Finite Group Databasebeta

← #26113019

Subgroups — #26113019

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254313C211yesmaximal, characteristic
2#20244220C211yesmaximal, characteristic
3#20242069C211yesmaximal, characteristic, complemented
4#20241947C211yesmaximal, characteristic
5#20241246C211yesmaximal, characteristic, complemented
6#20241217C211yesmaximal, characteristic, complemented
7C47×#2331C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689497C412yescomplemented
10#15689473C412yescomplemented
11#15689325—22no—
12#15689262—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C2×#13093952—21no—
17C2×#13093952—21no—
18#15688872—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×#10625336—24no—
29C2×#10625329—21no—
30#13094024C2×C411yescharacteristic
31#13094021C2×C411yescharacteristic
32#13094012—21no—
33#13093998—21no—
34#13093998—21no—
35C4×D188—21no—
36C4×D188—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41#13093952—42no—
42#13093952—42no—
43#13093948—24no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8C2×C412yes—
48C22×Q8×C47D412yes—
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—21no—
56C2×C4⋊C4×C47—22no—
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—82no—
69C2×Q8×C47C22⋊C414yes—
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—22no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—82no—
94C42×C47—21no—
95C42×C47—21no—
96C42×C47—21no—
97C42×C47—82no—
98C42×C47SD1612yescomplemented
99C42×C47C22×C411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102C42×C47#4111yescharacteristic
103#10625336—84no—
104(C2×D47)⋊C4—41no—
105(C2×D47)⋊C4—41no—
106C4⋊Dic47—21no—
107C4⋊Dic47—21no—
108#10625329—41no—
109#10625329—41no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C23×C47#12011yescharacteristic
114C22×D47—41no—
115Q8×C47—44no—
116Q8×C47—82no—
117C2×C4×C47—24no—
118C2×C4×C47—22no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47C2×C22⋊C411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12311yescharacteristic
137C2×C4×C47C2×SD1611yescharacteristic
138C2×C4×C47#13511yescharacteristic
139C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C47#13711yescharacteristic
141C22⋊D47—81no—
142C22⋊D47—81no—
143C2×Dic47—41no—
144D188—42no—
145D188—41no—
146D188—41no—
147D188—81no—
148C4×D47—81no—
149C4×D47—81no—
150Dic94—41no—
151Dic94—41no—
152C47⋊C8—42no—
153C47⋊C8—81no—
154C47⋊C8—81no—
155#9301—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47#38611yescharacteristic
159C22×C47#44811yescharacteristic
160C22×C47#44911yescharacteristic
161C2×D47—81no—
162C2×D47—81no—
163C2×D47—81no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—42no—
169C4×C47—44no—
170C4×C47—82no—
171C4×C47—81no—
172C4×C47C4×SD1611yescharacteristic
173C4×C47#44011yescharacteristic
174C4×C47#46011yescharacteristic
175C4×C47#46411yescharacteristic
176Dic47—81no—
177#3206—471no—
178#2331D4711yescharacteristic, complemented
179#1963—471no—
180#1548—471no—
181#1517—471no—
182#1286—471no—
183#1274—471no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47#188011yescharacteristic
187C2×C47#190511yescharacteristic
188C2×C47#191111yescharacteristic
189D47—161no—
190#532—471no—
191C2×C4⋊Q8C2×D4711yescharacteristic
192#492—472no—
193#487—472no—
194#461—942no—
195#448—942no—
196#431—471no—
197#431—471no—
198#423—471no—
199C2×#104—941no—
200C2×#104—941no—
201#391—22no—
202#389—21no—
203#387—22no—
204#382C2×D4711yescharacteristic
205C4×C4⋊C4—21no—
206#376C2×D4711yescharacteristic
207#333—942no—
208#327—942no—
209C47#930111yescharacteristic, complemented, Sylow
210C2×#41—941no—
211C22×Q8C22⋊D4712yes—
212C2×SD16—944no—
213C2×M16—941no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—471no—
217C4⋊Q8C4×D4712yes—
218C4⋊Q8D18812yes—
219#127—471no—
220#124—941no—
221#121—941no—
222#121—941no—
223C4×D4—941no—
224C4×D4—941no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C42—22no—
230C2×C42C22⋊D4711yescharacteristic
231C2×C42C22⋊D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233C4⋊C8—941no—
234C4⋊C8—941no—
235C4⋊C8—941no—
236C4⋊C8—941no—
237#104—1882no—
238#104—1882no—
239#103—944no—
240C22⋊C8—941no—
241C22⋊C8—941no—
242C4×C8—472no—
243C4×C8—941no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95C22⋊D4712yes—
250#41—1881no—
251#41—1881no—
252#41—1881no—
253#41—1881no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—941no—
257C2×Q8(C2×D47)⋊C414yes—
258C2×D4—941no—
259C2×D4—941no—
260C2×D4—941no—
261C2×D4—1881no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—1881no—
267C22×C4C2×C22⋊D4711yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270SD16—3764no—
271M16—1881no—
272M16—1881no—
273C2×C8—942no—
274C2×C8—1881no—
275C2×C8—1881no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—82no—
283C4⋊C4—941no—
284C4⋊C4—941no—
285C22⋊C4—1881no—
286C22⋊C4—1881no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42#1062533612yescomplemented
292C42C2×C4×D4711yescharacteristic
293C42C2×D18811yescharacteristic
294C42#1062535011yescharacteristic
295C42#1062535011yescharacteristic
296C23—1881no—
297C23#1309399311yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—1881no—
301Q8—1881no—
302D4—3761no—
303D4—3761no—
304D4—3761no—
305D4—1882no—
306D4—1881no—
307D4—1881no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—3761no—
323C2×C4—3761no—
324C2×C4—1881no—
325C2×C4C2×(C2×D47)⋊C411yescharacteristic
326C2×C4C4×C22⋊D4711yescharacteristic
327C2×C4C4×C22⋊D4711yescharacteristic
328C2×C4#1309399811yescharacteristic
329C2×C4#1309399811yescharacteristic
330C2×C4#1309400711yescharacteristic
331C2×C4C2×#1062533611yescharacteristic
332C2×C4#1309404411yescharacteristic
333C2×C4#1309404911yescharacteristic
334C2×C4#1309405511yescharacteristic
335C8—3761no—
336C8—3761no—
337C8—1882no—
338C22—21no—
339C22—21no—
340C22—3761no—
341C22—3761no—
342C22—3761no—
343C22#1568900211yescharacteristic
344C22#1568926211yescharacteristic
345C22#1568926811yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—3761no—
355C4C4×#1062533611yescharacteristic
356C4#1568922111yescharacteristic
357C4#1568932311yescharacteristic
358C4#1568934511yescharacteristic
359C2—21no—
360C2—21no—
361C2—7521no—
362C2#2024374311yescharacteristic
363C2#2024388211yescharacteristic
364C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2331C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2331D47
hypocenterC47#9301
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. #20242069C2semidirect
5. #20241246C2semidirect
6. #20241217C2semidirect
7. C47×#2331C2semidirect
9. #15689497C4semidirect
10. #15689473C4semidirect
98. C42×C47SD16semidirect
178. #2331D47semidirect
209. C47#9301semidirect
291. C42#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543132
#202442202
#202420692
#202419472
#202412462
#202412172
C47×#23312
#930147