Finite Group Databasebeta

← #26133668

Subgroups — #26133668

Order 12032.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093964C211yesmaximal, characteristic, complemented
2#20252969C211yesmaximal, characteristic
3C2×#15688932C212yesmaximal, complemented
4#20241495C212yesmaximal, complemented
5C47×#1807C211yesmaximal, characteristic, complemented
6C2×#13094046C2211yescharacteristic
7#15689634—21no—
8C2×#13093964—24no—
9C2×#13093964—21no—
10C2×#13093964C2212yes—
11C22×C47⋊C16—21no—
12#15689364C411yescharacteristic, complemented
13#15689364C411yescharacteristic, complemented
14#15689359C411yescharacteristic, complemented
15#15689359C411yescharacteristic, complemented
16#15689352—21no—
17#15689352—21no—
18#15689161C2211yescharacteristic
19#15689120—21no—
20#15688932—24no—
21C23×C8×C47C2211yescharacteristic
22C2×#106×C47C2212yes—
23C47×#380—21no—
24C47×#340—21no—
25C2×#10625350—21no—
26#13094046—22no—
27#13094046—22no—
28C2×Dic188C2×C411yescharacteristic
29C2×#10625325—21no—
30C2×D376C2×C411yescharacteristic
31#13094007—41no—
32#13094003—21no—
33#13094003—21no—
34#13093998—21no—
35#13093998—21no—
36#13093993—41no—
37C4×C22⋊D47—41no—
38C4×C22⋊D47—41no—
39#13093983—21no—
40#13093964—24no—
41#13093964—22no—
42#13093964D414yescomplemented
43C2×C47⋊C16—24no—
44C2×C47⋊C16—22no—
45#13093958C2×C411yescharacteristic
46#13093958C2×C411yescharacteristic
47#13093956—21no—
48#13093949—21no—
49#13093949—21no—
50#13093946—21no—
51#13093946—21no—
52C23×C4×C47D411yescharacteristic
53C22×C8×C47—22no—
54C22×C8×C47—22no—
55C22×C8×C47C2×C411yescharacteristic
56C22×C8×C47C2×C411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C22×C8×C47C2311yescharacteristic
61C2×C4⋊C4×C47—22no—
62C2×C22⋊C4×C47—41no—
63#106×C47—22no—
64#106×C47—22no—
65#95×C47—41no—
66C24×C47D811yescharacteristic
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—42no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47D811yescharacteristic
73C22×C4×C47SD1611yescharacteristic
74C22×C4×C47SD1611yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C2×C22⋊D47—41no—
77C2×C22⋊D47—41no—
78C2×C22⋊D47—41no—
79#10625350—42no—
80#10625350—42no—
81C2×D188—21no—
82C2×C4×D47—41no—
83C2×Dic94—21no—
84C2×C8×C47—22no—
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C2×C8×C47—21no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—21no—
91C2×C8×C47C22⋊C411yescharacteristic
92C2×C8×C47C22⋊C411yescharacteristic
93C2×C8×C47C22×C411yescharacteristic
94C2×C8×C47C2×D412yes—
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47#4112yes—
97C4⋊C4×C47—42no—
98C4⋊C4×C47—42no—
99C22⋊C4×C47—82no—
100C22⋊Dic47—41no—
101C22⋊Dic47—41no—
102C22⋊Dic47—41no—
103(C2×D47)⋊C4—41no—
104(C2×D47)⋊C4—41no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107Dic47⋊C4—41no—
108Dic47⋊C4—41no—
109C4×Dic47—41no—
110Dic188—22no—
111D376—22no—
112#10625325—41no—
113#10625325—41no—
114C47⋊C16—24no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—41no—
118C23×C47#10211yescharacteristic
119C23×C47#10211yescharacteristic
120C23×C47C2×D811yescharacteristic
121C22×D47—41no—
122C2×C4×C47—22no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—82no—
130C2×C4×C47—82no—
131C2×C4×C47—82no—
132C2×C4×C47#10211yescharacteristic
133C2×C4×C47#10211yescharacteristic
134C2×C4×C47C2×C22⋊C411yescharacteristic
135C2×C4×C47#12112yes—
136C2×C4×C47C2×SD1611yescharacteristic
137C2×C4×C47#13512yes—
138C22⋊D47—82no—
139C22⋊D47—82no—
140C22⋊D47—82no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144D188—41no—
145D188—41no—
146C4×D47—82no—
147Dic94—41no—
148Dic94—41no—
149C8×C47—21no—
150C8×C47—21no—
151C8×C47C4×D412yes—
152C8×C47#12011yescharacteristic
153C8×C47#12311yescharacteristic
154#13065—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—44no—
163C22×C47#36012yescomplemented
164C22×C47C2×#10211yescharacteristic
165C22×C47#46612yes—
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—162no—
172C4×C47#38611yescharacteristic
173C4×C47#41611yescharacteristic
174C4×C47#46512yes—
175Dic47—81no—
176Dic47—81no—
177Dic47—81no—
178C22×M32—471no—
179#3050—471no—
180C2×#360—472no—
181#1807D4711yescharacteristic, complemented
182#1386—472no—
183C2×C47—22no—
184C2×C47—22no—
185C2×C47—41no—
186C2×C47#189311yescharacteristic
187C2×C47C2×#36011yescharacteristic
188C2×C47#214511yescharacteristic
189D47—161no—
190C2×#135—471no—
191C23×C8C2×D4711yescharacteristic
192#525—941no—
193C2×M32—472no—
194C2×M32—944no—
195C2×M32—941no—
196C22×C16—941no—
197#467—471no—
198#467—471no—
199#466—471no—
200#466—471no—
201#465—941no—
202#465—941no—
203#427—471no—
204C2×#106C2×D4712yes—
205#416—941no—
206#380—21no—
207#360—944no—
208#340—21no—
209C47#1306511yescharacteristic, complemented, Sylow
210C2×#41—941no—
211C23×C4C22⋊D4711yescharacteristic
212#135—942no—
213#135—942no—
214C2×Q16—471no—
215C2×SD16—941no—
216C2×D8—471no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D4711yescharacteristic
220C22×C8C4×D4711yescharacteristic
221C22×C8D18811yescharacteristic
222C22×C8D18811yescharacteristic
223C22×C8C22⋊D4711yescharacteristic
224C22×C8C22×D4711yescharacteristic
225#123—1881no—
226C22⋊Q8—941no—
227C22⋊Q8—941no—
228#121—941no—
229#121—941no—
230#120—1881no—
231C4×D4—1881no—
232C4×D4—1881no—
233#117—941no—
234C2×C4⋊C4—22no—
235C2×C22⋊C4—41no—
236M32—474no—
237M32—944no—
238M32—942no—
239C2×C16—944no—
240C2×C16—942no—
241#107—471no—
242#107—471no—
243#106—22no—
244#106—22no—
245#106—941no—
246#103—941no—
247#103—941no—
248#102—941no—
249#102—941no—
250#95—41no—
251C24#1062533411yescharacteristic
252#41—1882no—
253#41—1882no—
254C2×Q8—941no—
255C2×D4—941no—
256C2×D4—1881no—
257C2×D4—1881no—
258C2×D4—1881no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—1881no—
263C22×C4(C2×D47)⋊C411yescharacteristic
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4#1062533411yescharacteristic
266C22×C4#1062533611yescharacteristic
267C22×C4#1062533611yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269Q16—942no—
270SD16—1881no—
271SD16—1881no—
272D8—942no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8(C2×D47)⋊C411yescharacteristic
281C2×C8(C2×D47)⋊C411yescharacteristic
282C2×C8C2×C4×D4711yescharacteristic
283C2×C8C2×D18811yescharacteristic
284C2×C8D4×D4712yes—
285C2×C8#1062535212yes—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—941no—
289C4⋊C4—941no—
290C4⋊C4—1881no—
291C4⋊C4—1881no—
292C22⋊C4—82no—
293C22⋊C4—1881no—
294C22⋊C4—1881no—
295C22⋊C4—1881no—
296C22⋊C4—1881no—
297C22⋊C4—1881no—
298C42—1881no—
299C16—944no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—1881no—
304C23#1309394411yescharacteristic
305C23#1309394411yescharacteristic
306C23C2×#1062533411yescharacteristic
307Q8—1881no—
308Q8—1881no—
309D4—3762no—
310D4—3762no—
311D4—3762no—
312D4—1881no—
313D4—1881no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—3762no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4#1309394411yescharacteristic
329C2×C4#1309394411yescharacteristic
330C2×C4C2×(C2×D47)⋊C411yescharacteristic
331C2×C4#1309399612yes—
332C2×C4C2×#1062533611yescharacteristic
333C2×C4#1309404412yes—
334C8—21no—
335C8—21no—
336C8(C22⋊D47)⋊C412yes—
337C8#1309399411yescharacteristic
338C8#1309400811yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—3761no—
348C22—3761no—
349C22—3761no—
350C22#1568893212yescomplemented
351C22C2×#1309394411yescharacteristic
352C22#1568935712yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—3761no—
357C4—3761no—
358C4—3761no—
359C4#1568900011yescharacteristic
360C4#1568911711yescharacteristic
361C4#1568935412yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—7521no—
366C2#2024381411yescharacteristic
367C2C2×#1568893211yescharacteristic
368C2#2024512311yescharacteristic
subgroupG/H
centerC22C2×#13093944
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1807C2
socleC22×C47C2×#102
prime radical#20252969C2
cosocleC22×C8×C47C23
hypercenter#1807D47
hypocenterC47#13065
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#13093964C2semidirect
3. C2×#15688932C2semidirect
4. #20241495C2semidirect
5. C47×#1807C2semidirect
12. #15689364C4semidirect
13. #15689364C4semidirect
14. #15689359C4semidirect
15. #15689359C4semidirect
42. #13093964D4semidirect
163. C22×C47#360semidirect
181. #1807D47semidirect
209. C47#13065semidirect
350. C22#15688932semidirect

Maximal subgroups

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

subgroupindex
C22×#130939642
#202529692
C2×#156889322
#202414952
C47×#18072
#1306547