Finite Group Databasebeta

← #26117119

Subgroups — #26117119

Order 12032.

734 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 105 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
1#20253388C212yesmaximal, complemented
2#20244220C211yesmaximal, characteristic, complemented
3#20241813C212yesmaximal, complemented
4#20241217C211yesmaximal, characteristic, complemented
5C47×#2397C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689436—22no—
8#15689403—22no—
9#15689318—22no—
10#15689257—22no—
11C4×#10625334—24no—
12#15689176C2212yes—
13#15689147C2211yescharacteristic
14C2×#13093952—22no—
15D4⋊(C47⋊C8)—24no—
16C2×C4×D4×C47—22no—
17C2×C4×D4×C47C2212yes—
18C47×#399—22no—
19C47×#392—22no—
20C47×#376C2211yescharacteristic
21C2×#10625350—21no—
22C2×#10625334—24no—
23C2×#10625329—21no—
24#13094024D411yescharacteristic, complemented
25#13094021D411yescharacteristic, complemented
26#13094012—21no—
27#13093998—22no—
28C4×D188—22no—
29#13093957—24no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093952—44no—
33#13093945—24no—
34#13093944—24no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8D414yescomplemented
37C22×D4×C47—22no—
38C22×D4×C47D412yes—
39C23×C4×C47—22no—
40C4×D4×C47—22no—
41C4×D4×C47—22no—
42C4×D4×C47—22no—
43C4×D4×C47—22no—
44C4×D4×C47—44no—
45C4×D4×C47—44no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47D412yes—
48C2×C22⋊C4×C47—24no—
49C2×C22⋊C4×C47—22no—
50C2×C42×C47D412yes—
51C2×C42×C47C2311yescharacteristic
52#95×C47—22no—
53C24×C47—22no—
54C2×D4×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47—44no—
57C2×D4×C47—44no—
58C2×D4×C47—48no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—44no—
64C22×C4×C47—44no—
65C22×C4×C47C2×D412yes—
66C22×C4×C47C2×D411yescharacteristic
67C2×C22⋊D47—41no—
68#10625350—42no—
69#10625350—42no—
70C2×D188—22no—
71C2×D188—21no—
72C2×C4×D47—41no—
73C2×Dic94—21no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—22no—
76C4⋊C4×C47—44no—
77C22⋊C4×C47—48no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—42no—
80C42×C47—22no—
81C42×C47D814yescomplemented
82C42×C47C2×D411yescharacteristic
83C42×C47C2×D412yes—
84C42×C47#4111yescharacteristic
85#10625334—48no—
86(C2×D47)⋊C4—42no—
87C4⋊Dic47—22no—
88#10625329—42no—
89C2×C47⋊C8—24no—
90C2×C47⋊C8—41no—
91C23×C47—24no—
92C23×C47—44no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—44no—
96C23×C47#12011yescharacteristic
97C22×D47—41no—
98D4×C47—24no—
99D4×C47—24no—
100D4×C47—42no—
101D4×C47—42no—
102D4×C47—416no—
103C2×C4×C47—24no—
104C2×C4×C47—42no—
105C2×C4×C47—44no—
106C2×C4×C47—42no—
107C2×C4×C47—44no—
108C2×C4×C47—84no—
109C2×C4×C47—82no—
110C2×C4×C47—82no—
111C2×C4×C47#12012yes—
112C2×C4×C47#12111yescharacteristic
113C2×C4×C47#12112yes—
114C2×C4×C47#12711yescharacteristic
115C2×C4×C47C2×D812yes—
116C2×C4×C47AGL(1,ℤ/8)12yes—
117C22⋊D47—82no—
118C2×Dic47—41no—
119D188—44no—
120D188—42no—
121C4×D47—81no—
122C4×D47—81no—
123Dic94—42no—
124C47⋊C8—44no—
125C47⋊C8—81no—
126#10012—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—48no—
129C22×C47—44no—
130C22×C47—84no—
131C22×C47—82no—
132C22×C47—82no—
133C22×C47#39211yescharacteristic
134C22×C47#44712yes—
135C2×D47—82no—
136C2×D47—81no—
137C4×C47—22no—
138C4×C47—22no—
139C4×C47—44no—
140C4×C47—82no—
141C4×C47#45912yes—
142C4×C47#49311yescharacteristic
143C4×C47#49611yescharacteristic
144Dic47—81no—
145#3100—472no—
146#2397D4711yescharacteristic, complemented
147#1963—471no—
148#1486—472no—
149#1274—471no—
150C2×C47—22no—
151C2×C47—84no—
152C2×C47#201012yes—
153C2×C47#201111yescharacteristic
154D47—161no—
155#532—471no—
156C2×C4×D4—22no—
157C2×C4×D4C2×D4712yes—
158#480—942no—
159#474—942no—
160#459—942no—
161#447—942no—
162C4×D8—944no—
163#431—472no—
164#423—471no—
165C2×#104—942no—
166#399—22no—
167#392—22no—
168#376C2×D4711yescharacteristic
169D4⋊C8—944no—
170C47#1001211yescharacteristic, complemented, Sylow
171C2×#41—941no—
172C22×D4—22no—
173C22×D4C22⋊D4712yes—
174C23×C4—22no—
175C2×D8—944no—
176C2×M16—941no—
177C4⋊Q8—471no—
178#127—471no—
179#124—941no—
180#121—942no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—942no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊D4712yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C42C22⋊D4712yes—
193C2×C42C22×D4711yescharacteristic
194#107—944no—
195C4⋊C8—942no—
196C4⋊C8—942no—
197#104—1884no—
198#102—944no—
199#102—944no—
200C22⋊C8—942no—
201C4×C8—474no—
202#95—22no—
203C24—22no—
204#41—1882no—
205#41—1882no—
206C2×Q8—941no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—942no—
213C2×D4—941no—
214C2×D4—1881no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—1881no—
222C22×C4C2×C22⋊D4712yes—
223C22×C4C2×C22⋊D4711yescharacteristic
224D8—1888no—
225M16—1882no—
226C2×C8—944no—
227C2×C8—1881no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—44no—
231C4⋊C4—942no—
232C22⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—1882no—
236C42—22no—
237C42#1062533414yescomplemented
238C42D4×D4711yescharacteristic
239C42D4×D4712yes—
240C42#1062535211yescharacteristic
241C23—24no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—42no—
246C23—1881no—
247C23#1309399311yescharacteristic
248Q8—1882no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—416no—
254D4—3762no—
255D4—1884no—
256D4—1882no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—3761no—
266C2×C4—3761no—
267C2×C4—1881no—
268C2×C4#1309399212yes—
269C2×C4#1309399511yescharacteristic
270C2×C4#1309399612yes—
271C2×C4#1309402011yescharacteristic
272C2×C4C2×#1062533412yes—
273C2×C4#1309405012yes—
274C8—3761no—
275C8—1884no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—3762no—
283C22—3761no—
284C22#1568903111yescharacteristic
285C22#1568925712yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—3761no—
291C4#1568931512yes—
292C4#1568949911yescharacteristic
293C4#1568951011yescharacteristic
294C2—22no—
295C2—84no—
296C2—7521no—
297C2#2024445312yes—
298C2#2024445911yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2397C2
socleC22×C47#392
hypercenter#2397D47
hypocenterC47#10012
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #20253388C2semidirect
2. #20244220C2semidirect
3. #20241813C2semidirect
4. #20241217C2semidirect
5. C47×#2397C2semidirect
24. #13094024D4semidirect
25. #13094021D4semidirect
36. C4×C47⋊C8D4semidirect
81. C42×C47D8semidirect
146. #2397D47semidirect
170. C47#10012semidirect
237. C42#10625334semidirect

Maximal subgroups

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

subgroupindex
#202533882
#202442202
#202418132
#202412172
C47×#23972
#1001247