Finite Group Databasebeta

← (#4177⋊C3)⋊C4 #14516922

Subgroups — (#4177⋊C3)⋊C4 #14516922

Order 1944.

228 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 42 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×#4177⋊C3C211yesmaximal, characteristic
2#66331⋊C3C211yesmaximal, characteristic, complemented
3#11757587C211yesmaximal, characteristic, complemented
4(C3⋊AGL(1,ℤ/9))⋊C4—31nomaximal
5C3×((C3⋊D3)⋊C3)⋊C4—31nomaximal
6#4177⋊C4C311yesmaximal, characteristic, complemented
7C2×#6910C2211yescharacteristic
8#4177⋊C3C412yescomplemented
9C2×C3⋊AGL(1,ℤ/9)D311yescharacteristic
10C2×C3×(C3⋊D3)⋊C3—31no—
11C3×#1084D311yescharacteristic
12Dic3×3−1+2—31no—
13C3×3−1+2⋊C4—31no—
14C3×3−1+2⋊C4—61no—
15C3×(C3⋊Dic3)⋊C3—31no—
16C2×#4177—21no—
17C2×#4177C2×C311yescharacteristic
18#66331C2×C311yescharacteristic, complemented
19#66323—21no—
20#66323C2×C311yescharacteristic, complemented
21#6910C2×C411yescharacteristic, complemented, Sylow
22C3×#624—91no—
23C3×D3×Dic3—91no—
24C4×AGL(1,ℤ/9)—91no—
25((C3⋊D3)⋊C3)⋊C4—31no—
26(C3⋊D9)⋊C4—31no—
27C2×C3×3−1+2—21no—
28C2×C3×3−1+2C2×D311yescharacteristic
29C2×C3×H(3,𝔽3)C2×D311yescharacteristic
30C3⋊AGL(1,ℤ/9)Dic312yescomplemented
31C3×(C3⋊D3)⋊C3—32no—
32C2×#731—21no—
33C2×#731—21no—
34C2×#731—41no—
35C2×#731C22×C311yescharacteristic
36#4177—22no—
37#4177C3×C412yescomplemented
38C2×C3×C3⋊D3—31no—
39C2×C32×D3—91no—
40C3×C3⋊Dic3—31no—
41C32×Dic3—31no—
42C32×Dic3—91no—
43C32×Dic3—91no—
44C2×C3⋊D9C3×D311yescharacteristic
45C2×AGL(1,ℤ/9)—91no—
46C2×(C3⋊D3)⋊C3—31no—
47C2×(C3⋊D3)⋊C3—61no—
48C4×3−1+2—91no—
49#1084C3×D311yescharacteristic
503−1+2⋊C4—31no—
513−1+2⋊C4—61no—
523−1+2⋊C4—61no—
533−1+2⋊C4—61no—
543−1+2⋊C4—61no—
55(C3⋊Dic3)⋊C3—31no—
56(C3⋊Dic3)⋊C3—61no—
57C9×Dic3—31no—
58C9×Dic3—61no—
59C3×Dic9—31no—
60C3×Dic9—61no—
61C3×3−1+2—21no—
62C3×3−1+2C2×Dic311yescharacteristic, complemented
63C3×H(3,𝔽3)C4×D311yescharacteristic
64#731—21no—
65#731—21no—
66#731—41no—
67#731C2×C3×C411yescharacteristic, complemented
68C2×C3×Dic3—271no—
69C3×C4×D3—271no—
70C3×C4×D3—271no—
71#624—91no—
72D3×Dic3—91no—
73C4×D9—91no—
74C2×C33—31no—
75C2×C33D3211yescharacteristic
76C3×C3⋊D3—32no—
77C32×D3—92no—
78C2×3−1+2—21no—
79C2×3−1+2—31no—
80C2×3−1+2—41no—
81C2×3−1+2—61no—
82C2×3−1+2—61no—
83C2×3−1+2—61no—
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—61no—
86C2×H(3,𝔽3)C2×C3×D311yescharacteristic
87C2×C3×C9—21no—
88C2×C3×C9—61no—
89C2×C3×C9C2×C3×D311yescharacteristic
90C3⋊D9C3×Dic312yescomplemented
91AGL(1,ℤ/9)—92no—
92(C3⋊D3)⋊C3—32no—
93(C3⋊D3)⋊C3—62no—
94C22×C32—271no—
95C2×C3⋊D3—31no—
96C2×C3×D3—91no—
97C2×C3×D3—91no—
98C2×C3×D3—181no—
99C2×C3×D3—271no—
100C32×C4—91no—
101C32×C4—271no—
102C3⋊Dic3—31no—
103C3×Dic3—31no—
104C3×Dic3—91no—
105C3×Dic3—91no—
106C3×Dic3—91no—
107C3×Dic3—91no—
108C3×Dic3—91no—
109C3×Dic3—181no—
110C3×Dic3—181no—
111C3×Dic3—271no—
112C2×D9—91no—
113C4×C9—91no—
114C4×C9—181no—
115Dic9—31no—
116Dic9—61no—
117C33—31no—
118C33D3×Dic311yescharacteristic
1193−1+2—21no—
1203−1+2—31no—
1213−1+2—41no—
1223−1+2—61no—
1233−1+2—61no—
1243−1+2—61no—
125H(3,𝔽3)—21no—
126H(3,𝔽3)—61no—
127H(3,𝔽3)C3×C4×D311yescharacteristic
128C3×C9—21no—
129C3×C9—61no—
130C3×C9C2×C3×Dic311yescharacteristic, complemented
131C2×C3×C4—811no—
132C2×Dic3—271no—
133C4×D3—271no—
134C4×D3—271no—
135C2×C32—21no—
136C2×C32—31no—
137C2×C32—61no—
138C2×C32—91no—
139C2×C32—91no—
140C2×C32—181no—
141C2×C32—272no—
142C2×C32((C3⋊D3)⋊C3)⋊C211yescharacteristic
143C2×C32C3×D3211yescharacteristic
144C3⋊D3—32no—
145C3×D3—92no—
146C3×D3—92no—
147C3×D3—182no—
148C3×D3—272no—
149C2×C9—31no—
150C2×C9—61no—
151C2×C9—61no—
152C2×C9—121no—
153D9—92no—
154C22×C3—271no—
155C22×C3—271no—
156C22×C3—541no—
157C2×D3—91no—
158C2×D3—271no—
159C3×C4—91no—
160C3×C4—271no—
161C3×C4—271no—
162C3×C4—271no—
163C3×C4—541no—
164Dic3—31no—
165Dic3—91no—
166Dic3—91no—
167Dic3—271no—
168C32—21no—
169C32—31no—
170C32—61no—
171C32—91no—
172C32—91no—
173C32—181no—
174C32((C3⋊D3)⋊C3)⋊C411yescharacteristic
175C32C3×D3×Dic311yescharacteristic
176C9—31no—
177C9—61no—
178C9—61no—
179C9—121no—
180C2×C4—811noSylow
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—61no—
184C2×C3—91no—
185C2×C3—181no—
186C2×C3—272no—
187C2×C3—272no—
188C2×C3—542no—
189C2×C3C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
190D3—92no—
191D3—272no—
192C22—271no—
193C4—91no—
194C4—271no—
195C3—31no—
196C3—31no—
197C3—61no—
198C3—91no—
199C3—181no—
200C3C3×((C3⋊D3)⋊C3)⋊C411yescharacteristic
201C2—272no—
202C2(#4176⋊C2)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2(#4176⋊C2)⋊C3
commutator, hypocenter#731C2×C3×C4
FrattiniC2×C32C3×D32
FittingC2×#6910C22
socleC2×C3C3×((C3⋊D3)⋊C3)⋊C2
prime radicalC2×#4177⋊C3C2
cosocleC2×#731C22×C3
LCSG ▷ #731
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #66331⋊C3C2semidirect
3. #11757587C2semidirect
6. #4177⋊C4C3semidirect
8. #4177⋊C3C4semidirect
18. #66331C2×C3semidirect
20. #66323C2×C3semidirect
21. #6910C2×C4semidirect
30. C3⋊AGL(1,ℤ/9)Dic3semidirect
37. #4177C3×C4semidirect
62. C3×3−1+2C2×Dic3semidirect
67. #731C2×C3×C4semidirect
90. C3⋊D9C3×Dic3semidirect
130. C3×C9C2×C3×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×#4177⋊C32
#66331⋊C32
#117575872
(C3⋊AGL(1,ℤ/9))⋊C43
C3×((C3⋊D3)⋊C3)⋊C43
#4177⋊C43