Finite Group Databasebeta

← #79604

Subgroups — #79604

Order 384.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#911C212yesmaximal, complemented
2#5554C212yesmaximal, complemented
3C3×C4×D8C211yesmaximal, characteristic, complemented
4#5016C212yesmaximal, complemented
5#4886C212yesmaximal, complemented
6#4783C212yesmaximal, complemented
7#4773C212yesmaximal, complemented
8#4696C211yesmaximal, characteristic, complemented
9#4688C211yesmaximal, characteristic, complemented
10#3273—31nomaximal, Sylow
11C2×#237C2212yes—
12C2×D3×D4C2212yescomplemented
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yescomplemented
15C2×#214C2212yescomplemented
16C2×#213C2212yes—
17#930—22no—
18C2×(C2×D3)⋊C4—22no—
19#911—28no—
20C2×Dic12C2211yescharacteristic, complemented
21C2×D24C2211yescharacteristic
22C2×#204C2212yescomplemented
23C2×ΓL(1,𝔽25)C2212yescomplemented
24#897C2212yescomplemented
25#895C2212yes—
26#886—22no—
27#882—22no—
28#875C2212yescomplemented
29C4×D12C2211yescharacteristic
30C3×#107C2211yescharacteristic, complemented
31C3×#102C2212yescomplemented
32C3×C4×C8C2211yescharacteristic, complemented
33(C2×D3)⋊C8C2212yescomplemented
34#810C2212yescomplemented
35#809C2212yescomplemented
36C4⋊(C3⋊C8)C2211yescharacteristic, complemented
37C2×AGL(1,ℤ/8)—32no—
38#546—32no—
39#495—31no—
40#468—32no—
41#463—32no—
42#452—32no—
43#447—32no—
44C4×D8D311yescharacteristic, complemented
45#436—31no—
46C23×D3—22no—
47C2×C3×D4C2312yes—
48C22×C3×C4—22no—
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52#237—24no—
53#237—42no—
54D3×D4—24no—
55D3×D4—42no—
56C2×D12—22no—
57C2×D12D411yescharacteristic
58C2×D12C2312yes—
59C2×C4×D3C2312yes—
60C2×Dic6C2312yes—
61C3×D8D414yescomplemented
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66C3×C42C2311yescharacteristic
67#214—24no—
68#213—24no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4D412yes—
74C4⋊Dic3—22no—
75C4⋊Dic3D411yescharacteristic
76C2×C3⋊C8C2312yes—
77Dic12—41no—
78D24—41no—
79#204—24no—
80ΓL(1,𝔽25)—24no—
81C2×#41—32no—
82C22×D4—32no—
83AGL(1,ℤ/8)—68no—
84C2×Q16—31no—
85C2×SD16—32no—
86C2×SD16—32no—
87C2×D8—32no—
88C2×D8—31no—
89C2×D8C2×D311yescharacteristic
90C2×M16—32no—
91#124—32no—
92#123—62no—
93C22⋊Q8—32no—
94#121—32no—
95#121—62no—
96#120—62no—
97C4×D4—31no—
98C4×D4C2×D312yescomplemented
99C2×C22⋊C4—62no—
100#107C2×D311yescharacteristic, complemented
101C4⋊C8—31no—
102#103—32no—
103#102—32no—
104#102C2×D312yescomplemented
105C22⋊C8—32no—
106C4×C8C2×D311yescharacteristic, complemented
107C23×C3—22no—
108C22×D3—22no—
109C22×D3—44no—
110C22×D3—44no—
111C22×D3—82no—
112C22×D3C2×D412yes—
113C3×D4—22no—
114C3×D4C2×D414yes—
115C2×C3×C4—22no—
116C2×C3×C4—82no—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—44no—
121#67—44no—
122C2×Dic3—22no—
123C2×Dic3—44no—
124C2×Dic3C2×D412yes—
125D12—22no—
126D12—22no—
127D12—41no—
128D12—82no—
129C4×D3—22no—
130C4×D3—22no—
131Dic6—22no—
132Dic6—22no—
133C3×C8—21no—
134C3×C8C2×D412yes—
135C3⋊C8—22no—
136C24—62no—
137#41—64no—
138#41—122no—
139C2×Q8—32no—
140C2×D4—31no—
141C2×D4—32no—
142C2×D4—62no—
143C2×D4—62no—
144C2×D4—64no—
145C2×D4—62no—
146C2×D4—122no—
147C2×D4C22×D312yes—
148C22×C4—22no—
149C22×C4—32no—
150C22×C4—62no—
151Q16—121no—
152SD16—64no—
153SD16—64no—
154D8—64no—
155D8—121no—
156D8D1214yescomplemented
157M16—64no—
158C2×C8—32no—
159C2×C8C22×D311yescharacteristic
160C2×C8C22×D311yescharacteristic
161C4⋊C4—31no—
162C4⋊C4—62no—
163C4⋊C4C22×D312yes—
164C22⋊C4—22no—
165C22⋊C4—32no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C42C22×D311yescharacteristic
171C22×C3—24no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3C22×D411yescharacteristic
175C2×D3—22no—
176C2×D3—22no—
177C2×D3—22no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C2×D3—88no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—42no—
185C3×C4C22×D411yescharacteristic
186C3×C4H(5,𝔽2)11yescharacteristic
187Dic3—22no—
188Dic3—42no—
189C23—22no—
190C23—32no—
191C23—62no—
192C23—124no—
193C23—124no—
194C23—242no—
195Q8—62no—
196Q8—62no—
197D4—22no—
198D4—62no—
199D4—62no—
200D4—121no—
201D4—124no—
202D4—124no—
203D4—242no—
204D4C2×D1214yes—
205C2×C4—22no—
206C2×C4—32no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—82no—
211C2×C4—124no—
212C2×C4D3×D411yescharacteristic
213C2×C4D3×D411yescharacteristic
214C2×C4C23×D311yescharacteristic
215C8—21no—
216C8—62no—
217C8C2×D1212yes—
218C2×C3—44no—
219C2×C3D4211yescharacteristic
220C2×C3#57511yescharacteristic
221C2×C3#57611yescharacteristic
222D3—42no—
223D3—82no—
224C22—24no—
225C22—42no—
226C22—42no—
227C22—62no—
228C22—62no—
229C22—62no—
230C22—122no—
231C22—122no—
232C22—122no—
233C22—248no—
234C22C2×D3×D411yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—62no—
239C4—122no—
240C4C22×D1211yescharacteristic
241C4#100911yescharacteristic
242C3#327311yescharacteristic, complemented, Sylow
243C2—44no—
244C2—122no—
245C2—242no—
246C2D4×D1211yescharacteristic
247C2#575711yescharacteristic
248C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3273
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. #5554C2semidirect
3. C3×C4×D8C2semidirect
4. #5016C2semidirect
5. #4886C2semidirect
6. #4783C2semidirect
7. #4773C2semidirect
8. #4696C2semidirect
9. #4688C2semidirect
12. C2×D3×D4C22semidirect
14. C3×C4×D4C22semidirect
15. C2×#214C22semidirect
20. C2×Dic12C22semidirect
22. C2×#204C22semidirect
23. C2×ΓL(1,𝔽25)C22semidirect
24. #897C22semidirect
28. #875C22semidirect
30. C3×#107C22semidirect
31. C3×#102C22semidirect
32. C3×C4×C8C22semidirect
33. (C2×D3)⋊C8C22semidirect
34. #810C22semidirect
35. #809C22semidirect
36. C4⋊(C3⋊C8)C22semidirect
44. C4×D8D3semidirect
61. C3×D8D4semidirect
98. C4×D4C2×D3semidirect
100. #107C2×D3semidirect
104. #102C2×D3semidirect
106. C4×C8C2×D3semidirect
156. D8D12semidirect
242. C3#3273semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#55542
C3×C4×D82
#50162
#48862
#47832
#47732
#46962
#46882
#32733