Finite Group Databasebeta

← C2×((C3⋊D3)⋊C3)⋊Dic3 #13048715

Subgroups — C2×((C3⋊D3)⋊C3)⋊Dic3 #13048715

Order 1296.

673 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 150 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×C3×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C3⋊#1084C211yesmaximal, characteristic, complemented
3C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4((C3⋊D3)⋊C3)⋊Dic3C214yesmaximal, direct product factor
5C2×#6294—31nomaximal
6C2×D3×C3⋊Dic3—31nomaximal
7C2×((C3⋊D3)⋊C3)⋊C4—33nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic
9C2×C3×(C3⋊D3)⋊C3C414yescomplemented
10C2×C3×(C3⋊D3)⋊C3C2212yes—
11C3⋊#1084C2212yescomplemented
12C3⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
13C22×C3×C3⋊D3D311yescharacteristic
14C22×C32×D3—31no—
15C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
16C2×C3×C3⋊Dic3—31no—
17C2×C3×C3⋊Dic3—31no—
18C2×C3×C3⋊Dic3—31no—
19C2×C3×C3⋊Dic3—61no—
20#6294—34no—
21D3×C3⋊Dic3—34no—
22C22×(C3⋊D3)⋊C3D313yes—
23C2×#1084—33no—
24C2×#1084—63no—
25C2×(C3⋊Dic3)⋊C3—33no—
26((C3⋊D3)⋊C3)⋊C4—312no—
27C2×C3×H(3,𝔽3)C2×C412yescomplemented
28C2×C3×H(3,𝔽3)C2311yescharacteristic
29C3×(C3⋊D3)⋊C3C2×C414yescomplemented
30C22×C3⋊Dic3—91no—
31C2×#624—91no—
32C2×D3×Dic3—93no—
33C2×D3×Dic3—91no—
34C2×D3×Dic3—91no—
35C22×C33—21no—
36C22×C33C2×D311yescharacteristic
37C22×C33C2×D311yescharacteristic
38C2×C3×C3⋊D3Dic314yescomplemented
39C2×C3×C3⋊D3C2×D312yes—
40C2×C32×D3—34no—
41C2×C32×D3—32no—
42C3⋊(C3⋊Dic3)C2×D312yescomplemented
43C3×C3⋊Dic3—32no—
44C3×C3⋊Dic3—32no—
45C3×C3⋊Dic3—32no—
46C3×C3⋊Dic3—62no—
47C22×H(3,𝔽3)—23no—
48C22×H(3,𝔽3)C2×D313yes—
49C2×(C3⋊D3)⋊C3Dic3112yescomplemented
50C2×(C3⋊D3)⋊C3C2×D316yes—
51#1084—36no—
52#1084—66no—
53(C3⋊Dic3)⋊C3—36no—
54C3×H(3,𝔽3)C22×C411yescharacteristic, complemented, Sylow
55C23×C32—91no—
56C22×C3⋊D3C3⋊D311yescharacteristic
57C22×C3×D3—33no—
58C22×C3×D3—31no—
59C22×C3×D3—91no—
60C2×C3⋊Dic3—31no—
61C2×C3⋊Dic3—31no—
62C2×C3⋊Dic3—31no—
63C2×C3⋊Dic3—33no—
64C2×C3⋊Dic3—61no—
65C2×C3⋊Dic3—91no—
66C2×C3⋊Dic3—94no—
67C2×C3⋊Dic3—181no—
68C2×C3×Dic3—91no—
69C2×C3×Dic3—93no—
70C2×C3×Dic3—91no—
71C2×C3×Dic3—91no—
72C2×C3×Dic3—91no—
73C2×C3×Dic3—183no—
74C2×C3×Dic3—181no—
75C2×C3×Dic3—181no—
76#624—94no—
77D3×Dic3—912no—
78D3×Dic3—94no—
79D3×Dic3—94no—
80C2×C33—22no—
81C2×C33—21no—
82C2×C33C4×D312yescomplemented
83C2×C33C2×Dic312yescomplemented
84C2×C33C22×D311yescharacteristic
85C2×C33C22×D311yescharacteristic
86C3×C3⋊D3C2×Dic314yescomplemented
87C32×D3—34no—
88C2×H(3,𝔽3)—26no—
89C2×H(3,𝔽3)—23no—
90C2×H(3,𝔽3)C2×Dic316yescomplemented
91C2×H(3,𝔽3)C22×D313yes—
92(C3⋊D3)⋊C3C2×Dic3112yescomplemented
93C22×Dic3—271no—
94C22×Dic3—273no—
95C2×C4×D3—271no—
96C2×C4×D3—271no—
97C22×C32—21no—
98C22×C32—23no—
99C22×C32—31no—
100C22×C32—31no—
101C22×C32—61no—
102C22×C32—61no—
103C22×C32—61no—
104C22×C32—92no—
105C22×C32—94no—
106C22×C32—121no—
107C22×C32D3211yescharacteristic
108C22×C32D3213yes—
109C22×C32C2×C3⋊D311yescharacteristic
110C2×C3⋊D3C3⋊Dic314yescomplemented
111C2×C3⋊D3C2×C3⋊D312yes—
112C2×C3×D3—34no—
113C2×C3×D3—312no—
114C2×C3×D3—36no—
115C2×C3×D3—32no—
116C2×C3×D3—94no—
117C2×C3×D3—92no—
118C3⋊Dic3—32no—
119C3⋊Dic3—32no—
120C3⋊Dic3—32no—
121C3⋊Dic3—36no—
122C3⋊Dic3—62no—
123C3⋊Dic3—92no—
124C3⋊Dic3—182no—
125C3×Dic3—92no—
126C3×Dic3—96no—
127C3×Dic3—92no—
128C3×Dic3—92no—
129C3×Dic3—92no—
130C3×Dic3—186no—
131C3×Dic3—182no—
132C3×Dic3—182no—
133C33—21no—
134C33C2×C4×D311yescharacteristic, complemented
135C33C22×Dic311yescharacteristic, complemented
136H(3,𝔽3)—23no—
137H(3,𝔽3)C22×Dic313yescomplemented
138C23×C3—93no—
139C23×C3—91no—
140C22×D3—31no—
141C22×D3—91no—
142C2×C3×C4—271no—
143C2×C3×C4—271no—
144C2×Dic3—91no—
145C2×Dic3—93no—
146C2×Dic3—91no—
147C2×Dic3—91no—
148C2×Dic3—91no—
149C2×Dic3—181no—
150C2×Dic3—181no—
151C2×Dic3—183no—
152C2×Dic3—273no—
153C2×Dic3—274no—
154C2×Dic3—2712no—
155C4×D3—274no—
156C4×D3—274no—
157C2×C32—22no—
158C2×C32—26no—
159C2×C32—21no—
160C2×C32—23no—
161C2×C32—32no—
162C2×C32—31no—
163C2×C32—32no—
164C2×C32—31no—
165C2×C32—62no—
166C2×C32—61no—
167C2×C32—62no—
168C2×C32—61no—
169C2×C32—62no—
170C2×C32—61no—
171C2×C32—94no—
172C2×C32—121no—
173C2×C32—122no—
174C2×C32D3×Dic312yes—
175C2×C32D3×Dic316yescomplemented
176C2×C32C2×C3⋊Dic312yescomplemented
177C2×C32C2×D3211yescharacteristic
178C2×C32C2×D3213yes—
179C2×C32C22×C3⋊D311yescharacteristic
180C3⋊D3C2×C3⋊Dic314yescomplemented
181C3×D3—34no—
182C3×D3—312no—
183C3×D3—94no—
184C22×C4—811noSylow
185C22×C3—21no—
186C22×C3—33no—
187C22×C3—31no—
188C22×C3—61no—
189C22×C3—63no—
190C22×C3—96no—
191C22×C3—912no—
192C22×C3—94no—
193C22×C3—92no—
194C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C22×C3D3×C3⋊D311yescharacteristic
196C2×D3—34no—
197C2×D3—32no—
198C2×D3—92no—
199C2×D3—94no—
200C3×C4—272no—
201C3×C4—272no—
202Dic3—92no—
203Dic3—96no—
204Dic3—92no—
205Dic3—92no—
206Dic3—92no—
207Dic3—182no—
208Dic3—182no—
209Dic3—186no—
210Dic3—276no—
211C32—21no—
212C32—23no—
213C32—31no—
214C32—31no—
215C32—61no—
216C32—61no—
217C32—61no—
218C32—121no—
219C32C2×D3×Dic311yescharacteristic
220C32C2×D3×Dic313yescomplemented
221C32C22×C3⋊Dic311yescharacteristic, complemented
222C23—91no—
223C2×C4—271no—
224C2×C4—271no—
225C2×C4—814no—
226C2×C3—21no—
227C2×C3—22no—
228C2×C3—33no—
229C2×C3—31no—
230C2×C3—32no—
231C2×C3—36no—
232C2×C3—61no—
233C2×C3—63no—
234C2×C3—62no—
235C2×C3—66no—
236C2×C3—912no—
237C2×C3—94no—
238C2×C3((C3⋊D3)⋊C3)⋊C412yescomplemented
239C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
240C2×C3D3×C3⋊Dic312yes—
241C2×C3C2×D3×C3⋊D311yescharacteristic
242D3—34no—
243D3—94no—
244C22—92no—
245C22—94no—
246C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
247C4—272no—
248C4—272no—
249C3—21no—
250C3—31no—
251C3—33no—
252C3—61no—
253C3—63no—
254C3C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
255C3C2×D3×C3⋊Dic311yescharacteristic
256C2—94no—
257C2((C3⋊D3)⋊C3)⋊Dic312yesdirect product factor
258C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22((C3⋊D3)⋊C3)⋊D3
commutator, hypocenterC3×H(3,𝔽3)C22×C4
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C3×H(3,𝔽3)C22
socleC22×C32D32
prime radicalC22×C3×(C3⋊D3)⋊C3C2
cosocleC2×C3×H(3,𝔽3)C23
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#1084C2semidirect
3. C2×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
4. ((C3⋊D3)⋊C3)⋊Dic3C2direct (N × H)
9. C2×C3×(C3⋊D3)⋊C3C4semidirect
11. C3⋊#1084C22semidirect
12. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
15. C2×C3⋊(C3⋊Dic3)D3semidirect
27. C2×C3×H(3,𝔽3)C2×C4semidirect
29. C3×(C3⋊D3)⋊C3C2×C4semidirect
38. C2×C3×C3⋊D3Dic3semidirect
42. C3⋊(C3⋊Dic3)C2×D3semidirect
49. C2×(C3⋊D3)⋊C3Dic3semidirect
54. C3×H(3,𝔽3)C22×C4semidirect
82. C2×C33C4×D3semidirect
83. C2×C33C2×Dic3semidirect
86. C3×C3⋊D3C2×Dic3semidirect
90. C2×H(3,𝔽3)C2×Dic3semidirect
92. (C3⋊D3)⋊C3C2×Dic3semidirect
110. C2×C3⋊D3C3⋊Dic3semidirect
134. C33C2×C4×D3semidirect
135. C33C22×Dic3semidirect
137. H(3,𝔽3)C22×Dic3semidirect
175. C2×C32D3×Dic3semidirect
176. C2×C32C2×C3⋊Dic3semidirect
180. C3⋊D3C2×C3⋊Dic3semidirect
220. C32C2×D3×Dic3semidirect
221. C32C22×C3⋊Dic3semidirect
238. C2×C3((C3⋊D3)⋊C3)⋊C4semidirect
254. C3C2×((C3⋊D3)⋊C3)⋊C4semidirect
257. C2((C3⋊D3)⋊C3)⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×(C3⋊D3)⋊C32
C2×C3⋊#10842
C2×C3⋊((C3⋊Dic3)⋊C3)2
((C3⋊D3)⋊C3)⋊Dic32
C2×#62943
C2×D3×C3⋊Dic33
C2×((C3⋊D3)⋊C3)⋊C43