Finite Group Databasebeta

← C2×C3⋊(C3⋊(#131⋊C3)) #13173181

Subgroups — C2×C3⋊(C3⋊(#131⋊C3)) #13173181

Order 1728.

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

normal subgroups only

This group has 560 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic
2C3⋊(C3⋊(#131⋊C3))C212yesmaximal, direct product factor
3C2×#64157C311yesmaximal, characteristic, complemented
4C2×C3⋊(#131⋊C3)—34nomaximal
5C32×#41⋊C3C411yescharacteristic
6C32×#41⋊C3C2211yescharacteristic
7C2×C32×SL(2,𝔽3)C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)—41nomaximal
9C2×C32×#41C2×C311yescharacteristic
10C2×C3×#41⋊C3—24no—
11C2×C3×#41⋊C3D314yes—
12#64157—32no—
13#64157C2×C312yescomplemented
14C2×#3748—31no—
15C22×C3⋊(C3⋊C8)—31no—
16C3⋊(#131⋊C3)—38no—
17C2×C33×C4—41no—
18C32×SL(2,𝔽3)C2×C411yescharacteristic
19C3×C3⋊(C3⋊C8)—42no—
20C2×#948—34no—
21C2×#131⋊C3—91no—
22C32×#41—31no—
23C32×#41—31no—
24C32×#41C3×C411yescharacteristic
25C32×#41C22×C311yescharacteristic
26C2×C32×Q8C3×C411yescharacteristic
27C2×C32×D4—31no—
28C22×C32×C4—31no—
29C3×#41⋊C3—24no—
30C3×#41⋊C3—24no—
31C3×#41⋊C3Dic314yes—
32C3×#41⋊C3C2×D314yes—
33C2×C3×SL(2,𝔽3)—24no—
34C2×C3×SL(2,𝔽3)Dic314yes—
35#3748—32no—
36#3748—32no—
37C2×C3⋊(C3⋊C8)—31no—
38C2×C3⋊(C3⋊C8)—32no—
39C2×C3⋊(C3⋊C8)—32no—
40C2×C3⋊(C3⋊C8)A411yescharacteristic
41C2×C3×C3⋊C8—124no—
42C22×C33—41no—
43C33×C4—41no—
44C33×C4—41no—
45C2×C3×#41C3×D314yes—
46C2×#41⋊C3—24no—
47C2×#41⋊C3C3⋊D311yescharacteristic
48#948—38no—
49#948—98no—
50C2×#208—94no—
51C22×C3⋊C8—94no—
52#131⋊C3—92no—
53C23×C32—31no—
54C32×Q8—31no—
55C32×Q8C2×C3×C411yescharacteristic
56C32×D4—31no—
57C32×D4—31no—
58C32×D4—32no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—44no—
65C2×C32×C4—84no—
66C2×C32×C4C2×A411yescharacteristic
67C3×SL(2,𝔽3)—24no—
68C3×SL(2,𝔽3)C2×Dic314yes—
69C3⋊(C3⋊C8)—32no—
70C3⋊(C3⋊C8)C2×A412yes—
71C3×C3⋊C8—128no—
72C2×#131—91noSylow
73C2×C33—42no—
74C2×C33—41no—
75C3×#41—34no—
76C3×#41—34no—
77C3×#41C3×Dic314yes—
78C3×#41C2×C3×D314yes—
79C2×C3×Q8C3×Dic314yes—
80C2×C3×D4—34no—
81C22×C3×C4—34no—
82#41⋊C3—24no—
83#41⋊C3—24no—
84#41⋊C3C3⋊Dic311yescharacteristic
85#41⋊C3C2×C3⋊D311yescharacteristic
86C2×SL(2,𝔽3)—24no—
87C2×SL(2,𝔽3)C3⋊Dic311yescharacteristic
88C2×C3×C8—361no—
89#208—98no—
90#208—98no—
91C2×C3⋊C8—34no—
92C2×C3⋊C8—98no—
93C2×C3⋊C8—98no—
94C2×C3⋊C8—94no—
95C22×C32—31no—
96C22×C32—31no—
97C22×C32—44no—
98C22×C32—62no—
99C22×C32—84no—
100C22×C32C4×A411yescharacteristic
101C32×C4—31no—
102C32×C4—31no—
103C32×C4—44no—
104C32×C4—44no—
105C32×C4—84no—
106C32×C4—84no—
107C32×C4C4×A411yescharacteristic
108C32×C4C22×A411yescharacteristic
109C2×#41C3×C3⋊D311yescharacteristic
110#131—92no—
111#131—272no—
112C2×M16—271no—
113C22×C8—271no—
114C33—41noSylow
115C23×C3—34no—
116C3×Q8—34no—
117C3×Q8C2×C3×Dic314yes—
118C3×D4—34no—
119C3×D4—34no—
120C3×D4—38no—
121C2×C3×C4—34no—
122C2×C3×C4—34no—
123C2×C3×C4—34no—
124C2×C3×C4—34no—
125C2×C3×C4—34no—
126C2×C3×C4—41no—
127C2×C3×C4—84no—
128C2×C3×C4D3×A414yes—
129SL(2,𝔽3)—24no—
130SL(2,𝔽3)C2×C3⋊Dic311yescharacteristic
131C3×C8—362no—
132C3⋊C8—38no—
133C3⋊C8—98no—
134C2×C32—48no—
135C2×C32—44no—
136C2×C32—61no—
137C2×C32—61no—
138C2×C32—88no—
139C2×C32—84no—
140C2×C32#131⋊C312yescomplemented
141C2×C32C2×C4×A411yescharacteristic
142#41—31no—
143#41—31no—
144#41C3×C3⋊Dic311yescharacteristic
145#41C2×C3×C3⋊D311yescharacteristic
146C2×Q8C3×C3⋊Dic311yescharacteristic
147C2×D4—31no—
148C22×C4—31no—
149M16—272no—
150M16—272no—
151C2×C8—91no—
152C2×C8—272no—
153C2×C8—272no—
154C2×C8—271no—
155C22×C3—34no—
156C22×C3—34no—
157C22×C3—41no—
158C22×C3—68no—
159C22×C3—84no—
160C22×C3Dic3×A414yes—
161C3×C4—34no—
162C3×C4—34no—
163C3×C4—41no—
164C3×C4—41no—
165C3×C4—84no—
166C3×C4—84no—
167C3×C4Dic3×A414yes—
168C3×C4C2×D3×A414yes—
169C32—44no—
170C32—84no—
171C32C2×#131⋊C311yescharacteristic, complemented
172C23—31no—
173Q8—31no—
174Q8C2×C3×C3⋊Dic311yescharacteristic
175D4—31no—
176D4—31no—
177D4—32no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4A4×C3⋊D311yescharacteristic
184C8—92no—
185C8—272no—
186C2×C3—42no—
187C2×C3—41no—
188C2×C3—64no—
189C2×C3—64no—
190C2×C3—88no—
191C2×C3—84no—
192C2×C3C3⋊(#131⋊C3)18yescomplemented
193C2×C3C2×Dic3×A414yes—
194C22—31no—
195C22—31no—
196C22—62no—
197C22A4×C3⋊Dic311yescharacteristic
198C4—31no—
199C4—31no—
200C4A4×C3⋊Dic311yescharacteristic
201C4C2×A4×C3⋊D311yescharacteristic
202C3—41no—
203C3—84no—
204C3C2×C3⋊(#131⋊C3)14yescomplemented
205C2—61no—
206C2—61no—
207C2C3⋊(C3⋊(#131⋊C3))12yesdirect product factor
208C2C2×A4×C3⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C4A4×C3⋊D3
commutator, hypocenterC32×Q8C2×C3×C4
FrattiniC4C2×A4×C3⋊D3
FittingC2×C32×#41C2×C3
socleC22×C32C4×A4
prime radicalC2×C32×#41⋊C3C2
cosocleC32×#41C22×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(#131⋊C3))C2direct (N × H)
3. C2×#64157C3semidirect
13. #64157C2×C3semidirect
140. C2×C32#131⋊C3semidirect
171. C32C2×#131⋊C3semidirect
192. C2×C3C3⋊(#131⋊C3)semidirect
204. C3C2×C3⋊(#131⋊C3)semidirect
207. C2C3⋊(C3⋊(#131⋊C3))direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3⋊(C3⋊(#131⋊C3))2
C2×#641573
C2×C3⋊(#131⋊C3)3
C2×C3×C3⋊(C3⋊C8)4