Finite Group Databasebeta

← #26104365

Subgroups — #26104365

Order 12032.

630 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 117 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#391×D47C211yesmaximal, characteristic
2#20243455C212yesmaximal
3#20241214C211yesmaximal, characteristic, complemented
4#20241213C212yesmaximal, complemented
5C47×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D47C411yescharacteristic
7C2×C4⋊C4×D47C411yescharacteristic
8C2×C42×D47C2211yescharacteristic
9C2×(C2×D47)⋊C8—22no—
10#15689030C411yescharacteristic
11#15689029C2211yescharacteristic
12#15689028C411yescharacteristic
13#15689027—22no—
14#95×D47—22no—
15#15688878C2212yes—
16#15688877—22no—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94—21no—
20C22×Dic94C2×C411yescharacteristic
21C2×Q8×D47—22no—
22C2×Q8×D47—42no—
23C2×Q8×D47—42no—
24C2×Q8×D47C814yescomplemented
25C22×C4×D47—21no—
26C22×C4×D47—21no—
27C22×C4×D47C2×C411yescharacteristic
28C22×C4×D47D412yes—
29C22×C47⋊C8—22no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C4⋊C4×D47—42no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35C42×D47—42no—
36(C2×D47)⋊C8—44no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C22×Q8×C47C2×C411yescharacteristic
41C22×C8×C47—22no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C42×C47C2311yescharacteristic
44#95×C47—22no—
45C23×D47C22⋊C411yescharacteristic
46C2×Q8×C47—21no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47C2×C812yes—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47C22×C411yescharacteristic
53C22×C4×C47C2×D412yes—
54C22×Dic47—21no—
55C22×Dic47—21no—
56C22×Dic47C22⋊C412yes—
57C22×Dic47C22⋊C411yescharacteristic
58Q8×D47—48no—
59Q8×D47—48no—
60C2×C4×D47—22no—
61C2×C4×D47—22no—
62C2×C4×D47—24no—
63C2×C4×D47—24no—
64C2×C4×D47—22no—
65C2×C4×D47—22no—
66C2×C4×D47—42no—
67C2×C4×D47—42no—
68C2×C4×D47—42no—
69C2×C4×D47—44no—
70C2×C4×D47—42no—
71C2×C4×D47C2×C812yes—
72C2×C4×D47M1612yes—
73C2×Dic94—21no—
74C2×Dic94—24no—
75C2×Dic94—42no—
76C2×Dic94—42no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94C2×C812yes—
80C2×C8×C47—42no—
81C2×C8×C47—42no—
82C2×C8×C47—42no—
83C4⋊C4×C47—41no—
84C42×C47—41no—
85C4⋊Dic47—41no—
86Dic47⋊C4—42no—
87C4×Dic47—41no—
88C4×Dic47—42no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C23×C47C2×C22⋊C411yescharacteristic
93C22×D47—22no—
94C22×D47—22no—
95C22×D47C22⋊C812yes—
96C22×D47#9912yes—
97C22×D47#10012yes—
98Q8×C47—42no—
99Q8×C47—42no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—42no—
109C2×C4×C47C22×C811yescharacteristic
110C2×C4×C47C2×M1611yescharacteristic
111C2×C4×C47#13712yes—
112C2×C4×C47#13712yes—
113C2×Dic47—21no—
114C2×Dic47—21no—
115C2×Dic47—22no—
116C2×Dic47—22no—
117C2×Dic47—22no—
118C2×Dic47—22no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—42no—
123C2×Dic47C22⋊C812yes—
124C4×D47—24no—
125C4×D47—24no—
126C4×D47—44no—
127C4×D47—44no—
128C4×D47—48no—
129C4×D47—84no—
130Dic94—48no—
131Dic94—42no—
132Dic94—42no—
133C8×C47—82no—
134C47⋊C8—82no—
135#7595—471nomaximal, Sylow
136C22×C47C2×C22⋊C811yescharacteristic
137C22×C47C2×#9911yescharacteristic
138C22×C47C2×#10011yescharacteristic
139C22×C47#41912yes—
140C22×C47#42112yes—
141C2×D47—22no—
142C2×D47—24no—
143C2×D47—24no—
144C2×D47—22no—
145C2×D47#32314yes—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—42no—
151C4×C47—81no—
152Dic47—22no—
153Dic47—42no—
154Dic47—42no—
155Dic47—81no—
156C2×#391—471no—
157#1817—472no—
158#1272—471no—
159#1272—472no—
160#1272D4711yescharacteristic, complemented
161C2×C47C2×#32311yescharacteristic
162C2×C47#147912yes—
163C2×C47#151112yes—
164C2×C47#153311yescharacteristic
165C2×C47#155811yescharacteristic
166D47—24no—
167C23×Q8—471no—
168C22×C4⋊C4—471no—
169C22×C42—471no—
170C2×C22⋊C8—942no—
171#391—471no—
172#391—471no—
173#391—471no—
174#391—942no—
175#391C2×D4711yescharacteristic
176C2×#95—942no—
177#336—472no—
178#336—942no—
179#336C2×D4712yes—
180C47#759511yescharacteristic, complemented, Sylow
181C22×Q8—474no—
182C22×Q8—471no—
183C22×Q8—941no—
184C22×Q8—942no—
185C22×Q8—1882no—
186C22×Q8—1882no—
187C22×Q8C4×D4711yescharacteristic
188C23×C4—471no—
189C23×C4—472no—
190C23×C4—941no—
191C23×C4—941no—
192C22×C8—22no—
193C22×C8—942no—
194C2×C4⋊C4—471no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—1882no—
197C2×C4⋊C4C4×D4711yescharacteristic
198C2×C42—471no—
199C2×C42—472no—
200C2×C42—1882no—
201C2×C42C22×D4711yescharacteristic
202C22⋊C8—1884no—
203#95—22no—
204#95—942no—
205#95—942no—
206#95—942no—
207C24—471no—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—472no—
212C2×Q8—944no—
213C2×Q8—941no—
214C2×Q8—1881no—
215C2×Q8—1881no—
216C2×Q8—1882no—
217C2×Q8—1882no—
218C2×Q8—1888no—
219C2×Q8—1888no—
220C2×Q8C8×D4712yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—471no—
224C22×C4—472no—
225C22×C4—472no—
226C22×C4—472no—
227C22×C4—942no—
228C22×C4—942no—
229C22×C4—944no—
230C22×C4—944no—
231C22×C4—941no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—941no—
235C22×C4—1882no—
236C22×C4—1882no—
237C22×C4—1884no—
238C22×C4—1882no—
239C22×C4—1882no—
240C22×C4C2×C4×D4711yescharacteristic
241C22×C4D4×D4712yes—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—1882no—
246C2×C8—1882no—
247C2×C8—1882no—
248C4⋊C4—41no—
249C4⋊C4—1881no—
250C4⋊C4—1882no—
251C42—41no—
252C42—1882no—
253C42—1881no—
254C23—472no—
255C23—472no—
256C23—472no—
257C23—942no—
258C23—942no—
259C23C22⋊C4×D4711yescharacteristic
260Q8—42no—
261Q8—42no—
262Q8—1882no—
263Q8—1882no—
264Q8—1888no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—472no—
275C2×C4—944no—
276C2×C4—944no—
277C2×C4—941no—
278C2×C4—941no—
279C2×C4—942no—
280C2×C4—942no—
281C2×C4—942no—
282C2×C4—942no—
283C2×C4—3764no—
284C2×C4—1884no—
285C2×C4—1884no—
286C2×C4—1888no—
287C2×C4—1881no—
288C2×C4—1882no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4C2×C8×D4711yescharacteristic
292C2×C4M16×D4711yescharacteristic
293C2×C4#1309405312yes—
294C2×C4#1309405612yes—
295C8—82no—
296C8—3762no—
297C22—474no—
298C22—942no—
299C22—944no—
300C22—942no—
301C22—944no—
302C22C22⋊C8×D4711yescharacteristic
303C22#99×D4711yescharacteristic
304C22#100×D4711yescharacteristic
305C22#1568913012yes—
306C22#1568913812yes—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—42no—
312C4—81no—
313C4—942no—
314C4—3761no—
315C4—1882no—
316C4—1882no—
317C2—944no—
318C2#323×D4711yescharacteristic
319C2#2024178112yes—
320C2#2024191912yes—
321C2#2024201011yescharacteristic
322C2#2024211211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7595
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241213C2semidirect
5. C47×#1272C2semidirect
24. C2×Q8×D47C8semidirect
160. #1272D47semidirect
180. C47#7595semidirect

Maximal subgroups

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

subgroupindex
#391×D472
#202434552
#202412142
#202412132
C47×#12722
#759547