Finite Group Databasebeta

← #26109458

Subgroups — #26109458

Order 12032.

432 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252354C211yesmaximal, characteristic
2#20243409C212yesmaximal
3#20242147C211yesmaximal, characteristic, complemented
4#20241291C212yesmaximal, complemented
5C4×#106×C47C211yesmaximal, characteristic, complemented
6C4×#10625350C2211yescharacteristic
7#15689253C411yescharacteristic, complemented
8#15689253C411yescharacteristic, complemented
9#15689202C411yescharacteristic, complemented
10#15689202C411yescharacteristic, complemented
11#15689194—21no—
12#15689194—21no—
13#15689176C2211yescharacteristic
14#15689174—21no—
15#15689147C2212yes—
16C2×#13093952—22no—
17C4×#10625324—21no—
18#15688883—22no—
19#15688876—21no—
20#15688876—21no—
21C2×#106×C47—21no—
22C2×C4×C8×C47C2211yescharacteristic
23C4×C4⋊C4×C47C2212yes—
24C47×#340—21no—
25C2×#10625350D411yescharacteristic
26C2×#10625329—22no—
27C2×#10625324—21no—
28C2×#10625324—21no—
29C4×Dic94C2×C411yescharacteristic
30C4×Dic94Q811yescharacteristic
31C4×D188C2×C411yescharacteristic
32C4×D188Q811yescharacteristic
33C4×C22⋊D47—21no—
34C4×C22⋊D47—21no—
35#13093983D411yescharacteristic
36#13093981—21no—
37C42×D47—21no—
38C4⋊(C47⋊C8)—22no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)C2×C411yescharacteristic
41C4⋊(C47⋊C8)C2×C411yescharacteristic
42Dic47⋊C8—21no—
43Dic47⋊C8—21no—
44#13093952—44no—
45C22⋊(C47⋊C8)—21no—
46(C2×D47)⋊C8—21no—
47(C2×D47)⋊C8—21no—
48#13093934—21no—
49#13093934—21no—
50C4×C47⋊C8—21no—
51C22×C8×C47D411yescharacteristic
52C22×C8×C47D411yescharacteristic
53C2×C4⋊C4×C47—22no—
54C2×C42×C47—22no—
55C2×C42×C47C2311yescharacteristic
56#106×C47—24no—
57C4×C8×C47C2×C411yescharacteristic
58C4×C8×C47C2×C411yescharacteristic
59C4×C8×C47D411yescharacteristic
60C4×C8×C47D411yescharacteristic
61#95×C47—22no—
62C22×C4×C47—42no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C2×C22⋊D47—21no—
67#10625350—22no—
68#10625350—22no—
69C2×D188C4⋊C411yescharacteristic
70C2×C4×D47—21no—
71C2×C4×D47—21no—
72C2×C4×D47—21no—
73C2×Dic94C4⋊C411yescharacteristic
74C2×C8×C47—22no—
75C2×C8×C47—41no—
76C2×C8×C47C22⋊C411yescharacteristic
77C2×C8×C47C22⋊C411yescharacteristic
78C2×C8×C47C22⋊C411yescharacteristic
79C2×C8×C47C22⋊C411yescharacteristic
80C4⋊C4×C47—24no—
81C4⋊C4×C47—42no—
82C42×C47—42no—
83C42×C47—42no—
84C42×C47C22×C411yescharacteristic
85C42×C47C2×D411yescharacteristic
86C42×C47C2×Q811yescharacteristic
87C42×C47#4111yescharacteristic
88C22⋊Dic47—21no—
89(C2×D47)⋊C4—21no—
90(C2×D47)⋊C4—21no—
91C4⋊Dic47C4⋊C411yescharacteristic
92C4⋊Dic47C4⋊C411yescharacteristic
93Dic47⋊C4—21no—
94Dic47⋊C4—21no—
95C4×Dic47—21no—
96C4×Dic47—21no—
97C4×Dic47—21no—
98#10625329—42no—
99#10625329—42no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—42no—
103#10625324—42no—
104#10625324—42no—
105C23×C47#12111yescharacteristic
106C22×D47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—44no—
111C2×C4×C47—82no—
112C2×C4×C47—82no—
113C2×C4×C47C2×C22⋊C411yescharacteristic
114C2×C4×C47C2×C4⋊C411yescharacteristic
115C2×C4×C47#11711yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47C22⋊Q811yescharacteristic
118C2×C4×C47C22⋊Q811yescharacteristic
119C2×C4×C47AGL(1,ℤ/8)12yes—
120C2×C4×C47#13712yes—
121C22⋊D47—42no—
122C2×Dic47—21no—
123C2×Dic47—21no—
124C2×Dic47—21no—
125D188—22no—
126C4×D47—42no—
127C4×D47—41no—
128C4×D47—41no—
129C4×D47—41no—
130C4×D47—41no—
131Dic94—22no—
132C8×C47—21no—
133C8×C47—21no—
134C8×C47—21no—
135C8×C47#10412yescomplemented
136C47⋊C8—41no—
137C47⋊C8—41no—
138C47⋊C8—82no—
139#8669—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#38011yescharacteristic
143C22×C47#46812yes—
144C2×D47—41no—
145C2×D47—41no—
146C2×D47—41no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—84no—
151C4×C47C2×#10411yescharacteristic
152C4×C47#42111yescharacteristic
153C4×C47#42812yes—
154C4×C47#46911yescharacteristic
155C4×C47#47011yescharacteristic
156Dic47—41no—
157Dic47—41no—
158Dic47—41no—
159#2965—471no—
160#1806—472no—
161C4×#106D4711yescharacteristic, complemented
162#1567—471no—
163#1303—472no—
164C2×C47—22no—
165C2×C47#180611yescharacteristic
166C2×C47#180911yescharacteristic
167C2×C47#181411yescharacteristic
168D47—81no—
169C4×#41—471no—
170#446—471no—
171#446—471no—
172#436—471no—
173#436—471no—
174#435—941no—
175#435—941no—
176#431—471no—
177#431—941no—
178C2×#106—21no—
179#423—472no—
180C2×#104—942no—
181C4×M16—941no—
182C2×C4×C8C2×D4711yescharacteristic
183C4×C4⋊C4C2×D4712yes—
184#340—21no—
185#339—942no—
186#335—941no—
187#335—941no—
188C47#866911yescharacteristic, complemented, Sylow
189C2×#41—471no—
190C2×M16—941no—
191C2×M16—941no—
192C2×M16—942no—
193C22×C8C22⋊D4711yescharacteristic
194C22×C8C22⋊D4711yescharacteristic
195C4×Q8—471no—
196C4×Q8—471no—
197C4×D4—471no—
198C4×D4—471no—
199C4×D4—941no—
200C4×D4—941no—
201#117—471no—
202#117—941no—
203C2×C4⋊C4—22no—
204C2×C42—22no—
205C2×C42—941no—
206C2×C42C22×D4711yescharacteristic
207#106—24no—
208C4⋊C8—471no—
209C4⋊C8—471no—
210C4⋊C8—941no—
211C4⋊C8—941no—
212C4⋊C8—942no—
213C4⋊C8—942no—
214#104—1884no—
215C22⋊C8—941no—
216C22⋊C8—941no—
217C22⋊C8—941no—
218#97—941no—
219#97—941no—
220C4×C8—941no—
221C4×C8C4×D4711yescharacteristic
222C4×C8C4×D4711yescharacteristic
223C4×C8D18811yescharacteristic
224C4×C8D18811yescharacteristic
225#95—22no—
226#41—942no—
227#41—942no—
228C2×Q8—471no—
229C2×D4—471no—
230C2×D4—941no—
231C22×C4—42no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4D4×D4711yescharacteristic
236C22×C4#1062535211yescharacteristic
237C22×C4C2×C22⋊D4711yescharacteristic
238M16—1882no—
239M16—1882no—
240M16—1882no—
241M16—1882no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—941no—
245C2×C8—941no—
246C2×C8—1882no—
247C2×C8(C2×D47)⋊C411yescharacteristic
248C2×C8(C2×D47)⋊C411yescharacteristic
249C2×C8(C2×D47)⋊C411yescharacteristic
250C2×C8(C2×D47)⋊C411yescharacteristic
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—471no—
254C4⋊C4—471no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C22⋊C4—941no—
258C22⋊C4—941no—
259C22⋊C4—941no—
260C42—42no—
261C42—42no—
262C42—941no—
263C42—941no—
264C42—941no—
265C42C2×C4×D4711yescharacteristic
266C42C2×D18811yescharacteristic
267C42Q8×D4711yescharacteristic
268C42#1062535411yescharacteristic
269C23—941no—
270C23#1309399511yescharacteristic
271Q8—942no—
272D4—942no—
273D4—1882no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—941no—
281C2×C4—941no—
282C2×C4—941no—
283C2×C4—1882no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4C2×(C2×D47)⋊C411yescharacteristic
289C2×C4C4⋊C4×D4711yescharacteristic
290C2×C4#1309398211yescharacteristic
291C2×C4#1309399911yescharacteristic
292C2×C4#1309400011yescharacteristic
293C2×C4#1309400411yescharacteristic
294C2×C4#1309405112yes—
295C2×C4#1309405312yes—
296C8—21no—
297C8—21no—
298C8—21no—
299C8—3762no—
300C8—1881no—
301C8—1881no—
302C8#1309395212yescomplemented
303C22—21no—
304C22—21no—
305C22—1881no—
306C22—1881no—
307C22—1881no—
308C22#1568898011yescharacteristic
309C22#1568936612yes—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—84no—
314C4—1881no—
315C4—1881no—
316C4—1881no—
317C4C2×#1309395211yescharacteristic
318C4#1568914111yescharacteristic
319C4#1568916212yes—
320C4#1568937711yescharacteristic
321C4#1568938211yescharacteristic
322C2—22no—
323C2—3761no—
324C2#2024340911yescharacteristic
325C2#2024341811yescharacteristic
326C2#2024343911yescharacteristic
subgroupG/H
centerC2×C4#13093999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#106×C47C2
socleC22×C47#380
prime radicalC2×#10625350D4
cosocleC2×C42×C47C23
hypercenterC4×#106D47
hypocenterC47#8669
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
3. #20242147C2semidirect
4. #20241291C2semidirect
5. C4×#106×C47C2semidirect
7. #15689253C4semidirect
8. #15689253C4semidirect
9. #15689202C4semidirect
10. #15689202C4semidirect
135. C8×C47#104semidirect
161. C4×#106D47semidirect
188. C47#8669semidirect
302. C8#13093952semidirect

Maximal subgroups

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

subgroupindex
#202523542
#202434092
#202421472
#202412912
C4×#106×C472
#866947