Finite Group Databasebeta

← #13166102

Subgroups — #13166102

Order 1728.

328 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 36 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×#64267C211yesmaximal, characteristic
2C3×#64265C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
4#10593224—31nomaximal
5#10593223—31nomaximal
6#10591681—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10#88811—41nomaximal
11#64267D311yescharacteristic
12#64265D311yescharacteristic, complemented
13C3×#986—31no—
14C3×#986—61no—
15C3×#984—31no—
16C3×#984—61no—
17#64157D311yescharacteristic, complemented
18C3×#951—31no—
19C3×#949—31no—
20C2×#3715—31no—
21C3⋊(#131⋊C3)—31no—
22C3⋊(#131⋊C3)—31no—
23C3⋊(#131⋊C3)—61no—
24#63578—31no—
25#63560—31no—
26#63558—31no—
27C3×C4⋊(C3⋊D3)—41no—
28C3×#634—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C3×C3⋊(C3⋊C8)—41no—
31#5406—91no—
32#5203—91no—
33#5200—91no—
34C32×#41C2×D311yescharacteristic
35C2×C3×D12—31no—
36C2×C3×Dic6—31no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3—21no—
39C3×#41⋊C3—21no—
40C3×#41⋊C3—41no—
41C3×#41⋊C3C2×D311yescharacteristic
42C3×#41⋊C3C2×D311yescharacteristic
43C3⋊GL(2,𝔽3)—21no—
44C3⋊#226—21no—
45C3×GL(2,𝔽3)—62no—
46C3×GL(2,𝔽3)—61no—
47C3×#226—62no—
48C3×#226—61no—
49#3748—31no—
50C2×C3⋊(C3⋊C8)—31no—
51C3×#216—61no—
52C3×#215—61no—
53C3×#214—61no—
54C3×#213—61no—
55C3×#208—31no—
56C3×#208—31no—
57#3717—121no—
58#3716—121no—
59#3715—61no—
60#3715—61no—
61C2×C3×C3⋊D3—81no—
62C33×C4—41no—
63C3×C3⋊Dic3—81no—
64#986—31no—
65#986—61no—
66#984—31no—
67#984—61no—
68C3×#137—91no—
69C3×AGL(1,ℤ/8)—91no—
70#951—31no—
71#949—31no—
72#948—31no—
73#948—31no—
74#948—61no—
75C2×#215—91no—
76C2×#214—91no—
77#131⋊C3—91no—
78#836—91no—
79#833—91no—
80#824—91no—
81#823—91no—
82#822—91no—
83#822—91no—
84C22×C3×D3—61no—
85C32×Q8D1211yescharacteristic
86C32×D4—31no—
87C2×C32×C4—31no—
88C4⋊(C3⋊D3)—41no—
89#634—41no—
90C2×C3×Dic3—61no—
91C3×D12—61no—
92C3×D12—61no—
93C3×D12—121no—
94C3×D12—241no—
95C3×Dic6—61no—
96C3×Dic6—61no—
97C3×Dic6—121no—
98C3×Dic6—241no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—41no—
103C3×SL(2,𝔽3)#6711yescharacteristic
104C3×SL(2,𝔽3)#6711yescharacteristic
105C3⋊(C3⋊C8)—31no—
106C3⋊(C3⋊C8)S411yescharacteristic
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—121no—
110C3×C3⋊C8—121no—
111C3×C3⋊C8—241no—
112#471—271noSylow
113C2×C33—41no—
114C3×C3⋊D3—161no—
115C3×#41—21no—
116C3×#41D3211yescharacteristic
117C3×#41D3211yescharacteristic
118C2×C3×Q8—91no—
119C2×C3×D4—91no—
120C2×D12—31no—
121C2×Dic6—31no—
122#41⋊C3—21no—
123#41⋊C3—21no—
124#41⋊C3—41no—
125#41⋊C3D3211yescharacteristic
126GL(2,𝔽3)—62no—
127GL(2,𝔽3)—61no—
128#226—62no—
129#226—61no—
130C3×Q16—181no—
131C3×SD16—181no—
132C3×SD16—181no—
133C3×D8—181no—
134C3×M16—91no—
135C3×M16—91no—
136#216—61no—
137#215—61no—
138#215—181no—
139#215—181no—
140#214—61no—
141#214—181no—
142#214—181no—
143#213—61no—
144#208—31no—
145#208—31no—
146#208—91no—
147#208—91no—
148#208—181no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—181no—
152#204—361no—
153C22×C32—31no—
154C2×C3⋊D3—81no—
155C2×C3×D3—121no—
156C2×C3×D3—241no—
157C2×C3×D3—242no—
158C2×C3×D3—241no—
159C32×C4—31no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—81no—
163C32×C4—81no—
164C32×C4—81no—
165C32×C4—161no—
166C32×C4C2×S411yescharacteristic
167C3⋊Dic3—81no—
168C3×Dic3—121no—
169C3×Dic3—241no—
170C3×Dic3—242no—
171#137—91no—
172AGL(1,ℤ/8)—91no—
173C2×SD16—271no—
174#131—91no—
175#108—271no—
176#101—271no—
177#100—271no—
178C33—41noSylow
179C23×C3—181no—
180C22×D3—61no—
181C3×Q8—21no—
182C3×Q8—181no—
183C3×Q8—181no—
184C3×Q8#62611yescharacteristic
185C3×Q8#62611yescharacteristic
186C3×D4—31no—
187C3×D4—31no—
188C3×D4—61no—
189C3×D4—181no—
190C3×D4—181no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×Dic3—61no—
196D12—61no—
197D12—61no—
198D12—121no—
199D12—241no—
200Dic6—61no—
201Dic6—61no—
202Dic6—121no—
203Dic6—241no—
204SL(2,𝔽3)—21no—
205SL(2,𝔽3)—21no—
206SL(2,𝔽3)—41no—
207SL(2,𝔽3)#62511yescharacteristic
208C3×C8—181no—
209C3×C8—181no—
210C3×C8—361no—
211C3⋊C8—31no—
212C3⋊C8—31no—
213C3⋊C8—61no—
214C3⋊C8—61no—
215C3⋊C8—61no—
216C3⋊C8—91no—
217C3⋊C8—91no—
218C3⋊C8—181no—
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—161no—
228C3×D3—241no—
229C3×D3—481no—
230C3×D3—482no—
231#41#111311yescharacteristic
232C2×Q8—91no—
233C2×D4—91no—
234Q16—181no—
235SD16—181no—
236SD16—181no—
237SD16—541no—
238SD16—541no—
239D8—181no—
240M16—91no—
241M16—91no—
242M16—271no—
243C2×C8—271no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—361no—
248C22×C3—721no—
249C2×D3—121no—
250C2×D3—241no—
251C2×D3—242no—
252C2×D3—241no—
253C3×C4—21no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—41no—
257C3×C4—61no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4—161no—
261C3×C4—361no—
262C3×C4D3×S411yescharacteristic
263C3×C4D3×S411yescharacteristic
264Dic3—121no—
265Dic3—241no—
266Dic3—242no—
267C32—41no—
268C32—41no—
269C32—81no—
270C32—81no—
271C32—81no—
272C32—161no—
273C32#540611yescharacteristic, complemented
274C23—181no—
275Q8—181no—
276Q8—181no—
277Q8#629511yescharacteristic
278D4—31no—
279D4—181no—
280D4—181no—
281C2×C4—31no—
282C2×C4—181no—
283C8—91no—
284C8—181no—
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—721no—
296C2×C3#6420711yescharacteristic
297C2×C3#6420711yescharacteristic
298D3—241no—
299D3—481no—
300D3—482no—
301C22—31no—
302C22—361no—
303C22—721no—
304C4—31no—
305C4—361no—
306C4#8910111yescharacteristic
307C3—21no—
308C3—41no—
309C3—81no—
310C3—81no—
311C3—161no—
312C3#1059322311yescharacteristic, complemented
313C3#1059322411yescharacteristic, complemented
314C2—61no—
315C2—721no—
316C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4#89101
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
prime radicalC3×#64267C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×#64265C2semidirect
3. C3⋊(C3⋊(#131⋊C3))C2semidirect
12. #64265D3semidirect
17. #64157D3semidirect
273. C32#5406semidirect
312. C3#10593223semidirect
313. C3#10593224semidirect

Maximal subgroups

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

subgroupindex
C3×#642672
C3×#642652
C3⋊(C3⋊(#131⋊C3))2
#105932243
#105932233
#105916813
#888114