Finite Group Databasebeta

← #26108403

Subgroups — #26108403

Order 12032.

868 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689839C211yesmaximal, characteristic
2#20243385C212yesmaximal, complemented
3C2×#15688872C211yesmaximal, characteristic, complemented
4C2×#15688852C212yesmaximal, complemented
5C47×#1956C211yesmaximal, characteristic, complemented
6#15689839—24no—
7#15689839—22no—
8#15689839C414yescomplemented
9C2×#13094013—22no—
10C2×#13093999—22no—
11C2×C4×D188C2212yes—
12C2×#13093982—22no—
13C2×C4⋊(C47⋊C8)C2212yes—
14C2×#13093944—22no—
15#15688885—22no—
16#15688872—22no—
17#15688852—24no—
18C2×C4⋊Q8×C47C2211yescharacteristic
19C2×#103×C47—22no—
20C2×C4×C8×C47C2211yescharacteristic
21C47×#384C2211yescharacteristic
22C22×D188—22no—
23C22×D188D412yes—
24C22×C4×D47—22no—
25C22×C47⋊C8—22no—
26#13094013—28no—
27#13094013—28no—
28#13093999—28no—
29#13093999—28no—
30C4×D188—22no—
31C4×D188—22no—
32C4×D188C2×C414yes—
33C4×D188D414yes—
34#13093982—28no—
35#13093982—28no—
36C2×C4⋊Dic47D412yes—
37C2×(C2×D47)⋊C4—24no—
38C2×(C2×D47)⋊C4—22no—
39C2×C4×Dic47—22no—
40C4⋊(C47⋊C8)—22no—
41C4⋊(C47⋊C8)—22no—
42#13093944—44no—
43C22×Q8×C47D412yes—
44C22×C8×C47—21no—
45C4⋊Q8×C47—22no—
46C4⋊Q8×C47C2×C412yes—
47C4⋊Q8×C47D412yes—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—21no—
51C2×C42×C47C2311yescharacteristic
52#103×C47—44no—
53C4×C8×C47—22no—
54C23×D47—22no—
55C2×Q8×C47—22no—
56C2×Q8×C47—42no—
57C2×Q8×C47—42no—
58C2×Q8×C47C22⋊C414yes—
59C22×C4×C47—22no—
60C22×C4×C47—41no—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—22no—
64C2×D188—22no—
65C2×D188—22no—
66C2×D188—22no—
67C2×D188—22no—
68C2×D188—22no—
69C2×D188—28no—
70C2×D188—82no—
71C2×D188—82no—
72C2×D188C22⋊C414yes—
73C2×C4×D47—24no—
74C2×C4×D47—24no—
75C2×C4×D47—42no—
76C2×C4×D47—42no—
77C2×C4×D47—42no—
78C2×C4×D47—42no—
79C2×C8×C47—22no—
80C2×C8×C47—41no—
81C2×C8×C47—41no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—42no—
87C42×C47C22×C411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D412yes—
90(C2×D47)⋊C4—216no—
91(C2×D47)⋊C4—24no—
92(C2×D47)⋊C4—24no—
93C4⋊Dic47—22no—
94C4⋊Dic47C22⋊C414yes—
95C4×Dic47—28no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C23×C47#12011yescharacteristic
100C22×D47—24no—
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—42no—
105C22×D47—42no—
106C22×D47—42no—
107Q8×C47—44no—
108Q8×C47—82no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—24no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—82no—
115C2×C4×C47—81no—
116C2×C4×C47C2×C22⋊C412yes—
117C2×C4×C47C2×C22⋊C411yescharacteristic
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12012yes—
120C2×C4×C47#13512yes—
121C2×C4×C47#13512yes—
122C2×C4×C47AGL(1,ℤ/8)12yes—
123C2×C4×C47#13712yes—
124C2×Dic47—24no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127D188—24no—
128D188—24no—
129D188—42no—
130D188—42no—
131D188—88no—
132C4×D47—44no—
133C4×D47—44no—
134C4×D47—44no—
135C4×D47—44no—
136C8×C47—42no—
137C47⋊C8—82no—
138#8452—471nomaximal, Sylow
139C22×C47#37911yescharacteristic
140C22×C47#44912yes—
141C22×C47#45212yes—
142C22×C47#45311yescharacteristic
143C22×C47#45511yescharacteristic
144C2×D47—44no—
145C2×D47—44no—
146C2×D47—44no—
147C2×D47—82no—
148C2×D47—82no—
149C2×D47—82no—
150C2×D47—82no—
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—44no—
154C4×C47—161no—
155C4×C47#41612yes—
156C4×C47#41712yes—
157C4×C47#44912yes—
158C4×C47#45212yes—
159Dic47—44no—
160C2×#555—471no—
161#1956D4711yescharacteristic, complemented
162#1802—472no—
163C2×#333—471no—
164C2×D4⋊C8—472no—
165C2×C47#162411yescharacteristic
166C2×C47#163411yescharacteristic
167C2×C47#164612yes—
168C2×C47#178812yes—
169C2×C47#179211yescharacteristic
170D47—84no—
171#555—474no—
172#555—944no—
173#555—942no—
174C2×C4⋊Q8C2×D4711yescharacteristic
175C2×#124—942no—
176C2×#121—942no—
177C2×C4×D4—472no—
178C2×#117—942no—
179C2×C4⋊C8—472no—
180C2×#103—22no—
181C2×#102—942no—
182C2×C4×C8C2×D4711yescharacteristic
183#384C2×D4711yescharacteristic
184#340—942no—
185#333—942no—
186D4⋊C8—944no—
187C47#845211yescharacteristic, complemented, Sylow
188C22×Q8C22⋊D4712yes—
189C22×D4—472no—
190C22×D4—942no—
191C23×C4—942no—
192C22×C8—21no—
193C22×C8—942no—
194C4⋊Q8—22no—
195C4⋊Q8C4×D4712yes—
196C4⋊Q8D18812yes—
197#124—948no—
198#124—948no—
199#121—948no—
200#121—948no—
201C4×D4—474no—
202C4×D4—474no—
203C4×D4—942no—
204C4×D4—942no—
205#117—948no—
206#117—948no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—472no—
211C2×C22⋊C4—944no—
212C2×C22⋊C4—942no—
213C2×C42—942no—
214C2×C42C22×D4711yescharacteristic
215C4⋊C8—942no—
216C4⋊C8—942no—
217#103—44no—
218#102—1884no—
219C4×C8—22no—
220C24—942no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8(C2×D47)⋊C414yes—
225C2×D4—474no—
226C2×D4—948no—
227C2×D4—942no—
228C2×D4—942no—
229C2×D4—942no—
230C2×D4—942no—
231C2×D4—942no—
232C2×D4—3762no—
233C2×D4—3762no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—942no—
237C22×C4—944no—
238C22×C4—944no—
239C22×C4—1882no—
240C22×C4—1882no—
241C22×C4—1882no—
242C22×C4—1882no—
243C22×C4D4×D4712yes—
244C22×C4C2×C22⋊D4711yescharacteristic
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—1882no—
249C2×C8—1882no—
250C2×C8—1882no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—474no—
257C4⋊C4—942no—
258C22⋊C4—9416no—
259C22⋊C4—944no—
260C22⋊C4—944no—
261C42—948no—
262C42C2×C4×D4711yescharacteristic
263C42C2×D18811yescharacteristic
264C42D4×D4712yes—
265C23—944no—
266C23—1882no—
267C23—1882no—
268C23—1882no—
269C23—1882no—
270C23—1882no—
271C23—1882no—
272C23#1309399211yescharacteristic
273Q8—44no—
274Q8—82no—
275D4—944no—
276D4—944no—
277D4—3768no—
278D4—1882no—
279D4—1882no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—944no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1884no—
291C2×C4—1884no—
292C2×C4—1884no—
293C2×C4—1884no—
294C2×C4C22⋊C4×D4712yes—
295C2×C4C2×(C2×D47)⋊C411yescharacteristic
296C2×C4#1309399211yescharacteristic
297C2×C4#1309399412yes—
298C2×C4#1309404512yes—
299C2×C4#1309404712yes—
300C2×C4#1309405112yes—
301C2×C4#1309405612yes—
302C8—42no—
303C8—3762no—
304C22—3762no—
305C22—3762no—
306C22—3762no—
307C22—3762no—
308C22—1884no—
309C22—1884no—
310C22—1884no—
311C22#1568897811yescharacteristic
312C22#1568926712yes—
313C22#1568928412yes—
314C22#1568929011yescharacteristic
315C22#1568930011yescharacteristic
316C4—21no—
317C4—21no—
318C4—44no—
319C4—161no—
320C4—1884no—
321C4#1568911812yes—
322C4#1568912312yes—
323C4#1568927112yes—
324C4#1568928812yes—
325C2—3764no—
326C2#2024246911yescharacteristic
327C2#2024253111yescharacteristic
328C2#2024261312yes—
329C2#2024332712yes—
330C2#2024334111yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1956C2
socleC23×C47#120
prime radicalC2×#15689839C2
cosocleC2×C42×C47C23
hypercenter#1956D47
hypocenterC47#8452
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. #20243385C2semidirect
3. C2×#15688872C2semidirect
4. C2×#15688852C2semidirect
5. C47×#1956C2semidirect
8. #15689839C4semidirect
161. #1956D47semidirect
187. C47#8452semidirect

Maximal subgroups

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

subgroupindex
C2×#156898392
#202433852
C2×#156888722
C2×#156888522
C47×#19562
#845247