Finite Group Databasebeta

← #13166049

Subgroups — #13166049

Order 1728.

456 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 48 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#11724471C211yesmaximal, characteristic, complemented
2C3×#64267C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#10593641—31nomaximal
5#10593225—33nomaximal
6#10591677—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9C3×C3⋊GL(2,𝔽3)—21no—
10#88791—41nomaximal
11#64267—31no—
12#64267—33no—
13#64267—61no—
14#64267—63no—
15#64267D311yescharacteristic, complemented
16C3×#986—33no—
17#64158—31no—
18C3×#949—31no—
19C3×#948D311yescharacteristic, complemented
20C3×#131⋊C3—31no—
21C2×#3714—31no—
22C3⋊(#131⋊C3)D313yescomplemented
23#63577—31no—
24#63559—31no—
25#63558—31no—
26C4⋊(C3⋊(C3⋊D3))—41no—
27C3×C4⋊(C3⋊D3)—41no—
28C32×SL(2,𝔽3)D411yescharacteristic
29C32×C3⋊C8—41no—
30#5407—93no—
31#5199—91no—
32#5142—91no—
33C32×#41C2×D311yescharacteristic
34C2×C4⋊(C3⋊D3)—31no—
35C2×C3×D12—31no—
36C3×#41⋊C3—21no—
37C3×#41⋊C3—23no—
38C3×#41⋊C3C2×D313yes—
39C3×#41⋊C3C2×D311yescharacteristic
40C3⋊GL(2,𝔽3)—21no—
41C3⋊GL(2,𝔽3)—62no—
42C3⋊GL(2,𝔽3)—66no—
43C3⋊GL(2,𝔽3)—63no—
44C3⋊GL(2,𝔽3)—61no—
45C3×GL(2,𝔽3)—63no—
46#3755—61no—
47#3753—61no—
48#3748—31no—
49#3745—121no—
50C3×#215—61no—
51C3×#213—61no—
52C3×#208—31no—
53C3×#208—31no—
54C2×C3×C3⋊C8—31no—
55#3714—61no—
56#3714—61no—
57#3714—123no—
58C2×C3⋊(C3⋊D3)—81no—
59C2×C3×C3⋊D3—81no—
60C33×C4—41no—
61#986—33no—
62#986—93no—
63#986—183no—
64C3×AGL(1,ℤ/8)—91no—
65C3×#131—31no—
66#949—31no—
67#949—91no—
68#949—91no—
69#949—181no—
70#948C3⋊D311yescharacteristic, complemented
71C2×#213—91no—
72C2×D24—91no—
73#131⋊C3—33no—
74#833—91no—
75#823—91no—
76#823—91no—
77#823—91no—
78#822—91no—
79#819—91no—
80C22×C3⋊D3—61no—
81C22×C3×D3—61no—
82C32×Q8D1211yescharacteristic
83C32×D4—31no—
84C2×C32×C4—31no—
85C4⋊(C3⋊D3)—41no—
86C4⋊(C3⋊D3)—61no—
87C4⋊(C3⋊D3)—61no—
88C4⋊(C3⋊D3)—121no—
89C4⋊(C3⋊D3)—123no—
90C4⋊(C3⋊D3)—241no—
91C4⋊(C3⋊D3)—243no—
92C3×D12—61no—
93C3×D12—61no—
94C3×D12—123no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—23no—
97C3×SL(2,𝔽3)D1213yes—
98C3×SL(2,𝔽3)#6711yescharacteristic
99C32×C8—121no—
100C3⋊(C3⋊C8)—61no—
101C3×C3⋊C8—31no—
102C3×C3⋊C8—43no—
103C3×C3⋊C8—61no—
104C3×C3⋊C8S411yescharacteristic
105#472—271noSylow
106C2×C33—41no—
107C3⋊(C3⋊D3)—161no—
108C3×C3⋊D3—161no—
109C3×#41—21no—
110C3×#41D3211yescharacteristic
111C3×#41C2×C3⋊D311yescharacteristic
112C2×C3×D4—91no—
113C2×D12—31no—
114C2×D12—91no—
115C2×D12—91no—
116C2×D12—181no—
117#41⋊C3—23no—
118#41⋊C3D3213yes—
119GL(2,𝔽3)—63no—
120GL(2,𝔽3)—186no—
121GL(2,𝔽3)—183no—
122C3×SD16—181no—
123C3×D8—181no—
124C3×M16—91no—
125C3×M16—91no—
126C2×C3×C8—91no—
127#215—61no—
128#215—182no—
129#215—181no—
130#215—181no—
131#213—61no—
132#213—181no—
133#213—181no—
134#213—181no—
135#213—182no—
136#213—181no—
137#208—31no—
138#208—31no—
139#208—91no—
140#208—91no—
141#208—181no—
142C2×C3⋊C8—31no—
143D24—181no—
144D24—181no—
145D24—363no—
146C22×C32—31no—
147C2×C3⋊D3—81no—
148C2×C3⋊D3—121no—
149C2×C3⋊D3—243no—
150C2×C3⋊D3—241no—
151C2×C3⋊D3—246no—
152C2×C3⋊D3—242no—
153C2×C3⋊D3—241no—
154C2×C3×D3—121no—
155C2×C3×D3—243no—
156C2×C3×D3—241no—
157C32×C4—31no—
158C32×C4—41no—
159C32×C4—43no—
160C32×C4—83no—
161C32×C4—81no—
162C32×C4C2×S411yescharacteristic
163AGL(1,ℤ/8)—91no—
164AGL(1,ℤ/8)—271no—
165C2×D8—271no—
166#131—31no—
167#108—271no—
168#100—271no—
169#100—271no—
170C33—41noSylow
171C23×C3—181no—
172C22×D3—61no—
173C22×D3—181no—
174C22×D3—181no—
175C22×D3—361no—
176C3×Q8—21no—
177C3×Q8#62611yescharacteristic
178C3×Q8C4⋊(C3⋊D3)11yescharacteristic
179C3×D4—31no—
180C3×D4—31no—
181C3×D4—61no—
182C3×D4—181no—
183C3×D4—181no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187D12—61no—
188D12—61no—
189D12—123no—
190D12—181no—
191D12—181no—
192D12—182no—
193D12—181no—
194D12—181no—
195D12—361no—
196D12—363no—
197D12—723no—
198SL(2,𝔽3)—23no—
199SL(2,𝔽3)#62613yes—
200C3×C8—31no—
201C3×C8—91no—
202C3×C8—123no—
203C3×C8—181no—
204C3⋊C8—31no—
205C3⋊C8—61no—
206C3⋊C8—181no—
207C3⋊C8—181no—
208C3⋊C8—182no—
209C3⋊C8C3⋊S411yescharacteristic
210C2×C32—41no—
211C2×C32—43no—
212C2×C32—61no—
213C2×C32—83no—
214C2×C32—81no—
215C2×C32C4⋊S411yescharacteristic
216C3⋊D3—161no—
217C3⋊D3—241no—
218C3⋊D3—483no—
219C3⋊D3—481no—
220C3⋊D3—486no—
221C3⋊D3—482no—
222C3×D3—241no—
223C3×D3—483no—
224#41D3×C3⋊D311yescharacteristic
225C2×D4—91no—
226C2×D4—271no—
227SD16—181no—
228SD16—541no—
229D8—181no—
230D8—541no—
231D8—541no—
232D8—541no—
233M16—91no—
234M16—91no—
235M16—271no—
236C2×C8—91no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—61no—
240C22×C3—361no—
241C22×C3—721no—
242C2×D3—121no—
243C2×D3—243no—
244C2×D3—241no—
245C2×D3—362no—
246C2×D3—361no—
247C2×D3—361no—
248C2×D3—723no—
249C2×D3—726no—
250C2×D3—722no—
251C2×D3—721no—
252C2×D3—721no—
253C3×C4—21no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—43no—
257C3×C4—61no—
258C3×C4—83no—
259C3×C4D3×S411yescharacteristic
260C3×C4C2×C3⋊S411yescharacteristic
261C32—41no—
262C32—43no—
263C32—83no—
264C32—81no—
265C32#540711yescharacteristic, complemented
266C23—181no—
267C23—541no—
268Q8#629211yescharacteristic
269D4—31no—
270D4—181no—
271D4—181no—
272D4—541no—
273D4—541no—
274C2×C4—31no—
275C8—31no—
276C8—91no—
277C8—181no—
278C8—541no—
279C2×C3—21no—
280C2×C3—43no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—83no—
284C2×C3—121no—
285C2×C3—721no—
286C2×C3#6420711yescharacteristic
287C2×C3C4⋊(C3⋊S4)11yescharacteristic
288D3—241no—
289D3—483no—
290D3—722no—
291D3—721no—
292D3—721no—
293D3—1443no—
294D3—1446no—
295C22—31no—
296C22—361no—
297C22—721no—
298C22—1081no—
299C22—2161no—
300C4—31no—
301C4D3×C3⋊S411yescharacteristic
302C3—21no—
303C3—43no—
304C3—83no—
305C3#1059322511yescharacteristic, complemented
306C3#1059364111yescharacteristic, complemented
307C2—61no—
308C2—721no—
309C2—2161no—
310C2#1172420211yescharacteristic
subgroupG/H
centerC2#11724202
commutator, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
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
1. #11724471C2semidirect
2. C3×#64267C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
15. #64267D3semidirect
19. C3×#948D3semidirect
22. C3⋊(#131⋊C3)D3semidirect
70. #948C3⋊D3semidirect
265. C32#5407semidirect
305. C3#10593225semidirect
306. C3#10593641semidirect

Maximal subgroups

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

subgroupindex
#117244712
C3×#642672
C3×C3⋊(#131⋊C3)2
#105936413
#105932253
#105916773
#887914