Finite Group Databasebeta

← C2×#10619977 #13048943

Subgroups — C2×#10619977 #13048943

Order 1296.

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

normal subgroups only

This group has 942 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
1C23×C3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#10619977C214yesmaximal, direct product factor
4C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#6312C311yesmaximal, characteristic, complemented
6C2×C3×#638—31nomaximal
7C2×#638⋊C3—33nomaximal
8C22×C3×H(3,𝔽3)—22no—
9C22×C3×H(3,𝔽3)C2211yescharacteristic
10C22×C3×H(3,𝔽3)C2212yescomplemented
11C2×C3⋊((C3⋊D3)⋊C3)—22no—
12C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
13C3⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
14C23×C33—21no—
15C23×C33D311yescharacteristic, complemented
16C23×C33C2×C311yescharacteristic, complemented
17C22×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
18C22×C3×C3⋊D3—31no—
19#6312C2×C314yescomplemented
20C2×C3⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
21C3×#638—34no—
22C2×C3×C3⋊Dic3—31no—
23C23×H(3,𝔽3)—23no—
24C23×H(3,𝔽3)D313yescomplemented
25C22×(C3⋊D3)⋊C3—33no—
26#638⋊C3—312no—
27C2×(C3⋊Dic3)⋊C3—33no—
28C2×C3×H(3,𝔽3)—22no—
29C2×C3×H(3,𝔽3)D412yescomplemented
30C2×C3×H(3,𝔽3)C2311yescharacteristic
31C3⋊((C3⋊D3)⋊C3)—22no—
32C2×#638—31no—
33C2×#638—33no—
34C2×#638—93no—
35C2×C3×#67—93no—
36C2×C3×#67—91no—
37C22×C33—24no—
38C22×C33—21no—
39C22×C33—22no—
40C22×C33—22no—
41C22×C33—22no—
42C22×C33C2×D311yescharacteristic
43C22×C33C2×D312yescomplemented
44C22×C33C22×C311yescharacteristic
45C22×C33C22×C312yescomplemented
46C2×C3⋊(C3⋊D3)—22no—
47C2×C3⋊(C3⋊D3)C22×C312yescomplemented
48C2×C3×C3⋊D3—32no—
49C2×C3×C3⋊D3—62no—
50C3⋊(C3⋊Dic3)C22×C312yescomplemented
51C3×C3⋊Dic3—32no—
52C22×H(3,𝔽3)—212no—
53C22×H(3,𝔽3)—23no—
54C22×H(3,𝔽3)—26no—
55C22×H(3,𝔽3)—26no—
56C22×H(3,𝔽3)C2×D313yes—
57C22×H(3,𝔽3)C2×D316yescomplemented
58C2×(C3⋊D3)⋊C3—36no—
59C2×(C3⋊D3)⋊C3—66no—
60(C3⋊Dic3)⋊C3—36no—
61C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
62C23×C32—21no—
63C23×C32—23no—
64C23×C32—33no—
65C23×C32—33no—
66C23×C32—63no—
67C23×C32C3×D311yescharacteristic
68C23×C32C3×D313yescomplemented
69C23×C32C3⋊D311yescharacteristic, complemented
70C22×C3⋊D3—31no—
71C22×C3⋊D3—33no—
72C22×C3⋊D3—93no—
73C22×C3×D3—93no—
74C22×C3×D3—91no—
75#638—34no—
76#638—312no—
77#638—912no—
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—33no—
80C2×C3⋊Dic3—93no—
81C3×#67—912no—
82C3×#67—94no—
83C2×C3×Dic3—93no—
84C2×C3×Dic3—91no—
85C2×C33—24no—
86C2×C33—22no—
87C2×C33—22no—
88C2×C33—22no—
89C2×C33—21no—
90C2×C33#6712yescomplemented
91C2×C33C3×D412yescomplemented
92C2×C33C22×D311yescharacteristic
93C2×C33C23×C311yescharacteristic
94C3⋊(C3⋊D3)—22no—
95C3×C3⋊D3—62no—
96C2×H(3,𝔽3)—212no—
97C2×H(3,𝔽3)—26no—
98C2×H(3,𝔽3)—26no—
99C2×H(3,𝔽3)—23no—
100C2×H(3,𝔽3)#6716yescomplemented
101C2×H(3,𝔽3)C22×D313yes—
102(C3⋊D3)⋊C3—66no—
103C2×C3×D4—271no—
104C2×#67—93no—
105C2×#67—91no—
106C2×#67—273no—
107C22×C32—22no—
108C22×C32—24no—
109C22×C32—212no—
110C22×C32—26no—
111C22×C32—22no—
112C22×C32—21no—
113C22×C32—22no—
114C22×C32—23no—
115C22×C32—26no—
116C22×C32—33no—
117C22×C32—36no—
118C22×C32—33no—
119C22×C32—36no—
120C22×C32—66no—
121C22×C32—66no—
122C22×C32—612no—
123C22×C32—63no—
124C22×C32—66no—
125C22×C32C2×C3×D313yes—
126C22×C32C2×C3×D311yescharacteristic
127C22×C32C2×C3×D312yes—
128C22×C32C2×C3×D316yescomplemented
129C22×C32C2×C3⋊D311yescharacteristic
130C22×C32C2×C3⋊D312yescomplemented
131C2×C3⋊D3—32no—
132C2×C3⋊D3—36no—
133C2×C3⋊D3—62no—
134C2×C3⋊D3—66no—
135C2×C3⋊D3—96no—
136C2×C3⋊D3—186no—
137C2×C3×D3—96no—
138C2×C3×D3—92no—
139C2×C3×D3—186no—
140C2×C3×D3—182no—
141C3⋊Dic3—32no—
142C3⋊Dic3—36no—
143C3⋊Dic3—96no—
144C3×Dic3—96no—
145C3×Dic3—92no—
146C33—21no—
147C33C2×#6711yescharacteristic, complemented
148C33C2×C3×D411yescharacteristic, complemented
149H(3,𝔽3)—23no—
150H(3,𝔽3)C2×#6713yescomplemented
151C23×C3—33no—
152C23×C3—31no—
153C23×C3—61no—
154C23×C3—63no—
155C23×C3—271no—
156C23×C3(C3⋊D3)⋊C313yescomplemented
157C23×C3C3×C3⋊D311yescharacteristic
158C22×D3—93no—
159C22×D3—91no—
160C22×D3—273no—
161C3×D4—274no—
162C2×C3×C4—271no—
163#67—912no—
164#67—94no—
165#67—2712no—
166C2×Dic3—93no—
167C2×Dic3—91no—
168C2×Dic3—273no—
169C2×C32—22no—
170C2×C32—24no—
171C2×C32—212no—
172C2×C32—26no—
173C2×C32—22no—
174C2×C32—22no—
175C2×C32—26no—
176C2×C32—21no—
177C2×C32—23no—
178C2×C32—36no—
179C2×C32—33no—
180C2×C32—36no—
181C2×C32—33no—
182C2×C32—66no—
183C2×C32—66no—
184C2×C32—612no—
185C2×C32—66no—
186C2×C32—63no—
187C2×C32C3×#6712yes—
188C2×C32C3×#6716yescomplemented
189C2×C32#63812yescomplemented
190C2×C32C22×C3×D313yes—
191C2×C32C22×C3×D311yescharacteristic
192C2×C32C22×C3⋊D311yescharacteristic
193C3⋊D3—62no—
194C3⋊D3—66no—
195C3⋊D3—186no—
196C3×D3—186no—
197C3×D3—182no—
198C2×D4—271noSylow
199C22×C3—26no—
200C22×C3—22no—
201C22×C3—33no—
202C22×C3—31no—
203C22×C3—36no—
204C22×C3—32no—
205C22×C3—66no—
206C22×C3—62no—
207C22×C3—64no—
208C22×C3—612no—
209C22×C3—61no—
210C22×C3—63no—
211C22×C3—62no—
212C22×C3—66no—
213C22×C3—272no—
214C22×C3—542no—
215C22×C3C2×(C3⋊D3)⋊C313yes—
216C22×C3C2×(C3⋊D3)⋊C316yescomplemented
217C22×C3C2×C3×C3⋊D311yescharacteristic
218C22×C3C2×C3×C3⋊D312yes—
219C2×D3—96no—
220C2×D3—92no—
221C2×D3—186no—
222C2×D3—182no—
223C2×D3—276no—
224C2×D3—546no—
225C3×C4—272no—
226Dic3—96no—
227Dic3—92no—
228Dic3—276no—
229C32—21no—
230C32—23no—
231C32—33no—
232C32—33no—
233C32—63no—
234C32C2×C3×#6711yescharacteristic
235C32C2×C3×#6713yescomplemented
236C32C2×#63811yescharacteristic, complemented
237C23—271no—
238C23C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
239D4—274no—
240C2×C4—271no—
241C2×C3—26no—
242C2×C3—22no—
243C2×C3—36no—
244C2×C3—32no—
245C2×C3—33no—
246C2×C3—31no—
247C2×C3—66no—
248C2×C3—62no—
249C2×C3—64no—
250C2×C3—612no—
251C2×C3—62no—
252C2×C3—66no—
253C2×C3—61no—
254C2×C3—63no—
255C2×C3—542no—
256C2×C3#638⋊C316yescomplemented
257C2×C3C22×(C3⋊D3)⋊C313yes—
258C2×C3C3×#63812yes—
259C2×C3C22×C3×C3⋊D311yescharacteristic
260D3—186no—
261D3—182no—
262D3—546no—
263C22—22no—
264C22—272no—
265C22—542no—
266C22C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
267C22C2×C3⋊((C3⋊D3)⋊C3)12yescomplemented
268C4—272no—
269C3—33no—
270C3—31no—
271C3—61no—
272C3—63no—
273C3C2×#638⋊C313yescomplemented
274C3C2×C3×#63811yescharacteristic
275C2—22no—
276C2—542no—
277C2#1061997712yesdirect product factor
278C2C22×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C33C23×C3
FrattiniC2×C3C22×C3×C3⋊D3
FittingC23×C3×H(3,𝔽3)C2
socleC22×C32C2×C3×D3
hypercenterC23C3⋊((C3⋊D3)⋊C3)
hypocenterC33C2×C3×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×H(3,𝔽3)C2semidirect
2. C22×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. #10619977C2direct (N × H)
4. C2×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#6312C3semidirect
10. C22×C3×H(3,𝔽3)C22semidirect
12. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
13. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
15. C23×C33D3semidirect
16. C23×C33C2×C3semidirect
17. C22×C3⋊(C3⋊D3)C2×C3semidirect
19. #6312C2×C3semidirect
20. C2×C3⋊(C3⋊Dic3)C2×C3semidirect
24. C23×H(3,𝔽3)D3semidirect
29. C2×C3×H(3,𝔽3)D4semidirect
43. C22×C33C2×D3semidirect
45. C22×C33C22×C3semidirect
47. C2×C3⋊(C3⋊D3)C22×C3semidirect
50. C3⋊(C3⋊Dic3)C22×C3semidirect
57. C22×H(3,𝔽3)C2×D3semidirect
61. C3×H(3,𝔽3)C2×D4semidirect
68. C23×C32C3×D3semidirect
69. C23×C32C3⋊D3semidirect
90. C2×C33#67semidirect
91. C2×C33C3×D4semidirect
100. C2×H(3,𝔽3)#67semidirect
128. C22×C32C2×C3×D3semidirect
130. C22×C32C2×C3⋊D3semidirect
147. C33C2×#67semidirect
148. C33C2×C3×D4semidirect
150. H(3,𝔽3)C2×#67semidirect
156. C23×C3(C3⋊D3)⋊C3semidirect
188. C2×C32C3×#67semidirect
189. C2×C32#638semidirect
216. C22×C3C2×(C3⋊D3)⋊C3semidirect
235. C32C2×C3×#67semidirect
236. C32C2×#638semidirect
238. C23C3⋊((C3⋊D3)⋊C3)semidirect
256. C2×C3#638⋊C3semidirect
267. C22C2×C3⋊((C3⋊D3)⋊C3)semidirect
273. C3C2×#638⋊C3semidirect
277. C2#10619977direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×H(3,𝔽3)2
C22×C3⋊((C3⋊D3)⋊C3)2
#106199772
C2×C3⋊((C3⋊Dic3)⋊C3)2
C2×#63123
C2×C3×#6383
C2×#638⋊C33