Finite Group Databasebeta

← #74346

Subgroups — #74346

Order 384.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#888C211yesmaximal, characteristic
3C3×#384C211yesmaximal, characteristic
4#4982C214yesmaximal
5#4979C214yesmaximal
6#4977C211yesmaximal, characteristic
7#4975C211yesmaximal, characteristic
8#4936C211yesmaximal, characteristic
9#4932C211yesmaximal, characteristic
10#2660—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2214yes—
15C2×C3×C42C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2214yes—
22C2×C4×Dic3C2211yescharacteristic
23C2×C4×Dic3C2211yescharacteristic
24#890—24no—
25#888—24no—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×#125—31no—
32#398—34no—
33#395—34no—
34#384—31no—
35#384D311yescharacteristic
36#382—31no—
37#382—31no—
38#376—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—24no—
49C2×Dic6—44no—
50C3×C4⋊C4—22no—
51C3×C4⋊C4—22no—
52C3×C4⋊C4—44no—
53C3×C42Q814yescomplemented
54C4⋊Dic3—22no—
55C4⋊Dic3—22no—
56C4⋊Dic3—24no—
57Dic3⋊C4—24no—
58Dic3⋊C4—24no—
59Dic3⋊C4—44no—
60C4×Dic3D414yes—
61C4×Dic3Q814yescomplemented
62C22×Q8—32no—
63C4⋊Q8—64no—
64#125—64no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—34no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D314yes—
74C2×C42—31no—
75C2×C42—31no—
76C2×C42C2×D311yescharacteristic
77#95—32no—
78#95—32no—
79#95—34no—
80#95—34no—
81C23×C3C2411yescharacteristic
82C2×C3×C4—22no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—24no—
87C2×C3×C4—24no—
88C2×C3×C4C2×D412yes—
89C2×C3×C4C2×Q814yes—
90C2×C3×C4C2×Q812yes—
91C2×C3×C4C2×Q812yes—
92C2×Dic3—22no—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—24no—
99C2×Dic3—24no—
100C2×Dic3C2×D412yes—
101C2×Dic3C2×D412yes—
102C2×Dic3C2×Q814yes—
103Dic6—48no—
104C2×Q8—64no—
105C2×Q8—124no—
106C22×C4—32no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—32no—
110C22×C4—32no—
111C22×C4C22×D312yes—
112C22×C4C22×D312yes—
113C22×C4C22×D312yes—
114C22×C4C22×D311yescharacteristic
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—44no—
118C4⋊C4—64no—
119C4⋊C4—64no—
120C4⋊C4—64no—
121C4⋊C4—62no—
122C4⋊C4—62no—
123C4⋊C4—124no—
124C42—34no—
125C42—34no—
126C42Dic614yescomplemented
127C22×C3C22×D411yescharacteristic
128C22×C3C22×Q811yescharacteristic
129C22×C3C22×Q811yescharacteristic
130C22×C3H(5,𝔽2)12yes—
131C22×C32−1+412yes—
132C3×C4—24no—
133C3×C4—42no—
134C3×C4—42no—
135C3×C4C4⋊Q814yes—
136Dic3—22no—
137Dic3—22no—
138Dic3—24no—
139Dic3—42no—
140Dic3—42no—
141C23C23×D311yescharacteristic
142Q8—128no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—34no—
152C2×C4—62no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4—64no—
159C2×C4—64no—
160C2×C4C2×Dic614yes—
161C2×C4C2×Dic612yes—
162C2×C4D3×D412yes—
163C2×C4D3×Q812yes—
164C2×C3C2×C4⋊Q811yescharacteristic
165C2×C3#53511yescharacteristic
166C2×C3#53611yescharacteristic
167C2×C3#54412yes—
168C2×C3#54412yes—
169C22C22×Dic611yescharacteristic
170C22C2×D3×D411yescharacteristic
171C22#100412yes—
172C22C2×D3×Q811yescharacteristic
173C22#100712yes—
174C4—24no—
175C4—42no—
176C4—42no—
177C4—62no—
178C4—62no—
179C4—64no—
180C4—122no—
181C4—122no—
182C4#88814yes—
183C3#266011yescharacteristic, complemented, Sylow
184C2C2×#88811yescharacteristic
185C2#550012yes—
186C2#571511yescharacteristic
187C2#572411yescharacteristic
188C2#572912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#384C2
hypercenter#384D3
hypocenterC3#2660
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
53. C3×C42Q8semidirect
61. C4×Dic3Q8semidirect
126. C42Dic6semidirect
183. C3#2660semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8882
C3×#3842
#49822
#49792
#49772
#49752
#49362
#49322
#26603