Finite Group Databasebeta

← #26134505

Subgroups — #26134505

Order 12032.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689508C211yesmaximal, characteristic
2C2×#15688927C211yesmaximal, characteristic, complemented
3C2×#15688925C211yesmaximal, characteristic
4C2×#15688923C211yesmaximal, characteristic
5C2×#15688898C211yesmaximal, characteristic, complemented
6#20243546C211yesmaximal, characteristic, complemented
7C2×C47×#358C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C22×D376C2211yescharacteristic
10C2×#13094013—21no—
11C22×C47⋊C16C2211yescharacteristic
12#15689508—22no—
13#15689508C414yescomplemented
14C2×#13093958C2211yescharacteristic
15C2×#13093957—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15688992—21no—
18#15688927—22no—
19#15688925—22no—
20#15688923—22no—
21#15688898—22no—
22C22×Q16×C47C2211yescharacteristic
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C47×#358—22no—
26C22×D188—21no—
27C2×#10625336—24no—
28C2×#10625336—42no—
29C2×#10625336—42no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376C2×C412yes—
36C22×C47⋊C8D411yescharacteristic
37#13094013—24no—
38#13094013—42no—
39C2×C4⋊Dic47—21no—
40C2×Dic47⋊C4—42no—
41C2×(C2×D47)⋊C4—42no—
42C2×C4×Dic47D411yescharacteristic
43C2×C47⋊C16—21no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16D412yes—
46#13093958—21no—
47#13093958—21no—
48#13093957—42no—
49C8×Dic47C2×C412yes—
50C8×Dic47D412yes—
51C22×Q8×C47—21no—
52C2×Q16×C47—21no—
53C2×Q16×C47—21no—
54C2×Q16×C47—21no—
55C2×Q16×C47—21no—
56C2×Q16×C47—21no—
57C2×Q16×C47C2×C412yes—
58C22×C8×C47C2311yescharacteristic
59C2×C4⋊C4×C47—21no—
60C2×C16×C47—21no—
61C2×C16×C47—21no—
62C2×C16×C47D412yes—
63#107×C47—21no—
64#107×C47—21no—
65C23×D47—41no—
66C2×Q8×C47—22no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—41no—
76C22×Dic47D812yes—
77C2×D188—22no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—41no—
83Q16×C47—22no—
84Q16×C47—22no—
85Q16×C47—41no—
86Q16×C47—41no—
87C2×C8×C47C22×C411yescharacteristic
88C2×C8×C47C2×D411yescharacteristic
89C2×C8×C47C2×D411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47#4111yescharacteristic
92C2×C8×C47#4111yescharacteristic
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95#10625336—48no—
96(C2×D47)⋊C4—48no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99Dic47⋊C4—84no—
100C4×Dic47C22⋊C412yes—
101C4×Dic47SD1612yes—
102C2×C47⋊C8C22⋊C412yes—
103C2×C47⋊C8D812yes—
104C2×C47⋊C8SD1612yes—
105D376—22no—
106D376—22no—
107D376—41no—
108D376—41no—
109C16×C47—22no—
110C47⋊C16—22no—
111C23×C47C2×D811yescharacteristic
112C22×D47—42no—
113C22×D47—81no—
114C22×D47—81no—
115C22×D47—82no—
116C22×D47—82no—
117Q8×C47—42no—
118Q8×C47—42no—
119Q8×C47—81no—
120Q8×C47—81no—
121C2×C4×C47—42no—
122C2×C4×C47—81no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47—82no—
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47#12411yescharacteristic
128C2×C4×C47#12711yescharacteristic
129C2×C4×C47C2×D811yescharacteristic
130C2×C4×C47C2×SD1611yescharacteristic
131C2×C4×C47C2×SD1611yescharacteristic
132C2×Dic47—22no—
133C2×Dic47—22no—
134C2×Dic47—81no—
135C2×Dic47—82no—
136C2×Dic47#10214yes—
137D188—42no—
138D188—42no—
139D188—81no—
140D188—81no—
141C8×C47C4×D411yescharacteristic
142C8×C47C4×D411yescharacteristic
143C8×C47#12111yescharacteristic
144C8×C47#12111yescharacteristic
145C47⋊C8#10214yes—
146#13205—471nomaximal, Sylow
147C22×C47C2×#10211yescharacteristic
148C22×C47#48611yescharacteristic
149C22×C47#49311yescharacteristic
150C22×C47#50912yes—
151C22×C47#51012yes—
152C2×D47—84no—
153C2×D47—82no—
154C2×D47—162no—
155C2×D47—161no—
156C2×D47—161no—
157C4×C47—82no—
158C4×C47—161no—
159C4×C47#39011yescharacteristic
160C4×C47C2×#10211yescharacteristic
161C4×C47#48611yescharacteristic
162C4×C47#49211yescharacteristic
163Dic47—24no—
164Dic47—161no—
165C2×#495—471no—
166C2×#358—471no—
167C2×#358D4711yescharacteristic, complemented
168C2×#357—471no—
169C2×#357—471no—
170C2×#346—471no—
171#1845—471no—
172C2×C47#195711yescharacteristic
173C2×C47#213912yes—
174C2×C47#224312yes—
175C2×C47#225011yescharacteristic
176C2×C47#225111yescharacteristic
177D47—162no—
178C22×Q16C2×D4711yescharacteristic
179C22×SD16—941no—
180C22×D8—471no—
181C2×#124—941no—
182C22×C16—471no—
183C22×C16C2×D4711yescharacteristic
184#495—474no—
185#495—942no—
186C2×#107—471no—
187C2×#107—941no—
188C2×#107C2×D4711yescharacteristic
189C2×C4×C8—471no—
190#384—941no—
191#358—22no—
192#358—942no—
193#357—942no—
194#357—942no—
195#346—942no—
196C47#1320511yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C22×D4—941no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16C4×D4712yes—
205C2×SD16—944no—
206C2×SD16—1882no—
207C2×SD16—1882no—
208C2×D8—472no—
209C2×D8—941no—
210C2×D8—941no—
211C2×D8—941no—
212C2×D8—941no—
213C2×D8—941no—
214C22×C8—471no—
215C22×C8C22×D4711yescharacteristic
216#124—944no—
217#124—1882no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—941no—
220C2×C4⋊C4—1882no—
221C2×C22⋊C4—1882no—
222C2×C42—471no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—472no—
226C2×C16—941no—
227C2×C16—941no—
228C2×C16C22⋊D4712yes—
229#107—21no—
230#107—21no—
231#107—941no—
232#107—941no—
233#107—1882no—
234C4×C8—472no—
235C4×C8—472no—
236C24—1881no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×D4—942no—
244C2×D4—1881no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4—1881no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—472no—
252C22×C4—1881no—
253C22×C4D4×D4711yescharacteristic
254Q16—22no—
255Q16—22no—
256Q16—41no—
257Q16—41no—
258SD16—1888no—
259D8—942no—
260D8—942no—
261D8—1881no—
262D8—1881no—
263C2×C8—472no—
264C2×C8—472no—
265C2×C8—472no—
266C2×C8C2×C4×D4711yescharacteristic
267C2×C8#1062535011yescharacteristic
268C2×C8D4×D4711yescharacteristic
269C2×C8D4×D4711yescharacteristic
270C2×C8#1062535211yescharacteristic
271C2×C8C2×C22⋊D4711yescharacteristic
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—3764no—
275C4⋊C4—1881no—
276C4⋊C4—1881no—
277C22⋊C4—1888no—
278C42—472no—
279C42—472no—
280C16—22no—
281C16—942no—
282C23—3761no—
283C23—3762no—
284C23—3762no—
285C23—3761no—
286C23—1882no—
287C23D8×D4711yescharacteristic
288Q8—42no—
289Q8—42no—
290Q8—81no—
291Q8—81no—
292D4—3761no—
293D4—3761no—
294D4—1882no—
295D4—1882no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—474no—
302C2×C4—942no—
303C2×C4—942no—
304C2×C4—3761no—
305C2×C4—3762no—
306C2×C4C22⋊C4×D4711yescharacteristic
307C2×C4#1309401011yescharacteristic
308C2×C4#1309402011yescharacteristic
309C2×C4D8×D4711yescharacteristic
310C2×C4SD16×D4711yescharacteristic
311C2×C4SD16×D4711yescharacteristic
312C8—474no—
313C8(C22⋊D47)⋊C411yescharacteristic
314C8C4×C22⋊D4711yescharacteristic
315C8#1309399611yescharacteristic
316C8#1309399711yescharacteristic
317C22—3764no—
318C22—3762no—
319C22—7522no—
320C22—7521no—
321C22—7521no—
322C22#102×D4711yescharacteristic
323C22#1568946811yescharacteristic
324C22#1568949811yescharacteristic
325C22#1568956012yes—
326C22#1568956512yes—
327C4—82no—
328C4—161no—
329C4—944no—
330C4—7521no—
331C4#1568902311yescharacteristic
332C4#102×D4711yescharacteristic
333C4#1568946811yescharacteristic
334C4#1568949411yescharacteristic
335C2—7522no—
336C2#2024419211yescharacteristic
337C2#2024509912yes—
338C2#2024562612yes—
339C2#2024565311yescharacteristic
340C2#2024565611yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#358C2
socleC23×C47C2×D8
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenterC2×#358D47
hypocenterC47#13205
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#15688927C2semidirect
5. C2×#15688898C2semidirect
6. #20243546C2semidirect
7. C2×C47×#358C2semidirect
13. #15689508C4semidirect
167. C2×#358D47semidirect
196. C47#13205semidirect

Maximal subgroups

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

subgroupindex
C2×#156895082
C2×#156889272
C2×#156889252
C2×#156889232
C2×#156888982
#202435462
C2×C47×#3582
#1320547