Finite Group Databasebeta

← #74338

Subgroups — #74338

Order 384.

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

normal subgroups only

This group has 452 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×#890C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic
3C3×#384C211yesmaximal, characteristic
4#4983C214yesmaximal
5#4980C214yesmaximal
6#4977C211yesmaximal, characteristic
7#4973C211yesmaximal, characteristic
8C4×Dic3⋊C4C211yesmaximal, characteristic
9#4929C211yesmaximal, characteristic
10#2565—31nomaximal, Sylow
11C22×Dic6C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2214yes—
15C2×C3×C42C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2212yes—
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2211yescharacteristic
22C2×C4×Dic3C2212yes—
23C2×C4×Dic3C2211yescharacteristic
24#890—24no—
25#887—24no—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×#125—31no—
31C2×C4×Q8—31no—
32#398—34no—
33#391—31no—
34#391—31no—
35#387—34no—
36#384D311yescharacteristic
37#382—31no—
38C4×C4⋊C4—31no—
39C22×C3×C4C2312yes—
40C22×C3×C4C2312yes—
41C22×C3×C4C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×Dic3C2312yes—
44C22×Dic3C2312yes—
45C22×Dic3C2311yescharacteristic
46C22×Dic3C2311yescharacteristic
47C22×Dic3C2312yes—
48C2×Dic6—22no—
49C2×Dic6—22no—
50C2×Dic6Q814yes—
51C3×C4⋊C4—22no—
52C3×C4⋊C4—22no—
53C3×C4⋊C4—28no—
54C3×C42—22no—
55C4⋊Dic3—22no—
56C4⋊Dic3—24no—
57Dic3⋊C4—24no—
58Dic3⋊C4—24no—
59Dic3⋊C4—24no—
60C4×Dic3—22no—
61C4×Dic3—24no—
62C4×Dic3D414yes—
63C22×Q8—31no—
64#125—64no—
65C4×Q8—64no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D314yes—
74C2×C42—31no—
75C2×C42—32no—
76C2×C42—31no—
77C2×C42C2×D311yescharacteristic
78#95—32no—
79#95—34no—
80#95—34no—
81#95—34no—
82C23×C3C2411yescharacteristic
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—24no—
88C2×C3×C4—24no—
89C2×C3×C4C2×D412yes—
90C2×C3×C4C2×Q812yes—
91C2×C3×C4#4112yes—
92C2×C3×C4#4114yes—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—24no—
101C2×Dic3—22no—
102C2×Dic3C2×D412yes—
103C2×Dic3C2×D412yes—
104C2×Dic3C2×Q814yes—
105Dic6—28no—
106C2×Q8—34no—
107C2×Q8—62no—
108C2×Q8—62no—
109C22×C4—32no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—32no—
113C22×C4—32no—
114C22×C4C22×D312yes—
115C22×C4C22×D312yes—
116C22×C4C22×D312yes—
117C22×C4C22×D311yescharacteristic
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—28no—
121C4⋊C4—64no—
122C4⋊C4—64no—
123C4⋊C4—64no—
124C4⋊C4—62no—
125C4⋊C4—64no—
126C42—22no—
127C42—34no—
128C42—64no—
129C42—62no—
130C22×C3C22×D411yescharacteristic
131C22×C3C22×Q811yescharacteristic
132C22×C3C2×#4111yescharacteristic
133C22×C3C2×#4112yes—
134C22×C32−1+412yes—
135C3×C4—24no—
136C3×C4—42no—
137C3×C4—42no—
138C3×C4C22⋊Q814yes—
139Dic3—24no—
140Dic3—22no—
141Dic3—22no—
142Dic3—42no—
143Dic3—42no—
144C23C23×D311yescharacteristic
145Q8—68no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—32no—
153C2×C4—32no—
154C2×C4—34no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4—62no—
159C2×C4—62no—
160C2×C4—62no—
161C2×C4—62no—
162C2×C4—64no—
163C2×C4—62no—
164C2×C4#23512yes—
165C2×C4D3×D412yes—
166C2×C4#23714yes—
167C2×C4D3×Q812yes—
168C2×C3C2×C22⋊Q811yescharacteristic
169C2×C3#53211yescharacteristic
170C2×C3#53611yescharacteristic
171C2×C3#55212yes—
172C2×C3#55712yes—
173C22C2×#23511yescharacteristic
174C22C2×D3×D411yescharacteristic
175C22C2×#23712yes—
176C22C2×D3×Q811yescharacteristic
177C22#100712yes—
178C4—24no—
179C4—42no—
180C4—42no—
181C4—62no—
182C4—62no—
183C4—64no—
184C4—122no—
185C4—122no—
186C4#89614yes—
187C3#256511yescharacteristic, complemented, Sylow
188C2C2×#89611yescharacteristic
189C2#551312yes—
190C2#571611yescharacteristic
191C2#572112yes—
192C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#384C2
hypercenter#384D3
hypocenterC3#2565
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
187. C3#2565semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8872
C3×#3842
#49832
#49802
#49772
#49732
C4×Dic3⋊C42
#49292
#25653