Finite Group Databasebeta

← C3⋊D3×#876 #13175422

Subgroups — C3⋊D3×#876 #13175422

Order 1728.

1716 conjugacy classes of subgroups; only the 256 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 256 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
1#11723917C211yesmaximal, characteristic, complemented
2C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#876C211yesmaximal, characteristic, complemented
4C3⋊D3×(C2×D3)⋊C4C213yesmaximal
5C3⋊D3×Dic3⋊C4C213yesmaximal, complemented
6#11723446C213yesmaximal, complemented
7#11723428C213yesmaximal, complemented
8C3⋊D3×#126—31nomaximal
9D3×#876—34nomaximal
10C22×D3×C3⋊D3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2213yes—
12C2×Dic3×C3⋊D3C2213yes—
13C33×C42C2211yescharacteristic, complemented
14#88859C2213yes—
15#88857C2213yescomplemented
16C3×C4×C3⋊Dic3C2213yescomplemented
17C32×(C2×D3)⋊C4C2213yes—
18C32×Dic3⋊C4C2213yescomplemented
19#88806C2213yescomplemented
20Dic3⋊(C3⋊Dic3)C2213yescomplemented
21#88804C2213yes—
22(C2×D3)⋊(C3⋊Dic3)C2213yes—
23C32×#126—31no—
24#64107—33no—
25C4⋊C4×C3⋊D3—33no—
26C22⋊C4×C3⋊D3—33no—
27#64088—33no—
28#64085—31no—
29#64085—34no—
30#64085—64no—
31C42×C3⋊D3D311yescharacteristic, complemented
32C3×#876D314yescomplemented
33C3×C42×D3—34no—
34D3×(C2×D3)⋊C4—312no—
35D3×Dic3⋊C4—312no—
36#63868—312no—
37#63859—312no—
38C22×C3⋊(C3⋊D3)C2311yescharacteristic
39C22×C3×C3⋊D3C2311yescharacteristic
40C22×C32×D3C2311yescharacteristic
41C2×D3×C3⋊D3—26no—
42C2×C33×C4C2313yes—
43C2×C3⋊(C3⋊Dic3)C2313yes—
44C2×C3×C3⋊Dic3C2313yes—
45C3×C4×C3⋊D3—26no—
46C2×C32×Dic3C2313yes—
47Dic3×C3⋊D3—26no—
48D3×#126—94no—
49C2×#876—91no—
50C23×C3⋊D3—31no—
51C22×D32—34no—
52C2×C4×C3⋊D3—33no—
53C2×C4×C3⋊D3C2×D313yes—
54C2×C3×C4×D3—312no—
55C2×D3×Dic3—312no—
56C32×C4⋊C4—33no—
57C32×C22⋊C4—33no—
58C32×C42—24no—
59C32×C42C2×D314yescomplemented
60C32×C42C2×D311yescharacteristic, complemented
61#3757—33no—
62#3752—33no—
63#3752—312no—
64#3752—33no—
65#3752—612no—
66C4⋊(C3⋊Dic3)—33no—
67#3750—33no—
68#3750—312no—
69#3750—33no—
70#3750—612no—
71C4×C3⋊Dic3C2×D313yescomplemented
72C3×(C2×D3)⋊C4C2×D3112yes—
73C3×Dic3⋊C4C2×D3112yescomplemented
74C3×C4×Dic3—312no—
75#3724—312no—
76Dic3⋊Dic3—312no—
77#3722—312no—
78(C2×D3)⋊Dic3—312no—
79C22×C33C2411yescharacteristic
80C2×C3⋊(C3⋊D3)—23no—
81C2×C3×C3⋊D3#4116yes—
82C2×C32×D3—23no—
83D3×C3⋊D3—44no—
84C33×C4—23no—
85C3⋊(C3⋊Dic3)—23no—
86C3×C3⋊Dic3—23no—
87C32×Dic3—23no—
88C3×#126—34no—
89C2×C3×C42—91no—
90C2×(C2×D3)⋊C4—93no—
91C2×Dic3⋊C4—93no—
92#898—912no—
93D3×C4⋊C4—912no—
94D3×C22⋊C4—912no—
95#879—912no—
96#876—94no—
97#876—91no—
98#876—93no—
99#876—93no—
100#876—184no—
101#876C3⋊D311yescharacteristic, direct product factor
102C42×D3—34no—
103C23×C32—31no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3—34no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—66no—
108C22×C3⋊D3C22×D311yescharacteristic
109C22×C3×D3—34no—
110C22×C3×D3C22×D314yes—
111C2×D32—624no—
112C2×C32×C4—212no—
113C2×C32×C4—33no—
114C2×C32×C4C22×D3112yes—
115C2×C32×C4C22×D313yes—
116C2×C3⋊Dic3—33no—
117C2×C3⋊Dic3—312no—
118C2×C3⋊Dic3—612no—
119C2×C3⋊Dic3C22×D313yes—
120C4×C3⋊D3—26no—
121C4×C3⋊D3—66no—
122C2×C3×Dic3—312no—
123C2×C3×Dic3C22×D3112yes—
124C3×C4×D3—624no—
125D3×Dic3—624no—
126C2×#126—271noSylow
127C2×C33C2×#4113yes—
128C3⋊(C3⋊D3)—41no—
129C3×C3⋊D3#12614yescomplemented
130C32×D3—41no—
131C23×D3—94no—
132C23×D3—91no—
133C22×C3×C4—93no—
134C22×Dic3—93no—
135C2×C4×D3—312no—
136C2×C4×D3—912no—
137C3×C4⋊C4—312no—
138C3×C22⋊C4—312no—
139C3×C42—24no—
140C3×C42—93no—
141C3×C42D3214yescomplemented
142C3×C42C2×C3⋊D311yescharacteristic, complemented
143#217—912no—
144(C2×D3)⋊C4—912no—
145(C2×D3)⋊C4—93no—
146(C2×D3)⋊C4—912no—
147(C2×D3)⋊C4—93no—
148(C2×D3)⋊C4—93no—
149(C2×D3)⋊C4—1812no—
150(C2×D3)⋊C4C2×C3⋊D313yes—
151C4⋊Dic3—912no—
152Dic3⋊C4—912no—
153Dic3⋊C4—912no—
154Dic3⋊C4—93no—
155Dic3⋊C4—93no—
156Dic3⋊C4—93no—
157Dic3⋊C4—1812no—
158Dic3⋊C4C2×C3⋊D313yescomplemented
159C4×Dic3—312no—
160C22×C32—24no—
161C22×C32—63no—
162C22×C32C23×D314yes—
163C22×C32C23×D311yescharacteristic
164C2×C3⋊D3—63no—
165C2×C3⋊D3—612no—
166C2×C3⋊D3—1212no—
167C2×C3⋊D3—124no—
168C2×C3⋊D3#23516yes—
169C2×C3×D3—212no—
170C2×C3×D3—324no—
171D32—1216no—
172C32×C4—23no—
173C32×C4—212no—
174C32×C4—412no—
175C32×C4—63no—
176C3⋊Dic3—23no—
177C3⋊Dic3—63no—
178C3⋊Dic3—612no—
179C3⋊Dic3—1212no—
180C3×Dic3—212no—
181C3×Dic3—612no—
182#126—31no—
183#126—273no—
184#126—273no—
185#126—271no—
186C2×C4⋊C4—273no—
187C2×C22⋊C4—273no—
188C2×C42—91no—
189C33C2×#12611yescharacteristic, complemented, Sylow
190C23×C3—34no—
191C23×C3—91no—
192C22×D3—34no—
193C22×D3—94no—
194C22×D3—91no—
195C22×D3—184no—
196C22×D3—186no—
197C22×D3—1824no—
198C22×D3C22×C3⋊D311yescharacteristic
199C2×C3×C4—212no—
200C2×C3×C4—312no—
201C2×C3×C4—93no—
202C2×C3×C4—186no—
203C2×C3×C4C2×D32112yes—
204C2×C3×C4C22×C3⋊D313yes—
205C2×Dic3—312no—
206C2×Dic3—912no—
207C2×Dic3—93no—
208C2×Dic3—1812no—
209C2×Dic3—186no—
210C2×Dic3C22×C3⋊D313yes—
211C4×D3—624no—
212C4×D3—1824no—
213C2×C32—212no—
214C2×C32—121no—
215C2×C32C2×#23513yes—
216C2×C32D3×#41112yes—
217C3⋊D3—121no—
218C3⋊D3—124no—
219C3⋊D3—244no—
220C3⋊D3#87614yesdirect product factor
221C3×D3—316no—
222C3×D3—44no—
223C24—271no—
224C22×C4—93no—
225C22×C4—273no—
226C4⋊C4—33no—
227C4⋊C4—273no—
228C4⋊C4—273no—
229C4⋊C4—273no—
230C22⋊C4—33no—
231C22⋊C4—273no—
232C22⋊C4—273no—
233C22⋊C4—273no—
234C42—93no—
235C42D3×C3⋊D311yescharacteristic, complemented
236C22×C3—24no—
237C22×C3—612no—
238C22×C3—96no—
239C22×C3C22×D3214yes—
240C22×C3C23×C3⋊D311yescharacteristic
241C2×D3—23no—
242C2×D3—324no—
243C2×D3—1812no—
244C2×D3—183no—
245C2×D3—3612no—
246C2×D3—3616no—
247C2×D3—364no—
248C3×C4—212no—
249C3×C4—23no—
250C3×C4—412no—
251C3×C4—612no—
252C3×C4—183no—
253Dic3—23no—
254Dic3—612no—
255Dic3—1812no—
256Dic3—183no—
257Dic3—3612no—
258C32—24no—
259C32C2×#87611yescharacteristic, complemented
260C32D3×#12614yescomplemented
261C23—31no—
262C23—91no—
263C23—271no—
264C23—546no—
265C2×C4—33no—
266C2×C4—93no—
267C2×C4—186no—
268C2×C4—273no—
269C2×C4—546no—
270C2×C4C2×D3×C3⋊D313yes—
271C2×C3—212no—
272C2×C3—94no—
273C2×C3—124no—
274C2×C3D3×#235112yes—
275C2×C3#41×C3⋊D313yes—
276D3—316no—
277D3—41no—
278D3—361no—
279D3—364no—
280D3—724no—
281C22—63no—
282C22—96no—
283C22—543no—
284C22—1084no—
285C22C22×D3×C3⋊D311yescharacteristic
286C4—23no—
287C4—63no—
288C4—183no—
289C4—543no—
290C3—24no—
291C3D3×#87614yescomplemented
292C3C3⋊D3×#12611yescharacteristic, complemented
293C2—94no—
294C2—121no—
295C2—1081no—
296C2C3⋊D3×#23513yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC22×D3×C3⋊D3C22
hypercenterC42D3×C3⋊D3
hypocenterC33C2×#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723917C2semidirect
2. C3×C42×C3⋊D3C2semidirect
3. C32×#876C2semidirect
5. C3⋊D3×Dic3⋊C4C2semidirect
6. #11723446C2semidirect
7. #11723428C2semidirect
13. C33×C42C22semidirect
15. #88857C22semidirect
16. C3×C4×C3⋊Dic3C22semidirect
18. C32×Dic3⋊C4C22semidirect
19. #88806C22semidirect
20. Dic3⋊(C3⋊Dic3)C22semidirect
31. C42×C3⋊D3D3semidirect
32. C3×#876D3semidirect
59. C32×C42C2×D3semidirect
60. C32×C42C2×D3semidirect
71. C4×C3⋊Dic3C2×D3semidirect
73. C3×Dic3⋊C4C2×D3semidirect
101. #876C3⋊D3direct (N × H)
129. C3×C3⋊D3#126semidirect
141. C3×C42D32semidirect
142. C3×C42C2×C3⋊D3semidirect
158. Dic3⋊C4C2×C3⋊D3semidirect
189. C33C2×#126semidirect
220. C3⋊D3#876direct (N × H)
235. C42D3×C3⋊D3semidirect
259. C32C2×#876semidirect
260. C32D3×#126semidirect
291. C3D3×#876semidirect
292. C3C3⋊D3×#126semidirect

Maximal subgroups

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

subgroupindex
#117239172
C3×C42×C3⋊D32
C32×#8762
C3⋊D3×(C2×D3)⋊C42
C3⋊D3×Dic3⋊C42
#117234462
#117234282
C3⋊D3×#1263
D3×#8763