Finite Group Databasebeta

← #26104590

Subgroups — #26104590

Order 12032.

468 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 99 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#20245935C211yesmaximal, characteristic
2#20243530C212yesmaximal, complemented
3#20241213C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#1270C211yesmaximal, characteristic
6C2×#13094011C411yescharacteristic
7C2×#13093989C411yescharacteristic
8C2×D4×Dic47C411yescharacteristic
9#15689029C2211yescharacteristic
10#15689011—22no—
11#15688988C411yescharacteristic
12C22⋊C4×Dic47—22no—
13C42×Dic47C2211yescharacteristic
14#15688878C2212yes—
15#15688877—22no—
16#15688877—22no—
17C47×#390C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94C2×C411yescharacteristic
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22#13094011—22no—
23#13094011—22no—
24#13093989—42no—
25D4×Dic47—42no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34C2×C4×Dic47D412yes—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C22×D4×C47C2×C411yescharacteristic
38C22×C8×C47—22no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C22⋊C4×C47—22no—
41C2×C42×C47C2311yescharacteristic
42C24×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47M1612yes—
47C22×C4×C47—21no—
48C22×C4×C47C22×C411yescharacteristic
49C22×C4×C47C2×D412yes—
50C22×Dic47—21no—
51C22×Dic47—21no—
52C22×Dic47—44no—
53C22×Dic47—42no—
54C22×Dic47C22⋊C412yes—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C411yescharacteristic
57C2×Dic94—21no—
58C2×Dic94—41no—
59C2×Dic94—41no—
60C2×Dic94M1612yes—
61C2×C8×C47—42no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C4⋊C4×C47—41no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C42×C47—41no—
68C22⋊Dic47—28no—
69C22⋊Dic47—42no—
70C4⋊Dic47—41no—
71Dic47⋊C4—42no—
72C4×Dic47—21no—
73C4×Dic47—21no—
74C4×Dic47—22no—
75C4×Dic47—22no—
76C4×Dic47—22no—
77C4×Dic47—42no—
78C4×Dic47—42no—
79C2×C47⋊C8—42no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C23×C47—22no—
83C23×C47—41no—
84C23×C47—41no—
85C23×C47—42no—
86C23×C47—42no—
87C23×C47C2×C22⋊C411yescharacteristic
88D4×C47—42no—
89D4×C47—42no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—41no—
95C2×C4×C47—42no—
96C2×C4×C47C2×M1611yescharacteristic
97C2×C4×C47C2×M1611yescharacteristic
98C2×C4×C47#13512yes—
99C2×C4×C47#13512yes—
100C2×Dic47—22no—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—21no—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47—84no—
112C2×Dic47#10012yes—
113Dic94—42no—
114Dic94—42no—
115C8×C47—82no—
116C47⋊C8—82no—
117#7649—471nomaximal, Sylow
118C22×C47—42no—
119C22×C47—44no—
120C22×C47—81no—
121C22×C47—82no—
122C22×C47—81no—
123C22×C47#40711yescharacteristic
124C22×C47#41011yescharacteristic
125C22×C47C2×#10011yescharacteristic
126C22×C47#41612yes—
127C22×C47#42112yes—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—22no—
131C4×C47—22no—
132C4×C47—81no—
133Dic47—22no—
134Dic47—41no—
135Dic47—42no—
136Dic47—42no—
137Dic47—81no—
138C4×#124—471no—
139#1839—472no—
140#1272—472no—
141#1270—471no—
142#1270D4711yescharacteristic
143C2×C47—82no—
144C2×C47#146611yescharacteristic
145C2×C47#148812yes—
146C2×C47#152112yes—
147C2×C47#153311yescharacteristic
148C2×C47#154311yescharacteristic
149C2×#124—471no—
150C2×C4×Q8—471no—
151C2×C4×D4—471no—
152#391—471no—
153#390C2×D4711yescharacteristic
154#388—942no—
155#382—471no—
156C4×C22⋊C4—942no—
157C43—471no—
158#336—472no—
159#336—942no—
160#336—942no—
161#336C2×D4712yes—
162C47#764911yescharacteristic, complemented, Sylow
163C22×Q8—471no—
164C22×D4C4×D4711yescharacteristic
165C23×C4—941no—
166C22×C8—22no—
167C22×C8—942no—
168#124—942no—
169#124—942no—
170C4×Q8—1882no—
171C4×D4—1882no—
172C2×C4⋊C4—471no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4C4×D4711yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—942no—
177C2×C22⋊C4—941no—
178C2×C42—471no—
179C2×C42—471no—
180C2×C42—472no—
181C2×C42—472no—
182C2×C42—941no—
183C2×C42C22×D4711yescharacteristic
184#95—942no—
185#95—942no—
186C24—21no—
187C2×Q8—472no—
188C2×Q8—941no—
189C2×Q8—1881no—
190C2×Q8—1881no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4#1062532412yes—
195C22×C4—21no—
196C22×C4—471no—
197C22×C4—472no—
198C22×C4—471no—
199C22×C4—941no—
200C22×C4—941no—
201C22×C4—1882no—
202C22×C4—1884no—
203C22×C4C2×C4×D4711yescharacteristic
204C22×C4D4×D4712yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—1882no—
209C2×C8—1882no—
210C2×C8—1882no—
211C4⋊C4—41no—
212C4⋊C4—1881no—
213C4⋊C4—1882no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—948no—
217C22⋊C4—1882no—
218C42—41no—
219C42—942no—
220C42—941no—
221C42—941no—
222C42—942no—
223C42—942no—
224C42—1882no—
225C42—1882no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23C22⋊C4×D4711yescharacteristic
232Q8—1882no—
233Q8—1882no—
234D4—42no—
235D4—42no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—472no—
243C2×C4—942no—
244C2×C4—942no—
245C2×C4—942no—
246C2×C4—941no—
247C2×C4—942no—
248C2×C4—942no—
249C2×C4—942no—
250C2×C4—3764no—
251C2×C4—1881no—
252C2×C4—1882no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4M16×D4711yescharacteristic
256C2×C4C2×#1062532411yescharacteristic
257C2×C4#1309404312yes—
258C2×C4#1309404512yes—
259C8—82no—
260C8—3762no—
261C22—42no—
262C22—44no—
263C22—82no—
264C22—81no—
265C22—81no—
266C22#1568908511yescharacteristic
267C22#1568909511yescharacteristic
268C22#100×D4711yescharacteristic
269C22#1568911612yes—
270C22#1568913812yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—81no—
276C4—942no—
277C4—3761no—
278C4—1882no—
279C4—1881no—
280C4—1882no—
281C2—82no—
282C2#2024172711yescharacteristic
283C2#2024181912yes—
284C2#2024196212yes—
285C2#2024201011yescharacteristic
286C2#2024204711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1270C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1270D47
hypocenterC47#7649
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20243530C2semidirect
3. #20241213C2semidirect
162. C47#7649semidirect

Maximal subgroups

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

subgroupindex
#202459352
#202435302
#202412132
#202412092
C47×#12702
#764947