Finite Group Databasebeta

← #26132283

Subgroups — #26132283

Order 12032.

480 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 101 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#20246101C211yesmaximal, characteristic
2C2×#15688920C211yesmaximal, characteristic
3#20243382C212yesmaximal, complemented
4#20241514C212yesmaximal, complemented
5C2×C47×#355C211yesmaximal, characteristic
6C2×Q8×Dic47C2211yescharacteristic
7C2×D4×Dic47C411yescharacteristic
8#15689146—22no—
9C2×#13093943C2212yes—
10#15689030C2211yescharacteristic
11#15689026C411yescharacteristic
12#15689016—22no—
13#15688998—22no—
14#15688993C411yescharacteristic
15#15688969C411yescharacteristic
16#15688920—22no—
17#15688891—22no—
18C2×#124×C47C2211yescharacteristic
19C2×#101×C47C2212yes—
20C47×#355—22no—
21C23×Dic47—21no—
22C2×#10625329—22no—
23Q8×Dic47—42no—
24D4×Dic47—42no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47D412yes—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—22no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34Dic47⋊C8—44no—
35#13093943—22no—
36#13093943—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C22×Q8×C47C2311yescharacteristic
41C22×D4×C47C2×C411yescharacteristic
42C2×M16×C47—22no—
43#124×C47—22no—
44#124×C47D412yes—
45#124×C47Q812yes—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47C2×C411yescharacteristic
48#101×C47—22no—
49#101×C47—22no—
50C24×C47—21no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47C2×D411yescharacteristic
54C2×Q8×C47C2×Q811yescharacteristic
55C2×Q8×C47#4112yes—
56C2×D4×C47—21no—
57C2×D4×C47—41no—
58C2×D4×C47—41no—
59C2×D4×C47C4⋊C412yes—
60C22×C4×C47—21no—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C22×Dic47—42no—
66C22×Dic47—82no—
67C22×Dic47C22⋊C411yescharacteristic
68C22×Dic47C22⋊C412yes—
69C22×Dic47C22⋊C411yescharacteristic
70M16×C47—42no—
71M16×C47—42no—
72C2×C8×C47—42no—
73C22⋊C4×C47—24no—
74C22⋊C4×C47—24no—
75C42×C47—21no—
76C42×C47C4⋊C412yes—
77C22⋊Dic47—42no—
78C22⋊Dic47—82no—
79C4⋊Dic47—41no—
80C4⋊Dic47—42no—
81C4⋊Dic47—41no—
82C4⋊Dic47—82no—
83C4×Dic47—24no—
84C4×Dic47—22no—
85C4×Dic47—41no—
86#10625329—42no—
87#10625329—42no—
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—42no—
91C23×C47—42no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47C2×C22⋊C411yescharacteristic
95Q8×C47—41no—
96Q8×C47—41no—
97Q8×C47—42no—
98D4×C47—42no—
99D4×C47—42no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47C2×C4⋊C411yescharacteristic
108C2×C4×C47#11711yescharacteristic
109C2×C4×C47#12112yes—
110C2×C4×C47C22⋊Q812yes—
111C2×Dic47—21no—
112C2×Dic47—22no—
113C2×Dic47—41no—
114C2×Dic47—42no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—84no—
122C2×Dic47#9912yes—
123C8×C47—82no—
124C47⋊C8—82no—
125#12803—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—44no—
129C22×C47—42no—
130C22×C47—82no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47#38011yescharacteristic
134C22×C47C2×#9911yescharacteristic
135C22×C47#41011yescharacteristic
136C4×C47—22no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—22no—
140C4×C47—42no—
141Dic47—22no—
142Dic47—42no—
143Dic47—41no—
144Dic47—81no—
145Dic47—81no—
146#2326—471no—
147C2×#355—471no—
148C2×#355D4711yescharacteristic
149#1801—472no—
150#1394—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—82no—
154C2×C47#179911yescharacteristic
155C2×C47#213311yescharacteristic
156C2×C47#213411yescharacteristic
157C2×#124C2×D4711yescharacteristic
158C2×C4×Q8—471no—
159C2×C4×D4—471no—
160#423—942no—
161C2×#101—472no—
162C2×#101C2×D4712yes—
163#391—471no—
164#390—471no—
165#388—942no—
166#385—942no—
167#384—471no—
168#376—471no—
169#355—22no—
170#355—942no—
171#343—942no—
172C47#1280311yescharacteristic, complemented, Sylow
173C22×Q8C22×D4711yescharacteristic
174C22×D4C4×D4711yescharacteristic
175C23×C4—941no—
176C2×M16—22no—
177C2×M16—942no—
178#124—22no—
179#124Dic9412yes—
180#124C22⋊D4712yes—
181C4×Q8—1882no—
182C4×D4—1882no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—472no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—942no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4—941no—
190C2×C42—471no—
191C2×C42—472no—
192C2×C42—471no—
193C2×C42C4×D4711yescharacteristic
194C4⋊C8—1884no—
195#101—22no—
196#101—22no—
197#101—942no—
198#101—942no—
199#95—941no—
200#95—941no—
201#95—942no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×Dic9411yescharacteristic
206C2×Q8#1062535212yes—
207C2×Q8C2×C22⋊D4711yescharacteristic
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4Dic47⋊C412yes—
212C22×C4—21no—
213C22×C4—471no—
214C22×C4—471no—
215C22×C4—472no—
216C22×C4—941no—
217C22×C4—941no—
218C22×C4—3762no—
219C22×C4—1882no—
220C22×C4C2×C4×D4711yescharacteristic
221C22×C4D4×D4712yes—
222M16—42no—
223M16—42no—
224M16—1882no—
225M16—1882no—
226C2×C8—42no—
227C2×C8—1882no—
228C4⋊C4—3762no—
229C4⋊C4—1881no—
230C4⋊C4—1882no—
231C4⋊C4—1881no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—3762no—
235C22⋊C4—1882no—
236C42—21no—
237C42—944no—
238C42—942no—
239C42—1881no—
240C42Dic47⋊C412yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C22⋊C4×D4711yescharacteristic
247Q8—41no—
248Q8—42no—
249Q8—41no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—472no—
260C2×C4—942no—
261C2×C4—941no—
262C2×C4—3764no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4—1882no—
266C2×C4—1881no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4—1882no—
270C2×C4—1881no—
271C2×C4C2×Dic47⋊C411yescharacteristic
272C2×C4#1309398011yescharacteristic
273C2×C4#1309399612yes—
274C2×C4#1309400112yes—
275C8—82no—
276C8—3762no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1568898111yescharacteristic
285C22#99×D4711yescharacteristic
286C22#1568909511yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—942no—
293C4—3761no—
294C4—3761no—
295C4—1881no—
296C4—1882no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2024337411yescharacteristic
301C2#2024507411yescharacteristic
302C2#2024507811yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#355C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC22×Q8×C47C23
hypercenterC2×#355D47
hypocenterC47#12803
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #20243382C2semidirect
4. #20241514C2semidirect
172. C47#12803semidirect

Maximal subgroups

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

subgroupindex
#202461012
C2×#156889202
#202433822
#202415142
C2×C47×#3552
#1280347