Finite Group Databasebeta

← #13164960

Subgroups — #13164960

Order 1728.

2981 conjugacy classes of subgroups; only the 954 normal subgroups are shown.

normal subgroups only

This group has 2981 conjugacy classes of subgroups — too many to draw. Only its 954 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
1H(3,𝔽3)×2−1+4C211yesmaximal, characteristic, complemented
2#11722638C2110yesmaximal, complemented
3#41×(C3⋊D3)⋊C3C2110yesmaximal, complemented
4#11722627C215yesmaximal, complemented
5C2×#88722C215yesmaximal, complemented
6#10596760C311yesmaximal, characteristic, complemented
7C3×#5968—31nomaximal
8#41×H(3,𝔽3)C22110yescomplemented
9C2×Q8×H(3,𝔽3)C2215yescomplemented
10#88722C22110yescomplemented
11#88722C22120yescomplemented
12Q8×(C3⋊D3)⋊C3C22110yescomplemented
13#3830⋊C3C22110yescomplemented
14D4×(C3⋊D3)⋊C3C22130yescomplemented
15#3828⋊C3C22130yescomplemented
16C2×C4⋊((C3⋊D3)⋊C3)C22115yescomplemented
17C2×C4×(C3⋊D3)⋊C3C22115yescomplemented
18C32×2−1+4—21no—
19C32×2−1+4D311yescharacteristic, complemented
20C32×2−1+4C2×C311yescharacteristic, complemented
21#64366C2×C3110yescomplemented
22#41×C3⋊D3C2×C3110yescomplemented
23#64364C2×C315yescomplemented
24C2×#3832C2×C315yescomplemented
25C3×#1009—310no—
26C3×D3×#41—310no—
27C3×#1007—35no—
28C2×C3×#239—35no—
29C22×(C3⋊D3)⋊C3C23115yes—
30Q8×H(3,𝔽3)C23110yes—
31D4×H(3,𝔽3)C23110yes—
32C2×C4×H(3,𝔽3)C23115yes—
33#638⋊C3C23120yes—
34C2×(C3⋊Dic3)⋊C3C2315yes—
35C4⋊((C3⋊D3)⋊C3)C23130yes—
36C4×(C3⋊D3)⋊C3C23110yes—
37C4×(C3⋊D3)⋊C3C23130yes—
38#634⋊C3C23110yes—
39C3×#585—91no—
40#5968—31no—
41#5968—91no—
42C32×#41—210no—
43C32×#41C2×D3110yescomplemented
44C32×#41C22×C3110yescomplemented
45C2×C32×Q8—25no—
46C2×C32×Q8C2×D315yescomplemented
47C2×C32×Q8C22×C315yescomplemented
48#3832C22×C3110yescomplemented
49#3832C22×C3120yescomplemented
50Q8×C3⋊D3C22×C3110yescomplemented
51#3830C22×C3110yescomplemented
52D4×C3⋊D3C22×C3130yescomplemented
53#3828C22×C3130yescomplemented
54C2×C4⋊(C3⋊D3)C22×C3115yescomplemented
55C2×C4×C3⋊D3C22×C3115yescomplemented
56C3×#239—310no—
57C3×#239—320no—
58C3×D3×Q8—310no—
59C3×#237—310no—
60C3×D3×D4—330no—
61C3×#235—330no—
62C2×C3×D12—315no—
63C2×C3×C4×D3—315no—
64C22×H(3,𝔽3)C2415yes—
65C2×(C3⋊D3)⋊C3—415no—
66C2×(C3⋊D3)⋊C3C24110yes—
67C4×H(3,𝔽3)C24110yes—
68(C3⋊Dic3)⋊C3C2415yes—
69(C3⋊Dic3)⋊C3C2411yescharacteristic
70C3×2−1+4—31no—
71C3×2−1+4—31no—
72C3×2−1+4—61no—
73C3×2−1+4—95no—
74C3×2−1+4C3×D311yescharacteristic
75C3×H(5,𝔽2)—910no—
76C2×C3×#41—95no—
77C2×C3×#41—910no—
78#1009—310no—
79#1009—910no—
80D3×#41—310no—
81D3×#41—910no—
82#1007—35no—
83#1007—95no—
84C2×#239—35no—
85C2×#239—95no—
86C22×C3⋊D3C23×C3115yes—
87C22×C3×D3—315no—
88C32×Q8—210no—
89C32×Q8C22×D3110yes—
90C32×Q8C23×C3110yes—
91C32×D4—210no—
92C32×D4C22×D3110yes—
93C32×D4C23×C3110yes—
94C2×C32×C4—215no—
95C2×C32×C4C22×D3115yes—
96C2×C32×C4C23×C3115yes—
97#638C23×C3120yes—
98C2×C3⋊Dic3C23×C315yes—
99C4⋊(C3⋊D3)C23×C3130yes—
100C4×C3⋊D3C23×C3110yes—
101C4×C3⋊D3C23×C3130yes—
102#634C23×C3110yes—
103C3×#67—320no—
104C2×C3×Dic3—35no—
105C3×D12—330no—
106C3×C4×D3—310no—
107C3×C4×D3—330no—
108C3×Dic6—310no—
109#585—91noSylow
110C2×H(3,𝔽3)—25no—
111C2×H(3,𝔽3)C2511yescharacteristic
112(C3⋊D3)⋊C3—210no—
113C3×#41—310no—
114C3×#41—310no—
115C3×#41—610no—
116C3×#41—930no—
117C3×#41—910no—
118C3×#41—910no—
119C3×#41—920no—
120C3×#41C2×C3×D3110yes—
121C2×C3×Q8—35no—
122C2×C3×Q8—35no—
123C2×C3×Q8—65no—
124C2×C3×Q8—910no—
125C2×C3×Q8C2×C3×D315yes—
126C2×C3×D4—915no—
127C2×C3×D4—930no—
128C22×C3×C4—915no—
129#239—310no—
130#239—320no—
131#239—910no—
132#239—920no—
133D3×Q8—310no—
134D3×Q8—910no—
135#237—310no—
136#237—910no—
137D3×D4—330no—
138D3×D4—930no—
139#235—330no—
140#235—930no—
141C2×D12—315no—
142C2×D12—915no—
143C2×C4×D3—315no—
144C2×C4×D3—915no—
145C22×C32—25no—
146C22×C32C23×D315yes—
147C22×C32C24×C315yes—
148C2×C3⋊D3—415no—
149C2×C3⋊D3C24×C3110yes—
150C2×C3×D3—310no—
151C2×C3×D3—1215no—
152C32×C4—210no—
153C32×C4C23×D3110yes—
154C32×C4C24×C3110yes—
155C3⋊Dic3C24×C315yes—
156C3⋊Dic3C24×C311yescharacteristic
157C3×Dic3—35no—
158C3×Dic3—31no—
1592−1+4—95no—
1602−1+4(C3⋊D3)⋊C311yescharacteristic, complemented
161H(5,𝔽2)—910no—
162C2×#41—910no—
163C2×#41—95no—
164H(3,𝔽3)#58511yescharacteristic, complemented, Sylow
165C23×C3—915no—
166C22×D3—315no—
167C22×D3—915no—
168C3×Q8—310no—
169C3×Q8—310no—
170C3×Q8—610no—
171C3×Q8—910no—
172C3×Q8C22×C3×D3110yes—
173C3×D4—310no—
174C3×D4—310no—
175C3×D4—610no—
176C3×D4—930no—
177C3×D4—920no—
178C3×D4C22×C3×D3110yes—
179C2×C3×C4—315no—
180C2×C3×C4—315no—
181C2×C3×C4—615no—
182C2×C3×C4—910no—
183C2×C3×C4—930no—
184C2×C3×C4—95no—
185C2×C3×C4C22×C3×D3115yes—
186#67—320no—
187#67—920no—
188C2×Dic3—35no—
189C2×Dic3—95no—
190D12—330no—
191D12—930no—
192C4×D3—310no—
193C4×D3—330no—
194C4×D3—910no—
195C4×D3—930no—
196Dic6—310no—
197Dic6—910no—
198C2×C32—25no—
199C2×C32—25no—
200C2×C32—21no—
201C2×C32—45no—
202C2×C32C24×D311yescharacteristic
203C2×C32C25×C311yescharacteristic
204C3⋊D3—210no—
205C3×D3—610no—
206#41—910no—
207#41—920no—
208#41—910no—
209#41—930no—
210#41C2×(C3⋊D3)⋊C3110yescomplemented
211C2×Q8—910no—
212C2×Q8C2×(C3⋊D3)⋊C315yescomplemented
213C2×D4—930no—
214C2×D4—915no—
215C22×C4—915no—
216C22×C3—35no—
217C22×C3—35no—
218C22×C3—65no—
219C22×C3—910no—
220C22×C3—3615no—
221C22×C3C23×C3×D315yes—
222C2×D3—310no—
223C2×D3—910no—
224C2×D3—1215no—
225C2×D3—3615no—
226C3×C4—310no—
227C3×C4—310no—
228C3×C4—610no—
229C3×C4—91no—
230C3×C4—95no—
231C3×C4C23×C3×D3110yes—
232Dic3—35no—
233Dic3—31no—
234Dic3—95no—
235Dic3—91no—
236C32—21no—
237C32#596811yescharacteristic, complemented
238C32C3×#58511yescharacteristic, complemented
239C23—915no—
240Q8—910no—
241Q8C22×(C3⋊D3)⋊C3110yes—
242D4—920no—
243D4—930no—
244D4C22×(C3⋊D3)⋊C3110yes—
245C2×C4—95no—
246C2×C4—910no—
247C2×C4—930no—
248C2×C4C22×(C3⋊D3)⋊C3115yes—
249C2×C3—25no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—65no—
253C2×C3—65no—
254C2×C3—61no—
255C2×C3—125no—
256C2×C3—1810no—
257C2×C3C24×C3×D311yescharacteristic
258D3—610no—
259D3—1810no—
260C22—910no—
261C22—3615no—
262C22C23×(C3⋊D3)⋊C315yes—
263C4—91no—
264C4—95no—
265C4C23×(C3⋊D3)⋊C3110yes—
266C3—31no—
267C3—31no—
268C3—61no—
269C3C3×#596811yescharacteristic
270C2—25no—
271C2—1810no—
272C2C24×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C24×(C3⋊D3)⋊C3
commutator, cosocleC2×C32C25×C3
Frattini, socleC2×C3C24×C3×D3
FittingH(3,𝔽3)×2−1+4C2
hypercenter2−1+4(C3⋊D3)⋊C3
hypocenterC32C3×#585
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×2−1+4C2semidirect
2. #11722638C2semidirect
3. #41×(C3⋊D3)⋊C3C2semidirect
4. #11722627C2semidirect
5. C2×#88722C2semidirect
6. #10596760C3semidirect
8. #41×H(3,𝔽3)C22semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
10. #88722C22semidirect
11. #88722C22semidirect
12. Q8×(C3⋊D3)⋊C3C22semidirect
13. #3830⋊C3C22semidirect
14. D4×(C3⋊D3)⋊C3C22semidirect
15. #3828⋊C3C22semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C32×2−1+4D3semidirect
20. C32×2−1+4C2×C3semidirect
21. #64366C2×C3semidirect
22. #41×C3⋊D3C2×C3semidirect
23. #64364C2×C3semidirect
24. C2×#3832C2×C3semidirect
43. C32×#41C2×D3semidirect
44. C32×#41C22×C3semidirect
46. C2×C32×Q8C2×D3semidirect
47. C2×C32×Q8C22×C3semidirect
48. #3832C22×C3semidirect
49. #3832C22×C3semidirect
50. Q8×C3⋊D3C22×C3semidirect
51. #3830C22×C3semidirect
52. D4×C3⋊D3C22×C3semidirect
53. #3828C22×C3semidirect
54. C2×C4⋊(C3⋊D3)C22×C3semidirect
55. C2×C4×C3⋊D3C22×C3semidirect
160. 2−1+4(C3⋊D3)⋊C3semidirect
164. H(3,𝔽3)#585semidirect
210. #41C2×(C3⋊D3)⋊C3semidirect
212. C2×Q8C2×(C3⋊D3)⋊C3semidirect
237. C32#5968semidirect
238. C32C3×#585semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×2−1+42
#117226382
#41×(C3⋊D3)⋊C32
#117226272
C2×#887222
#105967603
C3×#59683