Finite Group Databasebeta

← #26104247

Subgroups — #26104247

Order 12032.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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#20243134C212yesmaximal, complemented
2#20241214C212yesmaximal, complemented
3#20241209C212yesmaximal
4C4×#124×C47C211yesmaximal, characteristic
5C2×C4×C47⋊C8—22no—
6#15688878—22no—
7#15688878C2214yes—
8C2×#124×C47C411yescharacteristic
9C2×C4×Q8×C47C411yescharacteristic
10C2×C4×D4×C47C411yescharacteristic
11C47×#391C2211yescharacteristic
12C47×#390C2211yescharacteristic
13C47×#388—22no—
14C47×#382C411yescharacteristic
15C4×C22⋊C4×C47—22no—
16C43×C47C2211yescharacteristic
17C22×C47⋊C8—24no—
18C4×C47⋊C8—28no—
19C22×Q8×C47C2×C411yescharacteristic
20C22×D4×C47C2×C411yescharacteristic
21C23×C4×C47—21no—
22#124×C47—22no—
23#124×C47C814yescomplemented
24C4×Q8×C47—42no—
25C4×D4×C47—42no—
26C2×C4⋊C4×C47—21no—
27C2×C4⋊C4×C47C2×C411yescharacteristic
28C2×C4⋊C4×C47C2×C411yescharacteristic
29C2×C22⋊C4×C47—22no—
30C2×C22⋊C4×C47—21no—
31C2×C22⋊C4×C47—22no—
32C2×C42×C47—21no—
33C2×C42×C47C2×C411yescharacteristic
34C2×C42×C47C2×C411yescharacteristic
35C2×C42×C47D412yes—
36C2×C42×C47D412yes—
37C2×C42×C47C2311yescharacteristic
38#95×C47—22no—
39#95×C47—22no—
40C24×C47—21no—
41C2×Q8×C47—21no—
42C2×Q8×C47—41no—
43C2×Q8×C47—41no—
44C2×Q8×C47C2×C812yes—
45C2×D4×C47—21no—
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47C2×C812yes—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—42no—
53C22×C4×C47—44no—
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D412yes—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—41no—
62C22⋊C4×C47—28no—
63C22⋊C4×C47—42no—
64C22⋊C4×C47—44no—
65C42×C47—22no—
66C42×C47—24no—
67C42×C47—24no—
68C42×C47—41no—
69C42×C47—42no—
70C42×C47C2×C812yes—
71C42×C47M1612yes—
72C2×C47⋊C8—28no—
73C2×C47⋊C8—44no—
74C2×C47⋊C8—44no—
75C23×C47—22no—
76C23×C47—41no—
77C23×C47—41no—
78C23×C47—42no—
79C23×C47—42no—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—42no—
82Q8×C47—42no—
83D4×C47—42no—
84D4×C47—42no—
85C2×C4×C47—22no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—84no—
101C2×C4×C47C22⋊C814yes—
102C2×C4×C47#10012yes—
103C2×C4×C47#10414yes—
104C2×C4×C47C22×C811yescharacteristic
105C2×C4×C47C2×M1611yescharacteristic
106C2×C4×C47#13514yes—
107C47⋊C8—48no—
108#7567—471nomaximal, Sylow
109C22×C47—42no—
110C22×C47—44no—
111C22×C47—81no—
112C22×C47—82no—
113C22×C47—81no—
114C22×C47C2×C22⋊C811yescharacteristic
115C22×C47#41011yescharacteristic
116C22×C47C2×#10011yescharacteristic
117C22×C47#41612yes—
118C22×C47C2×#10412yes—
119C4×C47—21no—
120C4×C47—21no—
121C4×C47—24no—
122C4×C47—24no—
123C4×C47—41no—
124C4×C47—42no—
125C4×C47—81no—
126C4×C47—81no—
127C4×C47#32914yes—
128C4×#124D4711yescharacteristic
129#1744—472no—
130#1272—472no—
131#1270—472no—
132C2×C47—82no—
133C2×C47#146011yescharacteristic
134C2×C47#147612yes—
135C2×C47#150012yes—
136C2×C47C2×#32911yescharacteristic
137C2×C47#154311yescharacteristic
138C2×#124Dic4711yescharacteristic
139C2×C4×Q8Dic4711yescharacteristic
140C2×C4×D4Dic4711yescharacteristic
141C2×C4×C8—942no—
142#391C2×D4711yescharacteristic
143#390C2×D4711yescharacteristic
144#388—22no—
145#382Dic4711yescharacteristic
146C4×C22⋊C4—22no—
147C43C2×D4711yescharacteristic
148#336—474no—
149#336—942no—
150C47#756711yescharacteristic, complemented, Sylow
151C22×Q8C2×Dic4711yescharacteristic
152C22×D4C2×Dic4711yescharacteristic
153C23×C4—21no—
154C22×C8—944no—
155#124—22no—
156#124C47⋊C814yescomplemented
157C4×Q8—42no—
158C4×D4—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×Dic4711yescharacteristic
161C2×C4⋊C4C2×Dic4711yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42—21no—
166C2×C42C2×Dic4711yescharacteristic
167C2×C42C2×Dic4711yescharacteristic
168C2×C42C22⋊D4712yes—
169C2×C42C22⋊D4712yes—
170C2×C42C22×D4711yescharacteristic
171C4×C8—948no—
172#95—22no—
173#95—22no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×C47⋊C812yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×C47⋊C812yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4C22⋊Dic4711yescharacteristic
189C22×C4C22⋊Dic4712yes—
190C22×C4C22⋊Dic4711yescharacteristic
191C22×C4C22×Dic4711yescharacteristic
192C22×C4C2×C22⋊D4712yes—
193C2×C8—948no—
194C2×C8—1884no—
195C2×C8—1884no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C22⋊C4—28no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C42—22no—
203C42—24no—
204C42—24no—
205C42—41no—
206C42—42no—
207C42C2×C47⋊C812yes—
208C42#1062532912yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊Dic4711yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4C22⋊(C47⋊C8)14yes—
236C2×C4#1309394112yes—
237C2×C4#1309395114yes—
238C2×C4C22×C47⋊C811yescharacteristic
239C2×C4C2×#1062532911yescharacteristic
240C2×C4#1309404414yes—
241C8—1888no—
242C22—42no—
243C22—44no—
244C22—81no—
245C22—81no—
246C22—82no—
247C22C2×C22⋊(C47⋊C8)11yescharacteristic
248C22#1568909711yescharacteristic
249C22C2×#1309394111yescharacteristic
250C22#1568911912yes—
251C22C2×#1309395112yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—24no—
256C4—42no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4#1568886314yes—
261C2—82no—
262C2#2024170911yescharacteristic
263C2#2024177012yes—
264C2#2024187412yes—
265C2C2×#1568886311yescharacteristic
266C2#2024205011yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC4×#124×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC4×#124D47
hypocenterC47#7567
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #20243134C2semidirect
2. #20241214C2semidirect
23. #124×C47C8semidirect
150. C47#7567semidirect
156. #124C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202431342
#202412142
#202412092
C4×#124×C472
#756747