Finite Group Databasebeta

← D3×#64085 #13175431

Subgroups — D3×#64085 #13175431

Order 1728.

1716 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 296 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×#64085C211yesmaximal, characteristic, complemented
3C32×C42×D3C211yesmaximal, characteristic, complemented
4D3×#3752C213yesmaximal
5D3×#3750C213yesmaximal, complemented
6#11723453C213yesmaximal, complemented
7#11723438C213yesmaximal, complemented
8C2×#64085—31nomaximal
9D3×#876—34nomaximal
10C22×D3×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3C2213yes—
12C2×D3×C3⋊Dic3C2213yes—
13C33×C42C2211yescharacteristic, complemented
14#88859C2213yes—
15#88857C2213yescomplemented
16C3×#3752C2213yes—
17C3×#3750C2213yescomplemented
18C32×C4×Dic3C2213yescomplemented
19#88807C2213yescomplemented
20#88806C2213yescomplemented
21#88804C2213yes—
22#88803C2213yes—
23C2×C32×C42—31no—
24C2×#3752—33no—
25C2×#3750—33no—
26#64085—31no—
27#64085—33no—
28#64085—33no—
29#64085—34no—
30#64085—64no—
31#64085D311yescharacteristic, direct product factor
32C3×#876—34no—
33C3×C42×D3D314yescomplemented
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—
45C2×C32×Dic3C2313yes—
46C32×C4×D3—26no—
47D3×C3⋊Dic3—26no—
48D3×#126—91no—
49C2×#876—94no—
50C23×C3⋊D3—31no—
51C22×D32—34no—
52C22×C32×C4—33no—
53C22×C3⋊Dic3—33no—
54C2×C3×C4×D3C2×D3112yes—
55C2×D3×Dic3—312no—
56C32×C42—24no—
57C32×C42—33no—
58C32×C42C2×D314yescomplemented
59C32×C42C2×D311yescharacteristic, complemented
60#3752—33no—
61#3752—33no—
62#3752—312no—
63#3752—33no—
64#3752—612no—
65#3752C2×D313yes—
66#3750—33no—
67#3750—312no—
68#3750—33no—
69#3750—33no—
70#3750—612no—
71#3750C2×D313yescomplemented
72C3×(C2×D3)⋊C4—312no—
73C3×Dic3⋊C4—312no—
74C3×C4×Dic3C2×D3112yescomplemented
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—23no—
82C2×C32×D3#4116yes—
83D3×C3⋊D3—44no—
84C33×C4—23no—
85C3⋊(C3⋊Dic3)—23no—
86C3×C3⋊Dic3—23no—
87C32×Dic3—23no—
88C3×#126—91no—
89C2×C3×C42—34no—
90C2×(C2×D3)⋊C4—912no—
91C2×Dic3⋊C4—912no—
92#898—93no—
93D3×C4⋊C4—93no—
94D3×C22⋊C4—93no—
95#879—93no—
96#876—34no—
97#876—912no—
98#876—912no—
99#876—94no—
100#876—91no—
101#876—184no—
102C42×D3C3⋊D311yescharacteristic, complemented
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×C4—66no—
115C2×C32×C4C22×D3112yes—
116C2×C32×C4C22×D313yes—
117C2×C3⋊Dic3—33no—
118C2×C3⋊Dic3—312no—
119C2×C3⋊Dic3—612no—
120C2×C3⋊Dic3—66no—
121C2×C3⋊Dic3C22×D313yes—
122C2×C3×Dic3—312no—
123C2×C3×Dic3C22×D3112yes—
124C3×C4×D3—224no—
125D3×Dic3—624no—
126C2×#126—271noSylow
127C2×C33C2×#4113yes—
128C3⋊(C3⋊D3)—41no—
129C3×C3⋊D3—41no—
130C32×D3#12614yescomplemented
131C23×D3—94no—
132C23×D3—91no—
133C22×C3×C4—312no—
134C22×Dic3—912no—
135C2×C4×D3—93no—
136C2×C4×D3C2×C3⋊D313yes—
137C3×C4⋊C4—93no—
138C3×C22⋊C4—93no—
139C3×C42—24no—
140C3×C42—312no—
141C3×C42D3214yescomplemented
142C3×C42C2×C3⋊D311yescharacteristic, complemented
143#217—93no—
144(C2×D3)⋊C4—312no—
145(C2×D3)⋊C4—912no—
146(C2×D3)⋊C4—912no—
147(C2×D3)⋊C4—912no—
148(C2×D3)⋊C4—93no—
149(C2×D3)⋊C4—93no—
150(C2×D3)⋊C4—1812no—
151C4⋊Dic3—93no—
152Dic3⋊C4—312no—
153Dic3⋊C4—912no—
154Dic3⋊C4—912no—
155Dic3⋊C4—912no—
156Dic3⋊C4—93no—
157Dic3⋊C4—93no—
158Dic3⋊C4—1812no—
159C4×Dic3C2×C3⋊D313yescomplemented
160C22×C32—24no—
161C22×C32—36no—
162C22×C32C23×D314yes—
163C22×C32C23×D311yescharacteristic
164C2×C3⋊D3—23no—
165C2×C3⋊D3—63no—
166C2×C3⋊D3—612no—
167C2×C3⋊D3—1212no—
168C2×C3⋊D3—124no—
169C2×C3×D3—612no—
170C2×C3×D3#235124yes—
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—91no—
183#126—273no—
184#126—273no—
185#126—271no—
186C2×C4⋊C4—273no—
187C2×C22⋊C4—273no—
188C2×C42—31no—
189C33C2×#12611yescharacteristic, complemented, Sylow
190C23×C3—34no—
191C23×C3—91no—
192C22×D3—34no—
193C22×D3—94no—
194C22×D3—91no—
195C22×D3—184no—
196C22×D3—1824no—
197C22×D3—186no—
198C22×D3C22×C3⋊D311yescharacteristic
199C2×C3×C4—212no—
200C2×C3×C4—312no—
201C2×C3×C4—624no—
202C2×C3×C4—93no—
203C2×C3×C4C2×D32112yes—
204C2×C3×C4C22×C3⋊D313yes—
205C2×Dic3—312no—
206C2×Dic3—912no—
207C2×Dic3—93no—
208C2×Dic3—1812no—
209C2×Dic3—1824no—
210C2×Dic3C22×C3⋊D313yes—
211C4×D3—26no—
212C4×D3—186no—
213C2×C32—212no—
214C2×C32—34no—
215C2×C32C2×#235112yes—
216C2×C32D3×#4113yes—
217C3⋊D3—41no—
218C3⋊D3—121no—
219C3⋊D3—124no—
220C3⋊D3—244no—
221C3×D3—124no—
222C3×D3#876116yescomplemented
223C24—271no—
224C22×C4—33no—
225C22×C4—273no—
226C4⋊C4—93no—
227C4⋊C4—273no—
228C4⋊C4—273no—
229C4⋊C4—273no—
230C22⋊C4—93no—
231C22⋊C4—273no—
232C22⋊C4—273no—
233C22⋊C4—273no—
234C42—33no—
235C42D3×C3⋊D311yescharacteristic, complemented
236C22×C3—24no—
237C22×C3—324no—
238C22×C3—183no—
239C22×C3C22×D3214yes—
240C22×C3C23×C3⋊D311yescharacteristic
241C2×D3—612no—
242C2×D3—1812no—
243C2×D3—183no—
244C2×D3—3612no—
245C2×D3—3616no—
246C2×D3—364no—
247C2×D3#382816yes—
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×#87614yescomplemented
260C32D3×#12611yescharacteristic, complemented
261C23—31no—
262C23—91no—
263C23—271no—
264C23—546no—
265C2×C4—33no—
266C2×C4—66no—
267C2×C4—93no—
268C2×C4—273no—
269C2×C4—546no—
270C2×C4C2×D3×C3⋊D313yes—
271C2×C3—212no—
272C2×C3—316no—
273C2×C3—361no—
274C2×C3D3×#235112yes—
275C2×C3C2×#382813yes—
276D3—124no—
277D3—361no—
278D3—364no—
279D3—724no—
280D3#6408514yesdirect product factor
281C22—36no—
282C22—183no—
283C22—543no—
284C22—1084no—
285C22C22×D3×C3⋊D311yescharacteristic
286C4—23no—
287C4—63no—
288C4—183no—
289C4—543no—
290C3—24no—
291C3D3×#87614yescomplemented
292C3C2×#6408511yescharacteristic, complemented
293C2—34no—
294C2—361no—
295C2—1081no—
296C2D3×#382813yes—
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×#64085C2semidirect
3. C32×C42×D3C2semidirect
5. D3×#3750C2semidirect
6. #11723453C2semidirect
7. #11723438C2semidirect
13. C33×C42C22semidirect
15. #88857C22semidirect
17. C3×#3750C22semidirect
18. C32×C4×Dic3C22semidirect
19. #88807C22semidirect
20. #88806C22semidirect
31. #64085D3direct (N × H)
33. C3×C42×D3D3semidirect
58. C32×C42C2×D3semidirect
59. C32×C42C2×D3semidirect
71. #3750C2×D3semidirect
74. C3×C4×Dic3C2×D3semidirect
102. C42×D3C3⋊D3semidirect
130. C32×D3#126semidirect
141. C3×C42D32semidirect
142. C3×C42C2×C3⋊D3semidirect
159. C4×Dic3C2×C3⋊D3semidirect
189. C33C2×#126semidirect
222. C3×D3#876semidirect
235. C42D3×C3⋊D3semidirect
259. C32C2×#876semidirect
260. C32D3×#126semidirect
280. D3#64085direct (N × H)
291. C3D3×#876semidirect
292. C3C2×#64085semidirect

Maximal subgroups

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

subgroupindex
#117239172
C3×#640852
C32×C42×D32
D3×#37522
D3×#37502
#117234532
#117234382
C2×#640853
D3×#8763