Finite Group Databasebeta

← #10592416⋊C3 #13141538

Subgroups — #10592416⋊C3 #13141538

Order 1728.

404 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 108 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#64113⋊C3C211yesmaximal, characteristic
2#64110⋊C3C212yesmaximal, complemented
3H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#63632⋊C3C212yesmaximal, complemented
6#10592416C311yesmaximal, characteristic, complemented
7C3×#4717—31nomaximal
8#3828⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12#3748⋊C3C2212yes—
13#3745⋊C3C411yescharacteristic, complemented
14#3744⋊C3C412yescomplemented
15#3743⋊C3—22no—
16C8×(C3⋊D3)⋊C3—22no—
17#3746⋊C3C411yescharacteristic, complemented
18#64113C2×C311yescharacteristic
19#64110C2×C312yescomplemented
20C3×#904—31no—
21C3×#901—32no—
22C32×#97—21no—
23C32×#97D311yescharacteristic, complemented
24C32×#97C2×C311yescharacteristic, complemented
25#63646C2×C311yescharacteristic, complemented
26#63632C2×C312yescomplemented
27C3×#819—31no—
28C3×#805—32no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31#638⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)C2×C412yes—
33C4×(C3⋊D3)⋊C3—22no—
34#634⋊C3C2×C412yes—
35C8×H(3,𝔽3)C2×C411yescharacteristic
36C8×H(3,𝔽3)C2×C411yescharacteristic
37C8×H(3,𝔽3)D412yes—
38(C3⋊(C3⋊C8))⋊C3—22no—
39C3×#444—91no—
40#4717—31no—
41#4717—91no—
42#3828C22×C312yes—
43C3×#235—32no—
44C2×C32×C8—21no—
45C2×C32×C8—21no—
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C22×C311yescharacteristic
49C2×C32×C8C22×C311yescharacteristic
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C22×C311yescharacteristic
53#3748C22×C312yes—
54#3746C3×C411yescharacteristic, complemented
55#3745C3×C411yescharacteristic, complemented
56#3744C3×C412yescomplemented
57#3743—22no—
58C8×C3⋊D3—22no—
59C3×#208—32no—
60C3×Dic12—31no—
61C3×D24—31no—
62C3×#204—32no—
63C3×ΓL(1,𝔽25)—62no—
64C3×D3×C8—62no—
65C22×H(3,𝔽3)#4111yescharacteristic
66C2×(C3⋊D3)⋊C3—22no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)C22×C411yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70(C3⋊Dic3)⋊C3—22no—
71C3×#135—91no—
72C3×#131—92no—
73#904—31no—
74#904—91no—
75#901—32no—
76#901—92no—
77C3×#108—91no—
78C3×#104—92no—
79C3×#97—31no—
80C3×#97—31no—
81C3×#97—61no—
82C3×#97C3×D311yescharacteristic
83#819—31no—
84#819—91no—
85#805—32no—
86#805—92no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D311yescharacteristic
92C2×C32×C4C23×C311yescharacteristic
93#638—22no—
94C4⋊(C3⋊D3)C2×C3×C412yes—
95C4×C3⋊D3—22no—
96#634C2×C3×C412yes—
97C3×#67—62no—
98C3×D12—32no—
99C3×C4×D3—62no—
100C3×Dic6—32no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8—22no—
104C32×C8C4×D311yescharacteristic
105C32×C8C4×D311yescharacteristic
106C32×C8D1212yes—
107C32×C8C2×C3×C411yescharacteristic
108C32×C8C2×C3×C411yescharacteristic
109C32×C8C3×D412yes—
110C3⋊(C3⋊C8)—22no—
111C3×C3⋊C8—62no—
112#444—91noSylow
113C2×H(3,𝔽3)—21no—
114C2×H(3,𝔽3)C4×D411yescharacteristic
115(C3⋊D3)⋊C3—42no—
116C3×#41—92no—
117#235—32no—
118#235—92no—
119C3×Q16—91no—
120C3×SD16—92no—
121C3×D8—91no—
122C3×M16—92no—
123C3×M16—182no—
124C2×C3×C8—31no—
125C2×C3×C8—31no—
126C2×C3×C8—31no—
127C2×C3×C8—31no—
128C2×C3×C8—61no—
129C2×C3×C8—61no—
130C2×C3×C8—182no—
131C2×C3×C8C2×C3×D311yescharacteristic
132C2×C3×C8C2×C3×D311yescharacteristic
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—61no—
136C3×C42C2×C3×D311yescharacteristic
137#208—32no—
138#208—92no—
139Dic12—31no—
140Dic12—91no—
141D24—31no—
142D24—91no—
143#204—32no—
144#204—92no—
145ΓL(1,𝔽25)—62no—
146ΓL(1,𝔽25)—182no—
147D3×C8—62no—
148D3×C8—182no—
149C22×C32—21no—
150C22×C32#23511yescharacteristic
151C22×C32C3×#4111yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3×D3—62no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4C2×C4×D311yescharacteristic
160C32×C4C2×D1211yescharacteristic
161C32×C4C22×C3×C411yescharacteristic
162C32×C4C2×C3×D411yescharacteristic
163C3⋊Dic3—22no—
164C3×Dic3—62no—
165#135—91no—
166#131—92no—
167#108—91no—
168#104—92no—
169#97(C3⋊D3)⋊C311yescharacteristic, complemented
170H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
171C3×Q8—92no—
172C3×D4—92no—
173C3×D4—182no—
174C2×C3×C4—21no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—182no—
182C2×C3×C4C22×C3×D311yescharacteristic
183#67—62no—
184#67—182no—
185D12—32no—
186D12—92no—
187C4×D3—62no—
188C4×D3—182no—
189Dic6—32no—
190Dic6—92no—
191C3×C8—31no—
192C3×C8—31no—
193C3×C8—31no—
194C3×C8—31no—
195C3×C8—32no—
196C3×C8—32no—
197C3×C8—61no—
198C3×C8—61no—
199C3×C8—62no—
200C3×C8—182no—
201C3×C8C3×C4×D311yescharacteristic
202C3×C8C3×C4×D311yescharacteristic
203C3×C8C3×D1212yes—
204C3⋊C8—62no—
205C3⋊C8—182no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32C4×D1211yescharacteristic
211C2×C32C3×C4×D411yescharacteristic
212C3⋊D3—42no—
213C3×D3—122no—
214#41—92no—
215Q16—91no—
216SD16—92no—
217D8—91no—
218M16—92no—
219M16—182no—
220C2×C8—182no—
221C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
222C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
223C42C2×(C3⋊D3)⋊C311yescharacteristic
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—61no—
227C22×C3—182no—
228C22×C3C3×#23511yescharacteristic
229C2×D3—62no—
230C2×D3—182no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—41no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—121no—
239C3×C4—121no—
240C3×C4—122no—
241C3×C4—182no—
242C3×C4C2×C3×C4×D311yescharacteristic
243C3×C4C2×C3×D1211yescharacteristic
244Dic3—62no—
245Dic3—182no—
246C32—21no—
247C32#471711yescharacteristic, complemented
248C32C3×#44411yescharacteristic, complemented
249Q8—92no—
250D4—92no—
251D4—182no—
252C2×C4—21no—
253C2×C4—182no—
254C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
255C8—182no—
256C8C4×(C3⋊D3)⋊C311yescharacteristic
257C8C4×(C3⋊D3)⋊C311yescharacteristic
258C8C4⋊((C3⋊D3)⋊C3)12yes—
259C2×C3—21no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—61no—
263C2×C3—61no—
264C2×C3—62no—
265C2×C3—61no—
266C2×C3—362no—
267C2×C3C3×C4×D1211yescharacteristic
268D3—122no—
269D3—362no—
270C22—182no—
271C22#3828⋊C311yescharacteristic
272C4—41no—
273C4—182no—
274C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
275C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
276C3—31no—
277C3—31no—
278C3—61no—
279C3C3×#471711yescharacteristic
280C2—21no—
281C2—362no—
282C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
subgroupG/H
centerC4C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC2×C3C3×C4×D12
prime radical#64113⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #64110⋊C3C2semidirect
3. H(3,𝔽3)×#97C2semidirect
4. #63646⋊C3C2semidirect
5. #63632⋊C3C2semidirect
6. #10592416C3semidirect
13. #3745⋊C3C4semidirect
14. #3744⋊C3C4semidirect
17. #3746⋊C3C4semidirect
19. #64110C2×C3semidirect
23. C32×#97D3semidirect
24. C32×#97C2×C3semidirect
25. #63646C2×C3semidirect
26. #63632C2×C3semidirect
54. #3746C3×C4semidirect
55. #3745C3×C4semidirect
56. #3744C3×C4semidirect
169. #97(C3⋊D3)⋊C3semidirect
170. H(3,𝔽3)#444semidirect
247. C32#4717semidirect
248. C32C3×#444semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#64110⋊C32
H(3,𝔽3)×#972
#63646⋊C32
#63632⋊C32
#105924163
C3×#47173