Finite Group Databasebeta

← #11921080

Subgroups — #11921080

Order 1152.

312 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 73 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×C3×#131⋊C3C211yesmaximal, characteristic
2#10593660C212yesmaximal
3#73125—33nomaximal
4#72015—31nomaximal
5C2×C3×#41⋊C3C2211yescharacteristic
6C3×#131⋊C3C412yescomplemented
7C3⋊#859—22no—
8#63644—41nomaximal
9C2×C3×#131D311yescharacteristic
10C2×#131⋊C3D313yes—
11#5426—36no—
12#5232—32no—
13C2×#819—31no—
14C2×#817—31no—
15#4554—32no—
16C3×#41⋊C3C2×C411yescharacteristic
17C3×#41⋊C3Q811yescharacteristic
18C2×C3×SL(2,𝔽3)D411yescharacteristic
19C2×C32×C8—41no—
20C4⋊(C3⋊Dic3)—42no—
21#1815—91noSylow
22C2×C3×#41C2×D311yescharacteristic
23C2×#41⋊C3C2×D313yes—
24C3×#131—32no—
25C3×#131Dic312yescomplemented
26C2×C3×M16—31no—
27C22×C3×C8—31no—
28C2×C4⋊Dic3—32no—
29C2×#208—32no—
30#131⋊C3Dic316yescomplemented
31#859—66no—
32#835—62no—
33#832—62no—
34#819—62no—
35#817—61no—
36#817—61no—
37#817—123no—
38C2×C32×C4—41no—
39C2×C3⋊Dic3—82no—
40C3×SL(2,𝔽3)C4⋊C411yescharacteristic
41C32×C8—42no—
42C2×#131C3⋊D311yescharacteristic
43C2×#108—91no—
44C2×#106—91no—
45#417—92no—
46#343—92no—
47C3×#41—31no—
48C3×#41—31no—
49C3×#41Dic611yescharacteristic
50C3×#41C2×Dic311yescharacteristic
51C2×C3×Q8D1211yescharacteristic
52C2×C3×D4—31no—
53C22×C3×C4—31no—
54C22×Dic3—62no—
55#41⋊C3Dic613yes—
56#41⋊C3C2×Dic313yes—
57C2×SL(2,𝔽3)D1213yes—
58C3×M16—32no—
59C3×M16—61no—
60C2×C3×C8—31no—
61C2×C3×C8—32no—
62C2×C3×C8—32no—
63C2×C3×C8—43no—
64C2×C3×C8S411yescharacteristic
65C4⋊Dic3—62no—
66C4⋊Dic3—62no—
67C4⋊Dic3—126no—
68#208—62no—
69#208—62no—
70C2×C3⋊C8—62no—
71C22×C32—41no—
72C32×C4—41no—
73C32×C4—41no—
74C3⋊Dic3—162no—
75C2×#41C2×C3⋊D311yescharacteristic
76#131—32no—
77#131C3⋊Dic312yescomplemented
78C2×M16—31no—
79C2×M16—92no—
80C22×C8—31no—
81C2×C4⋊C4—92no—
82#108—182no—
83#106—181no—
84#106—181no—
85#103—182no—
86#102—182no—
87C23×C3—31no—
88C3×Q8—31no—
89C3×Q8C4⋊Dic311yescharacteristic
90C3×D4—31no—
91C3×D4—31no—
92C3×D4—61no—
93C2×C3×C4—31no—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C3×C4—31no—
97C2×C3×C4—31no—
98C2×C3×C4—43no—
99C2×C3×C4C2×S411yescharacteristic
100C2×Dic3—122no—
101C2×Dic3—246no—
102C2×Dic3—242no—
103SL(2,𝔽3)C4⋊Dic313yes—
104C3×C8—32no—
105C3×C8—46no—
106C3×C8SL(2,ℤ/4)12yes—
107C3⋊C8—122no—
108C2×C32—42no—
109C2×C32—41no—
110#41—31no—
111#41—31no—
112#41#63411yescharacteristic
113#41C2×C3⋊Dic311yescharacteristic
114C2×Q8C4⋊(C3⋊D3)11yescharacteristic
115C2×D4—31no—
116C22×C4—31no—
117C22×C4—182no—
118M16—32no—
119M16—61no—
120M16—182no—
121M16—182no—
122C2×C8—31no—
123C2×C8—32no—
124C2×C8—32no—
125C2×C8—182no—
126C2×C8C3⋊S411yescharacteristic
127C4⋊C4—182no—
128C4⋊C4—182no—
129C22×C3—31no—
130C22×C3—31no—
131C22×C3—43no—
132C22×C3—62no—
133C22×C3C4⋊S411yescharacteristic
134C3×C4—31no—
135C3×C4—31no—
136C3×C4—43no—
137C3×C4—43no—
138C3×C4A4⋊Q811yescharacteristic
139C3×C4C2×SL(2,ℤ/4)11yescharacteristic
140Dic3—242no—
141Dic3—486no—
142C32—41noSylow
143C23—31no—
144Q8—31no—
145Q8C4⋊(C3⋊Dic3)11yescharacteristic
146D4—31no—
147D4—31no—
148D4—61no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—362no—
155C2×C4—722no—
156C2×C4C2×C3⋊S411yescharacteristic
157C8—32no—
158C8—362no—
159C8C3⋊SL(2,ℤ/4)12yes—
160C2×C3—46no—
161C2×C3—43no—
162C2×C3—61no—
163C2×C3—61no—
164C2×C3#540612yes—
165C2×C3C4⋊SL(2,ℤ/4)11yescharacteristic
166C22—31no—
167C22—31no—
168C22—62no—
169C22C4⋊(C3⋊S4)11yescharacteristic
170C4—31no—
171C4—31no—
172C4—722no—
173C4#6425911yescharacteristic
174C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
175C3—43no—
176C3#7312511yescharacteristic, complemented
177C2—61no—
178C2—61no—
179C2#1059364012yes—
180C2C4⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊S4)
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC2×C3×#131D3
socleC22×C3C4⋊S4
prime radical, cosocleC2×C3×#41⋊C3C22
hypercenterC2×C8C3⋊S4
hypocenterC3×SL(2,𝔽3)C4⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
6. C3×#131⋊C3C4semidirect
25. C3×#131Dic3semidirect
30. #131⋊C3Dic3semidirect
77. #131C3⋊Dic3semidirect
176. C3#73125semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#131⋊C32
#105936602
#731253
#720153
#636444