Finite Group Databasebeta

← #26110923

Subgroups — #26110923

Order 12032.

586 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C4×#13094011C211yesmaximal, characteristic
2#20241782C212yesmaximal, complemented
3#20241709C211yesmaximal, characteristic
4#20241220C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic
6C2×#13094011C411yescharacteristic
7C2×C4×Dic94C2211yescharacteristic
8#15689176C2212yes—
9#15689028C411yescharacteristic
10#15689026C411yescharacteristic
11#15689017—22no—
12#15688988C411yescharacteristic
13C4×C22⋊Dic47—22no—
14C42×Dic47C2211yescharacteristic
15#15688853—24no—
16#15688847—22no—
17C2×C4×D4×C47C2211yescharacteristic
18C8×D4×C47—24no—
19C47×#431C2212yes—
20C2×C22⋊C8×C47—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094011—22no—
23#13094011C814yescomplemented
24C4×Dic94—22no—
25C4×Dic94—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C4⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C4×C47⋊C8—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C22×D4×C47C2×C411yescharacteristic
41C23×C4×C47D412yes—
42C22×C8×C47—24no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47D414yes—
49C22⋊C8×C47—24no—
50C22⋊C8×C47—22no—
51C4×C8×C47—22no—
52C24×C47C22⋊C412yes—
53C2×D4×C47—22no—
54C2×D4×C47—22no—
55C2×D4×C47C2×C812yes—
56C2×D4×C47M1612yes—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—44no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C2×Dic94—22no—
69C2×Dic94—22no—
70C2×Dic94—21no—
71C2×Dic94C2×C812yescomplemented
72C2×C8×C47—24no—
73C2×C8×C47—84no—
74C4⋊C4×C47—22no—
75C22⋊C4×C47—24no—
76C42×C47C2×D412yes—
77C42×C47#4112yes—
78C22⋊Dic47—28no—
79C22⋊Dic47—44no—
80C4⋊Dic47—22no—
81Dic47⋊C4—42no—
82C4×Dic47—21no—
83C4×Dic47—22no—
84C4×Dic47—42no—
85C4×Dic47—42no—
86C4×Dic47—41no—
87C4×Dic47—41no—
88C4×Dic47C2×C812yescomplemented
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47C22⋊C814yes—
95C23×C47C2×C22⋊C411yescharacteristic
96D4×C47—24no—
97D4×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—41no—
101C2×C4×C47—44no—
102C2×C4×C47—42no—
103C2×C4×C47—82no—
104C2×C4×C47C4×D412yes—
105C2×C4×C47#12011yescharacteristic
106C2×C4×C47#12112yes—
107C2×C4×C47#12311yescharacteristic
108C2×C4×C47C22×C811yescharacteristic
109C2×C4×C47C2×M1611yescharacteristic
110C2×C4×C47#13112yes—
111C2×Dic47—22no—
112C2×Dic47—22no—
113C2×Dic47—22no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—42no—
121C2×Dic47—41no—
122Dic94—24no—
123Dic94—42no—
124C8×C47—44no—
125C47⋊C8—82no—
126#8957—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—24no—
130C22×C47—44no—
131C22×C47—42no—
132C22×C47—42no—
133C22×C47—82no—
134C22×C47#38611yescharacteristic
135C22×C47C2×C22⋊C811yescharacteristic
136C22×C47#40811yescharacteristic
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47C8×D412yes—
141C4×C47#43512yes—
142C4×C47#45512yes—
143Dic47—42no—
144Dic47—42no—
145Dic47—42no—
146Dic47—81no—
147#2972D4711yescharacteristic
148C4×#124—471no—
149#1479—472no—
150#1460—471no—
151#1276—472no—
152C2×C47—21no—
153C2×C47—21no—
154C2×C47—44no—
155C2×C47#186911yescharacteristic
156C2×C47#188411yescharacteristic
157C2×C47#188511yescharacteristic
158C2×#124—471no—
159C2×C4×Q8—471no—
160C2×C4×D4C2×D4711yescharacteristic
161C8×D4—24no—
162#431—472no—
163#431C2×D4712yes—
164C2×C22⋊C8—22no—
165#391—471no—
166#390—471no—
167#388—942no—
168#382—471no—
169C4×C22⋊C4—942no—
170C43—471no—
171Q8⋊C8—944no—
172#323—942no—
173C47#895711yescharacteristic, complemented, Sylow
174C22×Q8—471no—
175C22×D4C4×D4711yescharacteristic
176C23×C4C22⋊D4712yes—
177C22×C8—24no—
178#124—474no—
179#124—942no—
180C4×Q8—942no—
181C4×Q8—942no—
182C4×D4—22no—
183C4×D4—22no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4C4×D4711yescharacteristic
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4D18812yes—
190C2×C42—471no—
191C2×C42—471no—
192C2×C42—941no—
193C2×C42—941no—
194C2×C42—941no—
195C2×C42C22×D4711yescharacteristic
196C4⋊C8—942no—
197C4⋊C8C22⋊D4714yes—
198C22⋊C8—24no—
199C22⋊C8—22no—
200C22⋊C8—942no—
201C4×C8—22no—
202C4×C8—942no—
203#95—942no—
204#95—942no—
205C24(C2×D47)⋊C412yes—
206C2×Q8—472no—
207C2×Q8—941no—
208C2×Q8—942no—
209C2×Q8—942no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C8×D4712yes—
213C2×D4#1062532412yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—44no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4(C2×D47)⋊C412yes—
222C22×C4C2×C4×D4711yescharacteristic
223C22×C4C2×D18811yescharacteristic
224C22×C4C2×C22⋊D4711yescharacteristic
225C2×C8—24no—
226C2×C8—84no—
227C2×C8—1882no—
228C4⋊C4—22no—
229C4⋊C4—942no—
230C4⋊C4—1882no—
231C22⋊C4—24no—
232C22⋊C4—948no—
233C22⋊C4—1884no—
234C42—472no—
235C42—941no—
236C42—942no—
237C42—1882no—
238C42—1882no—
239C42—1881no—
240C42—1881no—
241C42#1062535012yes—
242C42C2×C22⋊D4712yes—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23(C2×D47)⋊C814yes—
248C23C2×(C2×D47)⋊C411yescharacteristic
249Q8—944no—
250Q8—1882no—
251D4—24no—
252D4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—942no—
260C2×C4—942no—
261C2×C4—942no—
262C2×C4—1882no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4—1881no—
270C2×C4C4×C22⋊D4712yes—
271C2×C4#1309399311yescharacteristic
272C2×C4#1309399812yes—
273C2×C4#1309400711yescharacteristic
274C2×C4C2×C8×D4711yescharacteristic
275C2×C4C2×#1062532411yescharacteristic
276C2×C4#1309403012yes—
277C8—44no—
278C8—3762no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1568900211yescharacteristic
287C22C2×(C2×D47)⋊C811yescharacteristic
288C22#1568908911yescharacteristic
289C4—22no—
290C4—21no—
291C4—42no—
292C4—3761no—
293C4—1882no—
294C4—1882no—
295C4—1882no—
296C4C8×C22⋊D4712yes—
297C4#1568919412yes—
298C4#1568930112yes—
299C2—21no—
300C2—21no—
301C2—44no—
302C2#2024368011yescharacteristic
303C2#2024377111yescharacteristic
304C2#2024377611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#8957
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20241782C2semidirect
4. #20241220C2semidirect
23. #13094011C8semidirect
71. C2×Dic94C2×C8semidirect
88. C4×Dic47C2×C8semidirect
173. C47#8957semidirect

Maximal subgroups

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

subgroupindex
C4×#130940112
#202417822
#202417092
#202412202
C47×#29722
#895747