Finite Group Databasebeta

← #74339

Subgroups — #74339

Order 384.

548 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 157 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×#893C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic, complemented
3C3×#384C211yesmaximal, characteristic, complemented
4#4994C214yesmaximal, complemented
5#4989C214yesmaximal, complemented
6#4988C211yesmaximal, characteristic
7#4977C211yesmaximal, characteristic, complemented
8#4939C211yesmaximal, characteristic
9C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
10#2566—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×C4×D3C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17C2×(C2×D3)⋊C4C2214yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2214yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26#893—24no—
27#890—24no—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×#125—31no—
31C2×C4×D4—31no—
32#396—34no—
33#390—31no—
34#390—31no—
35#385—34no—
36#384D311yescharacteristic, complemented
37#382—31no—
38C4×C4⋊C4—31no—
39C23×D3C2312yes—
40C22×C3×C4C2312yes—
41C22×C3×C4C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×Dic3C2312yes—
45C22×Dic3C2312yes—
46C22×Dic3C2311yescharacteristic
47C22×Dic3C2311yescharacteristic
48C2×D12—22no—
49C2×D12—22no—
50C2×D12Q814yes—
51C2×C4×D3—24no—
52C2×C4×D3—44no—
53C3×C4⋊C4—22no—
54C3×C4⋊C4—22no—
55C3×C4⋊C4—44no—
56C3×C42—22no—
57(C2×D3)⋊C4—28no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4—44no—
60C4⋊Dic3—22no—
61C4⋊Dic3—24no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64C4×Dic3—22no—
65C4×Dic3D414yes—
66C22×D4—31no—
67C23×C4—32no—
68#125—64no—
69C4×D4—64no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4C2×D311yescharacteristic
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C4⋊C4C2×D314yes—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—34no—
79C2×C22⋊C4—34no—
80C2×C42—31no—
81C2×C42—31no—
82C2×C42C2×D311yescharacteristic
83#95—32no—
84#95—34no—
85C23×C3C2411yescharacteristic
86C22×D3—22no—
87C22×D3—24no—
88C22×D3—22no—
89C22×D3—24no—
90C22×D3C2×Q814yes—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—24no—
96C2×C3×C4—24no—
97C2×C3×C4C2×D412yes—
98C2×C3×C4C2×Q812yes—
99C2×C3×C4#4112yes—
100C2×C3×C4#4114yes—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—24no—
107C2×Dic3—22no—
108C2×Dic3C2×D412yes—
109C2×Dic3C2×D412yes—
110D12—28no—
111C4×D3—48no—
112C24—32no—
113C2×D4—34no—
114C2×D4—62no—
115C2×D4—62no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—64no—
121C22×C4—124no—
122C22×C4C22×D312yes—
123C22×C4C22×D312yes—
124C22×C4C22×D312yes—
125C22×C4C22×D311yescharacteristic
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—44no—
129C4⋊C4—64no—
130C4⋊C4—64no—
131C4⋊C4—62no—
132C4⋊C4—64no—
133C22⋊C4—64no—
134C22⋊C4—68no—
135C22⋊C4—124no—
136C42—22no—
137C42—34no—
138C42—62no—
139C22×C3C22×D411yescharacteristic
140C22×C3C22×Q811yescharacteristic
141C22×C3C2×#4111yescharacteristic
142C22×C3C2×#4112yes—
143C22×C3H(5,𝔽2)12yes—
144C2×D3—24no—
145C2×D3—28no—
146C2×D3—42no—
147C2×D3—42no—
148C2×D3—44no—
149C3×C4—24no—
150C3×C4—42no—
151C3×C4—42no—
152C3×C4C22⋊Q814yes—
153Dic3—22no—
154Dic3—22no—
155Dic3—42no—
156Dic3—42no—
157C23—34no—
158C23—62no—
159C23—62no—
160C23—64no—
161C23—64no—
162C23C23×D311yescharacteristic
163D4—68no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—64no—
178C2×C4—62no—
179C2×C4—128no—
180C2×C4#23512yes—
181C2×C4D3×D412yes—
182C2×C4D3×Q812yes—
183C2×C4#23914yes—
184C2×C3C2×C22⋊Q811yescharacteristic
185C2×C3#53211yescharacteristic
186C2×C3#53511yescharacteristic
187C2×C3#55312yes—
188C2×C3#55412yes—
189D3—44no—
190C22—68no—
191C22—64no—
192C22—122no—
193C22—122no—
194C22—124no—
195C22C2×#23511yescharacteristic
196C22C2×D3×D411yescharacteristic
197C22#100412yes—
198C22C2×D3×Q811yescharacteristic
199C22C2×#23912yes—
200C4—24no—
201C4—42no—
202C4—42no—
203C4—62no—
204C4—62no—
205C4—122no—
206C4—122no—
207C4#89614yes—
208C3#256611yescharacteristic, complemented, Sylow
209C2—124no—
210C2C2×#89611yescharacteristic
211C2#551412yes—
212C2#571511yescharacteristic
213C2#572511yescharacteristic
214C2#573012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#384C2
prime radicalC22×D12C22
hypercenter#384D3
hypocenterC3#2566
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#890C2semidirect
3. C3×#384C2semidirect
4. #4994C2semidirect
5. #4989C2semidirect
7. #4977C2semidirect
9. C4×Dic3⋊C4C2semidirect
36. #384D3semidirect
208. C3#2566semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×#8902
C3×#3842
#49942
#49892
#49882
#49772
#49392
C4×Dic3⋊C42
#25663