Finite Group Databasebeta

← #26369686

Subgroups — #26369686

Order 12032.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256521C211yesmaximal, characteristic
2#443×D47C212yesmaximal, complemented
3#20245274C211yesmaximal, characteristic, complemented
4#20245273C211yesmaximal, characteristic, complemented
5#20245271C212yesmaximal, complemented
6#20245270C211yesmaximal, characteristic, complemented
7#20245267C212yesmaximal, complemented
8#20245181C211yesmaximal, characteristic, complemented
9#20245114C212yesmaximal, complemented
10C4×#13093963C211yesmaximal, characteristic, complemented
11C47×#2178C211yesmaximal, characteristic, complemented
12#135×D47C2212yes—
13#131×D47—22no—
14#15689565C412yescomplemented
15#15689562C412yescomplemented
16#15689560C412yescomplemented
17#15689557C412yescomplemented
18#15689555C2211yescharacteristic
19#15689553C2212yes—
20C2×#13093963C2211yescharacteristic
21C2×#13093963C2211yescharacteristic
22#15689241C2212yescomplemented
23#15689240C2212yescomplemented
24#15689239C2212yescomplemented
25#15689238—22no—
26#15689237—22no—
27#108×D47C2212yes—
28#104×D47—22no—
29C4×C8×D47C2211yescharacteristic
30#15688953C2211yescharacteristic, complemented
31#15688952C2212yescomplemented
32#15688931C2212yescomplemented
33#15688930C2212yescomplemented
34#15688929C2212yescomplemented
35#15688898C2211yescharacteristic, complemented
36#15688898C2211yescharacteristic, complemented
37C4×C47⋊C16C2211yescharacteristic, complemented
38C47×#508C2211yescharacteristic
39C47×#443C2212yescomplemented
40C47×#368C2211yescharacteristic, complemented
41C47×#359C2212yescomplemented
42C4×C16×C47C2211yescharacteristic, complemented
43#41×D47—22no—
44#13094047C2×C412yes—
45#13094046C2312yes—
46#13094045—22no—
47#13094044—22no—
48#13094043C2×C412yes—
49Q16×D47C2×C412yes—
50SD16×D47—22no—
51D8×D47C2×C412yes—
52#13094032—22no—
53#13094032—42no—
54#13094031—22no—
55#13094030—22no—
56#13094030—42no—
57M16×D47—22no—
58M16×D47—42no—
59C2×C8×D47—42no—
60C2×C8×D47D411yescharacteristic
61C2×C8×D47C2311yescharacteristic
62C42×D47D411yescharacteristic
63Dic376C2×C411yescharacteristic, complemented
64#13093970C2×C412yescomplemented
65#13093969C2×C412yescomplemented
66#13093968C2×C412yescomplemented
67#13093967C2×C412yescomplemented
68D752C2×C411yescharacteristic, complemented
69#13093965C2×C412yescomplemented
70#13093963—22no—
71#13093963C2×C412yes—
72#13093963C2×C412yes—
73C2×C47⋊C16C2311yescharacteristic
74C2×C47⋊C16C2311yescharacteristic
75#13093960C2312yes—
76#13093959—22no—
77#13093952—22no—
78#13093951—22no—
79#13093950—22no—
80C4×C47⋊C8D411yescharacteristic
81C8×Dic47D411yescharacteristic
82C8×Dic47C2311yescharacteristic
83#135×C47C2312yes—
84#131×C47—22no—
85Q32×C47C2×C411yescharacteristic, complemented
86SD32×C47C2×C412yescomplemented
87D16×C47C2×C411yescharacteristic, complemented
88C2×C16×C47C2311yescharacteristic
89C2×C16×C47C2311yescharacteristic
90#108×C47C2312yes—
91#104×C47—22no—
92C4×C8×C47C2311yescharacteristic
93#41×C47—22no—
94#10625354—22no—
95Q8×D47—22no—
96#10625352—22no—
97#10625352—42no—
98D4×D47—22no—
99D4×D47—42no—
100#10625350—22no—
101#10625350—42no—
102C2×C4×D47—21no—
103C2×C4×D47—42no—
104C2×C4×D47C2×D411yescharacteristic
105Q16×C47C22×C412yes—
106SD16×C47—22no—
107D8×C47C22×C412yes—
108M16×C47—22no—
109M16×C47—42no—
110C2×C8×C47—42no—
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47C2411yescharacteristic
113C42×C47C2×D411yescharacteristic
114#10625337—22no—
115#10625336—22no—
116#10625335—22no—
117#10625334—22no—
118C4×Dic47—21no—
119C4×Dic47C2×D411yescharacteristic
120#10625329—22no—
121#10625329—42no—
122C2×C47⋊C8—42no—
123C2×C47⋊C8C2×D411yescharacteristic
124C2×C47⋊C8C2×D411yescharacteristic
125Dic188C22×C412yes—
126D376C22×C412yes—
127#10625325—22no—
128#10625324—42no—
129#10625324—42no—
130#10625324—42no—
131C8×D47—42no—
132C8×D47—42no—
133C8×D47—42no—
134C8×D47D812yes—
135C8×D47D812yes—
136C8×D47C22×C412yes—
137C8×D47#4112yes—
138C16×C47—21no—
139C16×C47C22×C411yescharacteristic
140C16×C47C22×C411yescharacteristic
141C47⋊C16—21no—
142C47⋊C16C22×C412yes—
143C22×D47—42no—
144C22×D47#13511yescharacteristic
145Q8×C47—22no—
146D4×C47—22no—
147D4×C47—42no—
148C2×C4×C47—21no—
149C2×C4×C47—42no—
150C2×C4×C47C22×D411yescharacteristic
151C22⋊D47—42no—
152C22⋊D47—42no—
153C22⋊D47—42no—
154C2×Dic47—21no—
155C2×Dic47—42no—
156C2×Dic47#13511yescharacteristic
157D188—22no—
158D188—42no—
159C4×D47—42no—
160C4×D47—42no—
161C4×D47—42no—
162C4×D47—42no—
163C4×D47C4×D412yes—
164C4×D47C2×D812yes—
165Dic94—22no—
166Dic94—42no—
167C8×C47—42no—
168C8×C47C2×D811yescharacteristic
169C8×C47C2×D811yescharacteristic
170C8×C47C23×C411yescharacteristic
171C8×C47C2×#4111yescharacteristic
172C47⋊C8—42no—
173C47⋊C8C4×D412yes—
174C47⋊C8C2×D812yes—
175#33463—471nomaximal, Sylow
176C22×C47—42no—
177C22×C47C2×#13511yescharacteristic
178C2×D47—41no—
179C2×D47—42no—
180C2×D47—84no—
181C2×D47C4×D812yes—
182C4×C47—22no—
183C4×C47—42no—
184C4×C47C2×C4×D411yescharacteristic
185C4×C47C22×D811yescharacteristic
186Dic47—41no—
187Dic47—42no—
188Dic47C4×D812yes—
189#3414—471no—
190C2×#443—472no—
191#2179—471no—
192#2179—471no—
193#2179—472no—
194#2178—472no—
195#2178—471no—
196#2178D4711yescharacteristic, complemented
197#2161—471no—
198#2143—472no—
199C4×M32—471no—
200C2×C47—21no—
201C2×C47—82no—
202C2×C47C2×C4×D811yescharacteristic
203D47—22no—
204D47—82no—
205C2×#135—472no—
206C2×#131—942no—
207#510—472no—
208#510—472no—
209#509—472no—
210#509—472no—
211#508—472no—
212#508—471no—
213#508C2×D4711yescharacteristic
214C2×M32—471no—
215C2×M32—471no—
216#443—472no—
217#443—472no—
218#443—472no—
219#443—942no—
220#443—942no—
221#443C2×D4712yescomplemented
222C2×#108—472no—
223C2×#104—942no—
224C2×C4×C8—471no—
225#368—471no—
226#368—472no—
227#368C2×D4711yescharacteristic, complemented
228#359—472no—
229#359—472no—
230#359—472no—
231#359C2×D4712yescomplemented
232#346—471no—
233#346—471no—
234C4×C16—471no—
235C4×C16C2×D4711yescharacteristic, complemented
236C47#3346311yescharacteristic, complemented, Sylow
237C2×#41—942no—
238#135—472no—
239#135—472no—
240#135—472no—
241#135—942no—
242#135—942no—
243#135C22×D4712yes—
244C2×Q16—472no—
245C2×SD16—942no—
246C2×D8—472no—
247#131—22no—
248#131—942no—
249#131—942no—
250#131—942no—
251#131—1882no—
252#131—1882no—
253C2×M16—942no—
254C2×M16—1882no—
255C22×C8—471no—
256C22×C8—471no—
257C22×C8—1882no—
258C2×C42—471no—
259Q32—472no—
260Q32—471no—
261Q32C4×D4711yescharacteristic, complemented
262SD32—472no—
263SD32—472no—
264SD32—472no—
265SD32C4×D4712yescomplemented
266D16—472no—
267D16—471no—
268D16C4×D4711yescharacteristic, complemented
269M32—472no—
270M32—472no—
271M32—942no—
272C2×C16—471no—
273C2×C16—471no—
274C2×C16C22×D4711yescharacteristic
275C2×C16C22×D4711yescharacteristic
276#108—472no—
277#108—942no—
278#108C22×D4712yes—
279#104—22no—
280#104—942no—
281#104—942no—
282#104—942no—
283C4×C8—471no—
284C4×C8—471no—
285C4×C8—471no—
286C4×C8C22×D4711yescharacteristic
287#41—22no—
288#41—942no—
289#41—942no—
290#41—942no—
291#41—1882no—
292#41—1882no—
293C2×Q8—942no—
294C2×D4—942no—
295C2×D4—1882no—
296C22×C4—471no—
297C22×C4—941no—
298C22×C4—1882no—
299Q16—472no—
300Q16—942no—
301Q16C2×C4×D4712yes—
302SD16—22no—
303SD16—942no—
304SD16—942no—
305SD16—942no—
306D8—472no—
307D8—942no—
308D8C2×C4×D4712yes—
309M16—22no—
310M16—42no—
311M16—942no—
312M16—1882no—
313M16—1882no—
314M16—1882no—
315M16—1882no—
316C2×C8—42no—
317C2×C8—471no—
318C2×C8—471no—
319C2×C8—472no—
320C2×C8—472no—
321C2×C8—472no—
322C2×C8—472no—
323C2×C8—1882no—
324C2×C8—1882no—
325C2×C8—1882no—
326C2×C8—1882no—
327C2×C8D4×D4711yescharacteristic
328C2×C8C23×D4711yescharacteristic
329C42—471no—
330C42—941no—
331C42D4×D4711yescharacteristic
332C16—21no—
333C16—472no—
334C16—941no—
335C16C2×C4×D4711yescharacteristic
336C16C2×C4×D4711yescharacteristic
337C23—471no—
338C23—1882no—
339Q8—22no—
340Q8—942no—
341Q8—1882no—
342D4—22no—
343D4—42no—
344D4—942no—
345D4—1882no—
346D4—1882no—
347D4—1882no—
348D4—1882no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—471no—
352C2×C4—472no—
353C2×C4—472no—
354C2×C4—941no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4—1882no—
360C2×C4C2×D4×D4711yescharacteristic
361C8—42no—
362C8—472no—
363C8—472no—
364C8—1882no—
365C8D8×D4711yescharacteristic
366C8D8×D4711yescharacteristic
367C8C22×C4×D4711yescharacteristic
368C8#41×D4711yescharacteristic
369C22—42no—
370C22—472no—
371C22—3764no—
372C22—1881no—
373C22—1882no—
374C22#135×D4711yescharacteristic
375C4—22no—
376C4—42no—
377C4—472no—
378C4—1881no—
379C4—1882no—
380C4C4×D4×D4711yescharacteristic
381C4C2×D8×D4711yescharacteristic
382C2—21no—
383C2—82no—
384C2—942no—
385C2—3762no—
386C2C4×D8×D4711yescharacteristic
subgroupG/H
centerC8D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2178C2
socleC2×C47C2×C4×D8
prime radical#20256521C2
cosocleC2×C8×C47C24
hypercenter#2178D47
hypocenterC47#33463
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ #2178

Complements

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

NH ≅ G/Ntype
2. #443×D47C2semidirect
3. #20245274C2semidirect
4. #20245273C2semidirect
5. #20245271C2semidirect
6. #20245270C2semidirect
7. #20245267C2semidirect
8. #20245181C2semidirect
9. #20245114C2semidirect
10. C4×#13093963C2semidirect
11. C47×#2178C2semidirect
14. #15689565C4semidirect
15. #15689562C4semidirect
16. #15689560C4semidirect
17. #15689557C4semidirect
22. #15689241C22semidirect
23. #15689240C22semidirect
24. #15689239C22semidirect
30. #15688953C22semidirect
31. #15688952C22semidirect
32. #15688931C22semidirect
33. #15688930C22semidirect
34. #15688929C22semidirect
35. #15688898C22semidirect
36. #15688898C22semidirect
37. C4×C47⋊C16C22semidirect
39. C47×#443C22semidirect
40. C47×#368C22semidirect
41. C47×#359C22semidirect
42. C4×C16×C47C22semidirect
63. Dic376C2×C4semidirect
64. #13093970C2×C4semidirect
65. #13093969C2×C4semidirect
66. #13093968C2×C4semidirect
67. #13093967C2×C4semidirect
68. D752C2×C4semidirect
69. #13093965C2×C4semidirect
85. Q32×C47C2×C4semidirect
86. SD32×C47C2×C4semidirect
87. D16×C47C2×C4semidirect
196. #2178D47semidirect
221. #443C2×D47semidirect
227. #368C2×D47semidirect
231. #359C2×D47semidirect
235. C4×C16C2×D47semidirect
236. C47#33463semidirect
261. Q32C4×D47semidirect
265. SD32C4×D47semidirect
268. D16C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565212
#443×D472
#202452742
#202452732
#202452712
#202452702
#202452672
#202451812
#202451142
C4×#130939632
C47×#21782
#3346347