Finite Group Databasebeta

← #26124144

Subgroups — #26124144

Order 12032.

1042 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 227 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
1C2×#15689087C212yesmaximal
2C2×#15688892C214yesmaximal, complemented
3C2×C47×#380C211yesmaximal, characteristic
4C22×#10625329C2214yes—
5#15689087—28no—
6C2×C22⋊(C47⋊C8)—22no—
7C2×C22⋊(C47⋊C8)—22no—
8#15688892—28no—
9C24×C4×C47C2211yescharacteristic
10C22×C4⋊C4×C47C2212yes—
11C22×C22⋊C4×C47C412yes—
12C47×#380—22no—
13C47×#380—22no—
14C2×#95×C47C412yes—
15C2×#10625329—24no—
16C2×#10625329—24no—
17C2×#10625329—24no—
18C2×#10625329—24no—
19C2×#10625329—24no—
20C2×#10625329D418yes—
21C22×C47⋊C8—24no—
22C22⋊(C47⋊C8)—28no—
23C22⋊(C47⋊C8)—28no—
24C25×C47C2×C411yescharacteristic
25C23×C4×C47—24no—
26C23×C4×C47—24no—
27C23×C4×C47—24no—
28C23×C4×C47C2×C411yescharacteristic
29C23×C4×C47C2×C414yes—
30C23×C4×C47D412yes—
31C23×C4×C47D412yes—
32C23×C4×C47C2311yescharacteristic
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—24no—
36C2×C4⋊C4×C47—24no—
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—44no—
40#95×C47—24no—
41C24×C47—22no—
42C24×C47—22no—
43C24×C47—22no—
44C24×C47—24no—
45C24×C47—42no—
46C24×C47C22⋊C411yescharacteristic
47C24×C47C22⋊C411yescharacteristic
48C24×C47C22×C411yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—24no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—24no—
54C22×C4×C47—24no—
55C22×C4×C47—24no—
56C22×C4×C47—24no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—24no—
60C22×C4×C47—28no—
61C22×C4×C47—44no—
62C22×C4×C47—44no—
63C22×C4×C47—44no—
64C22×C4×C47—44no—
65C22×C4×C47—44no—
66C22×C4×C47—44no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C2×D414yes—
69C22×C4×C47C2×D412yes—
70C22×C4×C47#4114yes—
71C22×C4×C47#4114yes—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—44no—
74C4⋊C4×C47—42no—
75C22⋊C4×C47—44no—
76C22⋊C4×C47—44no—
77#10625329—28no—
78#10625329—28no—
79#10625329—44no—
80#10625329—44no—
81C2×C47⋊C8—28no—
82C2×C47⋊C8—44no—
83C2×C47⋊C8—44no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—22no—
87C23×C47—28no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47—48no—
97C23×C47—48no—
98C23×C47—44no—
99C23×C47—44no—
100C23×C47#10012yes—
101C23×C47#10112yes—
102C23×C47C2×C22⋊C411yescharacteristic
103C23×C47#11712yes—
104C23×C47C4×D414yes—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—44no—
109C2×C4×C47—44no—
110C2×C4×C47—44no—
111C2×C4×C47—44no—
112C2×C4×C47—44no—
113C2×C4×C47—44no—
114C2×C4×C47—44no—
115C2×C4×C47—44no—
116C2×C4×C47—88no—
117C2×C4×C47#12012yes—
118C2×C4×C47#12114yes—
119C2×C4×C47#12114yes—
120C2×C4×C47#12312yes—
121C2×C4×C47#12314yes—
122C2×C4×C47#12314yes—
123C2×C4×C47#12414yes—
124C2×C4×C47#12614yes—
125C47⋊C8—48no—
126#11198—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—22no—
129C22×C47—24no—
130C22×C47—24no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—24no—
134C22×C47—24no—
135C22×C47—416no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47#38111yescharacteristic
143C22×C47#38612yes—
144C22×C47#38814yes—
145C22×C47C2×#10011yescharacteristic
146C22×C47C2×#10111yescharacteristic
147C22×C47#41312yes—
148C4×C47—42no—
149C4×C47—84no—
150C4×C47#39414yes—
151C4×C47#39714yes—
152C2×#407—472no—
153C2×#380D4711yescharacteristic
154C2×#343—474no—
155C2×C47—24no—
156C2×C47—42no—
157C2×C47—42no—
158C2×C47#144011yescharacteristic
159C2×C47#182111yescharacteristic
160C2×C47#182211yescharacteristic
161C2×C47#190012yes—
162C2×C47#190112yes—
163C24×C4C2×D4711yescharacteristic
164C22×M16—474no—
165C22×C4⋊C4C2×D4712yes—
166C22×C22⋊C4Dic4712yes—
167#407—948no—
168C2×C22⋊C8—942no—
169C2×C22⋊C8—942no—
170#380—22no—
171#380—22no—
172C2×#95Dic4712yes—
173#343—948no—
174C47#1119811yescharacteristic, complemented, Sylow
175C25C2×Dic4711yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2×Dic4711yescharacteristic
180C23×C4C2×Dic4714yes—
181C23×C4C22⋊D4712yes—
182C23×C4C22⋊D4712yes—
183C23×C4C22×D4711yescharacteristic
184C2×M16—478no—
185C2×M16—944no—
186C2×M16—944no—
187C2×M16—944no—
188C2×M16—944no—
189C2×M16—944no—
190C22×C8—944no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C22⋊C8—948no—
199C22⋊C8—948no—
200#95—24no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—24no—
205C24—42no—
206C24C22⋊Dic4711yescharacteristic
207C24C22⋊Dic4711yescharacteristic
208C24C22×Dic4711yescharacteristic
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4C22⋊Dic4712yes—
228C22×C4#1062535014yes—
229C22×C4D4×D4714yes—
230C22×C4#1062535214yes—
231C22×C4C2×C22⋊D4712yes—
232M16—948no—
233M16—948no—
234M16—1884no—
235M16—1884no—
236C2×C8—948no—
237C2×C8—1884no—
238C2×C8—1884no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23#1309394112yes—
261C23#1309394312yes—
262C23C2×C22⋊Dic4711yescharacteristic
263C23#1309398312yes—
264C23D4×Dic4714yes—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4#1309399212yes—
278C2×C4#1309399614yes—
279C2×C4#1309399714yes—
280C2×C4#1309400512yes—
281C2×C4#1309400614yes—
282C2×C4#1309400714yes—
283C2×C4#1309401014yes—
284C2×C4#1309401714yes—
285C8—1888no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—416no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1568898611yescharacteristic
302C22#1568900312yes—
303C22#1568901614yes—
304C22C2×#1309394111yescharacteristic
305C22C2×#1309394311yescharacteristic
306C22#1568910712yes—
307C4—42no—
308C4—84no—
309C4#1568903814yes—
310C4#1568905114yes—
311C2—24no—
312C2—42no—
313C2—42no—
314C2#2024163711yescharacteristic
315C2#2024346811yescharacteristic
316C2#2024347111yescharacteristic
317C2#2024385212yes—
318C2#2024385712yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#380C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#380D47
hypocenterC47#11198
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#15688892C2semidirect
174. C47#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#156890872
C2×#156888922
C2×C47×#3802
#1119847