Finite Group Databasebeta

← #13147745

Subgroups — #13147745

Order 1728.

846 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C42×C3⋊D9C211yesmaximal, characteristic, complemented
2C9×#876C211yesmaximal, characteristic, complemented
3C3×#63438C211yesmaximal, characteristic, complemented
4#11721048C213yesmaximal
5#11720980C213yesmaximal, complemented
6#11720966C213yesmaximal, complemented
7#11720950C213yesmaximal, complemented
8#10594020—31nomaximal
9D9×#126—31nomaximal
10C22×D3×D9C2211yescharacteristic
11C2×C4×C3⋊D9C2213yes—
12C2×(C3⋊D9)⋊C4C2213yes—
13C3×C42×C9C2211yescharacteristic, complemented
14C4×C3⋊Dic9C2213yescomplemented
15C9×(C2×D3)⋊C4C2213yes—
16C3×(C2×D9)⋊C4C2213yes—
17C9×Dic3⋊C4C2213yescomplemented
18C3×Dic9⋊C4C2213yescomplemented
19(C2×D3)⋊Dic9C2213yes—
20(C2×D9)⋊Dic3C2213yes—
21Dic3⋊Dic9C2213yescomplemented
22Dic9⋊Dic3C2213yescomplemented
23C42×C3⋊D3—31no—
24C3×#876—31no—
25C3×#876D311yescharacteristic
26#63921—33no—
27#63883—33no—
28#63862—33no—
29#63862—33no—
30C9×#126—31no—
31#63460—33no—
32C4⋊C4×D9—33no—
33C22⋊C4×D9—33no—
34#63441—33no—
35#63438D311yescharacteristic, complemented
36C42×D9—31no—
37C42×D9—61no—
38C22×C3⋊D9C2311yescharacteristic
39C22×D3×C9C2311yescharacteristic
40C22×C3×D9C2311yescharacteristic
41C2×D3×D9—26no—
42C2×C3×C4×C9C2313yes—
43C2×C3⋊Dic9C2313yes—
44C4×C3⋊D9—26no—
45C2×C9×Dic3C2313yes—
46C2×C3×Dic9C2313yes—
47(C3⋊D9)⋊C4—26no—
48D3×#126—91no—
49D3×#126—91no—
50C22×D32—31no—
51C2×C4×C3⋊D3—33no—
52C2×#624—33no—
53C23×D9—31no—
54C32×C42C2×D311yescharacteristic
55C4×C3⋊Dic3—33no—
56C3×(C2×D3)⋊C4—33no—
57C3×(C2×D3)⋊C4C2×D313yes—
58C3×Dic3⋊C4—33no—
59C3×Dic3⋊C4C2×D313yes—
60Dic3⋊Dic3—33no—
61Dic3⋊Dic3—33no—
62(C2×D3)⋊Dic3—33no—
63(C2×D3)⋊Dic3—33no—
64C2×C4×D9—33no—
65C2×C4×D9—33no—
66C2×C4×D9—63no—
67C9×C4⋊C4—33no—
68C9×C22⋊C4—33no—
69C42×C9—21no—
70C42×C9C2×D311yescharacteristic, complemented
71#3676—33no—
72(C2×D9)⋊C4—33no—
73(C2×D9)⋊C4C2×D313yes—
74C4⋊Dic9—33no—
75Dic9⋊C4—33no—
76Dic9⋊C4C2×D313yescomplemented
77C4×Dic9—33no—
78C4×Dic9—63no—
79C22×C3×C9C2411yescharacteristic
80C2×C3⋊D9#4116yes—
81C2×D3×C9—23no—
82C2×C3×D9—23no—
83D3×D9—44no—
84C3×C4×C9—23no—
85C3⋊Dic9—23no—
86C9×Dic3—23no—
87C3×Dic9—23no—
88C3×#126—31no—
89C3×#126—91no—
90#898—93no—
91#898—93no—
92D3×C4⋊C4—93no—
93D3×C4⋊C4—93no—
94D3×C22⋊C4—93no—
95D3×C22⋊C4—93no—
96#879—93no—
97#879—93no—
98#876—31no—
99#876D911yescharacteristic, complemented
100C42×D3—91no—
101C42×D3—91no—
102C42×D3—181no—
103C22×C3⋊D3—31no—
104C22×C3×D3—31no—
105C22×C3×D3C22×D311yescharacteristic
106C2×D32—66no—
107C2×C32×C4C22×D313yes—
108C2×C3⋊Dic3—33no—
109C4×C3⋊D3—66no—
110C2×C3×Dic3—33no—
111C2×C3×Dic3C22×D313yes—
112#624—66no—
113C23×C9—31no—
114C22×D9—31no—
115C22×D9—61no—
116C22×D9—66no—
117C22×D9C22×D311yescharacteristic
118C2×C4×C9—23no—
119C2×C4×C9—33no—
120C2×C4×C9C22×D313yes—
121C2×Dic9—33no—
122C2×Dic9—63no—
123C2×Dic9C22×D313yes—
124C4×D9—66no—
125C4×D9—66no—
126C4×D9—612no—
127C2×#126—271noSylow
128C2×C3×C9C2×#4113yes—
129C3⋊D9#12614yescomplemented
130D3×C9—41no—
131C3×D9—41no—
132C23×D3—91no—
133C23×D3—91no—
134C2×C4×D3—93no—
135C2×C4×D3—93no—
136C2×C4×D3—93no—
137C2×C4×D3—93no—
138C2×C4×D3—183no—
139C3×C4⋊C4—33no—
140C3×C4⋊C4—93no—
141C3×C22⋊C4—33no—
142C3×C22⋊C4—93no—
143C3×C42—21no—
144C3×C42C2×D911yescharacteristic, complemented
145C3×C42D3211yescharacteristic
146#217—93no—
147#217—93no—
148(C2×D3)⋊C4—33no—
149(C2×D3)⋊C4—93no—
150(C2×D3)⋊C4—93no—
151(C2×D3)⋊C4C2×D913yes—
152C4⋊Dic3—93no—
153C4⋊Dic3—93no—
154Dic3⋊C4—33no—
155Dic3⋊C4—93no—
156Dic3⋊C4—93no—
157Dic3⋊C4C2×D913yescomplemented
158C4×Dic3—93no—
159C4×Dic3—93no—
160C4×Dic3—183no—
161C22×C32C23×D311yescharacteristic
162C2×C3⋊D3—36no—
163C2×C3×D3—23no—
164C2×C3×D3—63no—
165D32—124no—
166C32×C4—23no—
167C3⋊Dic3—63no—
168C3×Dic3—23no—
169C3×Dic3—63no—
170C22×C9—21no—
171C22×C9—63no—
172C22×C9C23×D311yescharacteristic
173C2×D9—23no—
174C2×D9—36no—
175C2×D9—66no—
176C2×D9—124no—
177C4×C9—26no—
178C4×C9—23no—
179C4×C9—63no—
180Dic9—23no—
181Dic9—66no—
182Dic9—63no—
183#126—31no—
184#126—91no—
185#126—273no—
186#126—273no—
187C2×C4⋊C4—273no—
188C2×C22⋊C4—273no—
189C2×C42—271no—
190C3×C9C2×#12611yescharacteristic, complemented, Sylow
191C23×C3—31no—
192C23×C3—91no—
193C22×D3—31no—
194C22×D3—91no—
195C22×D3—91no—
196C22×D3—181no—
197C22×D3—186no—
198C22×D3—186no—
199C22×D3C22×D911yescharacteristic
200C2×C3×C4—23no—
201C2×C3×C4—33no—
202C2×C3×C4—93no—
203C2×C3×C4C22×D913yes—
204C2×C3×C4C2×D3213yes—
205C2×Dic3—33no—
206C2×Dic3—93no—
207C2×Dic3—93no—
208C2×Dic3—183no—
209C2×Dic3C22×D913yes—
210C4×D3—186no—
211C4×D3—186no—
212C4×D3—186no—
213C4×D3—186no—
214C4×D3—1812no—
215C2×C32D3×#4113yes—
216C3⋊D3—34no—
217C3×D3—41no—
218C3×D3—121no—
219C2×C9—23no—
220C2×C9—121no—
221C2×C9D3×#4113yes—
222D9—34no—
223D9—41no—
224D9—64no—
225C24—271no—
226C22×C4—273no—
227C22×C4—273no—
228C4⋊C4—33no—
229C4⋊C4—93no—
230C4⋊C4—273no—
231C4⋊C4—273no—
232C22⋊C4—33no—
233C22⋊C4—93no—
234C22⋊C4—273no—
235C22⋊C4—273no—
236C42—273no—
237C42D3×D911yescharacteristic, complemented
238C22×C3—21no—
239C22×C3—63no—
240C22×C3—183no—
241C22×C3C23×D911yescharacteristic
242C22×C3C22×D3211yescharacteristic
243C2×D3—23no—
244C2×D3—63no—
245C2×D3—96no—
246C2×D3—96no—
247C2×D3—186no—
248C2×D3—364no—
249C2×D3—364no—
250C3×C4—26no—
251C3×C4—23no—
252C3×C4—23no—
253C3×C4—63no—
254C3×C4—183no—
255Dic3—23no—
256Dic3—63no—
257Dic3—186no—
258Dic3—183no—
259Dic3—183no—
260C32D3×#12611yescharacteristic
261C9—21no—
262C9D3×#12611yescharacteristic, complemented
263C23—31no—
264C23—91no—
265C23—271no—
266C23—546no—
267C2×C4—33no—
268C2×C4—93no—
269C2×C4—273no—
270C2×C4—546no—
271C2×C4—546no—
272C2×C4C2×D3×D913yes—
273C2×C3—23no—
274C2×C3—121no—
275C2×C3—361no—
276C2×C3#41×D913yes—
277C2×C3#6430613yes—
278D3—41no—
279D3—94no—
280D3—94no—
281D3—121no—
282D3—184no—
283C22—63no—
284C22—183no—
285C22—276no—
286C22—1084no—
287C22C22×D3×D911yescharacteristic
288C4—23no—
289C4—63no—
290C4—183no—
291C4—543no—
292C3—21no—
293C3D9×#12611yescharacteristic, complemented
294C3#1059402011yescharacteristic
295C2—121no—
296C2—274no—
297C2—361no—
298C2#1172247513yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
hypercenterC42D3×D9
hypocenterC3×C9C2×#126
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C42×C3⋊D9C2semidirect
2. C9×#876C2semidirect
3. C3×#63438C2semidirect
5. #11720980C2semidirect
6. #11720966C2semidirect
7. #11720950C2semidirect
13. C3×C42×C9C22semidirect
14. C4×C3⋊Dic9C22semidirect
17. C9×Dic3⋊C4C22semidirect
18. C3×Dic9⋊C4C22semidirect
21. Dic3⋊Dic9C22semidirect
22. Dic9⋊Dic3C22semidirect
35. #63438D3semidirect
70. C42×C9C2×D3semidirect
76. Dic9⋊C4C2×D3semidirect
99. #876D9semidirect
129. C3⋊D9#126semidirect
144. C3×C42C2×D9semidirect
157. Dic3⋊C4C2×D9semidirect
190. C3×C9C2×#126semidirect
237. C42D3×D9semidirect
262. C9D3×#126semidirect
293. C3D9×#126semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊D92
C9×#8762
C3×#634382
#117210482
#117209802
#117209662
#117209502
#105940203
D9×#1263