Finite Group Databasebeta

← #13166046

Subgroups — #13166046

Order 1728.

437 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 50 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#11724470C211yesmaximal, characteristic
2C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C3×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593639—31nomaximal
5#10593222—33nomaximal
6#10591663—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))C411yescharacteristic, complemented
9C3⋊(C3⋊#226)C411yescharacteristic, complemented
10#88788—41nomaximal
11#64266—31no—
12#64266—33no—
13#64266—61no—
14#64266—63no—
15#64159—31no—
16C3×#948D311yescharacteristic, complemented
17C3×#131⋊C3—31no—
18#63817—31no—
19C3⋊(#131⋊C3)D313yescomplemented
20C3⋊GU(2,𝔽3)D311yescharacteristic, complemented
21C3×GU(2,𝔽3)—33no—
22C3×#837—31no—
23#63578—31no—
24#63570—31no—
25#63554—31no—
26C4×C3⋊(C3⋊D3)—41no—
27C32×SL(2,𝔽3)C2×C411yescharacteristic
28C3×C3⋊(C3⋊C8)—41no—
29C32×C3⋊C8—41no—
30#5405—93no—
31#5145—91no—
32#4823—91no—
33C32×#41C2×D311yescharacteristic
34#3828—31no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3—23no—
37C3×#41⋊C3C2×D313yes—
38C3×#41⋊C3C2×D311yescharacteristic
39C3⋊GL(2,𝔽3)—31no—
40C3⋊GL(2,𝔽3)—33no—
41C3⋊GL(2,𝔽3)—61no—
42C3⋊GL(2,𝔽3)—63no—
43C3⋊#226—31no—
44C3⋊#226—33no—
45C3⋊#226—61no—
46C3⋊#226—63no—
47#3756—31no—
48#3755—31no—
49#3754—31no—
50#3753—31no—
51C2×C3⋊(C3⋊C8)—31no—
52#3743—121no—
53C3×C4×Dic3—31no—
54C3×#208—31no—
55C3×#208—31no—
56C2×C3×C3⋊C8—31no—
57#3712—61no—
58#3712—123no—
59#3710—61no—
60C2×C3⋊(C3⋊D3)—41no—
61C33×C4—41no—
62C3⋊(C3⋊Dic3)—41no—
63#985—93no—
64#985—183no—
65C3×#131—31no—
66#950—91no—
67#950—91no—
68#950—181no—
69#948C3⋊D311yescharacteristic, complemented
70#907—91no—
71#901—91no—
72#131⋊C3—33no—
73GU(2,𝔽3)—33no—
74C3×#104—91no—
75#837—31no—
76#825—91no—
77#822—91no—
78#822—91no—
79#815—91no—
80#805—91no—
81#803—91no—
82C32×Q8C4×D311yescharacteristic
83C32×D4—31no—
84C2×C32×C4—31no—
85#638—61no—
86C4⋊(C3⋊D3)—31no—
87C4×C3⋊D3—61no—
88C4×C3⋊D3—121no—
89C4×C3⋊D3—123no—
90C4×C3⋊D3—241no—
91C4×C3⋊D3—243no—
92#634—31no—
93C2×C3×Dic3—61no—
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)—23no—
96C3×SL(2,𝔽3)C4×D313yes—
97C3×SL(2,𝔽3)C4×D311yescharacteristic
98C32×C8—121no—
99C3⋊(C3⋊C8)—31no—
100C3⋊(C3⋊C8)—31no—
101C3⋊(C3⋊C8)—41no—
102C3×C3⋊C8—31no—
103C3×C3⋊C8—43no—
104C3×C3⋊C8—61no—
105C3×C3⋊C8—123no—
106C3×C3⋊C8S411yescharacteristic
107#444—271noSylow
108C2×C33—41no—
109C3⋊(C3⋊D3)—81no—
110C3×#41—21no—
111C3×#41D3211yescharacteristic
112C3×#41C2×C3⋊D311yescharacteristic
113#235—91no—
114#235—91no—
115#235—181no—
116#41⋊C3—23no—
117#41⋊C3D3213yes—
118GL(2,𝔽3)—93no—
119GL(2,𝔽3)—183no—
120#226—93no—
121#226—183no—
122C3×M16—91no—
123C3×M16—91no—
124C2×C3×C8—91no—
125C3×C42—91no—
126#216—91no—
127#216—91no—
128#216—181no—
129#215—91no—
130#215—91no—
131#215—181no—
132#214—91no—
133#214—91no—
134#214—181no—
135#213—91no—
136#213—91no—
137#213—181no—
138C4×Dic3—31no—
139#208—31no—
140#208—31no—
141C2×C3⋊C8—31no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—181no—
145ΓL(1,𝔽25)—181no—
146ΓL(1,𝔽25)—181no—
147ΓL(1,𝔽25)—363no—
148D3×C8—181no—
149D3×C8—181no—
150C22×C32—31no—
151C2×C3⋊D3—61no—
152C2×C3⋊D3—121no—
153C2×C3⋊D3—123no—
154C2×C3⋊D3—241no—
155C2×C3⋊D3—243no—
156C32×C4—31no—
157C32×C4—41no—
158C32×C4—43no—
159C32×C4—83no—
160C32×C4—81no—
161C32×C4C2×S411yescharacteristic
162C3⋊Dic3—61no—
163C3⋊Dic3—121no—
164C3⋊Dic3—123no—
165C3⋊Dic3—241no—
166C3⋊Dic3—243no—
167C3×Dic3—121no—
168#135—271no—
169#131—31no—
170#131—271no—
171#108—271no—
172#104—91no—
173#104—271no—
174#97—271no—
175C33—41noSylow
176C3×Q8—21no—
177C3×Q8#62411yescharacteristic
178C3×Q8C4×C3⋊D311yescharacteristic
179C3×D4—31no—
180C3×D4—31no—
181C3×D4—61no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—181no—
186#67—181no—
187#67—181no—
188#67—361no—
189C2×Dic3—61no—
190D12—91no—
191D12—91no—
192D12—181no—
193C4×D3—181no—
194C4×D3—181no—
195C4×D3—361no—
196C4×D3—363no—
197C4×D3—723no—
198Dic6—91no—
199Dic6—91no—
200Dic6—181no—
201SL(2,𝔽3)—23no—
202SL(2,𝔽3)#62413yes—
203C3×C8—31no—
204C3×C8—91no—
205C3×C8—123no—
206C3×C8—181no—
207C3⋊C8—31no—
208C3⋊C8—61no—
209C3⋊C8—91no—
210C3⋊C8—91no—
211C3⋊C8—91no—
212C3⋊C8—91no—
213C3⋊C8—123no—
214C3⋊C8—181no—
215C3⋊C8—181no—
216C3⋊C8C3⋊S411yescharacteristic
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—61no—
220C2×C32—83no—
221C2×C32—81no—
222C2×C32C4×S411yescharacteristic
223C3⋊D3—121no—
224C3⋊D3—243no—
225C3⋊D3—241no—
226C3⋊D3—242no—
227C3⋊D3—246no—
228#41—271no—
229#41D3×C3⋊D311yescharacteristic
230Q16—271no—
231SD16—271no—
232SD16—271no—
233D8—271no—
234M16—91no—
235M16—91no—
236M16—541no—
237C2×C8—91no—
238C2×C8—271no—
239C2×C8—541no—
240C42—91no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—61no—
244C2×D3—181no—
245C2×D3—181no—
246C2×D3—361no—
247C2×D3—363no—
248C2×D3—723no—
249C3×C4—21no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—43no—
253C3×C4—61no—
254C3×C4—83no—
255C3×C4—361no—
256C3×C4D3×S411yescharacteristic
257C3×C4C2×C3⋊S411yescharacteristic
258Dic3—121no—
259Dic3—181no—
260Dic3—181no—
261Dic3—361no—
262Dic3—363no—
263Dic3—723no—
264C32—41no—
265C32—43no—
266C32—83no—
267C32—81no—
268C32#540511yescharacteristic, complemented
269Q8—271no—
270Q8#628911yescharacteristic
271D4—31no—
272D4—271no—
273D4—541no—
274C2×C4—31no—
275C2×C4—181no—
276C2×C4—541no—
277C8—31no—
278C8—91no—
279C8—181no—
280C8—271no—
281C8—271no—
282C2×C3—21no—
283C2×C3—43no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—83no—
287C2×C3—121no—
288C2×C3(C3⋊S4)⋊C411yescharacteristic
289C2×C3C4×C3⋊S411yescharacteristic
290D3—361no—
291D3—361no—
292D3—721no—
293D3—723no—
294D3—726no—
295C22—31no—
296C22—541no—
297C4—31no—
298C4—361no—
299C4—541no—
300C4D3×C3⋊S411yescharacteristic
301C3—21no—
302C3—43no—
303C3—83no—
304C3#1059322211yescharacteristic, complemented
305C3#1059363911yescharacteristic, complemented
306C2—61no—
307C2—1081no—
308C2#1172420111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radical#11724470C2
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⋊#226)C4semidirect
16. C3×#948D3semidirect
19. C3⋊(#131⋊C3)D3semidirect
20. C3⋊GU(2,𝔽3)D3semidirect
69. #948C3⋊D3semidirect
268. C32#5405semidirect
304. C3#10593222semidirect
305. C3#10593639semidirect

Maximal subgroups

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

subgroupindex
#117244702
C3×C3⋊(#131⋊C3)2
C3×C3⋊GU(2,𝔽3)2
#105936393
#105932223
#105916633
#887884