Finite Group Databasebeta

← #26136082

Subgroups — #26136082

Order 12032.

362 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688925C211yesmaximal, characteristic
2C2×#15688923C211yesmaximal, characteristic
3#20244252C211yesmaximal, characteristic
4#20243551C211yesmaximal, characteristic, complemented
5#20241479C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C47×#1380C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C22×C47⋊C16C2211yescharacteristic
10C2×#13093958C2211yescharacteristic
11C2×#13093955—21no—
12C2×#13093944—21no—
13C2×C8×Dic47C2211yescharacteristic
14#15689024—21no—
15#15688988—21no—
16#15688925—22no—
17#15688923—22no—
18C22×C16×C47C2211yescharacteristic
19C2×#107×C47C2211yescharacteristic
20C2×#107×C47C2211yescharacteristic
21C22×D188—21no—
22C2×D376—21no—
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376D412yes—
28C22×C47⋊C8D411yescharacteristic
29C2×C4⋊Dic47—21no—
30C2×(C2×D47)⋊C4—41no—
31C2×(C2×D47)⋊C4—41no—
32C2×C4×Dic47D411yescharacteristic
33C2×C47⋊C16—21no—
34C2×C47⋊C16—21no—
35C2×C47⋊C16—21no—
36#13093958—21no—
37#13093958—21no—
38#13093955—42no—
39#13093944—42no—
40C8×Dic47—21no—
41C8×Dic47—21no—
42C47⋊#95—41no—
43C47⋊#95—41no—
44C22×C8×C47C2311yescharacteristic
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C16×C47—21no—
48C2×C16×C47—21no—
49C2×C16×C47—21no—
50#107×C47—21no—
51#107×C47—21no—
52#107×C47—21no—
53#107×C47—21no—
54C23×D47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47C2×D411yescharacteristic
58C22×Dic47—41no—
59C22×Dic47D811yescharacteristic
60C22×Dic47D811yescharacteristic
61C2×D188—22no—
62C2×D188—41no—
63C2×D188—41no—
64C2×D188—41no—
65C2×D188—41no—
66C2×D188—41no—
67C2×C8×C47C2×D411yescharacteristic
68C2×C8×C47#4111yescharacteristic
69C2×C8×C47#4111yescharacteristic
70C2×C8×C47#4111yescharacteristic
71C2×C8×C47#4111yescharacteristic
72C2×C8×C47#4111yescharacteristic
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77(C2×D47)⋊C4—82no—
78(C2×D47)⋊C4—82no—
79C4⋊Dic47—41no—
80C4⋊Dic47—41no—
81C4×Dic47—21no—
82C4×Dic47—21no—
83C2×C47⋊C8—21no—
84C2×C47⋊C8—21no—
85C2×C47⋊C8SD1612yes—
86D376—22no—
87D376—22no—
88D376—41no—
89D376—41no—
90C16×C47—41no—
91C47⋊C16—41no—
92C23×C47C2×D811yescharacteristic
93C22×D47—42no—
94C22×D47—81no—
95C22×D47—81no—
96C22×D47—81no—
97C22×D47—81no—
98C22×D47—81no—
99C22×D47—81no—
100C2×C4×C47—81no—
101C2×C4×C47—81no—
102C2×C4×C47—81no—
103C2×C4×C47—81no—
104C2×C4×C47—81no—
105C2×C4×C47—81no—
106C2×C4×C47#12111yescharacteristic
107C2×C4×C47#12311yescharacteristic
108C2×C4×C47#12411yescharacteristic
109C2×C4×C47C2×SD1611yescharacteristic
110C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C47#13711yescharacteristic
112C2×Dic47—21no—
113C2×Dic47—21no—
114C2×Dic47—21no—
115C2×Dic47—21no—
116C2×Dic47—21no—
117C2×Dic47—21no—
118C2×Dic47—81no—
119C2×Dic47—81no—
120C2×Dic47—81no—
121D188—42no—
122D188—42no—
123D188—81no—
124D188—81no—
125C8×C47#12311yescharacteristic
126C8×C47#12311yescharacteristic
127C8×C47#12611yescharacteristic
128C8×C47#12611yescharacteristic
129C47⋊C8—22no—
130#13482—471nomaximal, Sylow
131C22×C47#45911yescharacteristic
132C22×C47#48011yescharacteristic
133C22×C47#48611yescharacteristic
134C22×C47#50811yescharacteristic
135C22×C47#50811yescharacteristic
136C22×C47#50911yescharacteristic
137C22×C47#50911yescharacteristic
138C2×D47—82no—
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—161no—
142C2×D47—161no—
143C2×D47—161no—
144C2×D47—161no—
145C4×C47—161no—
146C4×C47—161no—
147C4×C47#39711yescharacteristic
148C4×C47#46111yescharacteristic
149C4×C47#48111yescharacteristic
150C4×C47#48911yescharacteristic
151Dic47—41no—
152Dic47—41no—
153Dic47—161no—
154C2×#357—471no—
155C2×#357—471no—
156#1970—471no—
157#1846—471no—
158#1380—471no—
159#1380D4711yescharacteristic, complemented
160#1374—471no—
161C2×C47#207111yescharacteristic
162C2×C47#221011yescharacteristic
163C2×C47#221011yescharacteristic
164C2×C47#223411yescharacteristic
165C2×C47#223411yescharacteristic
166C2×C47#224211yescharacteristic
167C2×C47#224411yescharacteristic
168D47—162no—
169C22×D8—471no—
170C22×C16—471no—
171C22×C16C2×D4711yescharacteristic
172C2×#107—471no—
173C2×#107C2×D4711yescharacteristic
174C2×#107C2×D4711yescharacteristic
175C2×#106—941no—
176C2×#102—941no—
177C2×C4×C8—471no—
178#390—941no—
179#382—941no—
180#357—942no—
181#357—942no—
182C47#1348211yescharacteristic, complemented, Sylow
183C22×D4—941no—
184C2×D8—472no—
185C2×D8—941no—
186C2×D8—941no—
187C2×D8—941no—
188C2×D8—941no—
189C2×D8—941no—
190C22×C8—471no—
191C22×C8C22×D4711yescharacteristic
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—941no—
195C2×C22⋊C4—1881no—
196C2×C22⋊C4—1881no—
197C2×C42—471no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—941no—
202C2×C16—941no—
203C2×C16—941no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—941no—
209#107—941no—
210#106—1882no—
211#102—1882no—
212C4×C8—941no—
213C4×C8—941no—
214#95—1881no—
215#95—1881no—
216C24—1881no—
217C2×D4—942no—
218C2×D4—1881no—
219C2×D4—1881no—
220C2×D4—1881no—
221C2×D4—1881no—
222C2×D4—1881no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—471no—
226C22×C4—471no—
227C22×C4—1881no—
228C22×C4D4×D4711yescharacteristic
229D8—942no—
230D8—942no—
231D8—1881no—
232D8—1881no—
233C2×C8—472no—
234C2×C8—941no—
235C2×C8—941no—
236C2×C8#1062535011yescharacteristic
237C2×C8#1062535011yescharacteristic
238C2×C8#1062535011yescharacteristic
239C2×C8D4×D4711yescharacteristic
240C2×C8#1062535211yescharacteristic
241C2×C8#1062535411yescharacteristic
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—1881no—
247C4⋊C4—1881no—
248C22⋊C4—3762no—
249C22⋊C4—3762no—
250C42—941no—
251C42—941no—
252C16—41no—
253C16—1881no—
254C23—3761no—
255C23—3761no—
256C23—3761no—
257C23—3761no—
258C23—3761no—
259C23—3761no—
260C23—1882no—
261C23D8×D4711yescharacteristic
262D4—3761no—
263D4—3761no—
264D4—1882no—
265D4—1882no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—941no—
273C2×C4—941no—
274C2×C4—941no—
275C2×C4—941no—
276C2×C4—941no—
277C2×C4—941no—
278C2×C4—3761no—
279C2×C4—3761no—
280C2×C4—3761no—
281C2×C4#1309399711yescharacteristic
282C2×C4#1309400911yescharacteristic
283C2×C4#1309401011yescharacteristic
284C2×C4SD16×D4711yescharacteristic
285C2×C4#1309405111yescharacteristic
286C2×C4#1309405611yescharacteristic
287C8—942no—
288C8#1309400611yescharacteristic
289C8#1309400911yescharacteristic
290C8#1309401811yescharacteristic
291C8#1309401911yescharacteristic
292C22—3762no—
293C22—3762no—
294C22—3762no—
295C22—7521no—
296C22—7521no—
297C22—7521no—
298C22—7521no—
299C22#1568931711yescharacteristic
300C22#1568943811yescharacteristic
301C22#1568946811yescharacteristic
302C22#1568955411yescharacteristic
303C22#1568955611yescharacteristic
304C22#1568955711yescharacteristic
305C22#1568956011yescharacteristic
306C4—161no—
307C4—161no—
308C4—1881no—
309C4—1881no—
310C4—7521no—
311C4#1568905211yescharacteristic
312C4#1568932711yescharacteristic
313C4#1568944311yescharacteristic
314C4#1568948411yescharacteristic
315C2—7522no—
316C2#2024479511yescharacteristic
317C2#2024544211yescharacteristic
318C2#2024544611yescharacteristic
319C2#2024558111yescharacteristic
320C2#2024558211yescharacteristic
321C2#2024562111yescharacteristic
322C2#2024562811yescharacteristic
subgroupG/H
centerC23D8×D47
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#1380C2
socleC23×C47C2×D8
prime radicalC22×D376C22
hypercenter#1380D47
hypocenterC47#13482
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
4. #20243551C2semidirect
5. #20241479C2semidirect
6. #20241465C2semidirect
7. C47×#1380C2semidirect
159. #1380D47semidirect
182. C47#13482semidirect

Maximal subgroups

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

subgroupindex
C2×#156889252
C2×#156889232
#202442522
#202435512
#202414792
#202414652
C47×#13802
#1348247