Finite Group Databasebeta

← #26126900

Subgroups — #26126900

Order 12032.

516 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20248341C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241413C214yesmaximal, complemented
4C47×#1465C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094003—21no—
7C2×#13094003—21no—
8C2×#13093939—21no—
9#15689059C412yescomplemented
10#15689059C2212yes—
11#15689026—21no—
12#15688992—21no—
13C4×C4⋊Dic47C411yescharacteristic, complemented
14C4×C22⋊Dic47—21no—
15#15688890C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094003—44no—
24#13094003—44no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C412yes—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—41no—
35#13093939—24no—
36C47⋊#95—41no—
37C47⋊#95C2×C412yes—
38C22×D4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C2×M16×C47—41no—
42C4×D4×C47—41no—
43C4×D4×C47—41no—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C22⋊C8×C47—22no—
49C22⋊C8×C47—41no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—22no—
53C2×D4×C47—22no—
54C2×D4×C47—41no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—42no—
69C2×Dic94—42no—
70C2×Dic94—42no—
71C2×Dic94—41no—
72C2×Dic94—41no—
73M16×C47—82no—
74C2×C8×C47—42no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47—21no—
80C22⋊Dic47—24no—
81C22⋊Dic47—44no—
82C22⋊Dic47—44no—
83C22⋊Dic47—42no—
84C4⋊Dic47—24no—
85C4⋊Dic47—42no—
86C4⋊Dic47—42no—
87Dic47⋊C4—44no—
88Dic47⋊C4—44no—
89C4×Dic47—42no—
90C4×Dic47—82no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47C2×C22⋊C411yescharacteristic
101C23×C47#13712yes—
102D4×C47—44no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—81no—
115C2×C4×C47—81no—
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#9911yescharacteristic
118C2×C4×C47#10412yes—
119C2×Dic47—24no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—44no—
125C2×Dic47—42no—
126C2×Dic47—82no—
127C2×Dic47—82no—
128Dic94—82no—
129Dic94—82no—
130C8×C47—82no—
131#11788—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—22no—
135C22×C47—42no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47—81no—
143C22×C47—81no—
144C22×C47C2×#9911yescharacteristic
145C22×C47#41911yescharacteristic
146C22×C47C2×#10411yescharacteristic
147C4×C47—21no—
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47#35312yes—
153Dic47—44no—
154Dic47—84no—
155Dic47—82no—
156#2542—471no—
157C4×#99—471no—
158#1465D4711yescharacteristic
159#1352—474no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—42no—
163C2×C47—81no—
164C2×C47#150811yescharacteristic
165C2×C47#212311yescharacteristic
166C2×C47C2×#35311yescharacteristic
167C2×C4⋊Q8—471no—
168C2×C22⋊Q8—941no—
169C2×C22⋊Q8—941no—
170C2×C4×D4C2×D4711yescharacteristic
171C2×#99—941no—
172#407—21no—
173#399—472no—
174#399—472no—
175#390—941no—
176#384—941no—
177C4×C4⋊C4—471no—
178C4×C22⋊C4—941no—
179#342—472no—
180#324—21no—
181#323C2×D4712yes—
182C47#1178811yescharacteristic, complemented, Sylow
183C22×Q8—941no—
184C22×D4D18811yescharacteristic
185C23×C4D18811yescharacteristic
186C23×C4C22⋊D4711yescharacteristic
187C2×M16—41no—
188C4⋊Q8—1881no—
189C4⋊Q8—1881no—
190C22⋊Q8—1884no—
191C22⋊Q8—1884no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—472no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—942no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—942no—
199C2×C4⋊C4C22⋊D4711yescharacteristic
200C2×C22⋊C4—942no—
201C2×C22⋊C4—942no—
202C2×C22⋊C4—942no—
203C2×C22⋊C4C4×D4711yescharacteristic
204C2×C22⋊C4C22×D4711yescharacteristic
205C2×C42—941no—
206C2×C42—1881no—
207C2×C42C4×D4711yescharacteristic
208#99—944no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211#95—472no—
212#95—1881no—
213C24(C2×D47)⋊C411yescharacteristic
214C24C2×D18811yescharacteristic
215C2×Q8—1882no—
216C2×Q8—1882no—
217C2×Q8—1881no—
218C2×Q8—1881no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—942no—
229C22×C4—942no—
230C22×C4—1882no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236M16—82no—
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—944no—
240C4⋊C4—1882no—
241C4⋊C4—1882no—
242C4⋊C4—1884no—
243C4⋊C4—1884no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—944no—
247C22⋊C4—1884no—
248C22⋊C4—1884no—
249C22⋊C4—1882no—
250C42—21no—
251C42—21no—
252C42—3762no—
253C42—1882no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×(C2×D47)⋊C411yescharacteristic
264C23#1309405712yes—
265Q8—3762no—
266Q8—3762no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—944no—
282C2×C4—3762no—
283C2×C4—3762no—
284C2×C4—1884no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4#1309393811yescharacteristic
291C2×C4#1309393811yescharacteristic
292C2×C4#1309395012yes—
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#1309393811yescharacteristic
307C22#1568913311yescharacteristic
308C22C2×#1309395011yescharacteristic
309C4—21no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—3764no—
315C4—3762no—
316C4—1884no—
317C4#1568891512yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#2024190911yescharacteristic
323C2#2024504311yescharacteristic
324C2C2×#1568891511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1465C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1465D47
hypocenterC47#11788
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20241413C2semidirect
5. C2×#13094024C4semidirect
9. #15689059C4semidirect
13. C4×C4⋊Dic47C4semidirect
182. C47#11788semidirect

Maximal subgroups

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

subgroupindex
#202483412
C4×#130939392
#202414132
C47×#14652
#1178847