Finite Group Databasebeta

← #26126865

Subgroups — #26126865

Order 12032.

464 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13093941C211yesmaximal, characteristic
2#20241726C211yesmaximal, characteristic
3#20241394C214yesmaximal, complemented
4C47×#2486C211yesmaximal, characteristic
5C2×#13093941—21no—
6#15689087—21no—
7C4×#10625329—21no—
8#15688892C2212yes—
9#15688849C2212yes—
10#15688847—21no—
11C2×#123×C47—21no—
12C2×C4×D4×C47C2211yescharacteristic
13C22×C4⋊C4×C47C411yescharacteristic
14C47×#398C2212yes—
15C47×#397C412yes—
16C47×#389—21no—
17C47×#381—21no—
18C47×#380—21no—
19C47×#380C411yescharacteristic
20C2×#10625329—22no—
21C2×#10625329—41no—
22#13093941—24no—
23C22⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—41no—
25#13093935—41no—
26C4×C47⋊C8—41no—
27C22×D4×C47D411yescharacteristic
28C23×C4×C47—21no—
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47D411yescharacteristic
31#123×C47—44no—
32C4×D4×C47—41no—
33C4×D4×C47—41no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47—41no—
38C2×C4⋊C4×C47—41no—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46#95×C47—22no—
47#95×C47—22no—
48#95×C47C2×C412yes—
49C24×C47C22⋊C411yescharacteristic
50C24×C47C22⋊C411yescharacteristic
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C4⋊C4×C47—41no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—44no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—44no—
80C42×C47—21no—
81C42×C47—21no—
82#10625329—24no—
83#10625329—42no—
84#10625329—82no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8—42no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47#10412yes—
97C23×C47C2×C22⋊C411yescharacteristic
98D4×C47—44no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—81no—
118C2×C4×C47—84no—
119C2×C4×C47#9911yescharacteristic
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47AGL(1,ℤ/8)12yes—
122C47⋊C8—44no—
123C47⋊C8—82no—
124#11780—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—22no—
128C22×C47—42no—
129C22×C47—42no—
130C22×C47—42no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—81no—
137C22×C47C2×#9911yescharacteristic
138C22×C47#41811yescharacteristic
139C22×C47C2×#10411yescharacteristic
140C4×C47—21no—
141C4×C47—22no—
142C4×C47—41no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47—82no—
146C4×C47—82no—
147C4×C47#35212yes—
148#2486D4711yescharacteristic
149C4×#100—471no—
150#1465—471no—
151#1346—474no—
152C2×C47—21no—
153C2×C47—21no—
154C2×C47—42no—
155C2×C47—81no—
156C2×C47#150711yescharacteristic
157C2×C47C2×#35211yescharacteristic
158C2×C47#213511yescharacteristic
159C2×#123—21no—
160C2×C4×D4C2×D4711yescharacteristic
161C22×C4⋊C4Dic4711yescharacteristic
162C2×#100—941no—
163#407—941no—
164C4×M16—941no—
165#398C2×D4712yes—
166#397Dic4712yes—
167#389—21no—
168#381—21no—
169#380—21no—
170#380Dic4711yescharacteristic
171#343—472no—
172#324—472no—
173#323—941no—
174C47#1178011yescharacteristic, complemented, Sylow
175C22×D4C22⋊D4711yescharacteristic
176C23×C4—21no—
177C23×C4C2×Dic4711yescharacteristic
178C23×C4C22⋊D4711yescharacteristic
179C2×M16—942no—
180C2×M16—1881no—
181#123—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×Dic4712yes—
190C2×C4⋊C4C22×D4711yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C2×Dic4711yescharacteristic
194C2×C22⋊C4C22⋊D4711yescharacteristic
195C2×C42C22⋊D4711yescharacteristic
196#100—944no—
197C22⋊C8—942no—
198C22⋊C8—1881no—
199#97—1881no—
200C4×C8—1881no—
201#95—22no—
202#95—22no—
203#95C2×Dic4712yes—
204C24C22⋊Dic4711yescharacteristic
205C24C22⋊Dic4711yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊Dic4711yescharacteristic
221C22×C4C22⋊Dic4711yescharacteristic
222C22×C4C22×Dic4711yescharacteristic
223C22×C4C2×C22⋊D4711yescharacteristic
224C22×C4C2×C22⋊D4711yescharacteristic
225M16—944no—
226M16—3762no—
227M16—1882no—
228C2×C8—1882no—
229C2×C8—1882no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1309395112yes—
252C23C2×C22⋊Dic4711yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4#1309393911yescharacteristic
275C2×C4#1309393911yescharacteristic
276C2×C4#1309405012yes—
277C8—3762no—
278C8—1884no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#1309393911yescharacteristic
292C22#1568912811yescharacteristic
293C22C2×#1309395111yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4#1568891312yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2024190411yescharacteristic
307C2C2×#1568891311yescharacteristic
308C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2486C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#2486D47
hypocenterC47#11780
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #20241394C2semidirect
174. C47#11780semidirect

Maximal subgroups

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

subgroupindex
C4×#130939412
#202417262
#202413942
C47×#24862
#1178047