Finite Group Databasebeta

← (A4⋊((C3⋊D3)⋊C3))⋊C3 #14519202

Subgroups — (A4⋊((C3⋊D3)⋊C3))⋊C3 #14519202

Order 1944.

524 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 56 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⋊H(5,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊((C3⋊S4)⋊C3)C311yesmaximal, characteristic, complemented
3A4⋊((C3⋊D3)⋊C3)C313yesmaximal, complemented
4C3×(C3⋊S4)⋊C3—33nomaximal
5C3×#638⋊C3—31nomaximal
6#92391—41nomaximal
7C22×C3×H(3,𝔽3)D311yescharacteristic, complemented
8C2×C3×(C3⋊D3)⋊C3—31no—
9C3×C22⋊H(3,𝔽3)—21no—
10C3×C22⋊H(3,𝔽3)—23no—
11C3×C22⋊H(3,𝔽3)D313yescomplemented
12C3×C22⋊H(3,𝔽3)C2×C311yescharacteristic, complemented
13A4×H(3,𝔽3)—23no—
14A4×H(3,𝔽3)—23no—
15A4×H(3,𝔽3)—26no—
16A4×H(3,𝔽3)C2×C313yescomplemented
17C3×(C3⋊Dic3)⋊C3—31no—
18H(5,𝔽3)—41noSylow
19C3⋊(C3⋊S4)C3211yescharacteristic, complemented
20C3×C3⋊S4—33no—
21C3×#638—31no—
22C32×#67—91no—
23(C3⋊S4)⋊C3—36no—
24(C3⋊S4)⋊C3—33no—
25#638⋊C3—33no—
26C2×C3×H(3,𝔽3)—31no—
27C3⋊((C3⋊D3)⋊C3)—43no—
28C3⋊((C3⋊D3)⋊C3)—41no—
29C3×(C3⋊D3)⋊C3—61no—
30C3×(C3⋊D3)⋊C3—123no—
31C22×C33—21no—
32C22×C33C3×D311yescharacteristic, complemented
33C22×C33C3⋊D311yescharacteristic, complemented
34C2×C3×C3⋊D3—31no—
35C2×C32×D3—91no—
36C32×A4—23no—
37C32×A4—23no—
38C32×A4—23no—
39C32×A4—21no—
40C32×A4—23no—
41C32×A4—23no—
42C32×A4C3×D313yescomplemented
43C32×A4C2×C3211yescharacteristic, complemented
44C3×C3⋊Dic3—31no—
45C32×Dic3—91no—
46C22×H(3,𝔽3)—23no—
47C22×H(3,𝔽3)C3×D313yescomplemented
48C2×(C3⋊D3)⋊C3—33no—
49C22⋊H(3,𝔽3)—21no—
50C22⋊H(3,𝔽3)—26no—
51C22⋊H(3,𝔽3)—22no—
52C22⋊H(3,𝔽3)—26no—
53C22⋊H(3,𝔽3)—23no—
54C22⋊H(3,𝔽3)—26no—
55C22⋊H(3,𝔽3)—23no—
56C22⋊H(3,𝔽3)—23no—
57C22⋊H(3,𝔽3)—26no—
58C22⋊H(3,𝔽3)C3×D316yescomplemented
59C22⋊H(3,𝔽3)C3×D313yescomplemented
60(C3⋊Dic3)⋊C3—33no—
61C3×H(3,𝔽3)—41no—
62C3×H(3,𝔽3)—43no—
63C3×H(3,𝔽3)—43no—
64C3×H(3,𝔽3)—86no—
65C3×H(3,𝔽3)—83no—
66C3×H(3,𝔽3)—83no—
67C3×H(3,𝔽3)—81no—
68C3×H(3,𝔽3)—83no—
69C3×H(3,𝔽3)S411yescharacteristic, complemented
70C3⋊S4—33no—
71C3⋊S4—271no—
72C3×S4—273no—
73C32×D4—271no—
74#638—31no—
75C3×#67—93no—
76C3×#67—91no—
77C3×#67—271no—
78C2×C33—31no—
79C2×C33—31no—
80C2×C33—61no—
81C3⋊(C3⋊D3)—41no—
82C3×C3⋊D3—61no—
83C3×C3⋊D3—123no—
84C32×D3—181no—
85C2×H(3,𝔽3)—33no—
86C2×H(3,𝔽3)—63no—
87(C3⋊D3)⋊C3—63no—
88(C3⋊D3)⋊C3—126no—
89(C3⋊D3)⋊C3—123no—
90C22×C32—21no—
91C22×C32—23no—
92C22×C32—91no—
93C22×C32—91no—
94C22×C32—181no—
95C22×C32—271no—
96C22×C32C32×D311yescharacteristic
97C22×C32C3×C3⋊D311yescharacteristic
98C22×C32C3×C3⋊D313yes—
99C2×C3⋊D3—31no—
100C2×C3×D3—93no—
101C2×C3×D3—91no—
102C2×C3×D3—271no—
103C3×A4—23no—
104C3×A4—26no—
105C3×A4—23no—
106C3×A4—91no—
107C3×A4—93no—
108C3×A4—183no—
109C3×A4—181no—
110C3×A4—183no—
111C3×A4—183no—
112C3×A4—183no—
113C3×A4—183no—
114C3×A4C32×D313yes—
115C32×C4—271no—
116C3⋊Dic3—31no—
117C3×Dic3—93no—
118C3×Dic3—91no—
119C3×Dic3—271no—
120C33—21no—
121C33—41no—
122C33—43no—
123C33—83no—
124C33—83no—
125C33—83no—
126C33—83no—
127C33—81no—
128C33—83no—
129C33C3×S411yescharacteristic, complemented
130C33C3⋊S411yescharacteristic, complemented
131H(3,𝔽3)—23no—
132H(3,𝔽3)—46no—
133H(3,𝔽3)—43no—
134H(3,𝔽3)—83no—
135H(3,𝔽3)—86no—
136H(3,𝔽3)—82no—
137H(3,𝔽3)—81no—
138H(3,𝔽3)—86no—
139H(3,𝔽3)—86no—
140H(3,𝔽3)—83no—
141H(3,𝔽3)—86no—
142H(3,𝔽3)—83no—
143H(3,𝔽3)C3×S413yescomplemented
144S4—273no—
145C3×D4—271no—
146C3×D4—273no—
147#67—91no—
148#67—271no—
149C2×C32—31no—
150C2×C32—31no—
151C2×C32—33no—
152C2×C32—63no—
153C2×C32—61no—
154C2×C32—91no—
155C2×C32—91no—
156C2×C32—181no—
157C2×C32—181no—
158C2×C32—181no—
159C2×C32—182no—
160C2×C32—541no—
161C3⋊D3—61no—
162C3⋊D3—123no—
163C3⋊D3—1081no—
164C3×D3—183no—
165C3×D3—181no—
166C3×D3—541no—
167C3×D3—1083no—
168C22×C3—31no—
169C22×C3—31no—
170C22×C3—33no—
171C22×C3—61no—
172C22×C3—63no—
173C22×C3—271no—
174C22×C3—273no—
175C22×C3C32×C3⋊D311yescharacteristic
176C2×D3—91no—
177C2×D3—271no—
178A4—33no—
179A4—63no—
180A4—66no—
181A4—63no—
182C3×C4—271no—
183C3×C4—273no—
184Dic3—91no—
185Dic3—271no—
186C32—21no—
187C32—23no—
188C32—43no—
189C32—83no—
190C32—86no—
191C32—83no—
192C32—91no—
193C32—91no—
194C32—181no—
195C32—361no—
196C32—363no—
197C32—721no—
198C32—723no—
199C32—723no—
200C32—723no—
201C32—723no—
202C32—723no—
203C32C32×S411yescharacteristic
204C32C3×C3⋊S413yes—
205C32C3×C3⋊S411yescharacteristic
206D4—271noSylow
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—61no—
210C2×C3—91no—
211C2×C3—93no—
212C2×C3—181no—
213C2×C3—183no—
214C2×C3—541no—
215C2×C3—543no—
216D3—181no—
217D3—541no—
218D3—1083no—
219C22—271no—
220C22#9239111yescharacteristic, complemented
221C4—271no—
222C3—31no—
223C3—31no—
224C3—33no—
225C3—61no—
226C3—63no—
227C3—123no—
228C3—243no—
229C3—243no—
230C3—246no—
231C3C32×C3⋊S411yescharacteristic
232C2—31no—
233C2—541no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×A4C2×C32
FrattiniC3C32×C3⋊S4
FittingC22×C3×H(3,𝔽3)D3
socleC22×C3C32×C3⋊D3
LCSG ▷ C32×A4
UCSC1

Complements

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

NH ≅ G/Ntype
1. C22⋊H(5,𝔽3)C2semidirect
2. C3⋊((C3⋊S4)⋊C3)C3semidirect
3. A4⋊((C3⋊D3)⋊C3)C3semidirect
7. C22×C3×H(3,𝔽3)D3semidirect
11. C3×C22⋊H(3,𝔽3)D3semidirect
12. C3×C22⋊H(3,𝔽3)C2×C3semidirect
16. A4×H(3,𝔽3)C2×C3semidirect
19. C3⋊(C3⋊S4)C32semidirect
32. C22×C33C3×D3semidirect
33. C22×C33C3⋊D3semidirect
42. C32×A4C3×D3semidirect
43. C32×A4C2×C32semidirect
47. C22×H(3,𝔽3)C3×D3semidirect
58. C22⋊H(3,𝔽3)C3×D3semidirect
59. C22⋊H(3,𝔽3)C3×D3semidirect
69. C3×H(3,𝔽3)S4semidirect
129. C33C3×S4semidirect
130. C33C3⋊S4semidirect
143. H(3,𝔽3)C3×S4semidirect
220. C22#92391semidirect

Maximal subgroups

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

subgroupindex
C22⋊H(5,𝔽3)2
C3⋊((C3⋊S4)⋊C3)3
A4⋊((C3⋊D3)⋊C3)3
C3×(C3⋊S4)⋊C33
C3×#638⋊C33
#923914