Finite Group Databasebeta

← #13166036

Subgroups — #13166036

Order 1728.

454 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 60 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×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593620—31nomaximal
5#10593212—33nomaximal
6#10591544—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6293—41nomaximal
9C3×Q8×Dic3D311yescharacteristic
10C3×C4×SL(2,𝔽3)—31no—
11#63918—31no—
12Dic3×SL(2,𝔽3)D313yes—
13C3⋊#859—31no—
14C3⋊#859—33no—
15C3⋊#859—61no—
16C3⋊#859—63no—
17C3⋊#859D311yescharacteristic
18C3×#859—33no—
19#63662—31no—
20C3×#835—31no—
21#63574—31no—
22#63573—31no—
23Dic3⋊(C3⋊C8)—31no—
24C2×C3⋊(C3⋊Dic3)—41no—
25C2×C3×C3⋊Dic3—41no—
26C2×C32×Dic3—41no—
27C32×SL(2,𝔽3)Q811yescharacteristic, complemented
28#6293—44no—
29#5386—93no—
30#5020—91no—
31#4791—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×#634—121no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)—23no—
36C2×C3×SL(2,𝔽3)C2×D313yes—
37C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
38C2×#627—61no—
39C2×#627—123no—
40C4⋊(C3⋊Dic3)—31no—
41C2×C3⋊(C3⋊C8)—31no—
42C3×C4⋊Dic3—31no—
43C3×C4⋊Dic3—31no—
44C3×C4×Dic3—31no—
45C2×C3×C3⋊C8—31no—
46Dic3⋊Dic3—61no—
47C22×C33—41no—
48C3⋊(C3⋊Dic3)—42no—
49C3×C3⋊Dic3—42no—
50C32×Dic3—42no—
51C3×C4×Q8—31no—
52Q8×Dic3C3⋊D311yescharacteristic
53#888—91no—
54#869—91no—
55C4×SL(2,𝔽3)—33no—
56#859—33no—
57#859—93no—
58#859—183no—
59C3×#103—91no—
60#835—31no—
61#835—91no—
62#835—91no—
63#835—181no—
64#817—91no—
65Dic3⋊C8—91no—
66#808—91no—
67#808—91no—
68#808—91no—
69C4⋊(C3⋊C8)—91no—
70C32×Q8—31no—
71C32×Q8Dic611yescharacteristic, complemented
72C2×C32×C4—31no—
73C2×C32×C4—121no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—61no—
76C2×C3⋊Dic3—121no—
77C2×C3⋊Dic3—123no—
78C2×C3⋊Dic3—241no—
79C2×C3⋊Dic3—243no—
80#634—124no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—43no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—123no—
85C2×C3×Dic3S411yescharacteristic
86C3×SL(2,𝔽3)—21no—
87C3×SL(2,𝔽3)—23no—
88C3×SL(2,𝔽3)Dic613yescomplemented
89C3×SL(2,𝔽3)Dic611yescharacteristic, complemented
90#627—1212no—
91#627—122no—
92C3⋊(C3⋊C8)—61no—
93C3×C3⋊C8—61no—
94#475—271noSylow
95C2×C33—41no—
96C2×C33—41no—
97C2×C33—41no—
98C2×C3×Q8—21no—
99C2×C3×Q8D3211yescharacteristic
100C2×C3×Q8C2×C3⋊D311yescharacteristic
101C2×Dic6—181no—
102C2×Dic6—181no—
103C2×Dic6—363no—
104C2×SL(2,𝔽3)—23no—
105C2×SL(2,𝔽3)D3213yes—
106C2×C3×C8—91no—
107C3×C4⋊C4—91no—
108C3×C4⋊C4—91no—
109C3×C42—91no—
110C4⋊Dic3—31no—
111C4⋊Dic3—31no—
112C4⋊Dic3—91no—
113C4⋊Dic3—91no—
114C4⋊Dic3—181no—
115C4⋊Dic3—181no—
116Dic3⋊C4—181no—
117C4×Dic3—31no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—181no—
122C22×C32—41no—
123C22×C32—43no—
124C22×C32—83no—
125C22×C32—81no—
126C22×C32C2×S411yescharacteristic
127C32×C4—31no—
128C32×C4—31no—
129C32×C4—122no—
130C3⋊Dic3—42no—
131C3⋊Dic3—122no—
132C3⋊Dic3—126no—
133C3⋊Dic3—121no—
134C3⋊Dic3—246no—
135C3⋊Dic3—242no—
136C3×Dic3—46no—
137C3×Dic3—61no—
138C3×Dic3—126no—
139C3×Dic3—121no—
140C3×Dic3GL(2,𝔽3)12yes—
141C4⋊Q8—271no—
142C4×Q8—31no—
143#106—271no—
144#106—271no—
145C4⋊C8—271no—
146#103—91no—
147#103—271no—
148C33—41noSylow
149C3×Q8—21no—
150C3×Q8—31no—
151C3×Q8—31no—
152C3×Q8—61no—
153C3×Q8#62711yescharacteristic, complemented
154C3×Q8#63411yescharacteristic, complemented
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—61no—
159C2×C3×C4—91no—
160C2×C3×C4—123no—
161C2×C3×C4—181no—
162C2×Dic3—31no—
163C2×Dic3—61no—
164C2×Dic3—123no—
165C2×Dic3—181no—
166C2×Dic3—181no—
167C2×Dic3—363no—
168C2×Dic3—361no—
169C2×Dic3—723no—
170C2×Dic3C3⋊S411yescharacteristic
171Dic6—362no—
172Dic6—3612no—
173Dic6—362no—
174SL(2,𝔽3)—23no—
175SL(2,𝔽3)#62713yescomplemented
176C3×C8—181no—
177C3⋊C8—61no—
178C3⋊C8—181no—
179C3⋊C8—181no—
180C3⋊C8—361no—
181C2×C32—41no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32—43no—
185C2×C32—43no—
186C2×C32—43no—
187C2×C32—83no—
188C2×C32—83no—
189C2×C32—83no—
190C2×C32—81no—
191C2×C32—81no—
192C2×C32—81no—
193C2×C32A4⋊Q811yescharacteristic
194C2×C32C2×GL(2,𝔽3)11yescharacteristic
195C2×C32#98411yescharacteristic
196C2×Q8—541no—
197C2×Q8D3×C3⋊D311yescharacteristic
198C2×C8—91no—
199C2×C8—271no—
200C4⋊C4—91no—
201C4⋊C4—91no—
202C4⋊C4—271no—
203C4⋊C4—541no—
204C42—91no—
205C22×C3—21no—
206C22×C3—43no—
207C22×C3—83no—
208C22×C3D3×S411yescharacteristic
209C22×C3C2×C3⋊S411yescharacteristic
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—32no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—61no—
216C3×C4—61no—
217C3×C4—126no—
218C3×C4—181no—
219C3×C4—361no—
220Dic3—61no—
221Dic3—126no—
222Dic3—121no—
223Dic3—366no—
224Dic3—361no—
225Dic3—361no—
226Dic3—721no—
227Dic3—726no—
228Dic3C3⋊GL(2,𝔽3)12yes—
229C32—41no—
230C32—43no—
231C32—83no—
232C32—81no—
233C32#538611yescharacteristic, complemented
234Q8—31no—
235Q8—1082no—
236Q8#629311yescharacteristic, complemented
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—91no—
240C2×C4—181no—
241C2×C4—541no—
242C8—181no—
243C8—541no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—43no—
248C2×C3—43no—
249C2×C3—43no—
250C2×C3—83no—
251C2×C3—83no—
252C2×C3—83no—
253C2×C3#6419611yescharacteristic
254C2×C3D3×GL(2,𝔽3)11yescharacteristic
255C2×C3#6420411yescharacteristic
256C2×C3#6425911yescharacteristic
257C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
258C2×C3#6426511yescharacteristic
259C22D3×C3⋊S411yescharacteristic
260C4—32no—
261C4—31no—
262C4—31no—
263C4—181no—
264C4—361no—
265C4—1081no—
266C3—21no—
267C3—43no—
268C3—83no—
269C3#1059321211yescharacteristic, complemented
270C3#1059362011yescharacteristic, complemented
271C2#1172419111yescharacteristic
272C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
273C2#1172419911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)Q8
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
27. C32×SL(2,𝔽3)Q8semidirect
71. C32×Q8Dic6semidirect
88. C3×SL(2,𝔽3)Dic6semidirect
89. C3×SL(2,𝔽3)Dic6semidirect
153. C3×Q8#627semidirect
154. C3×Q8#634semidirect
175. SL(2,𝔽3)#627semidirect
233. C32#5386semidirect
236. Q8#6293semidirect
269. C3#10593212semidirect
270. C3#10593620semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C3×C3⋊#8592
#105936203
#105932123
#105915443
C2×#62934