Finite Group Databasebeta

← #26126594

Subgroups — #26126594

Order 12032.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249209C211yesmaximal, characteristic
2#20243127C211yesmaximal, characteristic
3#20241416C214yesmaximal, complemented
4C47×#1458C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×#13094024C411yescharacteristic
7#15689062C2212yes—
8#15689059C412yes—
9#15689043—21no—
10#15689035—21no—
11#15689021—21no—
12#15688989C411yescharacteristic
13#15688983—21no—
14#15688892—21no—
15#15688892C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C2×#10625329—24no—
22C2×#10625329—22no—
23C2×#10625329—41no—
24C2×#10625329—42no—
25C22×C47⋊C8—41no—
26#13094024—41no—
27#13094024—41no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C47⋊#95—21no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C23×C4×C47D411yescharacteristic
38#117×C47—41no—
39#117×C47—41no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C4⋊C8×C47—41no—
47C22⋊C8×C47—22no—
48C4×C8×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47SD1612yes—
57C22×C4×C47Q1612yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—22no—
63C2×Dic94—42no—
64C2×Dic94—82no—
65C2×C8×C47—42no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—41no—
68C22⋊C4×C47—41no—
69C22⋊C4×C47—41no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47—21no—
73C42×C47—21no—
74C22⋊Dic47—82no—
75C4⋊Dic47—41no—
76C4⋊Dic47—41no—
77C4⋊Dic47—42no—
78C4⋊Dic47—42no—
79Dic47⋊C4—82no—
80#10625329—24no—
81#10625329—44no—
82#10625329—42no—
83C2×C47⋊C8—42no—
84C2×C47⋊C8—44no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—41no—
88C23×C47—41no—
89C23×C47#10312yes—
90C23×C47C2×C22⋊C411yescharacteristic
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47#9911yescharacteristic
106C2×C4×C47#9911yescharacteristic
107C2×C4×C47#10312yes—
108C2×C4×C47#10412yes—
109C2×C4×C47C2×SD1611yescharacteristic
110C2×C4×C47C2×Q1611yescharacteristic
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117Dic94—84no—
118C8×C47—82no—
119C47⋊C8—44no—
120#11723—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—22no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—81no—
128C22×C47C2×#9911yescharacteristic
129C22×C47C2×#10311yescharacteristic
130C22×C47C2×#10411yescharacteristic
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—41no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47#32812yes—
138C4×C47#32812yes—
139Dic47—82no—
140Dic47—82no—
141#2628—471no—
142#1742—471no—
143#1458D4711yescharacteristic
144#1353—474no—
145C2×C47—21no—
146C2×C47—21no—
147C2×C47—42no—
148C2×C47C2×#32811yescharacteristic
149C2×C47#212311yescharacteristic
150C2×C47#213411yescharacteristic
151C22×M16—941no—
152C2×C4⋊Q8—471no—
153C2×#117C2×D4711yescharacteristic
154#431—21no—
155#400—472no—
156#399—472no—
157#395—941no—
158#393—941no—
159#389—941no—
160#382—471no—
161#380—941no—
162#343—472no—
163#343—941no—
164#324C2×D4712yes—
165#323—21no—
166C47#1172311yescharacteristic, complemented, Sylow
167C22×Q8—941no—
168C23×C4D18811yescharacteristic
169C2×M16—944no—
170C2×M16—942no—
171C2×M16—1881no—
172C2×M16—1882no—
173C22×C8—1881no—
174C4⋊Q8—1881no—
175C4⋊Q8—1881no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—472no—
179C2×C4⋊C4—942no—
180C2×C4⋊C4—941no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4D18811yescharacteristic
183C2×C4⋊C4C22×D4711yescharacteristic
184C2×C22⋊C4—942no—
185C2×C22⋊C4C4×D4711yescharacteristic
186C2×C22⋊C4C22⋊D4711yescharacteristic
187C2×C42C4×D4711yescharacteristic
188C2×C42C22⋊D4711yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C4×C8—41no—
192#95—472no—
193#95—941no—
194#95—942no—
195#95—942no—
196C24(C2×D47)⋊C411yescharacteristic
197C2×Q8—3762no—
198C2×Q8—1882no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—942no—
203C22×C4—942no—
204C22×C4#1062532512yes—
205C22×C4Dic18812yes—
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4C2×C4×D4711yescharacteristic
210C22×C4C2×D18811yescharacteristic
211C22×C4C2×C22⋊D4711yescharacteristic
212M16—944no—
213M16—1884no—
214M16—1882no—
215C2×C8—42no—
216C2×C8—1882no—
217C2×C8—1884no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—3762no—
221C4⋊C4—1882no—
222C4⋊C4—1881no—
223C4⋊C4—1881no—
224C4⋊C4—1882no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—3762no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1309394912yes—
237C23C2×(C2×D47)⋊C411yescharacteristic
238Q8—3764no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4#1309393811yescharacteristic
260C2×C4#1309393811yescharacteristic
261C2×C4#1309394912yes—
262C2×C4#1309395012yes—
263C2×C4C2×#1062532511yescharacteristic
264C2×C4C2×Dic18811yescharacteristic
265C8—82no—
266C8—1884no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1309393811yescharacteristic
275C22C2×#1309394911yescharacteristic
276C22C2×#1309395011yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—3762no—
284C4—3762no—
285C4#1568886112yes—
286C4#1568886112yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1568886111yescharacteristic
291C2#2024504111yescharacteristic
292C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11723
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20241416C2semidirect
166. C47#11723semidirect

Maximal subgroups

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

subgroupindex
#202492092
#202431272
#202414162
C47×#14582
#1172347