Finite Group Databasebeta

← #78339

Subgroups — #78339

Order 384.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#4986C211yesmaximal, characteristic
3#4979C212yesmaximal
4#4977C211yesmaximal, characteristic
5#4659C212yesmaximal
6#4658C212yesmaximal
7#4656C212yesmaximal
8#4656C212yesmaximal
9#4650C212yesmaximal
10#2844—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic3C2212yes—
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×Dic3⋊C4C2212yes—
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C3×#95C2212yes—
22C3×#95C2212yes—
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#400—31no—
33#400—32no—
34#400—32no—
35#400D311yescharacteristic
36#395—32no—
37#395—32no—
38#395—32no—
39#382—31no—
40#382—32no—
41C22×C3×C4C2311yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×Dic6—42no—
51C2×Dic6—42no—
52C2×Dic6—42no—
53C3×C4⋊C4—41no—
54C3×C4⋊C4—41no—
55C4⋊Dic3—41no—
56C4⋊Dic3—42no—
57C4⋊Dic3—41no—
58Dic3⋊C4—42no—
59Dic3⋊C4—42no—
60Dic3⋊C4—42no—
61C4×Dic3Q814yescomplemented
62C22×Q8—32no—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C42—31no—
72#95—32no—
73#95—32no—
74#95—32no—
75#95—32no—
76#95—32no—
77#95—32no—
78#95—32no—
79#95—31no—
80#95C2×D312yes—
81#95C2×D312yes—
82#95C2×D311yescharacteristic
83C23×C3C2411yescharacteristic
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4C2×Q812yes—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3C2×Q814yes—
109Dic6—48no—
110C2×Q8—122no—
111C2×Q8—122no—
112C2×Q8—122no—
113C22×C4—32no—
114C22×C4—32no—
115C22×C4—32no—
116C22×C4—32no—
117C22×C4C22×D311yescharacteristic
118C22×C4C22×D311yescharacteristic
119C22×C4C22×D312yes—
120C22×C4C22×D312yes—
121C22×C4C22×D311yescharacteristic
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—122no—
125C4⋊C4—122no—
126C4⋊C4—122no—
127C4⋊C4—121no—
128C4⋊C4—121no—
129C4⋊C4—122no—
130C42—34no—
131C22×C3C22×Q811yescharacteristic
132C22×C3H(5,𝔽2)11yescharacteristic
133C22×C3H(5,𝔽2)12yes—
134C22×C32−1+412yes—
135C22×C32−1+411yescharacteristic
136C3×C4—22no—
137C3×C4—42no—
138C3×C4—41no—
139C3×C4—41no—
140C3×C4—42no—
141Dic3—24no—
142Dic3—42no—
143Dic3—42no—
144Dic3—42no—
145C23C23×D311yescharacteristic
146Q8—128no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—34no—
160C2×C4—62no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4D3×Q812yes—
172C2×C3#54412yes—
173C2×C3#54411yescharacteristic
174C2×C3H(3,𝔽4)11yescharacteristic
175C2×C3#56311yescharacteristic
176C2×C3#56312yes—
177C22#100411yescharacteristic
178C22C2×D3×Q811yescharacteristic
179C22#100912yes—
180C22#101012yes—
181C22#101011yescharacteristic
182C4—22no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—42no—
187C4—64no—
188C4—122no—
189C4—122no—
190C4—122no—
191C3#284411yescharacteristic, complemented, Sylow
192C2#562311yescharacteristic
193C2#565111yescharacteristic
194C2#568311yescharacteristic
195C2#568912yes—
196C2#571212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2844
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
61. C4×Dic3Q8semidirect
191. C3#2844semidirect

Maximal subgroups

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

subgroupindex
C3×#4002
#49862
#49792
#49772
#46592
#46582
#46562
#46502
#28443