Finite Group Databasebeta

← #26114808

Subgroups — #26114808

Order 12032.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244307C211yesmaximal, characteristic
2#20244163C211yesmaximal, characteristic, complemented
3#20242035C211yesmaximal, characteristic
4#20241840C211yesmaximal, characteristic
5#20241290C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C47×#1451C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9C2×#13093959—21no—
10#15689147C2211yescharacteristic
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×#13093950—21no—
13C2×#13093950—21no—
14C4×C4⋊Dic47C2211yescharacteristic
15#15688887—21no—
16#15688868C411yescharacteristic, complemented
17#15688868C411yescharacteristic, complemented
18#15688866C411yescharacteristic, complemented
19#15688866C411yescharacteristic, complemented
20#15688852—21no—
21#15688852—21no—
22C47×#431C2211yescharacteristic
23C4×M16×C47C2211yescharacteristic
24C2×C4×C8×C47C2211yescharacteristic
25C47×#322—21no—
26C47×#322—21no—
27C2×#10625350—21no—
28C2×#10625329—21no—
29C22×C47⋊C8—21no—
30#13094024C2×C411yescharacteristic
31#13094021C2×C411yescharacteristic
32#13094012—21no—
33#13093998—21no—
34#13093998—21no—
35C4×D188—21no—
36C4×D188—21no—
37C2×C4⋊Dic47—21no—
38C2×C4×Dic47—21no—
39#13093959—42no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)C2×C411yescharacteristic
44C4⋊(C47⋊C8)C2×C411yescharacteristic
45#13093950—42no—
46#13093950—42no—
47C47⋊#95—21no—
48C2×M16×C47D411yescharacteristic
49C2×M16×C47D411yescharacteristic
50C22×C8×C47—21no—
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47C2×C411yescharacteristic
53C4⋊C8×C47C2×C411yescharacteristic
54C22⋊C8×C47—21no—
55#97×C47—21no—
56C4×C8×C47—21no—
57C4×C8×C47—21no—
58C4×C8×C47—21no—
59C4×C8×C47D411yescharacteristic
60C4×C8×C47D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47#4111yescharacteristic
63C22×C4×C47#4111yescharacteristic
64C2×C22⋊D47—41no—
65C22×Dic47—41no—
66#10625350—41no—
67#10625350—41no—
68#10625350—41no—
69#10625350—41no—
70C2×D188—21no—
71C2×D188—21no—
72C2×D188—21no—
73C2×C4×D47—41no—
74C2×Dic94—21no—
75M16×C47—21no—
76M16×C47—21no—
77M16×C47—21no—
78M16×C47—21no—
79C2×C8×C47—21no—
80C2×C8×C47—21no—
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—42no—
86C42×C47C22×C411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47#4111yescharacteristic
89C42×C47#4111yescharacteristic
90(C2×D47)⋊C4—41no—
91(C2×D47)⋊C4—41no—
92C4⋊Dic47—22no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—41no—
96C4×Dic47—41no—
97C4×Dic47—41no—
98#10625329—41no—
99#10625329—41no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C23×C47#12411yescharacteristic
106C22×D47—41no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47#11711yescharacteristic
111C2×C4×C47C4×D411yescharacteristic
112C2×C4×C47C4×D411yescharacteristic
113C2×C4×C47#12111yescharacteristic
114C2×C4×C47#12311yescharacteristic
115C2×C4×C47#12611yescharacteristic
116C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C22⋊D47—81no—
121C22⋊D47—81no—
122C2×Dic47—42no—
123C2×Dic47—41no—
124C2×Dic47—81no—
125C2×Dic47—81no—
126D188—41no—
127D188—41no—
128D188—41no—
129D188—41no—
130D188—81no—
131C4×D47—81no—
132C4×D47—81no—
133Dic94—41no—
134Dic94—41no—
135C8×C47—22no—
136C8×C47—41no—
137C8×C47—41no—
138C8×C47—41no—
139C8×C47—41no—
140C8×C47—41no—
141C47⋊C8—41no—
142C47⋊C8—41no—
143C47⋊C8—81no—
144#9592—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47#38811yescharacteristic
148C22×C47#48911yescharacteristic
149C22×C47#49011yescharacteristic
150C2×D47—81no—
151C2×D47—81no—
152C2×D47—81no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47#44011yescharacteristic
157C4×C47#44211yescharacteristic
158C4×C47#44311yescharacteristic
159C4×C47#44411yescharacteristic
160C4×C47#46811yescharacteristic
161C4×C47#46911yescharacteristic
162Dic47—82no—
163Dic47—81no—
164#1980—471no—
165#1950—471no—
166#1540—471no—
167#1492—471no—
168#1451D4711yescharacteristic, complemented
169#1303—471no—
170#1279—471no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47#192911yescharacteristic
174C2×C47#193211yescharacteristic
175C2×C47#193611yescharacteristic
176D47—161no—
177#532—471no—
178#431C2×D4711yescharacteristic
179C2×#108—941no—
180#423—471no—
181C2×C4⋊C8—471no—
182C2×#104—941no—
183C2×#104—941no—
184C4×M16C2×D4711yescharacteristic
185C2×C4×C8C2×D4711yescharacteristic
186C4×C4⋊C4—471no—
187#341—941no—
188#332—471no—
189#332—471no—
190#331—471no—
191#331—471no—
192D4⋊C8—941no—
193D4⋊C8—941no—
194#322—21no—
195#322—21no—
196C47#959211yescharacteristic, complemented, Sylow
197C2×#41—941no—
198C2×M16—941no—
199C2×M16D18811yescharacteristic
200C2×M16D18811yescharacteristic
201C22×C8—21no—
202C22×C8—941no—
203C4⋊Q8—471no—
204#127—471no—
205#124—941no—
206#121—941no—
207#121—941no—
208C4×D4—941no—
209C4×D4—941no—
210C2×C4⋊C4—941no—
211C2×C42—941no—
212C2×C42C22×D4711yescharacteristic
213#108—1882no—
214C4⋊C8—471no—
215C4⋊C8—471no—
216C4⋊C8—941no—
217C4⋊C8—941no—
218C4⋊C8—941no—
219C4⋊C8C4×D4711yescharacteristic
220C4⋊C8C4×D4711yescharacteristic
221#104—1882no—
222#104—1882no—
223C22⋊C8—21no—
224#97—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8C22⋊D4711yescharacteristic
229C4×C8C22⋊D4711yescharacteristic
230#95—941no—
231#41—1881no—
232#41—1881no—
233#41—1881no—
234#41—1881no—
235C2×Q8—941no—
236C2×D4—941no—
237C2×D4—941no—
238C2×D4—941no—
239C2×D4—1881no—
240C22×C4—1881no—
241C22×C4—1881no—
242C22×C4C2×D18811yescharacteristic
243C22×C4#1062535011yescharacteristic
244C22×C4#1062535011yescharacteristic
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—1881no—
250M16—1881no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—42no—
258C2×C8—941no—
259C2×C8—941no—
260C2×C8—1881no—
261C2×C8—1881no—
262C2×C8—1881no—
263C4⋊C4—942no—
264C4⋊C4—941no—
265C4⋊C4—941no—
266C4⋊C4—1881no—
267C22⋊C4—1881no—
268C22⋊C4—1881no—
269C42—1881no—
270C42—1881no—
271C42C2×C4×D4711yescharacteristic
272C42#1062535011yescharacteristic
273C42#1062535011yescharacteristic
274C42C2×C22⋊D4711yescharacteristic
275C23—1881no—
276C23#1309401211yescharacteristic
277Q8—1881no—
278Q8—1881no—
279D4—3761no—
280D4—3761no—
281D4—3761no—
282D4—1881no—
283D4—1881no—
284D4—1881no—
285D4—1881no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—3761no—
290C2×C4—3761no—
291C2×C4—3761no—
292C2×C4—3761no—
293C2×C4—1882no—
294C2×C4—1881no—
295C2×C4#1309398111yescharacteristic
296C2×C4C4×C22⋊D4711yescharacteristic
297C2×C4C4×D18811yescharacteristic
298C2×C4#1309399811yescharacteristic
299C2×C4#1309400711yescharacteristic
300C2×C4#1309401811yescharacteristic
301C2×C4#1309405211yescharacteristic
302C2×C4#1309405211yescharacteristic
303C2×C4#1309405211yescharacteristic
304C2×C4#1309405711yescharacteristic
305C8—22no—
306C8—41no—
307C8—41no—
308C8—41no—
309C8—41no—
310C8—41no—
311C8—3761no—
312C8—1881no—
313C8—1881no—
314C22—21no—
315C22—21no—
316C22—3761no—
317C22—3761no—
318C22—3761no—
319C22#1568901411yescharacteristic
320C22#1568948311yescharacteristic
321C22#1568948811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—3762no—
326C4—3761no—
327C4#1568922511yescharacteristic
328C4#1568923611yescharacteristic
329C4#1568923811yescharacteristic
330C4#1568924311yescharacteristic
331C4#1568936911yescharacteristic
332C4#1568937611yescharacteristic
333C2—21no—
334C2—21no—
335C2—7521no—
336C2#2024403611yescharacteristic
337C2#2024405711yescharacteristic
338C2#2024408511yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1451C2
socleC22×C47#388
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1451D47
hypocenterC47#9592
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1451

Complements

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

NH ≅ G/Ntype
2. #20244163C2semidirect
5. #20241290C2semidirect
6. #20241228C2semidirect
7. C47×#1451C2semidirect
16. #15688868C4semidirect
17. #15688868C4semidirect
18. #15688866C4semidirect
19. #15688866C4semidirect
168. #1451D47semidirect
196. C47#9592semidirect

Maximal subgroups

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

subgroupindex
#202443072
#202441632
#202420352
#202418402
#202412902
#202412282
C47×#14512
#959247