Finite Group Databasebeta

← #26133671

Subgroups — #26133671

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
2C2×#15688933C212yesmaximal, complemented
3#20243413C211yesmaximal, characteristic, complemented
4#20241493C212yesmaximal, complemented
5C47×#3050C211yesmaximal, characteristic
6C2×#13093964—24no—
7C2×#13093964—21no—
8C2×#13093964C2212yes—
9C22×C47⋊C16—21no—
10C2×#13093956C2212yes—
11#15688983—21no—
12#15688933—24no—
13#15688886—21no—
14C2×#135×C47C2211yescharacteristic
15C23×C8×C47C2211yescharacteristic
16C47×#525—21no—
17C47×#467C411yescharacteristic, complemented
18C47×#467C411yescharacteristic, complemented
19C47×#466C411yescharacteristic, complemented
20C47×#466C411yescharacteristic, complemented
21C47×#465—21no—
22C47×#465—21no—
23C47×#427C2211yescharacteristic
24C47×#416—21no—
25C2×C4⋊Dic47—22no—
26C2×C22⋊Dic47—41no—
27#13093964—24no—
28#13093964—22no—
29#13093964D414yescomplemented
30C2×C47⋊C16—24no—
31C2×C47⋊C16—22no—
32#13093956—22no—
33#13093956—22no—
34C47⋊#95—41no—
35C2×#41×C47—21no—
36C23×C4×C47D411yescharacteristic
37#135×C47—22no—
38#135×C47—22no—
39C2×Q16×C47C2×C411yescharacteristic
40C2×SD16×C47—21no—
41C2×D8×C47C2×C411yescharacteristic
42C22×C8×C47—22no—
43C22×C8×C47—22no—
44C22×C8×C47C2×C411yescharacteristic
45C22×C8×C47C2×C411yescharacteristic
46C22×C8×C47D411yescharacteristic
47C22×C8×C47D411yescharacteristic
48C22×C8×C47D411yescharacteristic
49C22×C8×C47C2311yescharacteristic
50#123×C47—41no—
51C22⋊Q8×C47—21no—
52C22⋊Q8×C47—21no—
53#121×C47—21no—
54#121×C47—21no—
55#120×C47—41no—
56C4×D4×C47—41no—
57C4×D4×C47—41no—
58#117×C47—21no—
59#107×C47C2×C411yescharacteristic
60#107×C47C2×C411yescharacteristic
61#106×C47—21no—
62#103×C47—21no—
63#103×C47—21no—
64#102×C47—21no—
65#102×C47—21no—
66C24×C47D811yescharacteristic
67#41×C47—42no—
68#41×C47—42no—
69C2×Q8×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—41no—
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47D811yescharacteristic
80C22×C4×C47SD1611yescharacteristic
81C22×C4×C47SD1611yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×Dic47—42no—
84Q16×C47—22no—
85SD16×C47—41no—
86SD16×C47—41no—
87D8×C47—22no—
88C2×C8×C47—22no—
89C2×C8×C47—22no—
90C2×C8×C47—22no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C2×D412yes—
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47#4112yes—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—41no—
110C42×C47—41no—
111C22⋊Dic47—82no—
112C4⋊Dic47—42no—
113C4⋊Dic47—42no—
114C47⋊C16—24no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47#10211yescharacteristic
120C23×C47#10211yescharacteristic
121C23×C47C2×D811yescharacteristic
122Q8×C47—41no—
123Q8×C47—41no—
124D4×C47—41no—
125D4×C47—41no—
126D4×C47—82no—
127D4×C47—82no—
128D4×C47—82no—
129C2×C4×C47—22no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—82no—
140C2×C4×C47#10211yescharacteristic
141C2×C4×C47#10211yescharacteristic
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47#12112yes—
144C2×C4×C47C2×SD1611yescharacteristic
145C2×C4×C47#13512yes—
146C2×Dic47—82no—
147C2×Dic47—82no—
148C2×Dic47—82no—
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—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47#36012yescomplemented
167C22×C47C2×#10211yescharacteristic
168C22×C47#46612yes—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—81no—
172C4×C47—81no—
173C4×C47—81no—
174C4×C47#38611yescharacteristic
175C4×C47#41611yescharacteristic
176C4×C47#46512yes—
177Dic47—162no—
178C22×M32—471no—
179#3050D4711yescharacteristic
180C2×#360—472no—
181#1807—471no—
182#1386—472no—
183C2×C47—22no—
184C2×C47—22no—
185C2×C47—41no—
186C2×C47—161no—
187C2×C47#189311yescharacteristic
188C2×C47C2×#36011yescharacteristic
189C2×C47#214511yescharacteristic
190C2×#135C2×D4711yescharacteristic
191C23×C8C2×D4711yescharacteristic
192#525—21no—
193C2×M32—472no—
194C2×M32—944no—
195C2×M32—941no—
196C22×C16—941no—
197#467Dic4711yescharacteristic, complemented
198#467Dic4711yescharacteristic, complemented
199#466Dic4711yescharacteristic, complemented
200#466Dic4711yescharacteristic, complemented
201#465—21no—
202#465—21no—
203#427C2×D4711yescharacteristic
204C2×#106—472no—
205#416—21no—
206#380—941no—
207#360—944no—
208#340—941no—
209C47#1306511yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C23×C4C22⋊D4711yescharacteristic
212#135—22no—
213#135—22no—
214C2×Q16C2×Dic4711yescharacteristic
215C2×SD16—21no—
216C2×D8C2×Dic4711yescharacteristic
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C2×Dic4711yescharacteristic
220C22×C8C2×Dic4711yescharacteristic
221C22×C8C22⋊D4711yescharacteristic
222C22×C8C22⋊D4711yescharacteristic
223C22×C8C22⋊D4711yescharacteristic
224C22×C8C22×D4711yescharacteristic
225#123—41no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#121—21no—
229#121—21no—
230#120—41no—
231C4×D4—41no—
232C4×D4—41no—
233#117—21no—
234C2×C4⋊C4—942no—
235C2×C22⋊C4—1881no—
236M32—474no—
237M32—944no—
238M32—942no—
239C2×C16—944no—
240C2×C16—942no—
241#107C2×Dic4711yescharacteristic
242#107C2×Dic4711yescharacteristic
243#106—21no—
244#106—942no—
245#106—942no—
246#103—21no—
247#103—21no—
248#102—21no—
249#102—21no—
250#95—1881no—
251C24#1062533411yescharacteristic
252#41—42no—
253#41—42no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—1882no—
263C22×C4#1062533411yescharacteristic
264C22×C4#1062533511yescharacteristic
265C22×C4#1062533511yescharacteristic
266C22×C4C22⋊Dic4711yescharacteristic
267C22×C4C22⋊Dic4711yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269Q16—22no—
270SD16—41no—
271SD16—41no—
272D8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22⋊Dic4711yescharacteristic
281C2×C8C22⋊Dic4711yescharacteristic
282C2×C8D4×D4712yes—
283C2×C8#1062535212yes—
284C2×C8C22×Dic4711yescharacteristic
285C2×C8C2×C22⋊D4711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—1882no—
291C4⋊C4—1882no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—3762no—
298C42—41no—
299C16—944no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23#1309394511yescharacteristic
305C23#1309394511yescharacteristic
306C23C2×#1062533411yescharacteristic
307Q8—41no—
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—82no—
312D4—82no—
313D4—82no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—3762no—
326C2×C4—3762no—
327C2×C4—3762no—
328C2×C4#1309394511yescharacteristic
329C2×C4#1309394511yescharacteristic
330C2×C4C2×C22⋊Dic4711yescharacteristic
331C2×C4#1309399612yes—
332C2×C4C2×#1062533511yescharacteristic
333C2×C4#1309404412yes—
334C8—21no—
335C8—21no—
336C8D4×Dic4712yes—
337C8#1309399211yescharacteristic
338C8#1309400511yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#1568893312yescomplemented
351C22C2×#1309394511yescharacteristic
352C22#1568935712yes—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—7522no—
359C4#1568900311yescharacteristic
360C4#1568911911yescharacteristic
361C4#1568935012yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—161no—
366C2#2024381711yescharacteristic
367C2C2×#1568893311yescharacteristic
368C2#2024512411yescharacteristic
subgroupG/H
centerC22C2×#13093945
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC47×#3050C2
socleC22×C47C2×#102
cosocleC22×C8×C47C23
hypercenter#3050D47
hypocenterC47#13065
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#13093964C2semidirect
2. C2×#15688933C2semidirect
3. #20243413C2semidirect
4. #20241493C2semidirect
17. C47×#467C4semidirect
18. C47×#467C4semidirect
19. C47×#466C4semidirect
20. C47×#466C4semidirect
29. #13093964D4semidirect
166. C22×C47#360semidirect
197. #467Dic47semidirect
198. #467Dic47semidirect
199. #466Dic47semidirect
200. #466Dic47semidirect
209. C47#13065semidirect
350. C22#15688933semidirect

Maximal subgroups

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

subgroupindex
C22×#130939642
C2×#156889332
#202434132
#202414932
C47×#30502
#1306547