Finite Group Databasebeta

← #26104799

Subgroups — #26104799

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243456C211yesmaximal, characteristic
2#20243082C211yesmaximal, characteristic
3#20241214C212yesmaximal, complemented
4#20241211C212yesmaximal, complemented
5C47×#2855C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688878—21no—
9#15688878—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C47×#391C411yescharacteristic
15C47×#391C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#385—21no—
21C47×#381—21no—
22C47×#380C411yescharacteristic
23C2×#10625329—24no—
24C2×#10625329—42no—
25C2×#10625329—42no—
26C22×C47⋊C8—22no—
27C22×C47⋊C8—22no—
28C22⋊(C47⋊C8)—42no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C22⋊Q8×C47—42no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C42×C47—42no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47C2311yescharacteristic
47#95×C47—21no—
48#95×C47—21no—
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54C24×C47C22⋊C411yescharacteristic
55C2×Q8×C47—22no—
56C2×Q8×C47—41no—
57C2×Q8×C47—41no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—22no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C42×C47—41no—
88C42×C47—41no—
89C42×C47—41no—
90C42×C47—41no—
91#10625329—48no—
92C2×C47⋊C8—24no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47#9912yes—
105C23×C47C2×C22⋊C411yescharacteristic
106Q8×C47—82no—
107C2×C4×C47—24no—
108C2×C4×C47—24no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—44no—
124C2×C4×C47—44no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47#10412yes—
138C2×C4×C47#13112yes—
139C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C47#13711yescharacteristic
141C2×C4×C47#13712yes—
142C47⋊C8—44no—
143C47⋊C8—82no—
144#7695—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47#40811yescharacteristic
153C22×C47C2×#9911yescharacteristic
154C22×C47#41311yescharacteristic
155C22×C47#41711yescharacteristic
156C22×C47#41911yescharacteristic
157C22×C47C2×#10411yescharacteristic
158C22×C47#42111yescharacteristic
159C4×C47—22no—
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—41no—
165C4×C47—41no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47—81no—
169C4×C47—81no—
170#2855D4711yescharacteristic
171#1817—471no—
172#1728—471no—
173#1272—472no—
174#1271—472no—
175C2×C47—42no—
176C2×C47#146811yescharacteristic
177C2×C47#148511yescharacteristic
178C2×C47#149511yescharacteristic
179C2×C47#150411yescharacteristic
180C2×C47#151111yescharacteristic
181C2×C47#152911yescharacteristic
182C2×C47#155911yescharacteristic
183C22×M16—941no—
184C2×C22⋊Q8Dic4711yescharacteristic
185C22×C42C2×D4711yescharacteristic
186C2×C22⋊C8—941no—
187#391Dic4711yescharacteristic
188#391C2×D4711yescharacteristic
189#389Dic4711yescharacteristic
190#389C2×D4711yescharacteristic
191#387—21no—
192#387—21no—
193#385—21no—
194#381—21no—
195#380Dic4711yescharacteristic
196#336—472no—
197#336—472no—
198#336—941no—
199#336—941no—
200C47#769511yescharacteristic, complemented, Sylow
201C22×Q8C2×Dic4711yescharacteristic
202C23×C4C2×Dic4711yescharacteristic
203C23×C4C22⋊D4711yescharacteristic
204C23×C4C22⋊D4711yescharacteristic
205C2×M16—944no—
206C2×M16—1882no—
207C2×M16—1882no—
208C22×C8—942no—
209C22×C8—942no—
210C22⋊Q8—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic4711yescharacteristic
215C2×C4⋊C4C2×Dic4711yescharacteristic
216C2×C4⋊C4C2×Dic4711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—42no—
220C2×C42C2×Dic4711yescharacteristic
221C2×C42C22⋊D4711yescharacteristic
222C2×C42C22⋊D4711yescharacteristic
223C2×C42C22×D4711yescharacteristic
224C22⋊C8—1882no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24C22⋊Dic4711yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C22⋊Dic4711yescharacteristic
251C22×C4C22⋊Dic4711yescharacteristic
252C22×C4C22⋊Dic4711yescharacteristic
253C22×C4C22×Dic4711yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255C22×C4C2×C22⋊D4711yescharacteristic
256M16—1888no—
257C2×C8—944no—
258C2×C8—1882no—
259C2×C8—1882no—
260C2×C8—1882no—
261C2×C8—1882no—
262C2×C8—1882no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#1309393912yes—
283C23C2×C22⋊Dic4711yescharacteristic
284Q8—82no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#1309395112yes—
316C2×C4#1309403112yes—
317C2×C4#1309404911yescharacteristic
318C2×C4#1309405411yescharacteristic
319C2×C4#1309405512yes—
320C8—3762no—
321C8—1884no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#1568909011yescharacteristic
330C22C2×#1309393911yescharacteristic
331C22#1568910711yescharacteristic
332C22#1568912411yescharacteristic
333C22#1568913211yescharacteristic
334C22C2×#1309395111yescharacteristic
335C22#1568914011yescharacteristic
336C4—22no—
337C4—22no—
338C4—21no—
339C4—22no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—42no—
348C2#2024173611yescharacteristic
349C2#2024180911yescharacteristic
350C2#2024185311yescharacteristic
351C2#2024189211yescharacteristic
352C2#2024192111yescharacteristic
353C2#2024199511yescharacteristic
354C2#2024211811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2855D47
hypocenterC47#7695
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241211C2semidirect
200. C47#7695semidirect

Maximal subgroups

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

subgroupindex
#202434562
#202430822
#202412142
#202412112
C47×#28552
#769547