Finite Group Databasebeta

← #82511⋊C3 #11921271

Subgroups — #82511⋊C3 #11921271

Order 1152.

325 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊(#131⋊C3)C212yesmaximal
3#82511C311yesmaximal, characteristic, complemented
4#2875⋊C3—31nomaximal
5C2×C3×#41⋊C3C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C412yes—
7C3⋊(#131⋊C3)C414yescomplemented
8C3×#803—41nomaximal
9C3×C4×#41C2×C311yescharacteristic
10C2×#948C2×C312yes—
11C2×#131⋊C3—32no—
12C4×#41⋊C3—21no—
13C4×#41⋊C3D311yescharacteristic
14#4997—32no—
15C4×#208—31no—
16C2×#803—31no—
17C3×#41⋊C3C2×C412yes—
18C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
19C32×C42—41no—
20C2×C3×C3⋊C8—42no—
21#2875—31noSylow
22C2×C3×#41C22×C311yescharacteristic
23C2×#41⋊C3—21no—
24C2×#41⋊C3C2×D311yescharacteristic
25C3×C4×Q8C3×C412yes—
26C3×C4×D4—32no—
27C3×#117—31no—
28C2×C3×C42—31no—
29#948—34no—
30#948C3×C414yescomplemented
31C2×#208—32no—
32C22×C3⋊C8—32no—
33#131⋊C3—34no—
34C4×SL(2,𝔽3)—22no—
35C4×SL(2,𝔽3)Dic312yes—
36C3×#97—121no—
37#803—31no—
38#803—32no—
39#803A411yescharacteristic
40C4×C3⋊C8—32no—
41C2×C32×C4—42no—
42C2×C32×C4—41no—
43C3×SL(2,𝔽3)C4211yescharacteristic
44C3×C3⋊C8—44no—
45C2×#131—32no—
46C4×#41C3×D311yescharacteristic
47#405—92no—
48C4×M16—91no—
49C2×#97—91no—
50C3×#41—32no—
51C3×#41C2×C3×C412yes—
52C2×C3×Q8C2×C3×C411yescharacteristic
53C2×C3×D4—31no—
54C22×C3×C4—31no—
55C22×C3×C4—32no—
56#41⋊C3—22no—
57#41⋊C3C4×D312yes—
58C2×SL(2,𝔽3)—21no—
59C2×SL(2,𝔽3)C2×Dic311yescharacteristic
60C2×C3×C8—122no—
61C3×C4⋊C4—32no—
62C3×C22⋊C4—32no—
63C3×C42—32no—
64C3×C42—31no—
65C3×C42—41no—
66C3×C42—81no—
67C3×C42C2×A411yescharacteristic
68#208—34no—
69#208—34no—
70C2×C3⋊C8—32no—
71C2×C3⋊C8—34no—
72C2×C3⋊C8—34no—
73C2×C3⋊C8C2×A412yes—
74C22×C32—41no—
75C32×C4—42no—
76C32×C4—82no—
77C2×#41C2×C3×D311yescharacteristic
78#131—34no—
79#131—94no—
80C2×M16—92no—
81C22×C8—92no—
82C4×Q8C3×Dic312yes—
83C4×D4—32no—
84#117—31no—
85C2×C42—31no—
86#97—31no—
87#97—92no—
88#97—91no—
89C4×C8—92no—
90C23×C3—31no—
91C3×Q8—31no—
92C3×Q8C3×C4211yescharacteristic
93C3×D4—32no—
94C3×D4—32no—
95C2×C3×C4—31no—
96C2×C3×C4—32no—
97C2×C3×C4—32no—
98C2×C3×C4—32no—
99C2×C3×C4—41no—
100C2×C3×C4—42no—
101C2×C3×C4—81no—
102C2×C3×C4—82no—
103C2×C3×C4—122no—
104C2×C3×C4C4×A412yes—
105C2×C3×C4C22×A411yescharacteristic
106SL(2,𝔽3)—21no—
107SL(2,𝔽3)C4×Dic311yescharacteristic
108C3×C8—124no—
109C3⋊C8—34no—
110C3⋊C8C4×A414yes—
111C2×C32—41no—
112C2×C32—42no—
113#41—32no—
114#41C3×C4×D312yes—
115C2×Q8C2×C3×Dic311yescharacteristic
116C2×D4—31no—
117C22×C4—31no—
118C22×C4—32no—
119M16—94no—
120M16—94no—
121C2×C8—32no—
122C2×C8—94no—
123C2×C8—94no—
124C2×C8—92no—
125C4⋊C4—32no—
126C22⋊C4—32no—
127C42—31no—
128C42—32no—
129C42D3×A411yescharacteristic
130C22×C3—32no—
131C22×C3—41no—
132C22×C3—62no—
133C22×C3—81no—
134C22×C3C2×C4×A411yescharacteristic
135C3×C4—22no—
136C3×C4—31no—
137C3×C4—31no—
138C3×C4—42no—
139C3×C4—62no—
140C3×C4—82no—
141C3×C4—84no—
142C3×C4—82no—
143C3×C4C2×C4×A412yes—
144C32—41noSylow
145C23—31no—
146Q8—31no—
147Q8C3×C4×Dic311yescharacteristic
148D4—32no—
149D4—32no—
150C2×C4—31no—
151C2×C4—32no—
152C2×C4—32no—
153C2×C4—32no—
154C2×C4—122no—
155C2×C4Dic3×A412yes—
156C2×C4C2×D3×A411yescharacteristic
157C8—34no—
158C8—94no—
159C2×C3—41no—
160C2×C3—42no—
161C2×C3—62no—
162C2×C3—81no—
163C2×C3—82no—
164C2×C3C42×A411yescharacteristic
165C2×C3#568⋊C312yes—
166C22—32no—
167C22—62no—
168C22C2×Dic3×A411yescharacteristic
169C4—22no—
170C4—31no—
171C4—31no—
172C4—62no—
173C4C4×D3×A412yes—
174C3—41no—
175C3—81no—
176C3#2875⋊C311yescharacteristic, complemented
177C2—62no—
178C2C4×Dic3×A411yescharacteristic
179C2#5819⋊C312yes—
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutator, hypocenterC3×Q8C3×C42
FittingC3×C4×#41C2×C3
socleC22×C3C2×C4×A4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×C3×#41C22×C3
hypercenterC42D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #82511C3semidirect
7. C3⋊(#131⋊C3)C4semidirect
30. #948C3×C4semidirect
176. C3#2875⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
C2×C3⋊(#131⋊C3)2
#825113
#2875⋊C33
C3×#8034