Finite Group Databasebeta

← #26126866

Subgroups — #26126866

Order 12032.

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

normal subgroups only

This group has 464 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#20247696C211yesmaximal, characteristic
2C4×#13093941C211yesmaximal, characteristic
3#20241395C214yesmaximal, complemented
4C47×#1465C211yesmaximal, characteristic
5C2×#13094005—21no—
6C22×C4⋊Dic47C411yescharacteristic
7C2×#13093941—21no—
8C4×#10625329—21no—
9#15689053C2212yes—
10#15689048C412yes—
11#15689021—21no—
12#15688984—21no—
13#15688983—21no—
14#15688983C411yescharacteristic
15#15688892C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C23×Dic47—21no—
21C2×#10625329—22no—
22#13094005—44no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47—41no—
27C2×C4⋊Dic47C2×C412yes—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—21no—
31#13093941—24no—
32#13093935—41no—
33C4×C47⋊C8—41no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C22×D4×C47D411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C2×M16×C47—41no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C22⋊C8×C47—22no—
48C22⋊C8×C47—41no—
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—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—42no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70M16×C47—82no—
71C2×C8×C47—42no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—21no—
77C22⋊Dic47—42no—
78C22⋊Dic47—44no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4⋊Dic47—42no—
83Dic47⋊C4—44no—
84#10625329—24no—
85#10625329—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—81no—
112C2×C4×C47#9911yescharacteristic
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47AGL(1,ℤ/8)12yes—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—84no—
122C8×C47—82no—
123C47⋊C8—44no—
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#35212yes—
146Dic47—82no—
147Dic47—82no—
148#2486—471no—
149C4×#100—471no—
150#1465D4711yescharacteristic
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—941no—
160C2×C4×D4C2×D4711yescharacteristic
161C22×C4⋊C4—471no—
162C2×#100—941no—
163#407—21no—
164C4×M16—941no—
165#398—472no—
166#397—472no—
167#389—941no—
168#381—941no—
169#380—471no—
170#380—941no—
171#343—472no—
172#324C2×D4712yes—
173#323—21no—
174C47#1178011yescharacteristic, complemented, Sylow
175C22×D4D18811yescharacteristic
176C23×C4—941no—
177C23×C4C4×D4711yescharacteristic
178C23×C4C22⋊D4711yescharacteristic
179C2×M16—41no—
180C2×M16—942no—
181#123—1884no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—472no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4—941no—
187C2×C4⋊C4—942no—
188C2×C4⋊C4—1881no—
189C2×C4⋊C4—1881no—
190C2×C4⋊C4C22×D4711yescharacteristic
191C2×C22⋊C4—941no—
192C2×C22⋊C4—942no—
193C2×C22⋊C4C4×D4711yescharacteristic
194C2×C22⋊C4C22⋊D4711yescharacteristic
195C2×C42D18811yescharacteristic
196#100—944no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199#97—1881no—
200C4×C8—1881no—
201#95—472no—
202#95—942no—
203#95—942no—
204C24(C2×D47)⋊C411yescharacteristic
205C24(C2×D47)⋊C411yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—942no—
216C22×C4—942no—
217C22×C4—1882no—
218C22×C4—1882no—
219C22×C4—1882no—
220C22×C4(C2×D47)⋊C411yescharacteristic
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4C2×C4×D4711yescharacteristic
223C22×C4C2×D18811yescharacteristic
224C22×C4C2×C22⋊D4711yescharacteristic
225M16—82no—
226M16—944no—
227M16—1882no—
228C2×C8—42no—
229C2×C8—1882no—
230C4⋊C4—41no—
231C4⋊C4—1882no—
232C4⋊C4—1882no—
233C4⋊C4—1881no—
234C4⋊C4—1881no—
235C4⋊C4—1884no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—1882no—
239C22⋊C4—1884no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1309395012yes—
252C23C2×(C2×D47)⋊C411yescharacteristic
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—81no—
267C2×C4—3764no—
268C2×C4—1882no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4#1309393811yescharacteristic
275C2×C4#1309393811yescharacteristic
276C2×C4#1309405212yes—
277C8—82no—
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×#1309393811yescharacteristic
292C22#1568912911yescharacteristic
293C22C2×#1309395011yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—3762no—
300C4—3762no—
301C4#1568891212yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#2024190511yescharacteristic
307C2C2×#1568891211yescharacteristic
308C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1465C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1465D47
hypocenterC47#11780
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. #20241395C2semidirect
174. C47#11780semidirect

Maximal subgroups

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

subgroupindex
#202476962
C4×#130939412
#202413952
C47×#14652
#1178047