Finite Group Databasebeta

← #78303

Subgroups — #78303

Order 384.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#400C211yesmaximal, characteristic, complemented
2#4994C212yesmaximal, complemented
3#4991C211yesmaximal, characteristic
4#4975C211yesmaximal, characteristic, complemented
5#4672C212yesmaximal
6#4670C212yesmaximal, complemented
7#4670C212yesmaximal, complemented
8#4669C212yesmaximal, complemented
9#4654C212yesmaximal, complemented
10#2795—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×(C2×D3)⋊C4C2212yes—
15C2×(C2×D3)⋊C4C2212yes—
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2212yes—
26C3×#95C2212yes—
27C3×#95C2211yescharacteristic
28C3×#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#400D311yescharacteristic, complemented
33#396—32no—
34#394—31no—
35#394—32no—
36#388—32no—
37#388—32no—
38#388—32no—
39#387—32no—
40#384—31no—
41C23×D3C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×D12—42no—
51C2×D12—41no—
52C3×C4⋊C4—41no—
53C3×C4⋊C4—41no—
54(C2×D3)⋊C4—24no—
55(C2×D3)⋊C4—24no—
56(C2×D3)⋊C4—24no—
57(C2×D3)⋊C4—42no—
58(C2×D3)⋊C4—42no—
59(C2×D3)⋊C4—42no—
60C4⋊Dic3—24no—
61C4⋊Dic3—41no—
62Dic3⋊C4—42no—
63Dic3⋊C4—42no—
64C4×Dic3—22no—
65C4×Dic3—42no—
66C22×D4—31no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C42—31no—
80C2×C42—32no—
81#95—31no—
82#95—32no—
83#95—32no—
84#95C2×D312yes—
85#95C2×D311yescharacteristic
86#95C2×D312yes—
87C23×C3C2411yescharacteristic
88C22×D3—22no—
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4#4112yes—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3#4114yes—
117D12—44no—
118C24—32no—
119C2×D4—121no—
120C2×D4—122no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4C22×D312yes—
125C22×C4C22×D312yes—
126C22×C4C22×D311yescharacteristic
127C22×C4C22×D311yescharacteristic
128C22×C4C22×D311yescharacteristic
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—64no—
132C4⋊C4—122no—
133C4⋊C4—122no—
134C4⋊C4—121no—
135C22⋊C4—64no—
136C22⋊C4—64no—
137C22⋊C4—64no—
138C22⋊C4—122no—
139C22⋊C4—122no—
140C22⋊C4—122no—
141C42—62no—
142C42—122no—
143C22×C3C2×#4111yescharacteristic
144C22×C3C2×#4112yes—
145C22×C3H(5,𝔽2)11yescharacteristic
146C22×C3H(5,𝔽2)12yes—
147C22×C32−1+411yescharacteristic
148C2×D3—42no—
149C2×D3—42no—
150C2×D3—42no—
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C3×C4—22no—
156C3×C4—42no—
157C3×C4—42no—
158C3×C4—41no—
159C3×C4—41no—
160Dic3—24no—
161Dic3—42no—
162Dic3—42no—
163C23—62no—
164C23—62no—
165C23—62no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—62no—
170C23C23×D311yescharacteristic
171D4—124no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—34no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4#23912yes—
194C2×C3#53911yescharacteristic
195C2×C3#54212yes—
196C2×C3#55312yes—
197C2×C3#55511yescharacteristic
198C2×C3#56011yescharacteristic
199D3—82no—
200C22—122no—
201C22—122no—
202C22—122no—
203C22—122no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22#100411yescharacteristic
208C22C2×#23911yescharacteristic
209C22D3×#4112yes—
210C22#100912yes—
211C22#101011yescharacteristic
212C4—22no—
213C4—41no—
214C4—41no—
215C4—42no—
216C4—42no—
217C4—64no—
218C4—122no—
219C4—122no—
220C3#279511yescharacteristic, complemented, Sylow
221C2—242no—
222C2#564511yescharacteristic
223C2#566611yescharacteristic
224C2#567011yescharacteristic
225C2#569512yes—
226C2#571012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2795
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C3×#400C2semidirect
2. #4994C2semidirect
4. #4975C2semidirect
6. #4670C2semidirect
7. #4670C2semidirect
8. #4669C2semidirect
9. #4654C2semidirect
32. #400D3semidirect
220. C3#2795semidirect

Maximal subgroups

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

subgroupindex
C3×#4002
#49942
#49912
#49752
#46722
#46702
#46692
#46542
#27953