Finite Group Databasebeta

← #10592593⋊C3 #13142268

Subgroups — #10592593⋊C3 #13142268

Order 1728.

394 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 102 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#64116⋊C3C212yesmaximal, complemented
2#64113⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#108C211yesmaximal, characteristic, complemented
4#63652⋊C3C212yesmaximal, complemented
5#63642⋊C3C211yesmaximal, characteristic, complemented
6#10592593C311yesmaximal, characteristic, complemented
7C3×#4894—31nomaximal
8#3828⋊C3C2212yes—
9M16×H(3,𝔽3)C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13#3745⋊C3C411yescharacteristic, complemented
14#3744⋊C3C412yescomplemented
15#3743⋊C3—22no—
16C8×(C3⋊D3)⋊C3—22no—
17#3746⋊C3C411yescharacteristic, complemented
18#64116C2×C312yescomplemented
19#64113C2×C311yescharacteristic
20C3×#907—32no—
21C3×#904—31no—
22C32×#108—21no—
23C32×#108D311yescharacteristic, complemented
24C32×#108C2×C311yescharacteristic, complemented
25#63652C2×C312yescomplemented
26#63642C2×C311yescharacteristic, complemented
27C3×#825—32no—
28C3×#815—31no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30#638⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3—21no—
32C4⋊((C3⋊D3)⋊C3)C2×C412yes—
33C4×(C3⋊D3)⋊C3—22no—
34#634⋊C3C2×C412yes—
35C8×H(3,𝔽3)—22no—
36C8×H(3,𝔽3)C2×C411yescharacteristic
37C8×H(3,𝔽3)C2×C411yescharacteristic
38(C3⋊(C3⋊C8))⋊C3D412yes—
39C3×#444—91no—
40#4894—31no—
41#4894—91no—
42#3828C22×C312yes—
43C3×#235—32no—
44C32×M16—22no—
45C32×M16C2×D312yes—
46C32×M16C22×C312yes—
47C2×C32×C8—21no—
48C2×C32×C8C2×D311yescharacteristic
49C2×C32×C8C22×C311yescharacteristic
50C4×C3⋊Dic3C22×C311yescharacteristic
51C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
52#3746C3×C411yescharacteristic, complemented
53#3745C3×C411yescharacteristic, complemented
54#3744C3×C412yescomplemented
55#3743—22no—
56C8×C3⋊D3—22no—
57C3×C4×Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C3×Dic12—31no—
60C3×D24—31no—
61C3×#204—32no—
62C3×ΓL(1,𝔽25)—62no—
63C3×D3×C8—62no—
64C22×H(3,𝔽3)#4111yescharacteristic
65C2×(C3⋊D3)⋊C3—22no—
66C4×H(3,𝔽3)C22×C411yescharacteristic
67C4×H(3,𝔽3)C2×D411yescharacteristic
68(C3⋊Dic3)⋊C3—22no—
69(C3⋊Dic3)⋊C3—41no—
70C3×#135—91no—
71C3×#131—92no—
72#907—32no—
73#907—92no—
74#904—31no—
75#904—91no—
76C3×#108—31no—
77C3×#108—31no—
78C3×#108—61no—
79C3×#108C3×D311yescharacteristic
80C3×#104—92no—
81C3×#97—91no—
82#825—32no—
83#825—92no—
84#815—31no—
85#815—91no—
86C2×C32×C4—21no—
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C23×C311yescharacteristic
89#638—22no—
90C2×C3⋊Dic3—21no—
91C4⋊(C3⋊D3)C2×C3×C412yes—
92C4×C3⋊D3—22no—
93#634C2×C3×C412yes—
94C3×#67—62no—
95C2×C3×Dic3—61no—
96C3×D12—32no—
97C3×C4×D3—62no—
98C3×Dic6—32no—
99C32×C8—22no—
100C32×C8—22no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8—42no—
104C32×C8C4×D311yescharacteristic
105C32×C8C4×D311yescharacteristic
106C32×C8C2×C3×C411yescharacteristic
107C32×C8C2×C3×C411yescharacteristic
108C3⋊(C3⋊C8)C3×D412yes—
109C3×C3⋊C8—32no—
110#444—91noSylow
111C2×H(3,𝔽3)—21no—
112C2×H(3,𝔽3)C4×D411yescharacteristic
113(C3⋊D3)⋊C3—42no—
114C3×#41—92no—
115#235—32no—
116#235—92no—
117C3×Q16—91no—
118C3×SD16—92no—
119C3×D8—91no—
120C3×M16—32no—
121C3×M16—32no—
122C3×M16—62no—
123C3×M16—182no—
124C3×M16C2×C3×D312yes—
125C2×C3×C8—31no—
126C2×C3×C8—31no—
127C2×C3×C8—61no—
128C2×C3×C8—91no—
129C2×C3×C8—182no—
130C2×C3×C8C2×C3×D311yescharacteristic
131C3×C42—91no—
132C4×Dic3—31no—
133C4×Dic3—91no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—91no—
136Dic12—31no—
137Dic12—91no—
138D24—31no—
139D24—91no—
140#204—32no—
141#204—92no—
142ΓL(1,𝔽25)—62no—
143ΓL(1,𝔽25)—182no—
144D3×C8—62no—
145D3×C8—182no—
146C22×C32—21no—
147C22×C32#23911yescharacteristic
148C22×C32C3×#4111yescharacteristic
149C2×C3⋊D3—22no—
150C2×C3×D3—62no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4C2×C4×D311yescharacteristic
154C32×C4D3×D411yescharacteristic
155C32×C4C22×C3×C411yescharacteristic
156C32×C4C2×C3×D411yescharacteristic
157C3⋊Dic3—22no—
158C3⋊Dic3—41no—
159C3×Dic3—62no—
160C3×Dic3—121no—
161#135—91no—
162#131—92no—
163#108(C3⋊D3)⋊C311yescharacteristic, complemented
164#104—92no—
165#97—91no—
166H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
167C3×Q8—92no—
168C3×D4—92no—
169C3×D4—182no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—182no—
174C2×C3×C4—181no—
175C2×C3×C4C22×C3×D311yescharacteristic
176#67—62no—
177#67—182no—
178C2×Dic3—61no—
179C2×Dic3—181no—
180D12—32no—
181D12—92no—
182C4×D3—62no—
183C4×D3—182no—
184Dic6—32no—
185Dic6—92no—
186C3×C8—22no—
187C3×C8—31no—
188C3×C8—31no—
189C3×C8—31no—
190C3×C8—31no—
191C3×C8—62no—
192C3×C8—62no—
193C3×C8—61no—
194C3×C8—61no—
195C3×C8—92no—
196C3×C8—122no—
197C3×C8C3×C4×D311yescharacteristic
198C3×C8C3×C4×D311yescharacteristic
199C3⋊C8—32no—
200C3⋊C8—92no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—41no—
205C2×C32#89311yescharacteristic
206C2×C32C3×C4×D411yescharacteristic
207C3⋊D3—42no—
208C3×D3—122no—
209#41—92no—
210Q16—91no—
211SD16—92no—
212D8—91no—
213M16—182no—
214M16C2×(C3⋊D3)⋊C312yes—
215C2×C8—91no—
216C2×C8—182no—
217C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
218C42—91no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—61no—
222C22×C3—182no—
223C22×C3C3×#23911yescharacteristic
224C2×D3—62no—
225C2×D3—182no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—182no—
233C3×C4—361no—
234C3×C4C2×C3×C4×D311yescharacteristic
235C3×C4C3×D3×D411yescharacteristic
236Dic3—62no—
237Dic3—121no—
238Dic3—182no—
239Dic3—361no—
240C32—21no—
241C32#489411yescharacteristic, complemented
242C32C3×#44411yescharacteristic, complemented
243Q8—92no—
244D4—92no—
245D4—182no—
246C2×C4—182no—
247C2×C4—181no—
248C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
249C8—22no—
250C8—92no—
251C8C4×(C3⋊D3)⋊C311yescharacteristic
252C8C4×(C3⋊D3)⋊C311yescharacteristic
253C2×C3—21no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—61no—
259C2×C3—121no—
260C2×C3—362no—
261C2×C3C3×#89311yescharacteristic
262D3—122no—
263D3—362no—
264C22—182no—
265C22#8872211yescharacteristic
266C4—182no—
267C4—361no—
268C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
269C4D4×(C3⋊D3)⋊C311yescharacteristic
270C3—31no—
271C3—31no—
272C3—61no—
273C3C3×#489411yescharacteristic
274C2—21no—
275C2—362no—
276C2#64102⋊C311yescharacteristic
subgroupG/H
centerC4D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#108C2
socleC2×C3C3×#893
prime radical#64113⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#108(C3⋊D3)⋊C3
hypocenterC32C3×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
1. #64116⋊C3C2semidirect
3. H(3,𝔽3)×#108C2semidirect
4. #63652⋊C3C2semidirect
5. #63642⋊C3C2semidirect
6. #10592593C3semidirect
13. #3745⋊C3C4semidirect
14. #3744⋊C3C4semidirect
17. #3746⋊C3C4semidirect
18. #64116C2×C3semidirect
23. C32×#108D3semidirect
24. C32×#108C2×C3semidirect
25. #63652C2×C3semidirect
26. #63642C2×C3semidirect
52. #3746C3×C4semidirect
53. #3745C3×C4semidirect
54. #3744C3×C4semidirect
163. #108(C3⋊D3)⋊C3semidirect
166. H(3,𝔽3)#444semidirect
241. C32#4894semidirect
242. C32C3×#444semidirect

Maximal subgroups

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

subgroupindex
#64116⋊C32
#64113⋊C32
H(3,𝔽3)×#1082
#63652⋊C32
#63642⋊C32
#105925933
C3×#48943