Finite Group Databasebeta

← #10592594⋊C3 #13142269

Subgroups — #10592594⋊C3 #13142269

Order 1728.

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

normal subgroups only

This group has 401 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
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#10592594C311yesmaximal, characteristic, complemented
7C3×#4895—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
26C8×C3⋊Dic3C2×C311yescharacteristic, complemented
27C3×#825—32no—
28C3×C8×Dic3—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))⋊C3D411yescharacteristic
39(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
40C3×#443—91no—
41#4895—31no—
42#4895—91no—
43#3828C22×C312yes—
44C3×#235—32no—
45C32×M16—22no—
46C32×M16C2×D312yes—
47C32×M16C22×C312yes—
48C2×C32×C8—21no—
49C2×C32×C8C2×D311yescharacteristic
50C2×C32×C8C22×C311yescharacteristic
51C4×C3⋊Dic3C22×C311yescharacteristic
52C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
53#3746C3×C411yescharacteristic, complemented
54#3745C3×C411yescharacteristic, complemented
55#3744C3×C412yescomplemented
56#3743—22no—
57C8×C3⋊D3—22no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8—31no—
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)C22×C411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69(C3⋊Dic3)⋊C3—22no—
70(C3⋊Dic3)⋊C3—22no—
71C3×#135—91no—
72C3×#131—92no—
73#907—32no—
74#907—92no—
75#904—31no—
76#904—91no—
77C3×#108—31no—
78C3×#108—31no—
79C3×#108—61no—
80C3×#108C3×D311yescharacteristic
81C3×#104—92no—
82C3×C4×C8—91no—
83#825—32no—
84#825—92no—
85C8×Dic3—31no—
86C8×Dic3—91no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C23×C311yescharacteristic
90#638—22no—
91C2×C3⋊Dic3—21no—
92C4⋊(C3⋊D3)C2×C3×C412yes—
93C4×C3⋊D3—22no—
94#634C2×C3×C412yes—
95C3×#67—62no—
96C2×C3×Dic3—61no—
97C3×D12—32no—
98C3×C4×D3—62no—
99C3×Dic6—32no—
100C32×C8—22no—
101C32×C8—22no—
102C32×C8—21no—
103C32×C8—21no—
104C32×C8—42no—
105C32×C8C4×D311yescharacteristic
106C32×C8C4×D311yescharacteristic
107C32×C8C2×C3×C411yescharacteristic
108C32×C8C2×C3×C411yescharacteristic
109C3⋊(C3⋊C8)C3×D411yescharacteristic
110C3⋊(C3⋊C8)C3×D411yescharacteristic
111C3×C3⋊C8—31no—
112C3×C3⋊C8—31no—
113#443—91noSylow
114C2×H(3,𝔽3)—21no—
115C2×H(3,𝔽3)C4×D411yescharacteristic
116(C3⋊D3)⋊C3—42no—
117C3×#41—92no—
118#235—32no—
119#235—92no—
120C3×Q16—91no—
121C3×SD16—92no—
122C3×D8—91no—
123C3×M16—32no—
124C3×M16—32no—
125C3×M16—62no—
126C3×M16—182no—
127C3×M16C2×C3×D312yes—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—61no—
131C2×C3×C8—91no—
132C2×C3×C8—182no—
133C2×C3×C8C2×C3×D311yescharacteristic
134C3×C42—91no—
135C4×Dic3—31no—
136C4×Dic3—91no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—91no—
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#23911yescharacteristic
151C22×C32C3×#4111yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3×D3—62no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4C2×C4×D311yescharacteristic
157C32×C4D3×D411yescharacteristic
158C32×C4C22×C3×C411yescharacteristic
159C32×C4C2×C3×D411yescharacteristic
160C3⋊Dic3—22no—
161C3⋊Dic3—22no—
162C3×Dic3—62no—
163C3×Dic3—62no—
164#135—91no—
165#131—92no—
166#108(C3⋊D3)⋊C311yescharacteristic, complemented
167#104—92no—
168C4×C8—91no—
169H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
170C3×Q8—92no—
171C3×D4—92no—
172C3×D4—182no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—182no—
177C2×C3×C4—181no—
178C2×C3×C4C22×C3×D311yescharacteristic
179#67—62no—
180#67—182no—
181C2×Dic3—61no—
182C2×Dic3—181no—
183D12—32no—
184D12—92no—
185C4×D3—62no—
186C4×D3—182no—
187Dic6—32no—
188Dic6—92no—
189C3×C8—22no—
190C3×C8—31no—
191C3×C8—31no—
192C3×C8—31no—
193C3×C8—31no—
194C3×C8—62no—
195C3×C8—62no—
196C3×C8—61no—
197C3×C8—61no—
198C3×C8—91no—
199C3×C8—91no—
200C3×C8—122no—
201C3×C8C3×C4×D311yescharacteristic
202C3×C8C3×C4×D311yescharacteristic
203C3⋊C8—31no—
204C3⋊C8—31no—
205C3⋊C8—91no—
206C3⋊C8—91no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—41no—
211C2×C32#89311yescharacteristic
212C2×C32C3×C4×D411yescharacteristic
213C3⋊D3—42no—
214C3×D3—122no—
215#41—92no—
216Q16—91no—
217SD16—92no—
218D8—91no—
219M16—182no—
220M16C2×(C3⋊D3)⋊C312yes—
221C2×C8—91no—
222C2×C8—182no—
223C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
224C42—91no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—61no—
228C22×C3—182no—
229C22×C3C3×#23911yescharacteristic
230C2×D3—62no—
231C2×D3—182no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—182no—
239C3×C4—182no—
240C3×C4C2×C3×C4×D311yescharacteristic
241C3×C4C3×D3×D411yescharacteristic
242Dic3—62no—
243Dic3—62no—
244Dic3—182no—
245Dic3—182no—
246C32—21no—
247C32#489511yescharacteristic, complemented
248C32C3×#44311yescharacteristic, complemented
249Q8—92no—
250D4—92no—
251D4—182no—
252C2×C4—181no—
253C2×C4—182no—
254C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
255C8—22no—
256C8—91no—
257C8—91no—
258C8C4×(C3⋊D3)⋊C311yescharacteristic
259C8C4×(C3⋊D3)⋊C311yescharacteristic
260C2×C3—21no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—121no—
267C2×C3—362no—
268C2×C3C3×#89311yescharacteristic
269D3—122no—
270D3—362no—
271C22—182no—
272C22#8872211yescharacteristic
273C4—182no—
274C4—182no—
275C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
276C4D4×(C3⋊D3)⋊C311yescharacteristic
277C3—31no—
278C3—31no—
279C3—61no—
280C3C3×#489511yescharacteristic
281C2—21no—
282C2—362no—
283C2#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×#443
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. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #10592594C3semidirect
13. #3745⋊C3C4semidirect
14. #3744⋊C3C4semidirect
17. #3746⋊C3C4semidirect
18. #64116C2×C3semidirect
23. C32×#108D3semidirect
24. C32×#108C2×C3semidirect
25. #63652C2×C3semidirect
26. C8×C3⋊Dic3C2×C3semidirect
53. #3746C3×C4semidirect
54. #3745C3×C4semidirect
55. #3744C3×C4semidirect
166. #108(C3⋊D3)⋊C3semidirect
169. H(3,𝔽3)#443semidirect
247. C32#4895semidirect
248. C32C3×#443semidirect

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
C8×(C3⋊Dic3)⋊C32
#105925943
C3×#48953