Finite Group Databasebeta

← #83063

Subgroups — #83063

Order 384.

532 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 153 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×C3×#125C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic
3#4991C212yesmaximal
4#4990C214yesmaximal, complemented
5#4987C212yesmaximal, complemented
6#4987C212yesmaximal, complemented
7#4986C212yesmaximal, complemented
8C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#2562—31nomaximal, Sylow
10C22×D12C2211yescharacteristic
11C22×C4×D3C2212yes—
12C3×#125—24no—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2214yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4×Dic3C2212yes—
24C4×D12—24no—
25C3⋊#95C2212yes—
26C3⋊#95C2214yes—
27C3⋊#95C2214yes—
28C2×#125D311yescharacteristic, complemented
29C2×C4×D4—31no—
30#400—32no—
31#394—32no—
32#388—34no—
33#380—32no—
34#380—32no—
35C4×C4⋊C4—31no—
36C23×D3C2312yes—
37C22×C3×C4C2312yes—
38C22×C3×C4C2312yes—
39C22×C3×C4C2311yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3C2312yes—
43C22×Dic3C2314yes—
44C2×D12—24no—
45C2×D12—22no—
46C2×C4×D3—24no—
47C2×C4×D3—24no—
48C2×C4×D3—24no—
49C3×C4⋊C4—24no—
50C3×C4⋊C4—24no—
51C3×C4⋊C4—22no—
52C3×C4⋊C4—22no—
53C3×C42D414yes—
54(C2×D3)⋊C4—28no—
55(C2×D3)⋊C4—24no—
56(C2×D3)⋊C4—44no—
57C4⋊Dic3—22no—
58C4⋊Dic3—24no—
59C4×Dic3—42no—
60C22×D4—31no—
61C23×C4—32no—
62#125—24no—
63C4×D4—64no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4C2×D311yescharacteristic
67C2×C4⋊C4C2×D311yescharacteristic
68C2×C4⋊C4C2×D312yes—
69C2×C4⋊C4C2×D312yes—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—34no—
72C2×C22⋊C4—34no—
73C2×C42—32no—
74C2×C42C2×D311yescharacteristic
75#95—32no—
76#95—34no—
77#95—34no—
78C23×C3C2411yescharacteristic
79C22×D3—22no—
80C22×D3—22no—
81C22×D3—22no—
82C22×D3—22no—
83C22×D3—22no—
84C22×D3—22no—
85C22×D3#4114yes—
86C2×C3×C4—22no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—24no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×D412yes—
96C2×C3×C4C2×D412yes—
97C2×C3×C4#4114yes—
98C2×Dic3—22no—
99C2×Dic3—24no—
100C2×Dic3—24no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—24no—
104D12—28no—
105C4×D3—48no—
106C24—32no—
107C2×D4—64no—
108C2×D4—62no—
109C22×C4—32no—
110C22×C4—34no—
111C22×C4—64no—
112C22×C4—64no—
113C22×C4—64no—
114C22×C4C22×D312yes—
115C22×C4C22×D312yes—
116C22×C4C22×D311yescharacteristic
117C22×C4C22×D311yescharacteristic
118C22×C4C22×D311yescharacteristic
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—64no—
124C4⋊C4—62no—
125C22⋊C4—68no—
126C22⋊C4—64no—
127C22⋊C4—124no—
128C42—122no—
129C42#6714yes—
130C22×C3C22×D411yescharacteristic
131C22×C3C2×#4112yes—
132C22×C3C2×#4112yes—
133C22×C3H(5,𝔽2)11yescharacteristic
134C22×C32−1+411yescharacteristic
135C2×D3—24no—
136C2×D3—24no—
137C2×D3—24no—
138C2×D3—42no—
139C2×D3—42no—
140C2×D3—42no—
141C2×D3—42no—
142C3×C4—22no—
143C3×C4—22no—
144C3×C4—42no—
145C3×C4—42no—
146C3×C4#12414yes—
147Dic3—42no—
148Dic3—44no—
149C23—34no—
150C23—62no—
151C23—62no—
152C23—62no—
153C23—62no—
154C23—62no—
155C23—62no—
156C23C23×D311yescharacteristic
157D4—68no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—64no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—64no—
170C2×C4—64no—
171C2×C4—62no—
172C2×C4—128no—
173C2×C4#23914yes—
174C2×C4C2×#6712yes—
175C2×C4C2×#6712yes—
176C2×C4C2×#6712yes—
177C2×C3#52511yescharacteristic
178C2×C3C2×#12411yescharacteristic
179C2×C3#53711yescharacteristic
180C2×C3#55212yes—
181C2×C3#55312yes—
182D3—44no—
183C22—64no—
184C22—64no—
185C22—64no—
186C22—122no—
187C22—122no—
188C22—122no—
189C22—122no—
190C22C2×#23912yes—
191C22D3×#4112yes—
192C22#100911yescharacteristic
193C22#101011yescharacteristic
194C22C22×#6711yescharacteristic
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—124no—
200C4—122no—
201C4#94714yes—
202C3#256211yescharacteristic, complemented, Sylow
203C2—124no—
204C2#569412yes—
205C2#569512yes—
206C2C2×#94711yescharacteristic
207C2#582811yescharacteristic
208C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#125C2
prime radicalC22×D12C22
hypercenterC2×#125D3
hypocenterC3#2562
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×#125C2semidirect
4. #4990C2semidirect
5. #4987C2semidirect
6. #4987C2semidirect
7. #4986C2semidirect
8. C4×C4⋊Dic3C2semidirect
28. C2×#125D3semidirect
202. C3#2562semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1252
C2×C4×D122
#49912
#49902
#49872
#49862
C4×C4⋊Dic32
#25623