Finite Group Databasebeta

← #81603

Subgroups — #81603

Order 384.

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

normal subgroups only

This group has 438 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#5725C211yesmaximal, characteristic, complemented
2#5725C211yesmaximal, characteristic, complemented
3C3×#487C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5#5085C211yesmaximal, characteristic, complemented
6#4819C214yesmaximal, complemented
7#4810C214yesmaximal, complemented
8#4695C211yesmaximal, characteristic, complemented
9#4685C211yesmaximal, characteristic, complemented
10#3207—31nomaximal, Sylow
11C2×#239—21no—
12C2×#239C2212yes—
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4⋊Q8C2211yescharacteristic
15#947—21no—
16#947C2212yes—
17C2×#215C2214yes—
18C2×D24C2214yes—
19#895—21no—
20#895—21no—
21#895—21no—
22#895C2212yes—
23#893—21no—
24#893—21no—
25#893—21no—
26#893C2212yes—
27#874C2211yescharacteristic
28#874C2211yescharacteristic
29C42×D3C2211yescharacteristic
30C3×#103C2214yes—
31C3×C4×C8C2211yescharacteristic
32(C2×D3)⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#532—31no—
37#532—31no—
38#493—31no—
39#492—31no—
40#487D311yescharacteristic, complemented
41#486—31no—
42#463—34no—
43#449—34no—
44#431—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47#239—44no—
48#239—44no—
49C2×D12—22no—
50C2×D12—22no—
51C2×D12—21no—
52C2×D12—21no—
53C2×D12—21no—
54C2×D12—21no—
55C2×D12C2312yes—
56C2×D12C2312yes—
57C2×C4×D3—22no—
58C2×C4×D3—22no—
59C2×C4×D3D411yescharacteristic
60C2×C4×D3D411yescharacteristic
61C2×C4×D3C2311yescharacteristic
62C2×C3×C8C2312yes—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2312yes—
67C3×C42C2311yescharacteristic
68#215—44no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73C4×Dic3D411yescharacteristic
74C4×Dic3D411yescharacteristic
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2312yes—
77D24—44no—
78C2×#41—32no—
79C2×#41—61no—
80C2×SD16—34no—
81C2×D8—34no—
82C4⋊Q8C2×D311yescharacteristic
83C4⋊Q8C2×D311yescharacteristic
84#127—31no—
85#127—31no—
86#124—32no—
87#124—61no—
88#121—32no—
89#121—61no—
90#121—61no—
91#121—61no—
92C4×D4—32no—
93C4×D4—61no—
94C4×D4—61no—
95C4×D4—61no—
96C2×C42—31no—
97C4⋊C8—32no—
98#103C2×D314yes—
99#102—34no—
100C22⋊C8—32no—
101C4×C8—31no—
102C4×C8C2×D311yescharacteristic
103C22×D3—22no—
104C22×D3—22no—
105C22×D3C2×D411yescharacteristic
106C3×Q8—24no—
107C3×Q8—42no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2411yescharacteristic
113C2×Dic3C2×D411yescharacteristic
114C2×Dic3C2×D411yescharacteristic
115C2×Dic3C2×D411yescharacteristic
116D12—24no—
117D12—24no—
118D12—44no—
119D12—44no—
120D12—42no—
121D12—42no—
122D12—42no—
123D12—42no—
124C4×D3—22no—
125C4×D3—22no—
126C4×D3—22no—
127C4×D3—44no—
128C4×D3—44no—
129C3×C8—22no—
130C3⋊C8—22no—
131#41—124no—
132#41—124no—
133C2×Q8—21no—
134C2×Q8C22×D312yes—
135C2×D4—32no—
136C2×D4—32no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4—61no—
140C2×D4—61no—
141C2×D4—61no—
142C2×D4—61no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—62no—
147C22×C4—62no—
148SD16—124no—
149D8—124no—
150C2×C8—32no—
151C2×C8C22×D312yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C22×D312yes—
156C22⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C42—31no—
161C42—31no—
162C42—31no—
163C42C22×D311yescharacteristic
164C22×C3C22×D411yescharacteristic
165C2×D3—21no—
166C2×D3—21no—
167C2×D3—21no—
168C2×D3—42no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4#13512yes—
177C3×C4AGL(1,ℤ/8)12yes—
178C3×C4H(5,𝔽2)12yes—
179Dic3—21no—
180Dic3—21no—
181Dic3—21no—
182C23—31no—
183C23—62no—
184C23—62no—
185Q8—24no—
186Q8—42no—
187D4—64no—
188D4—64no—
189D4—124no—
190D4—124no—
191D4—122no—
192D4—122no—
193D4—122no—
194D4—122no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—124no—
204C2×C4—124no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—22no—
209C8—62no—
210C2×C3#53511yescharacteristic
211C2×C3C2×#13511yescharacteristic
212C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
213D3—41no—
214D3—82no—
215D3—82no—
216C22—61no—
217C22—61no—
218C22—61no—
219C22—122no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22—122no—
225C22C2×D3×D411yescharacteristic
226C4—42no—
227C4—42no—
228C4—61no—
229C4—61no—
230C4—61no—
231C4#91412yes—
232C4#91912yes—
233C4#100912yes—
234C3#320711yescharacteristic, complemented, Sylow
235C2—121no—
236C2—242no—
237C2—242no—
238C2#567411yescharacteristic
239C2C2×#91411yescharacteristic
240C2C2×#91911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
hypercenter#487D3
hypocenterC3#3207
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #5725C2semidirect
2. #5725C2semidirect
3. C3×#487C2semidirect
4. #5088C2semidirect
5. #5085C2semidirect
6. #4819C2semidirect
7. #4810C2semidirect
8. #4695C2semidirect
9. #4685C2semidirect
40. #487D3semidirect
234. C3#3207semidirect

Maximal subgroups

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

subgroupindex
#57252
C3×#4872
#50882
#50852
#48192
#48102
#46952
#46852
#32073