
293C2×Q8—941no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—941no—
297C2×D4—1881no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—1882no—
301C22×C4—1881no—
302C22×C4D4×D4711yescharacteristic
303Q16—21no—
304Q16—472no—
305Q16—941no—
306Q16C2×C4×D4712yes—
307SD16—22no—
308SD16—22no—
309SD16—942no—
310SD16—942no—
311D8—21no—
312D8—472no—
313D8—941no—
314D8C2×C4×D4712yes—
315M16—941no—
316M16—941no—
317M16—942no—
318M16—1884no—
319M16—1881no—
320C2×C8—472no—
321C2×C8—1884no—
322C2×C8—1881no—
323C2×C8C2×C4×D4711yescharacteristic
324C2×C8C2×C4×D4711yescharacteristic
325C2×C8D4×D4711yescharacteristic
326C2×C8#1062535211yescharacteristic
327C2×C8#1062535211yescharacteristic
328C2×C8C23×D4711yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—941no—
332C4⋊C4—1882no—
333C22⋊C4—941no—
334C22⋊C4—1882no—
335C42—472no—
336C42—1882no—
337C16—22no—
338C16—474no—
339C23—41no—
340C23—1881no—
341C23#1309404811yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—942no—
345Q8—1881no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—942no—
351D4—1882no—
352D4—1882no—
353D4—1881no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—161no—
359C2×C4—942no—
360C2×C4—1881no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—7522no—
364C2×C4(C22⋊D47)⋊C411yescharacteristic
365C2×C4(C22⋊D47)⋊C411yescharacteristic
366C2×C4#1309404811yescharacteristic
367C2×C4#1309405111yescharacteristic
368C2×C4#1309405111yescharacteristic
369C2×C4C2×D4×D4711yescharacteristic
370C8—942no—
371C8—1882no—
372C8(C22⋊D47)⋊C411yescharacteristic
373C8(C22⋊D47)⋊C411yescharacteristic
374C8C22×C4×D4711yescharacteristic
375C8C2×#1062535211yescharacteristic
376C22—41no—
377C22—42no—
378C22—161no—
379C22—3761no—
380C22—3761no—
381C22—1882no—
382C22#1568923211yescharacteristic
383C22#1568923211yescharacteristic
384C22C2×#1309404811yescharacteristic
385C4—42no—
386C4—82no—
387C4—1882no—
388C4—1882no—
389C4#1568923211yescharacteristic
390C4#1568923211yescharacteristic
391C4C2×(C22⋊D47)⋊C411yescharacteristic
392C4C2×#1309405111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—3762no—
398C2C2×#1568923211yescharacteristic
subgroupG/H
centerC2C2×#15689232
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC2×C47×#360C2
socleC2×C47C2×#442
prime radical#20256522C2
cosocleC2×C8×C47C24
hypercenterC2×#360D47
hypocenterC47#33469
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
2. #20252305C2semidirect
3. #20252271C2semidirect
4. #20245283C2semidirect
5. #20245276C2semidirect
6. C2×#15688945C2semidirect
7. #20245138C2semidirect
8. #20244992C2semidirect
9. C2×C47×#360C2semidirect
13. #15689564C4semidirect
15. #15689559C4semidirect
17. #15689553C4semidirect
21. #15689234C22semidirect
22. #15689229C22semidirect
28. #15688945C22semidirect
29. #15688940C22semidirect
30. #15688937C22semidirect
31. #15688900C22semidirect
55. #13093970C2×C4semidirect
56. #13093968C2×C4semidirect
57. #13093967C2×C4semidirect
58. #13093965C2×C4semidirect
59. #13093964C2×C4semidirect
61. C2×C47⋊C16C2×C4semidirect
72. #135×C47C2×C4semidirect
74. C2×Q16×C47C2×C4semidirect
76. C2×D8×C47C2×C4semidirect
201. C2×#360D47semidirect
235. C47#33469semidirect
244. #135C4×D47semidirect
247. C2×Q16C4×D47semidirect
251. C2×D8C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565222
#202523052
#202522712
#202452832
#202452762
C2×#156889452
#202451382
#202449922
C2×C47×#3602
#3346947