Finite Group Databasebeta

← #26104018

Subgroups — #26104018

Order 12032.

348 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15688916C211yesmaximal, characteristic
2#20243787C211yesmaximal, characteristic
3C2×C47×#331C211yesmaximal, characteristic
4#15689598C411yescharacteristic, complemented
5#15689128—21no—
6C2×#13093941C411yescharacteristic, complemented
7C2×#13093939—21no—
8#15688916—21no—
9#15688916C412yescomplemented
10#15688883—21no—
11C2×#127×C47C2211yescharacteristic
12C2×C4⋊C8×C47—21no—
13C47×#331—21no—
14C47×#331C412yescomplemented
15C2×#10625329—21no—
16D4×Dic47—42no—
17#13093983—21no—
18C2×C22⋊Dic47—41no—
19C2×C22⋊Dic47—41no—
20#13093945—42no—
21#13093941—41no—
22#13093939—22no—
23#13093939—41no—
24C22×D4×C47—41no—
25C22×D4×C47C2×C411yescharacteristic
26C22×D4×C47D411yescharacteristic
27C22×C8×C47—41no—
28#127×C47—41no—
29#127×C47C2×C411yescharacteristic
30#127×C47C2×C411yescharacteristic
31#127×C47D411yescharacteristic
32#127×C47Q811yescharacteristic
33C2×C42×C47D411yescharacteristic
34C4⋊C8×C47—22no—
35C4⋊C8×C47—41no—
36C24×C47—21no—
37C24×C47—21no—
38C2×D4×C47—21no—
39C2×D4×C47—21no—
40C2×D4×C47—41no—
41C2×D4×C47—41no—
42C2×D4×C47—41no—
43C2×D4×C47—41no—
44C2×D4×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47—82no—
49C2×D4×C47—82no—
50C2×D4×C47C4211yescharacteristic
51C2×D4×C47C22⋊C411yescharacteristic
52C2×D4×C47C4⋊C411yescharacteristic
53C2×D4×C47C4⋊C411yescharacteristic
54C22×C4×C47—21no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×Dic47—42no—
57C22×Dic47—81no—
58C2×C8×C47—42no—
59C2×C8×C47—82no—
60C42×C47—21no—
61C42×C47C22⋊C411yescharacteristic
62C42×C47C4⋊C411yescharacteristic
63C22⋊Dic47—42no—
64C22⋊Dic47—41no—
65C22⋊Dic47—42no—
66C22⋊Dic47—81no—
67C4⋊Dic47—22no—
68C4×Dic47—41no—
69#10625329—41no—
70#10625329—41no—
71C2×C47⋊C8—41no—
72C23×C47—21no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—41no—
77C23×C47—41no—
78C23×C47—41no—
79C23×C47—41no—
80C23×C47—82no—
81C23×C47—82no—
82C23×C47#9911yescharacteristic
83D4×C47—41no—
84D4×C47—41no—
85D4×C47—41no—
86D4×C47—41no—
87D4×C47—42no—
88D4×C47—81no—
89D4×C47—81no—
90D4×C47—81no—
91D4×C47—82no—
92D4×C47—81no—
93D4×C47—81no—
94D4×C47—82no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47#9511yescharacteristic
102C2×C4×C47#9911yescharacteristic
103C2×Dic47—42no—
104C2×Dic47—82no—
105C2×Dic47—162no—
106C2×Dic47—162no—
107C8×C47—82no—
108C47⋊C8—81no—
109#7485—471nomaximal, Sylow
110C22×C47—22no—
111C22×C47—41no—
112C22×C47—41no—
113C22×C47—41no—
114C22×C47—41no—
115C22×C47—81no—
116C22×C47—81no—
117C22×C47—81no—
118C22×C47—81no—
119C22×C47—82no—
120C22×C47—82no—
121C22×C47—161no—
122C22×C47—161no—
123C22×C47#34211yescharacteristic
124C22×C47AΣL(1,𝔽16)11yescharacteristic
125C22×C47#35211yescharacteristic
126C4×C47—21no—
127C4×C47—22no—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—21no—
131Dic47—82no—
132Dic47—162no—
133C2×#353—471no—
134#1887—471no—
135C2×#331D4711yescharacteristic
136C2×C47—21no—
137C2×C47—21no—
138C2×C47—81no—
139C2×C47—81no—
140C2×C47—81no—
141C2×C47—81no—
142C2×C47#139011yescharacteristic
143C2×C47#140212yes—
144C2×#127C2×D4711yescharacteristic
145#518—471no—
146C2×C4⋊C8—21no—
147#418—941no—
148C2×#100—471no—
149C2×#99—941no—
150#353—472no—
151#353—941no—
152#339—941no—
153#331—21no—
154#331Dic4712yescomplemented
155C47#748511yescharacteristic, complemented, Sylow
156C22×D4—41no—
157C22×D4C4×D4711yescharacteristic
158C22×D4D18811yescharacteristic
159C2×M16—941no—
160C22×C8—41no—
161#127—41no—
162#127Dic9411yescharacteristic
163#127C4×D4711yescharacteristic
164#127C2×Dic4711yescharacteristic
165#127C22⋊D4711yescharacteristic
166C4×D4—1882no—
167#117—941no—
168C2×C22⋊C4—1881no—
169C2×C22⋊C4—1881no—
170C2×C42C22⋊D4711yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—41no—
173#102—1882no—
174#100—1881no—
175#99—942no—
176#99—1881no—
177C24—21no—
178C24—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—82no—
190C2×D4—82no—
191C2×D4C4×Dic4711yescharacteristic
192C2×D4Dic47⋊C411yescharacteristic
193C2×D4C4⋊Dic4711yescharacteristic
194C2×D4(C2×D47)⋊C411yescharacteristic
195C22×C4—21no—
196C22×C4—3761no—
197C22×C4—1882no—
198C22×C4(C2×D47)⋊C411yescharacteristic
199M16—1881no—
200M16—1881no—
201C2×C8—42no—
202C2×C8—82no—
203C2×C8—1881no—
204C4⋊C4—942no—
205C22⋊C4—3761no—
206C22⋊C4—1882no—
207C22⋊C4—1882no—
208C22⋊C4—1881no—
209C42—21no—
210C42—1881no—
211C42Dic47⋊C411yescharacteristic
212C42C22⋊Dic4711yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—82no—
222C23—82no—
223C23#1309393811yescharacteristic
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—42no—
229D4—81no—
230D4—81no—
231D4—81no—
232D4—82no—
233D4—82no—
234D4—81no—
235D4—81no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—3762no—
243C2×C4—1882no—
244C2×C4—7522no—
245C2×C4—7522no—
246C2×C4C47⋊#9511yescharacteristic
247C2×C4#1309393811yescharacteristic
248C8—82no—
249C8—3761no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—161no—
262C22—161no—
263C22#1568888911yescharacteristic
264C22#1568891111yescharacteristic
265C22#1568891411yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—3762no—
272C4—7522no—
273C2—21no—
274C2—21no—
275C2—81no—
276C2—81no—
277C2—81no—
278C2—81no—
279C2#2024150711yescharacteristic
280C2#2024153412yes—
subgroupG/H
centerC22#15688911
commutatorC2×D4×C47C42
FrattiniC2×#127C2×D47
FittingC2×C47×#331C2
socleC22×C47AΣL(1,𝔽16)
prime radical, cosocleC2×#127×C47C22
hypercenterC2×#331D47
hypocenterC47#7485
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
4. #15689598C4semidirect
6. C2×#13093941C4semidirect
9. #15688916C4semidirect
14. C47×#331C4semidirect
154. #331Dic47semidirect
155. C47#7485semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202437872
C2×C47×#3312
#748547