Finite Group Databasebeta

← #13173177

Subgroups — #13173177

Order 1728.

684 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×C3⋊Dic3C212yesmaximal, complemented
3#10596283C311yesmaximal, characteristic, complemented
4#5812⋊C3—34nomaximal
5C2×C32×SL(2,𝔽3)C412yescomplemented
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3×#3757—41nomaximal
8C22×C32×Q8C2×C311yescharacteristic
9C22×C3×SL(2,𝔽3)—24no—
10C22×C3×SL(2,𝔽3)D314yes—
11Q8×C3⋊Dic3—32no—
12Q8×C3⋊Dic3C2×C312yescomplemented
13#64135—31no—
14#64135—31no—
15Dic3×SL(2,𝔽3)—38no—
16C23×C33—41no—
17C2×C3×C3⋊Dic3—42no—
18C32×SL(2,𝔽3)C2×C411yescharacteristic
19#5812—34no—
20#519⋊C3—91no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8C3×C412yescomplemented
24C2×C32×Q8C22×C311yescharacteristic
25C22×C32×C4—31no—
26C2×C3×SL(2,𝔽3)—28no—
27C2×C3×SL(2,𝔽3)—24no—
28C2×C3×SL(2,𝔽3)Dic318yescomplemented
29C2×C3×SL(2,𝔽3)C2×D314yes—
30#3757—31no—
31#3757A411yescharacteristic
32C4⋊(C3⋊Dic3)—32no—
33C4⋊(C3⋊Dic3)—32no—
34C4×C3⋊Dic3—32no—
35C4×C3⋊Dic3—32no—
36C3×#217—124no—
37C22×C33—41no—
38C22×C33—42no—
39C22×C33—82no—
40C3×C3⋊Dic3—82no—
41C22×C3×Q8C3×D314yes—
42C22×SL(2,𝔽3)—24no—
43C22×SL(2,𝔽3)C3⋊D311yescharacteristic
44Q8×Dic3—38no—
45Q8×Dic3—98no—
46#926—94no—
47#926—94no—
48C4×SL(2,𝔽3)—92no—
49C23×C32—44no—
50C23×C32—84no—
51C23×C32C2×A411yescharacteristic
52C32×Q8—32no—
53C32×Q8—31no—
54C32×Q8—32no—
55C32×Q8C2×C3×C411yescharacteristic
56C2×C32×C4—32no—
57C2×C32×C4—32no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C3⋊Dic3—32no—
61C2×C3⋊Dic3C2×A412yes—
62C2×C3×Dic3—128no—
63C3×SL(2,𝔽3)—24no—
64C3×SL(2,𝔽3)C2×Dic314yes—
65#519—91noSylow
66C2×C33—41no—
67C2×C33—42no—
68C2×C33—82no—
69C2×C3×Q8—34no—
70C2×C3×Q8—38no—
71C2×C3×Q8C3×Dic318yescomplemented
72C2×C3×Q8C2×C3×D314yes—
73C22×C3×C4—34no—
74C2×SL(2,𝔽3)—28no—
75C2×SL(2,𝔽3)—24no—
76C2×SL(2,𝔽3)C3⋊Dic312yescomplemented
77C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
78C3×C22⋊C4—361no—
79#217—34no—
80#217—94no—
81C4⋊Dic3—98no—
82C4⋊Dic3—98no—
83C4×Dic3—98no—
84C4×Dic3—98no—
85C22×C32—22no—
86C22×C32—44no—
87C22×C32—48no—
88C22×C32—88no—
89C22×C32—816no—
90C22×C32—84no—
91C22×C32—88no—
92C22×C32C4×A412yes—
93C22×C32C22×A411yescharacteristic
94C32×C4—31no—
95C32×C4—32no—
96C32×C4—31no—
97C3⋊Dic3—22no—
98C3⋊Dic3—62no—
99C3×Dic3—248no—
100C22×Q8C3×C3⋊D311yescharacteristic
101C4×Q8—92no—
102C4×Q8—272no—
103#117—271no—
104#117—271no—
105C33—41noSylow
106C23×C3—41no—
107C23×C3—84no—
108C23×C3D3×A414yes—
109C3×Q8—38no—
110C3×Q8—34no—
111C3×Q8—38no—
112C3×Q8C2×C3×Dic314yes—
113C2×C3×C4—34no—
114C2×C3×C4—34no—
115C2×C3×C4—38no—
116C2×C3×C4—38no—
117C2×C3×C4—362no—
118C2×Dic3—38no—
119C2×Dic3—98no—
120SL(2,𝔽3)—24no—
121SL(2,𝔽3)C2×C3⋊Dic311yescharacteristic
122C2×C32—22no—
123C2×C32—48no—
124C2×C32—44no—
125C2×C32—88no—
126C2×C32—816no—
127C2×C32—88no—
128C2×C32—84no—
129C2×C32C2×C4×A411yescharacteristic
130C2×C322−1+4⋊C312yes—
131C2×Q8—31no—
132C2×Q8—32no—
133C2×Q8C3×C3⋊Dic312yescomplemented
134C2×Q8C2×C3×C3⋊D311yescharacteristic
135C22×C4—31no—
136C4⋊C4—272no—
137C4⋊C4—272no—
138C22⋊C4—91no—
139C22⋊C4—271no—
140C42—272no—
141C42—272no—
142C22×C3—28no—
143C22×C3—41no—
144C22×C3—42no—
145C22×C3—82no—
146C22×C3—816no—
147C22×C3—84no—
148C22×C3—88no—
149C22×C3Dic3×A418yes—
150C22×C3C2×D3×A414yes—
151C3×C4—34no—
152C3×C4—38no—
153C3×C4—34no—
154C3×C4—722no—
155Dic3—68no—
156Dic3—188no—
157C32—44no—
158C32—84no—
159C32#519⋊C311yescharacteristic, complemented
160C23A4×C3⋊D311yescharacteristic
161Q8—32no—
162Q8—31no—
163Q8—32no—
164Q8C2×C3×C3⋊Dic311yescharacteristic
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—32no—
168C2×C4—32no—
169C2×C4—92no—
170C2×C4—272no—
171C2×C3—28no—
172C2×C3—42no—
173C2×C3—41no—
174C2×C3—82no—
175C2×C3—816no—
176C2×C3—88no—
177C2×C3—84no—
178C2×C3#1007⋊C318yes—
179C2×C3C2×Dic3×A414yes—
180C22—22no—
181C22A4×C3⋊Dic312yes—
182C22C2×A4×C3⋊D311yescharacteristic
183C4—31no—
184C4—31no—
185C4—32no—
186C4—182no—
187C4—542no—
188C3—41no—
189C3—84no—
190C3#5812⋊C314yescomplemented
191C2—22no—
192C2#1172449012yes—
193C2C2×A4×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, hypocenterC32×Q8C2×C3×C4
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C32×Q8C22×C3
hypercenterC23A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. #10596283C3semidirect
5. C2×C32×SL(2,𝔽3)C4semidirect
12. Q8×C3⋊Dic3C2×C3semidirect
23. C2×C32×Q8C3×C4semidirect
28. C2×C3×SL(2,𝔽3)Dic3semidirect
71. C2×C3×Q8C3×Dic3semidirect
76. C2×SL(2,𝔽3)C3⋊Dic3semidirect
133. C2×Q8C3×C3⋊Dic3semidirect
159. C32#519⋊C3semidirect
190. C3#5812⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
#105962833
#5812⋊C33
C3×#37574