Finite Group Databasebeta

← #13166100

Subgroups — #13166100

Order 1728.

328 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C3×#64266C211yesmaximal, characteristic
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
3C3×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593220—31nomaximal
5#10593219—31nomaximal
6#10591675—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
9C3×C3⋊#226C411yescharacteristic, complemented
10#88808—41nomaximal
11#64266D311yescharacteristic
12C3×#985—31no—
13C3×#985—61no—
14#64157D311yescharacteristic, complemented
15C3×#950—31no—
16#63815—31no—
17C3⋊(#131⋊C3)—31no—
18C3⋊(#131⋊C3)—31no—
19C3⋊(#131⋊C3)—61no—
20C3⋊GU(2,𝔽3)D311yescharacteristic, complemented
21C3×GU(2,𝔽3)—31no—
22C3×GU(2,𝔽3)—61no—
23C3×#837—31no—
24#63577—31no—
25#63569—31no—
26Dic3×C3⋊C8—31no—
27C3×C4×C3⋊D3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C3×C3⋊(C3⋊C8)—41no—
30C3×C3⋊(C3⋊C8)—41no—
31#5404—91no—
32#5196—91no—
33#4824—91no—
34C32×#41C2×D311yescharacteristic
35C3×#235—31no—
36C3×#41⋊C3—21no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3—21no—
39C3×#41⋊C3—41no—
40C3×#41⋊C3C2×D311yescharacteristic
41C3×#41⋊C3C2×D311yescharacteristic
42C3⋊GL(2,𝔽3)Dic311yescharacteristic, complemented
43C3⋊#226Dic311yescharacteristic, complemented
44C3×GL(2,𝔽3)—31no—
45C3×GL(2,𝔽3)—61no—
46C3×#226—31no—
47C3×#226—61no—
48#3748—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#216—31no—
51C3×#215—31no—
52C3×#214—31no—
53C3×#213—31no—
54C3×C4×Dic3—31no—
55C3×#208—31no—
56C2×C3×C3⋊C8—31no—
57#3711—61no—
58#3710—121no—
59D3×C3⋊C8—61no—
60D3×C3⋊C8—121no—
61C2×C3×C3⋊D3—41no—
62C33×C4—41no—
63C3×C3⋊Dic3—41no—
64#985—31no—
65#985—61no—
66C3×#135—91no—
67#950—31no—
68#948—31no—
69#948—31no—
70#948—61no—
71#948—91no—
72#907—91no—
73#131⋊C3—91no—
74GU(2,𝔽3)—31no—
75GU(2,𝔽3)—61no—
76C3×#104—91no—
77#837—31no—
78#837—91no—
79#822—91no—
80#819—91no—
81C8×Dic3—91no—
82#805—91no—
83C4×C3⋊C8—91no—
84C32×Q8C4×D311yescharacteristic
85C32×D4—31no—
86C2×C32×C4—31no—
87C4×C3⋊D3—41no—
88C3×#67—61no—
89C2×C3×Dic3—61no—
90C3×D12—31no—
91C3×C4×D3—61no—
92C3×C4×D3—121no—
93C3×C4×D3—241no—
94C3×Dic6—31no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—41no—
99C3×SL(2,𝔽3)C4×D311yescharacteristic
100C3×SL(2,𝔽3)C2×Dic311yescharacteristic
101C3⋊(C3⋊C8)—31no—
102C3⋊(C3⋊C8)—41no—
103C3⋊(C3⋊C8)S411yescharacteristic
104C3×C3⋊C8—31no—
105C3×C3⋊C8—31no—
106C3×C3⋊C8—61no—
107C3×C3⋊C8—121no—
108C3×C3⋊C8—121no—
109C3×C3⋊C8—121no—
110C3×C3⋊C8—241no—
111C3×C3⋊C8—241no—
112#443—271noSylow
113C2×C33—41no—
114C3×C3⋊D3—42no—
115C3×#41—21no—
116C3×#41—91no—
117C3×#41D3211yescharacteristic
118C3×#41D3211yescharacteristic
119#235—31no—
120#41⋊C3—21no—
121#41⋊C3—21no—
122#41⋊C3—41no—
123#41⋊C3D3211yescharacteristic
124GL(2,𝔽3)—31no—
125GL(2,𝔽3)—61no—
126#226—31no—
127#226—61no—
128C3×Q16—91no—
129C3×SD16—91no—
130C3×SD16—91no—
131C3×D8—91no—
132C3×M16—91no—
133C2×C3×C8—91no—
134C3×C42—91no—
135#216—31no—
136#215—31no—
137#214—31no—
138#213—31no—
139C4×Dic3—31no—
140#208—31no—
141#208—91no—
142#208—91no—
143#208—181no—
144#208—181no—
145C2×C3⋊C8—31no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—181no—
149C2×C3⋊C8—181no—
150ΓL(1,𝔽25)—181no—
151D3×C8—181no—
152D3×C8—361no—
153C22×C32—31no—
154C2×C3⋊D3—41no—
155C2×C3×D3—61no—
156C2×C3×D3—121no—
157C2×C3×D3—241no—
158C32×C4—31no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4—81no—
162C32×C4—81no—
163C32×C4—81no—
164C32×C4—161no—
165C32×C4C2×S411yescharacteristic
166C3⋊Dic3—41no—
167C3×Dic3—62no—
168C3×Dic3—61no—
169C3×Dic3—121no—
170C3×Dic3—241no—
171#135—91no—
172#131—91no—
173#131—271no—
174#108—271no—
175#104—91no—
176#104—271no—
177C4×C8—271no—
178C33—41noSylow
179C3×Q8—21no—
180C3×Q8—91no—
181C3×Q8D3×Dic311yescharacteristic
182C3×Q8#62411yescharacteristic
183C3×D4—31no—
184C3×D4—31no—
185C3×D4—61no—
186C3×D4—91no—
187C3×D4—181no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193#67—61no—
194C2×Dic3—61no—
195D12—31no—
196C4×D3—61no—
197C4×D3—121no—
198C4×D3—241no—
199Dic6—31no—
200SL(2,𝔽3)—21no—
201SL(2,𝔽3)—21no—
202SL(2,𝔽3)—41no—
203SL(2,𝔽3)D3×Dic311yescharacteristic
204C3×C8—91no—
205C3×C8—91no—
206C3×C8—181no—
207C3×C8—361no—
208C3⋊C8—31no—
209C3⋊C8—31no—
210C3⋊C8—31no—
211C3⋊C8—31no—
212C3⋊C8—61no—
213C3⋊C8—61no—
214C3⋊C8—91no—
215C3⋊C8—91no—
216C3⋊C8—121no—
217C3⋊C8—181no—
218C3⋊C8—241no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—61no—
222C2×C32—81no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32—161no—
226C2×C32C4×S411yescharacteristic
227C3⋊D3—42no—
228C3×D3—122no—
229C3×D3—121no—
230C3×D3—242no—
231#41—91no—
232#41#111311yescharacteristic
233Q16—91no—
234SD16—91no—
235SD16—91no—
236D8—91no—
237M16—91no—
238M16—271no—
239M16—541no—
240C2×C8—91no—
241C2×C8—271no—
242C2×C8—541no—
243C42—91no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—181no—
248C2×D3—61no—
249C2×D3—121no—
250C2×D3—241no—
251C3×C4—21no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—41no—
255C3×C4—61no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—161no—
259C3×C4—182no—
260C3×C4—181no—
261C3×C4D3×S411yescharacteristic
262C3×C4D3×S411yescharacteristic
263Dic3—62no—
264Dic3—61no—
265Dic3—121no—
266Dic3—241no—
267C32—41no—
268C32—41no—
269C32—81no—
270C32—81no—
271C32—81no—
272C32—161no—
273C32#540411yescharacteristic, complemented
274Q8—91no—
275Q8#629411yescharacteristic
276D4—31no—
277D4—91no—
278D4—181no—
279C2×C4—31no—
280C2×C4—181no—
281C2×C4—181no—
282C8—91no—
283C8—91no—
284C8—91no—
285C8—181no—
286C8—271no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—121no—
294C2×C3—161no—
295C2×C3—361no—
296C2×C3Dic3×S411yescharacteristic
297C2×C3(C3⋊S4)⋊C411yescharacteristic
298D3—122no—
299D3—121no—
300D3—242no—
301C22—31no—
302C22—181no—
303C4—31no—
304C4—182no—
305C4—181no—
306C4#8910111yescharacteristic
307C3—21no—
308C3—41no—
309C3—81no—
310C3—81no—
311C3—161no—
312C3#1059321911yescharacteristic, complemented
313C3#1059322011yescharacteristic, complemented
314C2—61no—
315C2—361no—
316C2#1172421311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC3×#64266C2
cosocleC32×#41⋊C3C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(#131⋊C3))C2semidirect
3. C3×C3⋊GU(2,𝔽3)C2semidirect
8. C3×C3⋊GL(2,𝔽3)C4semidirect
9. C3×C3⋊#226C4semidirect
14. #64157D3semidirect
20. C3⋊GU(2,𝔽3)D3semidirect
42. C3⋊GL(2,𝔽3)Dic3semidirect
43. C3⋊#226Dic3semidirect
273. C32#5404semidirect
312. C3#10593219semidirect
313. C3#10593220semidirect

Maximal subgroups

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

subgroupindex
C3×#642662
C3⋊(C3⋊(#131⋊C3))2
C3×C3⋊GU(2,𝔽3)2
#105932203
#105932193
#105916753
#888084