Finite Group Databasebeta

← #79867

Subgroups — #79867

Order 384.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D3×D8C211yesmaximal, characteristic
2#5545C212yesmaximal, complemented
3C3×#442C211yesmaximal, characteristic, complemented
4#5152C211yesmaximal, characteristic, complemented
5C4×#213C212yesmaximal, complemented
6#4869C211yesmaximal, characteristic, complemented
7#4860C211yesmaximal, characteristic, complemented
8#4770C212yesmaximal, complemented
9#4756C212yesmaximal, complemented
10#4711C211yesmaximal, characteristic, complemented
11#4708C211yesmaximal, characteristic, complemented
12#2948—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C2×C3×D8C2211yescharacteristic
15C3×C4×D4C2212yescomplemented
16D4×Dic3C2212yescomplemented
17C2×#213C2212yes—
18C4×#67—22no—
19C2×(C2×D3)⋊C4—22no—
20D3×D8C418yescomplemented
21C2×D24C2211yescharacteristic
22C2×ΓL(1,𝔽25)C2211yescharacteristic
23C2×D3×C8C2211yescharacteristic
24#893C2212yescomplemented
25#892C2212yes—
26#67⋊C4—22no—
27D3×C22⋊C4—22no—
28C4×D12C2212yescomplemented
29#872C2211yescharacteristic
30C3×#106C2211yescharacteristic, complemented
31C3×#102C2212yescomplemented
32C3×#97C2211yescharacteristic, complemented
33#832C2212yescomplemented
34#821C2212yescomplemented
35#817C2211yescharacteristic, complemented
36#815C2211yescharacteristic, complemented
37C8×Dic3C2211yescharacteristic, complemented
38#809C2212yescomplemented
39#807C2212yescomplemented
40C4×C3⋊C8C2211yescharacteristic, complemented
41C22×D8—31no—
42#518—32no—
43#442—31no—
44#442—31no—
45#442—31no—
46#442D311yescharacteristic, complemented
47C4×D8—32no—
48C4×D8—32no—
49#427—31no—
50#418—32no—
51#405—31no—
52C23×D3—22no—
53C2×C3×D4C2312yes—
54C22×C3×C4—22no—
55C2×#67—22no—
56C22×Dic3—22no—
57D3×D4—24no—
58D3×D4C2×C418yescomplemented
59C2×D12C2312yes—
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×C4×D3C2311yescharacteristic
63C3×D8C2×C414yescomplemented
64C2×C3×C8C2311yescharacteristic
65C2×C3×C8C2311yescharacteristic
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—22no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#213C2×C418yescomplemented
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4D412yes—
75C4⋊Dic3C2312yes—
76Dic3⋊C4D412yes—
77C4×Dic3—22no—
78C4×Dic3C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80C2×C3⋊C8C2311yescharacteristic
81D24C2×C414yescomplemented
82ΓL(1,𝔽25)—22no—
83D3×C8C2×C414yescomplemented
84C22×D4—32no—
85C2×D8—32no—
86C2×D8—38no—
87C2×D8—31no—
88C2×D8C2×D311yescharacteristic
89C2×M16—31no—
90C22×C8—31no—
91C4×D4—32no—
92C4×D4—32no—
93C4×D4—32no—
94C4×D4—62no—
95C4×D4—62no—
96C4×D4C2×D312yescomplemented
97#117—31no—
98#117—32no—
99C2×C22⋊C4—62no—
100C2×C22⋊C4—62no—
101#107—32no—
102#106—31no—
103#106C2×D311yescharacteristic, complemented
104#102—32no—
105#102—32no—
106#102—32no—
107#102C2×D312yescomplemented
108#97—31no—
109#97C2×D311yescharacteristic, complemented
110C4×C8—31no—
111C4×C8—31no—
112C23×C3—22no—
113C22×D3—22no—
114C22×D3—28no—
115C22×D3—82no—
116C22×D3—82no—
117C22×D3C2×D411yescharacteristic
118C3×D4—22no—
119C3×D4C22×C414yes—
120C2×C3×C4—22no—
121C2×C3×C4—82no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2411yescharacteristic
124#67—28no—
125C2×Dic3—22no—
126C2×Dic3—82no—
127C2×Dic3C2×D412yes—
128C2×Dic3C2×D411yescharacteristic
129D12—22no—
130D12C22×C414yes—
131C4×D3—82no—
132C4×D3—82no—
133C4×D3C22×C412yes—
134C4×D3#4112yes—
135C3×C8—21no—
136C3×C8C22×C412yes—
137C3⋊C8—21no—
138C3⋊C8C22×C412yes—
139C24—62no—
140C2×D4—38no—
141C2×D4—32no—
142C2×D4—64no—
143C2×D4—62no—
144C2×D4C22×D312yes—
145C22×C4—22no—
146C22×C4—31no—
147C22×C4—62no—
148C22×C4—62no—
149C22×C4—62no—
150D8—38no—
151D8—34no—
152D8C4×D314yescomplemented
153M16—62no—
154C2×C8—31no—
155C2×C8—34no—
156C2×C8—31no—
157C2×C8C22×D311yescharacteristic
158C2×C8C22×D311yescharacteristic
159C4⋊C4—32no—
160C4⋊C4—32no—
161C4⋊C4C22×D312yes—
162C22⋊C4—22no—
163C22⋊C4—32no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C42—31no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3—24no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3C22×D411yescharacteristic
175C2×D3—22no—
176C2×D3—24no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—88no—
180C2×D3C4×D412yes—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4C23×C411yescharacteristic
184C3×C4C2×#4111yescharacteristic
185Dic3—22no—
186Dic3—42no—
187Dic3C4×D412yes—
188C23—22no—
189C23—31no—
190C23—68no—
191C23—62no—
192C23—242no—
193C23—242no—
194D4—22no—
195D4—34no—
196D4—68no—
197D4—62no—
198D4C2×C4×D314yes—
199C2×C4—22no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—62no—
205C2×C4—82no—
206C2×C4—242no—
207C2×C4—242no—
208C2×C4—242no—
209C2×C4D3×D412yes—
210C2×C4C23×D311yescharacteristic
211C8—21no—
212C8—32no—
213C8—61no—
214C8C2×C4×D312yes—
215C2×C3—44no—
216C2×C3C2×C4×D411yescharacteristic
217C2×C3#57612yes—
218D3—22no—
219D3—44no—
220C22—24no—
221C22—32no—
222C22—42no—
223C22—42no—
224C22—62no—
225C22—64no—
226C22—122no—
227C22—122no—
228C22—248no—
229C22C2×D3×D411yescharacteristic
230C4—22no—
231C4—32no—
232C4—42no—
233C4—62no—
234C4—122no—
235C4C22×C4×D311yescharacteristic
236C4D3×#4111yescharacteristic
237C3#294811yescharacteristic, complemented, Sylow
238C2—44no—
239C2—62no—
240C2—124no—
241C2C4×D3×D411yescharacteristic
242C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
prime radicalC2×D3×D8C2
cosocleC2×C3×C4C24
hypercenter#442D3
hypocenterC3#2948
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #5545C2semidirect
3. C3×#442C2semidirect
4. #5152C2semidirect
5. C4×#213C2semidirect
6. #4869C2semidirect
7. #4860C2semidirect
8. #4770C2semidirect
9. #4756C2semidirect
10. #4711C2semidirect
11. #4708C2semidirect
15. C3×C4×D4C22semidirect
16. D4×Dic3C22semidirect
20. D3×D8C4semidirect
24. #893C22semidirect
28. C4×D12C22semidirect
30. C3×#106C22semidirect
31. C3×#102C22semidirect
32. C3×#97C22semidirect
33. #832C22semidirect
34. #821C22semidirect
35. #817C22semidirect
36. #815C22semidirect
37. C8×Dic3C22semidirect
38. #809C22semidirect
39. #807C22semidirect
40. C4×C3⋊C8C22semidirect
46. #442D3semidirect
58. D3×D4C2×C4semidirect
63. C3×D8C2×C4semidirect
70. #213C2×C4semidirect
81. D24C2×C4semidirect
83. D3×C8C2×C4semidirect
96. C4×D4C2×D3semidirect
103. #106C2×D3semidirect
107. #102C2×D3semidirect
109. #97C2×D3semidirect
152. D8C4×D3semidirect
237. C3#2948semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D82
#55452
C3×#4422
#51522
C4×#2132
#48692
#48602
#47702
#47562
#47112
#47082
#29483