Finite Group Databasebeta

← #11920994

Subgroups — #11920994

Order 1152.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#10593638C211yesmaximal, characteristic
2C3×#504⋊C3C211yesmaximal, characteristic, complemented
3C3⋊#4628C211yesmaximal, characteristic, complemented
4#73038—33nomaximal
5#71452—31nomaximal
6#64266C411yescharacteristic
7C3×#131⋊C3C2211yescharacteristic
8C3⋊GU(2,𝔽3)C411yescharacteristic
9C16×C3⋊D3—41nomaximal
10#5404—33no—
11C3×#504D311yescharacteristic, complemented
12#5144—31no—
13#4901—31no—
14#504⋊C3D313yescomplemented
15#4628—33no—
16#4561—31no—
17#4515—31no—
18#4511—31no—
19Dic3×C16—31no—
20C3×#41⋊C3C2×C411yescharacteristic
21C3⋊GL(2,𝔽3)C811yescharacteristic
22C3⋊#226C811yescharacteristic
23C8×C3⋊D3—41no—
24C32×C16—41no—
25C3⋊(C3⋊C16)—41no—
26#2170—91noSylow
27#985—33no—
28C3×#131C2×D311yescharacteristic
29#950—31no—
30#901—31no—
31#131⋊C3C2×D313yes—
32GU(2,𝔽3)—33no—
33C3×M32—31no—
34C2×C3×C16—31no—
35#837—31no—
36#825—31no—
37#822—31no—
38C8×Dic3—31no—
39#812—31no—
40C2×C3⋊C16—31no—
41#798—61no—
42D3×C16—61no—
43D3×C16—123no—
44C4×C3⋊D3—41no—
45C3×SL(2,𝔽3)C2×C811yescharacteristic
46C32×C8—41no—
47C3⋊(C3⋊C8)—41no—
48#504—91no—
49#504C3⋊D311yescharacteristic, complemented
50#443—91no—
51#364—91no—
52#350—91no—
53#350—91no—
54C4×C16—91no—
55C3×#41C4×D311yescharacteristic
56#235—31no—
57#41⋊C3C4×D313yes—
58GL(2,𝔽3)—33no—
59#226—33no—
60C3×M16—31no—
61C2×C3×C8—31no—
62#216—31no—
63#215—31no—
64#214—31no—
65#213—31no—
66C4×Dic3—31no—
67#208—31no—
68C2×C3⋊C8—31no—
69ΓL(1,𝔽25)—61no—
70D3×C8—61no—
71D3×C8—123no—
72C3×C16—31no—
73C3×C16—43no—
74C3×C16S411yescharacteristic
75C3⋊C16—61no—
76C3⋊C16—61no—
77C3⋊C16—123no—
78C2×C3⋊D3—41no—
79C32×C4—41no—
80C3⋊Dic3—41no—
81#135—91no—
82#131—91no—
83#131C2×C3⋊D311yescharacteristic
84M32—31no—
85M32—91no—
86M32—181no—
87C2×C16—31no—
88C2×C16—91no—
89C2×C16—181no—
90#108—91no—
91#104—91no—
92#104—91no—
93C4×C8—91no—
94C3×Q8D3×C811yescharacteristic
95C3×D4—31no—
96C2×C3×C4—31no—
97#67—61no—
98C2×Dic3—61no—
99D12—31no—
100C4×D3—61no—
101C4×D3—123no—
102Dic6—31no—
103SL(2,𝔽3)D3×C813yes—
104C3×C8—31no—
105C3×C8—43no—
106C3×C8C2×S411yescharacteristic
107C3⋊C8—31no—
108C3⋊C8—31no—
109C3⋊C8—61no—
110C3⋊C8—123no—
111C2×C32—41no—
112C3⋊D3—42no—
113#41—91no—
114#41C4×C3⋊D311yescharacteristic
115Q16—91no—
116SD16—91no—
117SD16—91no—
118D8—91no—
119M16—31no—
120M16—91no—
121M16—181no—
122C2×C8—31no—
123C2×C8—91no—
124C2×C8—181no—
125C42—91no—
126C16—31no—
127C16—181no—
128C16—181no—
129C16C3⋊S411yescharacteristic
130C22×C3—31no—
131C2×D3—61no—
132C2×D3—123no—
133C3×C4—31no—
134C3×C4—43no—
135C3×C4C4×S411yescharacteristic
136Dic3—62no—
137Dic3—61no—
138Dic3—123no—
139C32—41noSylow
140Q8—91no—
141Q8C8×C3⋊D311yescharacteristic
142D4—31no—
143D4—91no—
144D4—181no—
145C2×C4—31no—
146C2×C4—181no—
147C2×C4—181no—
148C8—31no—
149C8—91no—
150C8—91no—
151C8—181no—
152C8C2×C3⋊S411yescharacteristic
153C2×C3—43no—
154C2×C3—61no—
155C2×C3C8×S411yescharacteristic
156D3—126no—
157D3—121no—
158C22—31no—
159C22—181no—
160C4—31no—
161C4—182no—
162C4—181no—
163C4C4×C3⋊S411yescharacteristic
164C3—43no—
165C3#7303811yescharacteristic, complemented
166C2—61no—
167C2—361no—
168C2C8×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC16C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C8
FrattiniC8C2×C3⋊S4
FittingC3×#504D3
socleC2×C3C8×S4
prime radical#64266C4
cosocleC3×#131⋊C3C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C16

Complements

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

NH ≅ G/Ntype
2. C3×#504⋊C3C2semidirect
3. C3⋊#4628C2semidirect
11. C3×#504D3semidirect
14. #504⋊C3D3semidirect
49. #504C3⋊D3semidirect
165. C3#73038semidirect

Maximal subgroups

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

subgroupindex
#105936382
C3×#504⋊C32
C3⋊#46282
#730383
#714523
C16×C3⋊D34