Finite Group Databasebeta

← C2×#10619885 #13048718

Subgroups — C2×#10619885 #13048718

Order 1296.

777 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 110 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⋊#272C211yesmaximal, characteristic
2C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×C3×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#10619885C214yesmaximal, direct product factor
5C2×#6294—31nomaximal
6C2×Dic3×C3⋊D3—31nomaximal
7C2×#6199—33nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic
9C2×C3⋊#272C414yescomplemented
10C2×C3⋊#272C2212yes—
11C3⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
12C3×(C3⋊Dic3)⋊C3C2212yescomplemented
13C22×C3×C3⋊D3—31no—
14C22×C3×C3⋊D3—31no—
15C22×C3×C3⋊D3—61no—
16C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
17C2×C3×C3⋊Dic3—31no—
18C2×C3×C3⋊Dic3D311yescharacteristic, complemented
19C2×C32×Dic3—31no—
20#6294—34no—
21Dic3×C3⋊D3—34no—
22C22×#272—33no—
23C22×#272—63no—
24C2×(C3⋊Dic3)⋊C3—33no—
25C2×(C3⋊Dic3)⋊C3D313yescomplemented
26#6199—312no—
27C2×C3×H(3,𝔽3)C2×C412yescomplemented
28C2×C3×H(3,𝔽3)C2311yescharacteristic
29C3⋊#272C2×C414yescomplemented
30C2×C4×C3⋊D3—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⋊D3—34no—
39C2×C3×C3⋊D3—34no—
40C2×C3×C3⋊D3—32no—
41C2×C3×C3⋊D3—32no—
42C2×C3×C3⋊D3—64no—
43C2×C3×C3⋊D3—62no—
44C3⋊(C3⋊Dic3)C2×D312yescomplemented
45C3×C3⋊Dic3—32no—
46C3×C3⋊Dic3C2×D312yescomplemented
47C32×Dic3—32no—
48C22×H(3,𝔽3)—23no—
49C22×H(3,𝔽3)C2×D313yes—
50C2×#272—312no—
51C2×#272—36no—
52C2×#272—612no—
53C2×#272—66no—
54(C3⋊Dic3)⋊C3—36no—
55(C3⋊Dic3)⋊C3C2×D316yescomplemented
56C3×H(3,𝔽3)C22×C411yescharacteristic, complemented, Sylow
57C22×C3⋊D3—31no—
58C22×C3⋊D3—31no—
59C22×C3⋊D3—61no—
60C22×C3×D3—93no—
61C22×C3×D3—91no—
62C22×C3×D3—91no—
63C22×C3×D3—183no—
64C22×C3×D3—181no—
65C2×C32×C4—91no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—33no—
68C2×C3⋊Dic3—91no—
69C2×C3⋊Dic3—181no—
70C2×C3⋊Dic3C3⋊D311yescharacteristic, complemented
71C4×C3⋊D3—94no—
72C2×C3×Dic3—33no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—91no—
75C2×C3×Dic3—91no—
76C2×C3×Dic3—91no—
77C2×C3×Dic3—181no—
78#624—94no—
79D3×Dic3—912no—
80D3×Dic3—94no—
81D3×Dic3—94no—
82C2×C33—22no—
83C2×C33—21no—
84C2×C33C4×D312yescomplemented
85C2×C33C4×D312yescomplemented
86C2×C33C22×D311yescharacteristic
87C2×C33C22×D311yescharacteristic
88C3×C3⋊D3—34no—
89C3×C3⋊D3—34no—
90C3×C3⋊D3—64no—
91C2×H(3,𝔽3)—26no—
92C2×H(3,𝔽3)—23no—
93C2×H(3,𝔽3)C4×D316yescomplemented
94C2×H(3,𝔽3)C22×D313yes—
95#272—312no—
96#272—612no—
97C22×Dic3—271no—
98C2×C4×D3—271no—
99C2×C4×D3—271no—
100C2×C4×D3—273no—
101C22×C32—21no—
102C22×C32—23no—
103C22×C32—31no—
104C22×C32—31no—
105C22×C32—61no—
106C22×C32—61no—
107C22×C32—61no—
108C22×C32—121no—
109C22×C32D3211yescharacteristic
110C22×C32D3213yes—
111C22×C32C2×C3⋊D311yescharacteristic
112C2×C3⋊D3—32no—
113C2×C3⋊D3—34no—
114C2×C3⋊D3—32no—
115C2×C3⋊D3—34no—
116C2×C3⋊D3—64no—
117C2×C3⋊D3—62no—
118C2×C3×D3—94no—
119C2×C3×D3—912no—
120C2×C3×D3—94no—
121C2×C3×D3—96no—
122C2×C3×D3—92no—
123C2×C3×D3—92no—
124C2×C3×D3—1812no—
125C2×C3×D3—186no—
126C2×C3×D3—184no—
127C2×C3×D3—182no—
128C32×C4—92no—
129C3⋊Dic3—32no—
130C3⋊Dic3—36no—
131C3⋊Dic3—92no—
132C3⋊Dic3—182no—
133C3⋊Dic3C2×C3⋊D312yescomplemented
134C3×Dic3—36no—
135C3×Dic3—32no—
136C3×Dic3—92no—
137C3×Dic3—92no—
138C3×Dic3—92no—
139C3×Dic3—182no—
140C33—21no—
141C33C2×C4×D311yescharacteristic, complemented
142C33C2×C4×D311yescharacteristic, complemented
143H(3,𝔽3)—23no—
144H(3,𝔽3)C2×C4×D313yescomplemented
145C23×C3—271no—
146C22×D3—91no—
147C22×D3—93no—
148C22×D3—91no—
149C22×D3—181no—
150C22×D3—183no—
151C2×C3×C4—93no—
152C2×C3×C4—91no—
153C2×C3×C4—271no—
154C2×Dic3—31no—
155C2×Dic3—91no—
156C2×Dic3—91no—
157C2×Dic3—91no—
158C2×Dic3—181no—
159C2×Dic3—273no—
160C2×Dic3—274no—
161C4×D3—274no—
162C4×D3—274no—
163C4×D3—2712no—
164C2×C32—22no—
165C2×C32—26no—
166C2×C32—21no—
167C2×C32—23no—
168C2×C32—32no—
169C2×C32—31no—
170C2×C32—32no—
171C2×C32—31no—
172C2×C32—62no—
173C2×C32—61no—
174C2×C32—62no—
175C2×C32—61no—
176C2×C32—62no—
177C2×C32—61no—
178C2×C32—121no—
179C2×C32—122no—
180C2×C32#62412yes—
181C2×C32#62416yescomplemented
182C2×C32C4×C3⋊D312yescomplemented
183C2×C32C2×D3211yescharacteristic
184C2×C32C2×D3213yes—
185C2×C32C22×C3⋊D311yescharacteristic
186C3⋊D3—34no—
187C3⋊D3—34no—
188C3⋊D3—64no—
189C3×D3—94no—
190C3×D3—912no—
191C3×D3—94no—
192C3×D3—1812no—
193C3×D3—184no—
194C22×C4—811noSylow
195C22×C3—21no—
196C22×C3—31no—
197C22×C3—33no—
198C22×C3—61no—
199C22×C3—63no—
200C22×C3—274no—
201C22×C3—272no—
202C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
203C22×C3D3×C3⋊D311yescharacteristic
204C2×D3—92no—
205C2×D3—96no—
206C2×D3—912no—
207C2×D3—94no—
208C2×D3—94no—
209C2×D3—92no—
210C2×D3—182no—
211C2×D3—184no—
212C2×D3—186no—
213C2×D3—1812no—
214C3×C4—96no—
215C3×C4—92no—
216C3×C4—272no—
217Dic3—32no—
218Dic3—92no—
219Dic3—92no—
220Dic3—92no—
221Dic3—182no—
222Dic3—276no—
223C32—21no—
224C32—23no—
225C32—31no—
226C32—31no—
227C32—61no—
228C32—61no—
229C32—61no—
230C32—121no—
231C32C2×#62411yescharacteristic
232C32C2×#62413yescomplemented
233C32C2×C4×C3⋊D311yescharacteristic, complemented
234C23—271no—
235C2×C4—91no—
236C2×C4—271no—
237C2×C4—814no—
238C2×C3—21no—
239C2×C3—22no—
240C2×C3—31no—
241C2×C3—33no—
242C2×C3—36no—
243C2×C3—32no—
244C2×C3—61no—
245C2×C3—63no—
246C2×C3—62no—
247C2×C3—66no—
248C2×C3—274no—
249C2×C3#619912yescomplemented
250C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
251C2×C3#628912yes—
252C2×C3C2×D3×C3⋊D311yescharacteristic
253D3—912no—
254D3—94no—
255D3—94no—
256D3—184no—
257D3—1812no—
258C22—272no—
259C22—274no—
260C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
261C4—92no—
262C4—272no—
263C3—21no—
264C3—33no—
265C3—31no—
266C3—61no—
267C3—63no—
268C3C2×#619911yescharacteristic, complemented
269C3C2×#628911yescharacteristic
270C2—274no—
271C2#1061988512yesdirect product factor
272C2C2×((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⋊#272C2
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⋊((C3⋊Dic3)⋊C3)C2semidirect
3. C2×C3×(C3⋊Dic3)⋊C3C2semidirect
4. #10619885C2direct (N × H)
9. C2×C3⋊#272C4semidirect
11. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
12. C3×(C3⋊Dic3)⋊C3C22semidirect
16. C2×C3⋊(C3⋊Dic3)D3semidirect
18. C2×C3×C3⋊Dic3D3semidirect
25. C2×(C3⋊Dic3)⋊C3D3semidirect
27. C2×C3×H(3,𝔽3)C2×C4semidirect
29. C3⋊#272C2×C4semidirect
44. C3⋊(C3⋊Dic3)C2×D3semidirect
46. C3×C3⋊Dic3C2×D3semidirect
55. (C3⋊Dic3)⋊C3C2×D3semidirect
56. C3×H(3,𝔽3)C22×C4semidirect
70. C2×C3⋊Dic3C3⋊D3semidirect
84. C2×C33C4×D3semidirect
85. C2×C33C4×D3semidirect
93. C2×H(3,𝔽3)C4×D3semidirect
133. C3⋊Dic3C2×C3⋊D3semidirect
141. C33C2×C4×D3semidirect
142. C33C2×C4×D3semidirect
144. H(3,𝔽3)C2×C4×D3semidirect
181. C2×C32#624semidirect
182. C2×C32C4×C3⋊D3semidirect
232. C32C2×#624semidirect
233. C32C2×C4×C3⋊D3semidirect
249. C2×C3#6199semidirect
268. C3C2×#6199semidirect
271. C2#10619885direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#2722
C2×C3⋊((C3⋊Dic3)⋊C3)2
C2×C3×(C3⋊Dic3)⋊C32
#106198852
C2×#62943
C2×Dic3×C3⋊D33
C2×#61993