Finite Group Databasebeta

← C2×#10619886 #13048720

Subgroups — C2×#10619886 #13048720

Order 1296.

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

normal subgroups only

This group has 777 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⋊#272C211yesmaximal, characteristic, complemented
2C22×C3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#10619886C214yesmaximal, direct product factor
5C2×#6295—31nomaximal
6C2×#6290—31nomaximal
7C2×#6200—33nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic
9C2×C3⋊#272—22no—
10C2×C3⋊#272C2212yescomplemented
11C2×C3×(C3⋊D3)⋊C3—22no—
12C2×C3×(C3⋊D3)⋊C3C2212yescomplemented
13C3⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
14C22×C3×C3⋊D3—31no—
15C22×C3×C3⋊D3—31no—
16C22×C3×C3⋊D3—61no—
17C22×C3×C3⋊D3D311yescharacteristic, complemented
18C22×C32×D3—31no—
19C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
20C2×C3×C3⋊Dic3—31no—
21#6295—34no—
22#6290—34no—
23C22×#272—33no—
24C22×#272—63no—
25C22×(C3⋊D3)⋊C3D313yescomplemented
26C2×(C3⋊Dic3)⋊C3—33no—
27#6200—312no—
28C2×C3×H(3,𝔽3)D412yescomplemented
29C2×C3×H(3,𝔽3)C2311yescharacteristic
30C3⋊#272—22no—
31C3×(C3⋊D3)⋊C3—22no—
32C2×#638—91no—
33C2×#626—91no—
34C2×#626—91no—
35C2×#625—93no—
36C2×#625—91no—
37C22×C33—21no—
38C22×C33C2×D311yescharacteristic
39C22×C33C2×D311yescharacteristic
40C2×C3×C3⋊D3—22no—
41C2×C3×C3⋊D3—32no—
42C2×C3×C3⋊D3—32no—
43C2×C3×C3⋊D3—62no—
44C2×C3×C3⋊D3—64no—
45C2×C3×C3⋊D3—62no—
46C2×C3×C3⋊D3—62no—
47C2×C3×C3⋊D3C2×D312yescomplemented
48C2×C32×D3—32no—
49C2×C32×D3—62no—
50C3⋊(C3⋊Dic3)C2×D312yescomplemented
51C3×C3⋊Dic3—32no—
52C22×H(3,𝔽3)—23no—
53C22×H(3,𝔽3)C2×D313yes—
54C2×#272—36no—
55C2×#272—66no—
56C2×#272—612no—
57C2×#272—66no—
58C2×(C3⋊D3)⋊C3—26no—
59C2×(C3⋊D3)⋊C3C2×D316yescomplemented
60(C3⋊Dic3)⋊C3—36no—
61C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
62C23×C32—91no—
63C22×C3⋊D3—31no—
64C22×C3⋊D3—31no—
65C22×C3⋊D3—61no—
66C22×C3⋊D3C3⋊D311yescharacteristic, complemented
67C22×C3×D3—33no—
68C22×C3×D3—31no—
69C22×C3×D3—91no—
70C22×C3×D3—93no—
71C22×C3×D3—91no—
72C22×C3×D3—91no—
73C22×C3×D3—183no—
74C22×C3×D3—181no—
75#638—94no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—33no—
78C2×C3⋊Dic3—91no—
79C2×C3⋊Dic3—181no—
80C2×C3×Dic3—91no—
81C2×C3×Dic3—91no—
82C2×C3×Dic3—181no—
83#626—94no—
84#626—94no—
85#625—912no—
86#625—94no—
87C2×C33—22no—
88C2×C33—21no—
89C2×C33D1212yescomplemented
90C2×C33#6712yescomplemented
91C2×C33C22×D311yescharacteristic
92C2×C33C22×D311yescharacteristic
93C3×C3⋊D3—22no—
94C3×C3⋊D3—64no—
95C3×C3⋊D3—62no—
96C3×C3⋊D3—62no—
97C32×D3—62no—
98C2×H(3,𝔽3)—26no—
99C2×H(3,𝔽3)—23no—
100C2×H(3,𝔽3)#6716yescomplemented
101C2×H(3,𝔽3)C22×D313yes—
102#272—612no—
103#272—66no—
104(C3⋊D3)⋊C3—26no—
105C2×#67—271no—
106C2×#67—271no—
107C2×#67—273no—
108C2×D12—271no—
109C22×C32—21no—
110C22×C32—23no—
111C22×C32—31no—
112C22×C32—31no—
113C22×C32—61no—
114C22×C32—61no—
115C22×C32—61no—
116C22×C32—92no—
117C22×C32—121no—
118C22×C32—182no—
119C22×C32D3211yescharacteristic
120C22×C32D3213yes—
121C22×C32C2×C3⋊D311yescharacteristic
122C2×C3⋊D3—22no—
123C2×C3⋊D3—32no—
124C2×C3⋊D3—32no—
125C2×C3⋊D3—62no—
126C2×C3⋊D3—62no—
127C2×C3⋊D3—62no—
128C2×C3⋊D3—64no—
129C2×C3⋊D3C2×C3⋊D312yescomplemented
130C2×C3×D3—36no—
131C2×C3×D3—32no—
132C2×C3×D3—62no—
133C2×C3×D3—66no—
134C2×C3×D3—92no—
135C2×C3×D3—96no—
136C2×C3×D3—92no—
137C2×C3×D3—92no—
138C2×C3×D3—182no—
139C2×C3×D3—186no—
140C2×C3×D3—182no—
141C2×C3×D3—184no—
142C2×C3×D3—1812no—
143C2×C3×D3—182no—
144C2×C3×D3—186no—
145C2×C3×D3—182no—
146C3⋊Dic3—32no—
147C3⋊Dic3—36no—
148C3⋊Dic3—92no—
149C3⋊Dic3—182no—
150C3×Dic3—92no—
151C3×Dic3—92no—
152C3×Dic3—182no—
153C33—21no—
154C33C2×D1211yescharacteristic, complemented
155C33C2×#6711yescharacteristic, complemented
156H(3,𝔽3)—23no—
157H(3,𝔽3)C2×#6713yescomplemented
158C23×C3—93no—
159C23×C3—91no—
160C23×C3—271no—
161C22×D3—31no—
162C22×D3—91no—
163C22×D3—91no—
164C22×D3—93no—
165C22×D3—91no—
166C22×D3—181no—
167C22×D3—183no—
168C2×C3×C4—271no—
169#67—274no—
170#67—274no—
171#67—2712no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—181no—
175C2×Dic3—273no—
176D12—274no—
177C2×C32—22no—
178C2×C32—26no—
179C2×C32—21no—
180C2×C32—23no—
181C2×C32—32no—
182C2×C32—31no—
183C2×C32—32no—
184C2×C32—31no—
185C2×C32—62no—
186C2×C32—61no—
187C2×C32—62no—
188C2×C32—61no—
189C2×C32—62no—
190C2×C32—61no—
191C2×C32—121no—
192C2×C32—122no—
193C2×C32—182no—
194C2×C32#62612yes—
195C2×C32#62616yescomplemented
196C2×C32#63812yescomplemented
197C2×C32C2×D3211yescharacteristic
198C2×C32C2×D3213yes—
199C2×C32C22×C3⋊D311yescharacteristic
200C3⋊D3—22no—
201C3⋊D3—62no—
202C3⋊D3—62no—
203C3⋊D3—64no—
204C3×D3—62no—
205C3×D3—66no—
206C3×D3—182no—
207C3×D3—184no—
208C3×D3—1812no—
209C3×D3—182no—
210C3×D3—186no—
211C3×D3—182no—
212C2×D4—811noSylow
213C22×C3—21no—
214C22×C3—33no—
215C22×C3—31no—
216C22×C3—61no—
217C22×C3—63no—
218C22×C3—96no—
219C22×C3—92no—
220C22×C3—186no—
221C22×C3—182no—
222C22×C3—272no—
223C22×C3—542no—
224C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C22×C3D3×C3⋊D311yescharacteristic
226C2×D3—32no—
227C2×D3—62no—
228C2×D3—92no—
229C2×D3—92no—
230C2×D3—96no—
231C2×D3—92no—
232C2×D3—182no—
233C2×D3—182no—
234C2×D3—186no—
235C2×D3—182no—
236C2×D3—186no—
237C2×D3—184no—
238C2×D3—1812no—
239C2×D3—182no—
240C3×C4—272no—
241Dic3—92no—
242Dic3—92no—
243Dic3—182no—
244Dic3—276no—
245C32—21no—
246C32—23no—
247C32—31no—
248C32—31no—
249C32—61no—
250C32—61no—
251C32—61no—
252C32—121no—
253C32C2×#62611yescharacteristic
254C32C2×#62613yescomplemented
255C32C2×#63811yescharacteristic, complemented
256C23—91no—
257C23—271no—
258D4—814no—
259C2×C4—271no—
260C2×C3—21no—
261C2×C3—22no—
262C2×C3—33no—
263C2×C3—31no—
264C2×C3—32no—
265C2×C3—36no—
266C2×C3—61no—
267C2×C3—63no—
268C2×C3—62no—
269C2×C3—66no—
270C2×C3—186no—
271C2×C3—182no—
272C2×C3—542no—
273C2×C3#620012yescomplemented
274C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C2×C3#629112yes—
276C2×C3C2×D3×C3⋊D311yescharacteristic
277D3—62no—
278D3—182no—
279D3—182no—
280D3—186no—
281D3—184no—
282D3—1812no—
283D3—182no—
284C22—92no—
285C22—182no—
286C22—272no—
287C22—542no—
288C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
289C4—272no—
290C3—21no—
291C3—31no—
292C3—33no—
293C3—61no—
294C3—63no—
295C3C2×#620011yescharacteristic, complemented
296C3C2×#629111yescharacteristic
297C2—182no—
298C2—542no—
299C2#1061988612yesdirect product factor
300C2C2×((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⋊#272C2semidirect
2. C22×C3×(C3⋊D3)⋊C3C2semidirect
3. C2×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #10619886C2direct (N × H)
10. C2×C3⋊#272C22semidirect
12. C2×C3×(C3⋊D3)⋊C3C22semidirect
13. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
17. C22×C3×C3⋊D3D3semidirect
19. C2×C3⋊(C3⋊Dic3)D3semidirect
25. C22×(C3⋊D3)⋊C3D3semidirect
28. C2×C3×H(3,𝔽3)D4semidirect
47. C2×C3×C3⋊D3C2×D3semidirect
50. C3⋊(C3⋊Dic3)C2×D3semidirect
59. C2×(C3⋊D3)⋊C3C2×D3semidirect
61. C3×H(3,𝔽3)C2×D4semidirect
66. C22×C3⋊D3C3⋊D3semidirect
89. C2×C33D12semidirect
90. C2×C33#67semidirect
100. C2×H(3,𝔽3)#67semidirect
129. C2×C3⋊D3C2×C3⋊D3semidirect
154. C33C2×D12semidirect
155. C33C2×#67semidirect
157. H(3,𝔽3)C2×#67semidirect
195. C2×C32#626semidirect
196. C2×C32#638semidirect
254. C32C2×#626semidirect
255. C32C2×#638semidirect
273. C2×C3#6200semidirect
295. C3C2×#6200semidirect
299. C2#10619886direct (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
C22×C3×(C3⋊D3)⋊C32
C2×C3⋊((C3⋊Dic3)⋊C3)2
#106198862
C2×#62953
C2×#62903
C2×#62003