Finite Group Databasebeta

← #70047

Subgroups — #70047

Order 384.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#5830C211yesmaximal, characteristic
2#5819C211yesmaximal, characteristic, complemented
3C3×#410C211yesmaximal, characteristic, complemented
4#5001C212yesmaximal, complemented
5#4531C212yesmaximal, complemented
6#1910—31nomaximal, Sylow
7C2×C3×#41C2211yescharacteristic
8C22×D12C2211yescharacteristic
9C3×#117C2212yes—
10#948—24no—
11#947C412yescomplemented
12#940C412yescomplemented
13#937—22no—
14#930—22no—
15#926C2211yescharacteristic
16C2×#208—22no—
17C2×#208C2212yes—
18C3×#99—22no—
19#809—24no—
20#568—31no—
21#535—31no—
22#418—32no—
23#410D311yescharacteristic, complemented
24H(3,ℤ/4)—32no—
25C23×D3—21no—
26C3×#41—22no—
27C3×#41—22no—
28C2×C3×Q8C2×C411yescharacteristic
29C2×C3×D4C2×C411yescharacteristic
30C2×C3×D4D412yes—
31C22×C3×C4D412yes—
32C22×C3×C4C2311yescharacteristic
33C2×#67—24no—
34C2×D12—22no—
35C2×D12—22no—
36C2×D12—21no—
37C2×D12C2×C412yes—
38C3×C4⋊C4—22no—
39C3×C22⋊C4—22no—
40C3×C42—22no—
41#217—21no—
42(C2×D3)⋊C4—24no—
43C4⋊Dic3—21no—
44C4×Dic3C2×C412yes—
45#208—24no—
46#208—22no—
47#208—22no—
48C2×C3⋊C8—24no—
49C2×C3⋊C8D414yescomplemented
50C2×#41C2×D311yescharacteristic
51C22×D4—31no—
52#131—64no—
53C2×M16—32no—
54C2×M16—62no—
55#127—32no—
56#124—32no—
57#121—62no—
58#120—62no—
59#117—31no—
60#117C2×D312yes—
61#102—64no—
62#99—22no—
63C23×C3C22⋊C412yes—
64C23×C3C2×D411yescharacteristic
65C22×D3—22no—
66C22×D3—42no—
67C22×D3—82no—
68C3×Q8—22no—
69C3×D4—24no—
70C3×D4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—42no—
74C2×C3×C4C22⋊C412yes—
75C2×C3×C4C22×C411yescharacteristic
76C2×C3×C4C2×D411yescharacteristic
77C2×C3×C4C2×D412yes—
78C2×C3×C4#4112yes—
79#67—84no—
80C2×Dic3—22no—
81D12—24no—
82D12—41no—
83D12—41no—
84C3⋊C8—24no—
85C24—61no—
86#41—22no—
87#41—22no—
88C2×Q8C4×D311yescharacteristic
89C2×D4—32no—
90C2×D4—64no—
91C2×D4—61no—
92C2×D4—62no—
93C2×D4—62no—
94C2×D4C4×D311yescharacteristic
95C2×D4#6712yes—
96C22×C4D1212yes—
97C22×C4C22×D311yescharacteristic
98M16—64no—
99M16—62no—
100M16—62no—
101C2×C8—34no—
102C2×C8—64no—
103C4⋊C4—22no—
104C4⋊C4—61no—
105C22⋊C4—22no—
106C22⋊C4—61no—
107C22⋊C4—64no—
108C42—22no—
109C42—32no—
110C22×C3—22no—
111C22×C3—41no—
112C22×C3—42no—
113C22×C3C2×C22⋊C411yescharacteristic
114C22×C3#12112yes—
115C2×D3—44no—
116C2×D3—42no—
117C2×D3—84no—
118C3×C4—22no—
119C3×C4—82no—
120C3×C4C4×D412yes—
121C3×C4#12011yescharacteristic
122C3×C4#12311yescharacteristic
123Dic3—42no—
124C23—62no—
125C23—122no—
126C23—242no—
127C23(C2×D3)⋊C412yes—
128C23C2×#6711yescharacteristic
129Q8—22no—
130D4—24no—
131D4—22no—
132D4—64no—
133D4—121no—
134D4—121no—
135D4—244no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—62no—
140C2×C4(C2×D3)⋊C412yes—
141C2×C4C2×C4×D311yescharacteristic
142C2×C4C2×D1211yescharacteristic
143C2×C4D3×D412yes—
144C2×C4#23712yes—
145C8—64no—
146C2×C3—21no—
147C2×C3—22no—
148C2×C3—42no—
149C2×C3#38611yescharacteristic
150D3—82no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—124no—
155C22—122no—
156C22—244no—
157C22C2×(C2×D3)⋊C411yescharacteristic
158C22#93912yes—
159C4—22no—
160C4—82no—
161C4—122no—
162C4#67⋊C412yes—
163C4#88211yescharacteristic
164C4#88611yescharacteristic
165C3#191011yescharacteristic, complemented, Sylow
166C2—22no—
167C2—21no—
168C2—42no—
169C2—242no—
170C2#495811yescharacteristic
subgroupG/H
centerC2#4958
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#410C2
socleC2×C3#386
prime radical#5830C2
cosocleC22×C3×C4C23
hypercenter#410D3
hypocenterC3#1910
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #5819C2semidirect
3. C3×#410C2semidirect
4. #5001C2semidirect
5. #4531C2semidirect
11. #947C4semidirect
12. #940C4semidirect
23. #410D3semidirect
49. C2×C3⋊C8D4semidirect
165. C3#1910semidirect

Maximal subgroups

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

subgroupindex
#58302
#58192
C3×#4102
#50012
#45312
#19103