Finite Group Databasebeta

← C2×#10619884 #13048717

Subgroups — C2×#10619884 #13048717

Order 1296.

697 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 102 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)⋊C3)C211yesmaximal, characteristic, complemented
2C22×C3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊#1084C211yesmaximal, characteristic, complemented
4#10619884C214yesmaximal, direct product factor
5C2×#6295—31nomaximal
6C2×#6291—31nomaximal
7C2×#6198—33nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)—22no—
10C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
11C2×C3×(C3⋊D3)⋊C3—22no—
12C2×C3×(C3⋊D3)⋊C3C2212yescomplemented
13C3⋊#1084C2212yescomplemented
14C22×C3⋊(C3⋊D3)D311yescharacteristic, complemented
15C22×C3×C3⋊D3—31no—
16C22×C3×C3⋊D3D311yescharacteristic, complemented
17C22×C32×D3—31no—
18C2×C3×C3⋊Dic3—31no—
19C2×C3×C3⋊Dic3—31no—
20C2×C3×C3⋊Dic3—61no—
21#6295—34no—
22#6291—34no—
23C22×(C3⋊D3)⋊C3—33no—
24C22×(C3⋊D3)⋊C3D313yescomplemented
25C2×#1084—33no—
26C2×#1084—63no—
27#6198—312no—
28C2×C3×H(3,𝔽3)D412yescomplemented
29C2×C3×H(3,𝔽3)C2311yescharacteristic
30C3⋊((C3⋊D3)⋊C3)—22no—
31C3×(C3⋊D3)⋊C3—22no—
32C2×#638—91no—
33C2×#626—93no—
34C2×#626—91no—
35C2×#626—91no—
36C2×#625—91no—
37C22×C33—21no—
38C22×C33C2×D311yescharacteristic
39C22×C33C2×D311yescharacteristic
40C2×C3⋊(C3⋊D3)—22no—
41C2×C3⋊(C3⋊D3)C2×D312yescomplemented
42C2×C3×C3⋊D3—22no—
43C2×C3×C3⋊D3—32no—
44C2×C3×C3⋊D3—62no—
45C2×C3×C3⋊D3C2×D312yescomplemented
46C2×C32×D3—32no—
47C2×C32×D3—62no—
48C3×C3⋊Dic3—32no—
49C3×C3⋊Dic3—32no—
50C3×C3⋊Dic3—62no—
51C22×H(3,𝔽3)—23no—
52C22×H(3,𝔽3)C2×D313yes—
53C2×(C3⋊D3)⋊C3—26no—
54C2×(C3⋊D3)⋊C3—36no—
55C2×(C3⋊D3)⋊C3—66no—
56C2×(C3⋊D3)⋊C3C2×D316yescomplemented
57#1084—36no—
58#1084—66no—
59C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
60C23×C32—91no—
61C22×C3⋊D3—31no—
62C22×C3⋊D3—33no—
63C22×C3⋊D3—91no—
64C22×C3⋊D3—181no—
65C22×C3⋊D3C3⋊D311yescharacteristic, complemented
66C22×C3×D3—33no—
67C22×C3×D3—31no—
68C22×C3×D3—91no—
69C22×C3×D3—91no—
70C22×C3×D3—91no—
71C22×C3×D3—181no—
72#638—94no—
73C2×C3⋊Dic3—31no—
74C2×C3⋊Dic3—31no—
75C2×C3⋊Dic3—61no—
76C2×C3×Dic3—93no—
77C2×C3×Dic3—91no—
78C2×C3×Dic3—91no—
79C2×C3×Dic3—183no—
80C2×C3×Dic3—181no—
81#626—912no—
82#626—94no—
83#626—94no—
84#625—94no—
85C2×C33—22no—
86C2×C33—21no—
87C2×C33#6712yescomplemented
88C2×C33#6712yescomplemented
89C2×C33C22×D311yescharacteristic
90C2×C33C22×D311yescharacteristic
91C3⋊(C3⋊D3)—22no—
92C3×C3⋊D3—22no—
93C3×C3⋊D3—62no—
94C32×D3—62no—
95C2×H(3,𝔽3)—26no—
96C2×H(3,𝔽3)—23no—
97C2×H(3,𝔽3)#6716yescomplemented
98C2×H(3,𝔽3)C22×D313yes—
99(C3⋊D3)⋊C3—26no—
100(C3⋊D3)⋊C3—66no—
101C2×#67—271no—
102C2×#67—271no—
103C2×#67—273no—
104C2×D12—271no—
105C22×C32—21no—
106C22×C32—23no—
107C22×C32—31no—
108C22×C32—31no—
109C22×C32—61no—
110C22×C32—61no—
111C22×C32—61no—
112C22×C32—92no—
113C22×C32—121no—
114C22×C32—182no—
115C22×C32D3211yescharacteristic
116C22×C32D3213yes—
117C22×C32C2×C3⋊D311yescharacteristic
118C2×C3⋊D3—22no—
119C2×C3⋊D3—32no—
120C2×C3⋊D3—36no—
121C2×C3⋊D3—62no—
122C2×C3⋊D3—66no—
123C2×C3⋊D3—92no—
124C2×C3⋊D3—182no—
125C2×C3⋊D3—182no—
126C2×C3⋊D3—184no—
127C2×C3⋊D3C2×C3⋊D312yescomplemented
128C2×C3×D3—36no—
129C2×C3×D3—32no—
130C2×C3×D3—62no—
131C2×C3×D3—66no—
132C2×C3×D3—92no—
133C2×C3×D3—92no—
134C2×C3×D3—92no—
135C2×C3×D3—182no—
136C2×C3×D3—182no—
137C2×C3×D3—182no—
138C2×C3×D3—184no—
139C2×C3×D3—182no—
140C3⋊Dic3—32no—
141C3⋊Dic3—32no—
142C3⋊Dic3—62no—
143C3×Dic3—96no—
144C3×Dic3—92no—
145C3×Dic3—92no—
146C3×Dic3—186no—
147C3×Dic3—182no—
148C33—21no—
149C33C2×#6711yescharacteristic, complemented
150C33C2×#6711yescharacteristic, complemented
151H(3,𝔽3)—23no—
152H(3,𝔽3)C2×#6713yescomplemented
153C23×C3—93no—
154C23×C3—91no—
155C23×C3—271no—
156C22×D3—31no—
157C22×D3—91no—
158C22×D3—91no—
159C22×D3—91no—
160C22×D3—181no—
161C22×D3—273no—
162C2×C3×C4—271no—
163#67—274no—
164#67—274no—
165#67—2712no—
166C2×Dic3—91no—
167C2×Dic3—93no—
168C2×Dic3—91no—
169C2×Dic3—181no—
170C2×Dic3—183no—
171D12—274no—
172C2×C32—22no—
173C2×C32—26no—
174C2×C32—21no—
175C2×C32—23no—
176C2×C32—32no—
177C2×C32—31no—
178C2×C32—32no—
179C2×C32—31no—
180C2×C32—62no—
181C2×C32—61no—
182C2×C32—62no—
183C2×C32—61no—
184C2×C32—62no—
185C2×C32—61no—
186C2×C32—121no—
187C2×C32—122no—
188C2×C32—182no—
189C2×C32#62512yes—
190C2×C32#62516yescomplemented
191C2×C32#63812yescomplemented
192C2×C32C2×D3211yescharacteristic
193C2×C32C2×D3213yes—
194C2×C32C22×C3⋊D311yescharacteristic
195C3⋊D3—22no—
196C3⋊D3—62no—
197C3⋊D3—66no—
198C3⋊D3—182no—
199C3⋊D3—184no—
200C3×D3—62no—
201C3×D3—66no—
202C3×D3—182no—
203C3×D3—182no—
204C3×D3—184no—
205C3×D3—182no—
206C2×D4—811noSylow
207C22×C3—21no—
208C22×C3—31no—
209C22×C3—33no—
210C22×C3—61no—
211C22×C3—63no—
212C22×C3—96no—
213C22×C3—92no—
214C22×C3—186no—
215C22×C3—182no—
216C22×C3—272no—
217C22×C3—542no—
218C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C22×C3D3×C3⋊D311yescharacteristic
220C2×D3—32no—
221C2×D3—62no—
222C2×D3—92no—
223C2×D3—92no—
224C2×D3—92no—
225C2×D3—182no—
226C2×D3—182no—
227C2×D3—184no—
228C2×D3—182no—
229C2×D3—182no—
230C2×D3—276no—
231C2×D3—546no—
232C3×C4—272no—
233Dic3—92no—
234Dic3—96no—
235Dic3—92no—
236Dic3—182no—
237Dic3—186no—
238C32—21no—
239C32—23no—
240C32—31no—
241C32—31no—
242C32—61no—
243C32—61no—
244C32—61no—
245C32—121no—
246C32C2×#62511yescharacteristic
247C32C2×#62513yescomplemented
248C32C2×#63811yescharacteristic, complemented
249C23—91no—
250C23—271no—
251D4—814no—
252C2×C4—271no—
253C2×C3—21no—
254C2×C3—22no—
255C2×C3—31no—
256C2×C3—33no—
257C2×C3—32no—
258C2×C3—36no—
259C2×C3—61no—
260C2×C3—63no—
261C2×C3—62no—
262C2×C3—66no—
263C2×C3—186no—
264C2×C3—182no—
265C2×C3—542no—
266C2×C3#619812yescomplemented
267C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C2×C3#629012yes—
269C2×C3C2×D3×C3⋊D311yescharacteristic
270D3—62no—
271D3—182no—
272D3—184no—
273D3—182no—
274D3—182no—
275D3—546no—
276C22—92no—
277C22—182no—
278C22—272no—
279C22—542no—
280C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
281C4—272no—
282C3—21no—
283C3—31no—
284C3—33no—
285C3—61no—
286C3—63no—
287C3C2×#619811yescharacteristic, complemented
288C3C2×#629011yescharacteristic
289C2—182no—
290C2—542no—
291C2#1061988412yesdirect product factor
292C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C3×H(3,𝔽3)C22
socleC22×C32D32
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊((C3⋊D3)⋊C3)C2semidirect
2. C22×C3×(C3⋊D3)⋊C3C2semidirect
3. C2×C3⋊#1084C2semidirect
4. #10619884C2direct (N × H)
10. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
12. C2×C3×(C3⋊D3)⋊C3C22semidirect
13. C3⋊#1084C22semidirect
14. C22×C3⋊(C3⋊D3)D3semidirect
16. C22×C3×C3⋊D3D3semidirect
24. C22×(C3⋊D3)⋊C3D3semidirect
28. C2×C3×H(3,𝔽3)D4semidirect
41. C2×C3⋊(C3⋊D3)C2×D3semidirect
45. C2×C3×C3⋊D3C2×D3semidirect
56. C2×(C3⋊D3)⋊C3C2×D3semidirect
59. C3×H(3,𝔽3)C2×D4semidirect
65. C22×C3⋊D3C3⋊D3semidirect
87. C2×C33#67semidirect
88. C2×C33#67semidirect
97. C2×H(3,𝔽3)#67semidirect
127. C2×C3⋊D3C2×C3⋊D3semidirect
149. C33C2×#67semidirect
150. C33C2×#67semidirect
152. H(3,𝔽3)C2×#67semidirect
190. C2×C32#625semidirect
191. C2×C32#638semidirect
247. C32C2×#625semidirect
248. C32C2×#638semidirect
266. C2×C3#6198semidirect
287. C3C2×#6198semidirect
291. C2#10619884direct (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)⋊C3)2
C22×C3×(C3⋊D3)⋊C32
C2×C3⋊#10842
#106198842
C2×#62953
C2×#62913
C2×#61983