Finite Group Databasebeta

← #11798434

Subgroups — #11798434

Order 1152.

648 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 117 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#10596223C211yesmaximal, characteristic
2#10592842C211yesmaximal, characteristic, complemented
3#10592608C211yesmaximal, characteristic, complemented
4C2×#63627C211yesmaximal, characteristic, complemented
5C32×#350C211yesmaximal, characteristic, complemented
6#10589382C211yesmaximal, characteristic, complemented
7#10589376C211yesmaximal, characteristic, complemented
8#69128—34nomaximal
9#64366—21no—
10C32×#131C2211yescharacteristic
11#64160—21no—
12#64117—21no—
13#64117—22no—
14C2×#3746C2211yescharacteristic, complemented
15#64113—21no—
16#64113C2211yescharacteristic
17C2×#3745—21no—
18C2×#3745C2211yescharacteristic
19C32×M32C2211yescharacteristic, complemented
20C2×C32×C16C2211yescharacteristic, complemented
21#63651—21no—
22#63646C2211yescharacteristic, complemented
23#63628—21no—
24#63627—22no—
25#63627—21no—
26#5752—34no—
27#5143—34no—
28#4909—34no—
29C2×#800—34no—
30C3×#350D314yescomplemented
31#4516—34no—
32#4510—34no—
33C32×#41D411yescharacteristic
34#3832—41no—
35D4×C3⋊D3—42no—
36D4×C3⋊D3—41no—
37#3828—22no—
38#3828—21no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×#634—21no—
42C32×M16—21no—
43C32×M16D411yescharacteristic
44C2×C32×C8—21no—
45C2×C32×C8C2311yescharacteristic
46#3756—41no—
47#3754—41no—
48#3748—21no—
49#3746—21no—
50#3746—21no—
51#3746D411yescharacteristic
52#3745—22no—
53#3745—21no—
54#3745D411yescharacteristic
55#3745D412yes—
56#3744—22no—
57#3744—21no—
58C32×C16—21no—
59C32×C16—21no—
60#2194—91noSylow
61#1009—64no—
62C3×#131C2×D314yes—
63#951—64no—
64#908—68no—
65#908—64no—
66C2×Dic12—34no—
67#904—34no—
68#904—64no—
69C2×D24—34no—
70C2×D24—64no—
71C3×M32C2×D314yescomplemented
72C2×C3×C16C2×D314yescomplemented
73#824—64no—
74#819—34no—
75Dic24—64no—
76#800—68no—
77#800—64no—
78C22×C3⋊D3—42no—
79C22×C3⋊D3—41no—
80C32×Q8D811yescharacteristic
81C32×D4—21no—
82C32×D4D811yescharacteristic
83C2×C32×C4—21no—
84C2×C32×C4C2×D411yescharacteristic
85#638—42no—
86#638—41no—
87C2×C3⋊Dic3—41no—
88C4⋊(C3⋊D3)—22no—
89C4⋊(C3⋊D3)—21no—
90C4⋊(C3⋊D3)—42no—
91C4⋊(C3⋊D3)—41no—
92C4×C3⋊D3—42no—
93C4×C3⋊D3—41no—
94#634—21no—
95#634—41no—
96#634—41no—
97C32×C8—21no—
98C32×C8C2×D411yescharacteristic
99C32×C8C2×D411yescharacteristic
100C3⋊(C3⋊C8)—41no—
101#576—91no—
102#510—91no—
103C2×SD32—91no—
104#473—91no—
105#361—91no—
106#359—91no—
107#350C3⋊D311yescharacteristic, complemented
108C3×#41D1214yes—
109#239—124no—
110D3×D4—128no—
111D3×D4—124no—
112#235—68no—
113#235—64no—
114C2×D12—68no—
115C2×D12—64no—
116C2×Dic6—64no—
117C3×M16—24no—
118C3×M16D1214yes—
119C2×C3×C8—24no—
120C2×C3×C8C22×D314yes—
121#216—124no—
122#214—124no—
123#208—64no—
124Dic12—34no—
125Dic12—64no—
126Dic12—64no—
127D24—38no—
128D24—34no—
129D24—68no—
130D24—64no—
131#204—68no—
132#204—64no—
133C3×C16—24no—
134C3×C16—24no—
135C22×C32—21no—
136C22×C32AGL(1,ℤ/8)11yescharacteristic
137C2×C3⋊D3—42no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—161no—
140C2×C3⋊D3—162no—
141C32×C4—21no—
142C32×C4#12011yescharacteristic
143C32×C4C2×D811yescharacteristic
144C3⋊Dic3—41no—
145C3⋊Dic3—81no—
146H(5,𝔽2)—181no—
147#137—181no—
148AGL(1,ℤ/8)—182no—
149AGL(1,ℤ/8)—181no—
150#135—91no—
151#135—181no—
152C2×Q16—91no—
153C2×D8—91no—
154C2×D8—181no—
155#131C2×C3⋊D311yescharacteristic
156Q32—181no—
157SD32—182no—
158SD32—181no—
159M32C2×C3⋊D311yescharacteristic, complemented
160C2×C16C2×C3⋊D311yescharacteristic, complemented
161#108—91no—
162#101—181no—
163C22×D3—128no—
164C22×D3—124no—
165C3×Q8D2414yes—
166C3×D4—24no—
167C3×D4D2414yes—
168C2×C3×C4—24no—
169C2×C3×C4C2×D1214yes—
170#67—128no—
171#67—124no—
172C2×Dic3—124no—
173D12—68no—
174D12—64no—
175D12—128no—
176D12—124no—
177C4×D3—128no—
178C4×D3—124no—
179Dic6—64no—
180Dic6—124no—
181Dic6—124no—
182C3×C8—24no—
183C3×C8D3×D414yes—
184C3×C8D3×D414yes—
185C3⋊C8—124no—
186C2×C32—21no—
187C2×C32—41no—
188C2×C32#44711yescharacteristic
189C3⋊D3—82no—
190C3⋊D3—81no—
191#41—182no—
192#41—181no—
193#41—361no—
194#41C4⋊(C3⋊D3)11yescharacteristic
195C2×Q8—181no—
196C2×D4—182no—
197C2×D4—181no—
198C2×D4—362no—
199C2×D4—361no—
200Q16—91no—
201Q16—181no—
202Q16—181no—
203Q16—361no—
204SD16—182no—
205SD16—181no—
206SD16—361no—
207D8—92no—
208D8—91no—
209D8—182no—
210D8—181no—
211M16—21no—
212M16—181no—
213M16C4⋊(C3⋊D3)11yescharacteristic
214C2×C8—21no—
215C2×C8C22×C3⋊D311yescharacteristic
216C16—21no—
217C16—21no—
218C22×C3—24no—
219C22×C3#90814yes—
220C2×D3—128no—
221C2×D3—124no—
222C2×D3—484no—
223C2×D3—488no—
224C3×C4—24no—
225C3×C4#88214yes—
226C3×C4C2×D2414yes—
227Dic3—124no—
228Dic3—244no—
229C32#219411yescharacteristic, complemented, Sylow
230C23—362no—
231C23—361no—
232Q8—181no—
233Q8—361no—
234Q8—361no—
235Q8#374511yescharacteristic
236D4—21no—
237D4—182no—
238D4—181no—
239D4—362no—
240D4—362no—
241D4—361no—
242D4—361no—
243D4#374511yescharacteristic
244C2×C4—21no—
245C2×C4—362no—
246C2×C4—361no—
247C2×C4—361no—
248C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
249C8—21no—
250C8—361no—
251C8D4×C3⋊D311yescharacteristic
252C8D4×C3⋊D311yescharacteristic
253C2×C3—24no—
254C2×C3—44no—
255C2×C3#473214yes—
256D3—248no—
257D3—244no—
258C22—21no—
259C22—362no—
260C22—361no—
261C22—1441no—
262C22—1442no—
263C22#6411711yescharacteristic
264C4—21no—
265C4—361no—
266C4—721no—
267C4#6409111yescharacteristic
268C4C2×#374511yescharacteristic
269C3#6912814yescomplemented
270C2—21no—
271C2—41no—
272C2—722no—
273C2—721no—
274C2#1059243111yescharacteristic
subgroupG/H
centerC2#10592431
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C22×C3⋊D3
FittingC32×#350C2
socleC2×C32#447
prime radical#10596223C2
hypercenter#350C3⋊D3
hypocenterC32#2194
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #10592842C2semidirect
3. #10592608C2semidirect
4. C2×#63627C2semidirect
5. C32×#350C2semidirect
6. #10589382C2semidirect
7. #10589376C2semidirect
14. C2×#3746C22semidirect
19. C32×M32C22semidirect
20. C2×C32×C16C22semidirect
22. #63646C22semidirect
30. C3×#350D3semidirect
71. C3×M32C2×D3semidirect
72. C2×C3×C16C2×D3semidirect
107. #350C3⋊D3semidirect
159. M32C2×C3⋊D3semidirect
160. C2×C16C2×C3⋊D3semidirect
229. C32#2194semidirect
269. C3#69128semidirect

Maximal subgroups

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

subgroupindex
#105962232
#105928422
#105926082
C2×#636272
C32×#3502
#105893822
#105893762
#691283