Finite Group Databasebeta

← #26109593

Subgroups — #26109593

Order 12032.

384 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 107 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#20252363C211yesmaximal, characteristic
2#20242151C211yesmaximal, characteristic, complemented
3#20241911C212yesmaximal
4#20241244C212yesmaximal, complemented
5C4×#106×C47C211yesmaximal, characteristic, complemented
6#15689657C2211yescharacteristic
7#15689253C411yescharacteristic, complemented
8C8×Dic94C411yescharacteristic, complemented
9#15689202C411yescharacteristic, complemented
10#15689194—21no—
11C8×D188C411yescharacteristic, complemented
12C8×C22⋊D47—21no—
13#15689182—21no—
14#15689180C2212yes—
15#15689180C2211yescharacteristic
16#15689077—21no—
17#15688876—21no—
18#15688874—21no—
19#15688859—22no—
20#15688856—22no—
21C2×#106×C47—21no—
22C2×C4×C8×C47C2211yescharacteristic
23C4×C4⋊C4×C47C2212yes—
24C47×#340—21no—
25C2×#10625324—21no—
26C2×C8×D47—21no—
27#13094018—21no—
28#13094016Q811yescharacteristic
29#13094012Q811yescharacteristic
30#13094007—21no—
31#13094003D411yescharacteristic
32#13093998D411yescharacteristic
33C4×Dic94C2×C411yescharacteristic
34C4×D188C2×C411yescharacteristic
35C4×C22⋊D47—21no—
36#13093981—21no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)C2×C411yescharacteristic
39C4⋊(C47⋊C8)C2×C411yescharacteristic
40Dic47⋊C8—21no—
41Dic47⋊C8—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—22no—
44(C2×D47)⋊C8—21no—
45(C2×D47)⋊C8—21no—
46#13093935—21no—
47#13093935—22no—
48#13093934—21no—
49C8×Dic47—21no—
50C22×C8×C47D411yescharacteristic
51C22×C8×C47D411yescharacteristic
52C2×C4⋊C4×C47—22no—
53C2×C42×C47—22no—
54C2×C42×C47C2311yescharacteristic
55#106×C47—24no—
56C4×C8×C47C2×C411yescharacteristic
57C4×C8×C47C2×C411yescharacteristic
58C4×C8×C47D411yescharacteristic
59C4×C8×C47D411yescharacteristic
60#95×C47—22no—
61C22×C4×C47—42no—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4111yescharacteristic
65C2×C22⋊D47—21no—
66C2×D188C4⋊C411yescharacteristic
67C2×C4×D47—21no—
68C2×Dic94C4⋊C411yescharacteristic
69C2×C8×C47—22no—
70C2×C8×C47—41no—
71C2×C8×C47C22⋊C411yescharacteristic
72C2×C8×C47C22⋊C411yescharacteristic
73C2×C8×C47C22⋊C411yescharacteristic
74C2×C8×C47C22⋊C411yescharacteristic
75C4⋊C4×C47—24no—
76C4⋊C4×C47—42no—
77C42×C47—82no—
78C42×C47C22×C411yescharacteristic
79C42×C47C2×D411yescharacteristic
80C42×C47C2×Q811yescharacteristic
81C42×C47#4111yescharacteristic
82C22⋊Dic47—21no—
83(C2×D47)⋊C4—21no—
84(C2×D47)⋊C4—21no—
85(C2×D47)⋊C4—21no—
86(C2×D47)⋊C4—21no—
87C4⋊Dic47C4⋊C411yescharacteristic
88C4⋊Dic47C4⋊C411yescharacteristic
89Dic47⋊C4—21no—
90Dic47⋊C4—21no—
91Dic47⋊C4—21no—
92Dic47⋊C4—21no—
93C4×Dic47—21no—
94C2×C47⋊C8—21no—
95C2×C47⋊C8—21no—
96C2×C47⋊C8—42no—
97#10625324—42no—
98C8×D47—42no—
99C23×C47#12111yescharacteristic
100C22×D47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—41no—
105C2×C4×C47—44no—
106C2×C4×C47—82no—
107C2×C4×C47—82no—
108C2×C4×C47C2×C22⋊C411yescharacteristic
109C2×C4×C47C2×C4⋊C411yescharacteristic
110C2×C4×C47#11711yescharacteristic
111C2×C4×C47#12111yescharacteristic
112C2×C4×C47C22⋊Q811yescharacteristic
113C2×C4×C47C22⋊Q811yescharacteristic
114C22⋊D47—42no—
115C2×Dic47—21no—
116C2×Dic47—21no—
117C2×Dic47—21no—
118D188—41no—
119C4×D47—41no—
120C4×D47—41no—
121Dic94—41no—
122C8×C47—21no—
123C8×C47—21no—
124C8×C47—21no—
125C8×C47#10412yes—
126C47⋊C8—41no—
127C47⋊C8—41no—
128C47⋊C8—82no—
129#8696—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47#38011yescharacteristic
133C22×C47#47112yes—
134C2×D47—41no—
135C2×D47—41no—
136C2×D47—41no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—84no—
143C4×C47C2×#10411yescharacteristic
144C4×C47#42111yescharacteristic
145Dic47—41no—
146Dic47—41no—
147Dic47—41no—
148#2966—471no—
149C4×#106D4711yescharacteristic, complemented
150#1568—471no—
151#1509—472no—
152#1285—472no—
153C2×C47—22no—
154C2×C47#180611yescharacteristic
155C2×C47#181211yescharacteristic
156C2×C47#181511yescharacteristic
157D47—81no—
158#529—471no—
159#446—471no—
160C8×Q8—471no—
161#436—471no—
162#435—941no—
163C8×D4—471no—
164C8×D4—941no—
165#433—941no—
166#432—471no—
167#432—472no—
168C2×#106—21no—
169#405—941no—
170C2×C4×C8C2×D4711yescharacteristic
171C4×C4⋊C4C2×D4712yes—
172#340—21no—
173#335—941no—
174#334—941no—
175#328—942no—
176#327—942no—
177C47#869611yescharacteristic, complemented, Sylow
178C2×M16—941no—
179C22×C8—941no—
180C22×C8C22⋊D4711yescharacteristic
181C22×C8C22⋊D4711yescharacteristic
182#126—941no—
183#125—471no—
184#124—471no—
185#123—941no—
186C22⋊Q8—471no—
187#121—471no—
188C4×Q8—471no—
189C4×D4—471no—
190C4×D4—941no—
191#117—941no—
192C2×C4⋊C4—22no—
193C2×C42—22no—
194C2×C42C22×D4711yescharacteristic
195#106—24no—
196C4⋊C8—471no—
197C4⋊C8—471no—
198C4⋊C8—941no—
199C4⋊C8—941no—
200C4⋊C8—942no—
201C22⋊C8—941no—
202C22⋊C8—941no—
203C22⋊C8—941no—
204C22⋊C8—942no—
205#97—941no—
206#97—941no—
207#97—942no—
208C4×C8—941no—
209C4×C8C4×D4711yescharacteristic
210C4×C8C4×D4711yescharacteristic
211C4×C8D18811yescharacteristic
212C4×C8D18811yescharacteristic
213#95—22no—
214C2×Q8—471no—
215C2×D4—471no—
216C2×D4—941no—
217C22×C4—42no—
218C22×C4—941no—
219C22×C4D4×D4711yescharacteristic
220C22×C4#1062535211yescharacteristic
221C22×C4C2×C22⋊D4711yescharacteristic
222M16—1882no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—941no—
226C2×C8—941no—
227C2×C8—1882no—
228C2×C8—1882no—
229C2×C8(C2×D47)⋊C411yescharacteristic
230C2×C8(C2×D47)⋊C411yescharacteristic
231C2×C8(C2×D47)⋊C411yescharacteristic
232C2×C8(C2×D47)⋊C411yescharacteristic
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—471no—
236C4⋊C4—471no—
237C4⋊C4—941no—
238C4⋊C4—941no—
239C4⋊C4—941no—
240C4⋊C4—941no—
241C22⋊C4—941no—
242C22⋊C4—941no—
243C22⋊C4—941no—
244C22⋊C4—941no—
245C22⋊C4—941no—
246C42—82no—
247C42—941no—
248C42C2×C4×D4711yescharacteristic
249C42C2×D18811yescharacteristic
250C42Q8×D4711yescharacteristic
251C42#1062535411yescharacteristic
252C23—941no—
253C23#1309399511yescharacteristic
254Q8—1881no—
255D4—1881no—
256D4—1882no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—941no—
265C2×C4—941no—
266C2×C4—941no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4C2×(C2×D47)⋊C411yescharacteristic
270C2×C4C4⋊C4×D4711yescharacteristic
271C2×C4#1309398211yescharacteristic
272C2×C4#1309399911yescharacteristic
273C2×C4#1309400011yescharacteristic
274C2×C4#1309400411yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—3762no—
279C8—1881no—
280C8—1881no—
281C8#1309395212yes—
282C22—21no—
283C22—21no—
284C22—1881no—
285C22—1881no—
286C22—1881no—
287C22#1568898011yescharacteristic
288C22#1568938312yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—84no—
295C4—1881no—
296C4—1881no—
297C4—1881no—
298C4C2×#1309395211yescharacteristic
299C4#1568914111yescharacteristic
300C2—22no—
301C2—3761no—
302C2#2024340911yescharacteristic
303C2#2024343111yescharacteristic
304C2#2024344811yescharacteristic
subgroupG/H
centerC22#15688980
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#106×C47C2
socleC22×C47#380
prime radical#13093998D4
cosocleC2×C42×C47C23
hypercenterC4×#106D47
hypocenterC47#8696
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #20242151C2semidirect
4. #20241244C2semidirect
5. C4×#106×C47C2semidirect
7. #15689253C4semidirect
8. C8×Dic94C4semidirect
9. #15689202C4semidirect
11. C8×D188C4semidirect
149. C4×#106D47semidirect
177. C47#8696semidirect

Maximal subgroups

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

subgroupindex
#202523632
#202421512
#202419112
#202412442
C4×#106×C472
#869647