Finite Group Databasebeta

← #26113026

Subgroups — #26113026

Order 12032.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254516C211yesmaximal, characteristic
2#20244220C211yesmaximal, characteristic
3#20242073C211yesmaximal, characteristic, complemented
4#20241947C211yesmaximal, characteristic
5#20241246C211yesmaximal, characteristic, complemented
6#20241217C211yesmaximal, characteristic, complemented
7C47×#2331C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689519C412yescomplemented
10#15689471C412yescomplemented
11#15689441—22no—
12#15689410—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C2×#13093952—21no—
17C2×#13093952—21no—
18#15688869—22no—
19#15688856—22no—
20C2×C4⋊Q8×C47C2211yescharacteristic
21C47×#391—22no—
22C47×#389—21no—
23C47×#387—22no—
24C47×#382C2211yescharacteristic
25C4×C4⋊C4×C47—21no—
26C47×#376C2211yescharacteristic
27C2×#10625350—21no—
28C2×#10625329—21no—
29#13094024C2×C411yescharacteristic
30#13094021C2×C411yescharacteristic
31#13094012—21no—
32#13093998—21no—
33#13093998—21no—
34C4×D188—21no—
35C4×D188—21no—
36#13093955—24no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41#13093952—42no—
42#13093952—42no—
43#13093944—24no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8C2×C412yes—
48C22×Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47C2×C412yes—
52C4⋊Q8×C47D412yes—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—22no—
56C2×C4⋊C4×C47D412yes—
57C2×C42×C47—22no—
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61#95×C47—22no—
62#95×C47—22no—
63#95×C47—21no—
64#95×C47—21no—
65#95×C47—21no—
66#95×C47D412yes—
67C2×Q8×C47—22no—
68C2×Q8×C47—22no—
69C2×Q8×C47—82no—
70C22×C4×C47—21no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C2×C22⋊D47—41no—
78#10625350—41no—
79#10625350—41no—
80#10625350—41no—
81#10625350—41no—
82C2×D188—21no—
83C2×D188—21no—
84C2×D188—21no—
85C2×C4×D47—41no—
86C2×Dic94—21no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—82no—
93C4⋊C4×C47C22⋊C414yes—
94C42×C47—21no—
95C42×C47—21no—
96C42×C47—21no—
97C42×C47—82no—
98C42×C47SD1612yes—
99C42×C47C22×C411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102C42×C47#4111yescharacteristic
103(C2×D47)⋊C4—41no—
104(C2×D47)⋊C4—41no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107#10625329—41no—
108#10625329—41no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C23×C47#12011yescharacteristic
113C22×D47—41no—
114Q8×C47—82no—
115Q8×C47—82no—
116C2×C4×C47—24no—
117C2×C4×C47—22no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47#12111yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12311yescharacteristic
136C2×C4×C47C2×SD1611yescharacteristic
137C2×C4×C47#13511yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47#13711yescharacteristic
140C22⋊D47—81no—
141C22⋊D47—81no—
142C2×Dic47—41no—
143D188—42no—
144D188—41no—
145D188—41no—
146D188—81no—
147C4×D47—81no—
148C4×D47—81no—
149Dic94—41no—
150Dic94—41no—
151C47⋊C8—42no—
152C47⋊C8—81no—
153C47⋊C8—81no—
154#9302—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47#38611yescharacteristic
158C22×C47#44811yescharacteristic
159C22×C47#44911yescharacteristic
160C2×D47—81no—
161C2×D47—81no—
162C2×D47—81no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—42no—
168C4×C47—44no—
169C4×C47—82no—
170C4×C47—81no—
171C4×C47C4×SD1611yescharacteristic
172C4×C47#44011yescharacteristic
173C4×C47#46011yescharacteristic
174C4×C47#46411yescharacteristic
175Dic47—81no—
176#3227—471no—
177#2331D4711yescharacteristic, complemented
178#1963—471no—
179#1549—471no—
180#1517—471no—
181#1286—471no—
182#1274—471no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47#188011yescharacteristic
186C2×C47#190411yescharacteristic
187C2×C47#191211yescharacteristic
188D47—161no—
189#532—471no—
190C2×C4⋊Q8C2×D4711yescharacteristic
191#498—472no—
192#486—472no—
193#481—942no—
194#476—942no—
195#431—471no—
196#431—471no—
197#423—471no—
198C2×#104—941no—
199C2×#104—941no—
200#391—22no—
201#389—21no—
202#387—22no—
203#382C2×D4711yescharacteristic
204C4×C4⋊C4—21no—
205#376C2×D4711yescharacteristic
206#332—942no—
207#327—942no—
208C47#930211yescharacteristic, complemented, Sylow
209C2×#41—941no—
210C22×Q8—21no—
211C2×M16—941no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—471no—
215C4⋊Q8C4×D4712yes—
216C4⋊Q8D18812yes—
217#127—471no—
218#124—941no—
219#121—941no—
220#121—941no—
221C4×D4—941no—
222C4×D4—941no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D4712yes—
227C2×C42—22no—
228C2×C42C22⋊D4711yescharacteristic
229C2×C42C22⋊D4711yescharacteristic
230C2×C42C22×D4711yescharacteristic
231#106—944no—
232C4⋊C8—941no—
233C4⋊C8—941no—
234C4⋊C8—941no—
235C4⋊C8—941no—
236#104—1882no—
237#104—1882no—
238#102—944no—
239C22⋊C8—941no—
240C22⋊C8—941no—
241C4×C8—472no—
242C4×C8—941no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C22⋊D4712yes—
249#41—1881no—
250#41—1881no—
251#41—1881no—
252#41—1881no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—941no—
257C2×D4—941no—
258C2×D4—941no—
259C2×D4—941no—
260C2×D4—1881no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—1881no—
266C22×C4C2×C22⋊D4711yescharacteristic
267C22×C4C2×C22⋊D4711yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269M16—1881no—
270M16—1881no—
271C2×C8—942no—
272C2×C8—1881no—
273C2×C8—1881no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—941no—
281C4⋊C4—941no—
282C4⋊C4(C2×D47)⋊C414yes—
283C22⋊C4—1881no—
284C22⋊C4—1881no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42#1062533612yes—
290C42C2×C4×D4711yescharacteristic
291C42C2×D18811yescharacteristic
292C42#1062535011yescharacteristic
293C42#1062535011yescharacteristic
294C23—1881no—
295C23#1309399311yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—1881no—
299Q8—1881no—
300D4—3761no—
301D4—3761no—
302D4—3761no—
303D4—1882no—
304D4—1881no—
305D4—1881no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—1881no—
323C2×C4C2×(C2×D47)⋊C411yescharacteristic
324C2×C4C4×C22⋊D4711yescharacteristic
325C2×C4C4×C22⋊D4711yescharacteristic
326C2×C4#1309399811yescharacteristic
327C2×C4#1309399811yescharacteristic
328C2×C4#1309400711yescharacteristic
329C2×C4C2×#1062533611yescharacteristic
330C2×C4#1309404411yescharacteristic
331C2×C4#1309404911yescharacteristic
332C2×C4#1309405511yescharacteristic
333C8—3761no—
334C8—3761no—
335C8—1882no—
336C22—21no—
337C22—21no—
338C22—3761no—
339C22—3761no—
340C22—3761no—
341C22#1568900211yescharacteristic
342C22#1568926211yescharacteristic
343C22#1568926811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—3761no—
353C4C4×#1062533611yescharacteristic
354C4#1568922111yescharacteristic
355C4#1568932311yescharacteristic
356C4#1568934511yescharacteristic
357C2—21no—
358C2—21no—
359C2—7521no—
360C2#2024374311yescharacteristic
361C2#2024387511yescharacteristic
362C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2331C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2331D47
hypocenterC47#9302
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #20242073C2semidirect
5. #20241246C2semidirect
6. #20241217C2semidirect
7. C47×#2331C2semidirect
9. #15689519C4semidirect
10. #15689471C4semidirect
177. #2331D47semidirect
208. C47#9302semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202442202
#202420732
#202419472
#202412462
#202412172
C47×#23312
#930247