Finite Group Databasebeta

← #26109588

Subgroups — #26109588

Order 12032.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252363C211yesmaximal, characteristic
2#20242151C211yesmaximal, characteristic, complemented
3#20241911C212yesmaximal
4#20241244C212yesmaximal, complemented
5C4×#107×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×#107×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#107×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—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—41no—
106C2×C4×C47—44no—
107C2×C4×C47—82no—
108C2×C4×C47—82no—
109C2×C4×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47C2×C4⋊C411yescharacteristic
111C2×C4×C47#11711yescharacteristic
112C2×C4×C47#12111yescharacteristic
113C2×C4×C47C22⋊Q811yescharacteristic
114C2×C4×C47C22⋊Q811yescharacteristic
115C22⋊D47—41no—
116C22⋊D47—41no—
117C2×Dic47—21no—
118C2×Dic47—21no—
119C2×Dic47—21no—
120D188—41no—
121C4×D47—41no—
122C4×D47—41no—
123Dic94—41no—
124C8×C47—21no—
125C8×C47—21no—
126C8×C47—21no—
127C8×C47#10412yes—
128C47⋊C8—41no—
129C47⋊C8—41no—
130C47⋊C8—82no—
131#8695—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47#38011yescharacteristic
135C22×C47#47211yescharacteristic
136C22×C47#47311yescharacteristic
137C2×D47—41no—
138C2×D47—41no—
139C2×D47—41no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—84no—
146C4×C47C2×#10411yescharacteristic
147C4×C47#42111yescharacteristic
148Dic47—41no—
149Dic47—41no—
150Dic47—41no—
151#2966—471no—
152C4×#107D4711yescharacteristic, complemented
153#1568—471no—
154#1509—472no—
155#1285—472no—
156C2×C47—21no—
157C2×C47—21no—
158C2×C47#180611yescharacteristic
159C2×C47#181211yescharacteristic
160C2×C47#181611yescharacteristic
161D47—81no—
162#529—471no—
163#446—471no—
164C8×Q8—471no—
165#436—471no—
166#435—941no—
167C8×D4—471no—
168C8×D4—941no—
169#433—941no—
170#432—471no—
171#432—472no—
172C2×#107—21no—
173#405—941no—
174C2×C4×C8C2×D4711yescharacteristic
175C4×C4⋊C4C2×D4712yes—
176#340—21no—
177#335—941no—
178#334—941no—
179#328—942no—
180#327—942no—
181C47#869511yescharacteristic, complemented, Sylow
182C2×M16—941no—
183C22×C8—941no—
184C22×C8C22⋊D4711yescharacteristic
185C22×C8C22⋊D4711yescharacteristic
186#126—941no—
187#125—471no—
188#124—471no—
189#123—941no—
190C22⋊Q8—471no—
191#121—471no—
192C4×Q8—471no—
193C4×D4—471no—
194C4×D4—941no—
195#117—941no—
196C2×C4⋊C4—22no—
197C2×C42—22no—
198C2×C42C22×D4711yescharacteristic
199#107—24no—
200C4⋊C8—471no—
201C4⋊C8—471no—
202C4⋊C8—941no—
203C4⋊C8—941no—
204C4⋊C8—942no—
205C22⋊C8—941no—
206C22⋊C8—941no—
207C22⋊C8—941no—
208C22⋊C8—942no—
209#97—941no—
210#97—941no—
211#97—942no—
212C4×C8—941no—
213C4×C8C4×D4711yescharacteristic
214C4×C8C4×D4711yescharacteristic
215C4×C8D18811yescharacteristic
216C4×C8D18811yescharacteristic
217#95—22no—
218C2×Q8—471no—
219C2×D4—471no—
220C2×D4—941no—
221C22×C4—42no—
222C22×C4—941no—
223C22×C4D4×D4711yescharacteristic
224C22×C4#1062535211yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226M16—1882no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—941no—
230C2×C8—941no—
231C2×C8—1882no—
232C2×C8—1882no—
233C2×C8(C2×D47)⋊C411yescharacteristic
234C2×C8(C2×D47)⋊C411yescharacteristic
235C2×C8(C2×D47)⋊C411yescharacteristic
236C2×C8(C2×D47)⋊C411yescharacteristic
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—471no—
240C4⋊C4—471no—
241C4⋊C4—941no—
242C4⋊C4—941no—
243C4⋊C4—941no—
244C4⋊C4—941no—
245C22⋊C4—941no—
246C22⋊C4—941no—
247C22⋊C4—941no—
248C22⋊C4—941no—
249C22⋊C4—941no—
250C42—82no—
251C42—941no—
252C42C2×C4×D4711yescharacteristic
253C42C2×D18811yescharacteristic
254C42Q8×D4711yescharacteristic
255C42#1062535411yescharacteristic
256C23—941no—
257C23#1309399511yescharacteristic
258Q8—1881no—
259D4—1881no—
260D4—1881no—
261D4—1881no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—941no—
271C2×C4—941no—
272C2×C4—941no—
273C2×C4—1881no—
274C2×C4—1881no—
275C2×C4C2×(C2×D47)⋊C411yescharacteristic
276C2×C4C4⋊C4×D4711yescharacteristic
277C2×C4#1309398211yescharacteristic
278C2×C4#1309399911yescharacteristic
279C2×C4#1309400011yescharacteristic
280C2×C4#1309400411yescharacteristic
281C8—21no—
282C8—21no—
283C8—21no—
284C8—3762no—
285C8—1881no—
286C8—1881no—
287C8#1309395212yes—
288C22—21no—
289C22—21no—
290C22—1881no—
291C22—1881no—
292C22—1881no—
293C22#1568898011yescharacteristic
294C22#1568939011yescharacteristic
295C22#1568939511yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—84no—
302C4—1881no—
303C4—1881no—
304C4—1881no—
305C4C2×#1309395211yescharacteristic
306C4#1568914111yescharacteristic
307C2—21no—
308C2—21no—
309C2—3761no—
310C2#2024340911yescharacteristic
311C2#2024343111yescharacteristic
312C2#2024345311yescharacteristic
subgroupG/H
centerC22#15688980
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#107×C47C2
socleC22×C47#380
prime radical#13093998D4
cosocleC2×C42×C47C23
hypercenterC4×#107D47
hypocenterC47#8695
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
2. #20242151C2semidirect
4. #20241244C2semidirect
5. C4×#107×C47C2semidirect
7. #15689253C4semidirect
8. C8×Dic94C4semidirect
9. #15689202C4semidirect
11. C8×D188C4semidirect
152. C4×#107D47semidirect
181. C47#8695semidirect

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×#107×C472
#869547