Finite Group Databasebeta

← #13153412

Subgroups — #13153412

Order 1728.

1000 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 242 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#11722638C212yesmaximal, complemented
2H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
3#64117⋊C3C216yesmaximal, complemented
4#64113⋊C3C213yesmaximal, complemented
5C2×#3745⋊C3C213yesmaximal, complemented
6#10596223C311yesmaximal, characteristic, complemented
7C3×#5752—31nomaximal
8#41×H(3,𝔽3)C2211yescharacteristic
9#88722—22no—
10D4×(C3⋊D3)⋊C3—26no—
11#3828⋊C3C2216yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2216yescomplemented
13M16×H(3,𝔽3)C2213yescomplemented
14C2×C8×H(3,𝔽3)C2213yescomplemented
15#3745⋊C3C2213yescomplemented
16#3745⋊C3C2216yescomplemented
17#3744⋊C3C2216yescomplemented
18#3746⋊C3C2211yescharacteristic, complemented
19#64366C2×C312yescomplemented
20C3×#1009—32no—
21C32×#131—21no—
22C32×#131D311yescharacteristic, complemented
23C32×#131C2×C311yescharacteristic, complemented
24#64117C2×C316yescomplemented
25#64113C2×C313yescomplemented
26C2×#3745C2×C313yescomplemented
27C3×#908—36no—
28C3×#904—33no—
29C2×C3×D24—33no—
30C22×(C3⋊D3)⋊C3—26no—
31Q8×H(3,𝔽3)D411yescharacteristic
32D4×H(3,𝔽3)D413yes—
33C2×C4×H(3,𝔽3)C2313yes—
34#638⋊C3—26no—
35C4⋊((C3⋊D3)⋊C3)—26no—
36C4⋊((C3⋊D3)⋊C3)C2316yes—
37C4×(C3⋊D3)⋊C3—26no—
38#634⋊C3C2312yes—
39C8×H(3,𝔽3)C2311yescharacteristic
40C8×H(3,𝔽3)C2313yes—
41C3×#576—91no—
42#5752—31no—
43#5752—91no—
44C32×#41—21no—
45C32×#41C2×D311yescharacteristic
46C32×#41C22×C311yescharacteristic
47#3832—22no—
48D4×C3⋊D3—26no—
49#3828C22×C316yescomplemented
50C2×C4⋊(C3⋊D3)C22×C316yescomplemented
51C3×#239—62no—
52C3×D3×D4—66no—
53C3×#235—36no—
54C2×C3×D12—36no—
55C32×M16—23no—
56C32×M16C2×D313yescomplemented
57C32×M16C22×C313yescomplemented
58C2×C32×C8—23no—
59C2×C32×C8C2×D313yescomplemented
60C2×C32×C8C22×C313yescomplemented
61#3746C22×C311yescharacteristic, complemented
62#3745C22×C313yescomplemented
63#3745C22×C316yescomplemented
64#3744C22×C316yescomplemented
65C3×Dic12—31no—
66C3×D24—33no—
67C3×D24—36no—
68C3×#204—36no—
69C22×H(3,𝔽3)C2×D413yes—
70C2×(C3⋊D3)⋊C3—26no—
71C2×(C3⋊D3)⋊C3—86no—
72C4×H(3,𝔽3)C2×D413yes—
73C4×H(3,𝔽3)C2411yescharacteristic
74(C3⋊Dic3)⋊C3—22no—
75C3×H(5,𝔽2)—92no—
76#1009—32no—
77#1009—92no—
78C3×AGL(1,ℤ/8)—96no—
79C3×#135—93no—
80C2×C3×D8—93no—
81C3×#131—31no—
82C3×#131—31no—
83C3×#131—61no—
84C3×#131C3×D311yescharacteristic
85#908—36no—
86#908—96no—
87#904—33no—
88#904—93no—
89C2×D24—33no—
90C2×D24—93no—
91C22×C3⋊D3—26no—
92C22×C3×D3—66no—
93C32×Q8—21no—
94C32×Q8D1211yescharacteristic
95C32×Q8C3×D411yescharacteristic
96C32×D4—23no—
97C32×D4D1213yes—
98C32×D4C3×D413yes—
99C2×C32×C4—23no—
100C2×C32×C4C22×D313yes—
101C2×C32×C4C23×C313yes—
102#638—26no—
103C4⋊(C3⋊D3)—26no—
104C4⋊(C3⋊D3)C23×C316yes—
105C4×C3⋊D3—26no—
106#634C23×C312yes—
107C3×#67—66no—
108C3×D12—36no—
109C3×D12—66no—
110C3×C4×D3—66no—
111C3×Dic6—32no—
112C32×C8—21no—
113C32×C8—23no—
114C32×C8C22×D313yes—
115C32×C8C22×D311yescharacteristic
116C32×C8C23×C313yes—
117C32×C8C23×C311yescharacteristic
118#576—91noSylow
119C2×H(3,𝔽3)—23no—
120C2×H(3,𝔽3)C22×D411yescharacteristic
121(C3⋊D3)⋊C3—46no—
122C3×#41—31no—
123C3×#41—31no—
124C3×#41—61no—
125C3×#41—96no—
126C3×#41—182no—
127C3×#41C2×C3×D311yescharacteristic
128C2×C3×D4—96no—
129C2×C3×D4—186no—
130#239—62no—
131#239—182no—
132D3×D4—66no—
133D3×D4—186no—
134#235—36no—
135#235—96no—
136C2×D12—36no—
137C2×D12—96no—
138C3×Q16—91no—
139C3×SD16—96no—
140C3×D8—93no—
141C3×D8—96no—
142C3×M16—33no—
143C3×M16—33no—
144C3×M16—63no—
145C3×M16C2×C3×D313yes—
146C2×C3×C8—33no—
147C2×C3×C8—33no—
148C2×C3×C8—63no—
149C2×C3×C8C2×C3×D313yes—
150Dic12—31no—
151Dic12—91no—
152D24—33no—
153D24—36no—
154D24—93no—
155D24—96no—
156#204—36no—
157#204—96no—
158C22×C32—23no—
159C22×C32C2×D1213yes—
160C22×C32C2×C3×D413yes—
161C2×C3⋊D3—26no—
162C2×C3⋊D3—86no—
163C2×C3×D3—66no—
164C2×C3×D3—246no—
165C32×C4—21no—
166C32×C4—23no—
167C32×C4C2×D1213yes—
168C32×C4C2×C3×D413yes—
169C32×C4C23×D311yescharacteristic
170C32×C4C24×C311yescharacteristic
171C3⋊Dic3—22no—
172C3×Dic3—62no—
173H(5,𝔽2)—92no—
174AGL(1,ℤ/8)—96no—
175#135—93no—
176C2×D8—93no—
177#131(C3⋊D3)⋊C311yescharacteristic, complemented
178H(3,𝔽3)#57611yescharacteristic, complemented, Sylow
179C23×C3—186no—
180C22×D3—66no—
181C22×D3—186no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8—92no—
186C3×Q8C3×D1211yescharacteristic
187C3×D4—33no—
188C3×D4—33no—
189C3×D4—63no—
190C3×D4—96no—
191C3×D4—186no—
192C3×D4—186no—
193C3×D4C3×D1213yes—
194C2×C3×C4—33no—
195C2×C3×C4—33no—
196C2×C3×C4—63no—
197C2×C3×C4—186no—
198C2×C3×C4C22×C3×D313yes—
199#67—66no—
200#67—186no—
201D12—36no—
202D12—66no—
203D12—96no—
204D12—186no—
205C4×D3—66no—
206C4×D3—186no—
207Dic6—32no—
208Dic6—92no—
209C3×C8—31no—
210C3×C8—33no—
211C3×C8—33no—
212C3×C8—31no—
213C3×C8—61no—
214C3×C8—63no—
215C3×C8C22×C3×D313yes—
216C3×C8C22×C3×D311yescharacteristic
217C2×C32—23no—
218C2×C32—23no—
219C2×C32—21no—
220C2×C32—43no—
221C2×C32C22×D1211yescharacteristic
222C2×C32C22×C3×D411yescharacteristic
223C3⋊D3—46no—
224C3×D3—126no—
225#41—96no—
226#41—182no—
227#41C2×(C3⋊D3)⋊C311yescharacteristic
228C2×D4—96no—
229C2×D4—186no—
230Q16—91no—
231SD16—96no—
232D8—93no—
233D8—96no—
234M16C2×(C3⋊D3)⋊C313yescomplemented
235C2×C8C2×(C3⋊D3)⋊C313yescomplemented
236C22×C3—33no—
237C22×C3—33no—
238C22×C3—63no—
239C22×C3—186no—
240C22×C3—726no—
241C22×C3C2×C3×D1213yes—
242C2×D3—66no—
243C2×D3—186no—
244C2×D3—246no—
245C2×D3—726no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—33no—
249C3×C4—33no—
250C3×C4—61no—
251C3×C4—63no—
252C3×C4—182no—
253C3×C4C2×C3×D1213yes—
254C3×C4C23×C3×D311yescharacteristic
255Dic3—62no—
256Dic3—182no—
257C32—21no—
258C32#575211yescharacteristic, complemented
259C32C3×#57611yescharacteristic, complemented
260C23—186no—
261Q8—92no—
262Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
263D4—96no—
264D4—186no—
265D4—186no—
266D4C4⋊((C3⋊D3)⋊C3)13yes—
267C2×C4—186no—
268C2×C4C22×(C3⋊D3)⋊C313yes—
269C8C22×(C3⋊D3)⋊C311yescharacteristic
270C8C22×(C3⋊D3)⋊C313yes—
271C2×C3—23no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—63no—
275C2×C3—63no—
276C2×C3—61no—
277C2×C3—123no—
278C2×C3—366no—
279C2×C3C22×C3×D1211yescharacteristic
280D3—126no—
281D3—366no—
282C22—186no—
283C22—726no—
284C22C2×C4⋊((C3⋊D3)⋊C3)13yes—
285C4—182no—
286C4C2×C4⋊((C3⋊D3)⋊C3)13yes—
287C4C23×(C3⋊D3)⋊C311yescharacteristic
288C3—31no—
289C3—31no—
290C3—61no—
291C3C3×#575211yescharacteristic
292C2—23no—
293C2—366no—
294C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
subgroupG/H
centerC2C22×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C22×C3×D12
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#576
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #11722638C2semidirect
2. H(3,𝔽3)×#131C2semidirect
3. #64117⋊C3C2semidirect
4. #64113⋊C3C2semidirect
5. C2×#3745⋊C3C2semidirect
6. #10596223C3semidirect
11. #3828⋊C3C22semidirect
12. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
13. M16×H(3,𝔽3)C22semidirect
14. C2×C8×H(3,𝔽3)C22semidirect
15. #3745⋊C3C22semidirect
16. #3745⋊C3C22semidirect
17. #3744⋊C3C22semidirect
18. #3746⋊C3C22semidirect
19. #64366C2×C3semidirect
22. C32×#131D3semidirect
23. C32×#131C2×C3semidirect
24. #64117C2×C3semidirect
25. #64113C2×C3semidirect
26. C2×#3745C2×C3semidirect
49. #3828C22×C3semidirect
50. C2×C4⋊(C3⋊D3)C22×C3semidirect
56. C32×M16C2×D3semidirect
57. C32×M16C22×C3semidirect
59. C2×C32×C8C2×D3semidirect
60. C2×C32×C8C22×C3semidirect
61. #3746C22×C3semidirect
62. #3745C22×C3semidirect
63. #3745C22×C3semidirect
64. #3744C22×C3semidirect
177. #131(C3⋊D3)⋊C3semidirect
178. H(3,𝔽3)#576semidirect
234. M16C2×(C3⋊D3)⋊C3semidirect
235. C2×C8C2×(C3⋊D3)⋊C3semidirect
258. C32#5752semidirect
259. C32C3×#576semidirect

Maximal subgroups

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

subgroupindex
#117226382
H(3,𝔽3)×#1312
#64117⋊C32
#64113⋊C32
C2×#3745⋊C32
#105962233
C3×#57523