Finite Group Databasebeta

← C3×C4⋊(C3⋊GL(2,𝔽3)) #13172995

Subgroups — C3×C4⋊(C3⋊GL(2,𝔽3)) #13172995

Order 1728.

600 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×C3×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊GL(2,𝔽3))C311yesmaximal, characteristic, direct product factor
4C3×C4⋊GL(2,𝔽3)—33nomaximal
5C3×#5027—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—41nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—22no—
9C2×C3⋊GL(2,𝔽3)C2×C312yescomplemented
10C2×C3×GL(2,𝔽3)—36no—
11C32×C4×Q8D311yescharacteristic, complemented
12C2×C3×#215—32no—
13C3×#874—31no—
14C3×C4×SL(2,𝔽3)—21no—
15C3×C4×SL(2,𝔽3)—23no—
16C3×C4×SL(2,𝔽3)D313yescomplemented
17C3×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#809—32no—
19C3×C4⋊(C3⋊C8)—31no—
20C22×C3×C3⋊D3—42no—
21C2×C33×C4—41no—
22C3×C4⋊(C3⋊D3)—44no—
23C32×SL(2,𝔽3)D411yescharacteristic, complemented
24C4⋊GL(2,𝔽3)—33no—
25C3×#460—91no—
26#5027—31no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C4⋊(C3⋊D3)—41no—
29C2×C3×D12—32no—
30C2×C3×D12—61no—
31C2×C3×D12—61no—
32C2×C3×D12—123no—
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)—23no—
35C2×C3×SL(2,𝔽3)C2×D313yes—
36C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
37C3⋊GL(2,𝔽3)—22no—
38C3×GL(2,𝔽3)—66no—
39C32×C4⋊C4—31no—
40C32×C42—31no—
41C3×#215—62no—
42C3×#215—62no—
43C2×C3×C3⋊C8—32no—
44C22×C33—41no—
45C2×C3×C3⋊D3—44no—
46C2×C3×C3⋊D3—82no—
47C2×C3×C3⋊D3—82no—
48C33×C4—42no—
49C2×GL(2,𝔽3)—36no—
50C2×C3×SD16—92no—
51C3×#127—91no—
52C3×C4×Q8—21no—
53C3×C4×Q8C3×D311yescharacteristic, complemented
54C3×C4×Q8C3⋊D311yescharacteristic, complemented
55C2×#215—32no—
56#874—31no—
57C4×SL(2,𝔽3)—23no—
58C4×SL(2,𝔽3)C3×D313yescomplemented
59C3×C4⋊C8—91no—
60C3×#102—92no—
61#809—32no—
62C4⋊(C3⋊C8)—31no—
63C22×C3⋊D3—42no—
64C22×C3×D3—62no—
65C22×C3×D3—126no—
66C32×Q8—31no—
67C32×Q8D1211yescharacteristic, complemented
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—41no—
71C2×C32×C4—43no—
72C2×C32×C4—83no—
73C2×C32×C4—81no—
74C2×C32×C4S411yescharacteristic
75C4⋊(C3⋊D3)—44no—
76C3×D12—62no—
77C3×D12—62no—
78C3×D12—1212no—
79C3×D12—122no—
80C3×D12—122no—
81C3×SL(2,𝔽3)—21no—
82C3×SL(2,𝔽3)—23no—
83C3×SL(2,𝔽3)D1213yescomplemented
84C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
85C3×C3⋊C8—62no—
86#460—91noSylow
87C2×C33—41no—
88C2×C33—41no—
89C2×C33—41no—
90C3×C3⋊D3—84no—
91C2×C3×Q8—21no—
92C2×C3×Q8C2×C3×D311yescharacteristic
93C2×C3×Q8C2×C3⋊D311yescharacteristic
94C2×C3×D4—92no—
95C2×C3×D4—181no—
96C2×C3×D4—181no—
97C2×D12—32no—
98C2×D12—61no—
99C2×D12—61no—
100C2×D12—123no—
101C2×SL(2,𝔽3)—23no—
102C2×SL(2,𝔽3)C2×C3×D313yes—
103GL(2,𝔽3)—66no—
104C3×SD16—182no—
105C3×SD16—182no—
106C2×C3×C8—92no—
107C3×C4⋊C4—31no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—61no—
110C3×C42—31no—
111C3×C42—31no—
112C3×C42—61no—
113#215—62no—
114#215—62no—
115C2×C3⋊C8—32no—
116C22×C32—41no—
117C22×C32—43no—
118C22×C32—83no—
119C22×C32—81no—
120C22×C32C2×S411yescharacteristic
121C2×C3⋊D3—44no—
122C2×C3⋊D3—82no—
123C2×C3⋊D3—82no—
124C2×C3×D3—1212no—
125C2×C3×D3—122no—
126C2×C3×D3—122no—
127C2×C3×D3—122no—
128C2×C3×D3—246no—
129C2×C3×D3—246no—
130C32×C4—31no—
131C32×C4—31no—
132C32×C4—42no—
133C32×C4—46no—
134C32×C4—61no—
135C32×C4—86no—
136C32×C4—82no—
137C32×C4GL(2,𝔽3)12yescomplemented
138C2×SD16—92no—
139#127—91no—
140C4×Q8C3×C3⋊D311yescharacteristic, complemented
141C4⋊C8—91no—
142#102—92no—
143C33—41noSylow
144C23×C3—182no—
145C22×D3—62no—
146C22×D3—126no—
147C3×Q8—21no—
148C3×Q8—31no—
149C3×Q8—31no—
150C3×Q8—61no—
151C3×Q8C3×D1211yescharacteristic, complemented
152C3×Q8C4⋊(C3⋊D3)11yescharacteristic, complemented
153C3×D4—182no—
154C3×D4—182no—
155C3×D4—362no—
156C3×D4—362no—
157C2×C3×C4—21no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—43no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—83no—
166C2×C3×C4C3×S411yescharacteristic
167C2×C3×C4C3⋊S411yescharacteristic
168D12—62no—
169D12—62no—
170D12—1212no—
171D12—122no—
172D12—122no—
173SL(2,𝔽3)—23no—
174SL(2,𝔽3)C3×D1213yescomplemented
175C3×C8—182no—
176C3⋊C8—62no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—43no—
181C2×C32—43no—
182C2×C32—43no—
183C2×C32—83no—
184C2×C32—83no—
185C2×C32—83no—
186C2×C32—81no—
187C2×C32—81no—
188C2×C32—81no—
189C2×C32C4⋊S411yescharacteristic
190C2×C32C2×GL(2,𝔽3)11yescharacteristic
191C2×C32#98611yescharacteristic
192C3⋊D3—84no—
193C3×D3—2412no—
194C3×D3—242no—
195C2×Q8C2×C3×C3⋊D311yescharacteristic
196C2×D4—92no—
197C2×D4—181no—
198C2×D4—181no—
199SD16—182no—
200SD16—182no—
201C2×C8—92no—
202C4⋊C4—31no—
203C42—31no—
204C22×C3—21no—
205C22×C3—43no—
206C22×C3—83no—
207C22×C3—362no—
208C22×C3—362no—
209C22×C3—362no—
210C22×C3C2×C3×S411yescharacteristic
211C22×C3C2×C3⋊S411yescharacteristic
212C2×D3—1212no—
213C2×D3—122no—
214C2×D3—122no—
215C2×D3—122no—
216C2×D3—246no—
217C2×D3—246no—
218C3×C4—22no—
219C3×C4—31no—
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—46no—
224C3×C4—61no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—86no—
229C3×C4—121no—
230C3×C4C3×GL(2,𝔽3)12yescomplemented
231C3×C4C3⋊GL(2,𝔽3)12yescomplemented
232C32—41no—
233C32—43no—
234C32—83no—
235C32—81no—
236C32C4⋊GL(2,𝔽3)11yescharacteristic, complemented
237C23—182no—
238Q8—31no—
239Q8C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
240D4—182no—
241D4—182no—
242D4—362no—
243D4—362no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4C3×C3⋊S411yescharacteristic
247C8—182no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—43no—
252C2×C3—43no—
253C2×C3—43no—
254C2×C3—83no—
255C2×C3—83no—
256C2×C3—83no—
257C2×C3—722no—
258C2×C3C3×C4⋊S411yescharacteristic
259C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
260C2×C3C3×#98611yescharacteristic
261C2×C3C4⋊(C3⋊S4)11yescharacteristic
262C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
263C2×C3#6426711yescharacteristic
264D3—2412no—
265D3—242no—
266C22—362no—
267C22—362no—
268C22—362no—
269C22C2×C3×C3⋊S411yescharacteristic
270C4—31no—
271C4—31no—
272C4—61no—
273C4C3×C3⋊GL(2,𝔽3)12yescomplemented
274C3—21no—
275C3—43no—
276C3—83no—
277C3C3×C4⋊GL(2,𝔽3)11yescharacteristic, complemented
278C3C4⋊(C3⋊GL(2,𝔽3))11yescharacteristic, direct product factor
279C2—722no—
280C2C3×C4⋊(C3⋊S4)11yescharacteristic
281C2C2×C3×C3⋊GL(2,𝔽3)11yescharacteristic
282C2C3×#6426711yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
2. C32×C4×SL(2,𝔽3)C2semidirect
3. C4⋊(C3⋊GL(2,𝔽3))C3direct (N × H)
9. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
11. C32×C4×Q8D3semidirect
16. C3×C4×SL(2,𝔽3)D3semidirect
17. C3×C4×SL(2,𝔽3)C2×C3semidirect
23. C32×SL(2,𝔽3)D4semidirect
53. C3×C4×Q8C3×D3semidirect
54. C3×C4×Q8C3⋊D3semidirect
58. C4×SL(2,𝔽3)C3×D3semidirect
67. C32×Q8D12semidirect
83. C3×SL(2,𝔽3)D12semidirect
84. C3×SL(2,𝔽3)C3×D4semidirect
137. C32×C4GL(2,𝔽3)semidirect
140. C4×Q8C3×C3⋊D3semidirect
151. C3×Q8C3×D12semidirect
152. C3×Q8C4⋊(C3⋊D3)semidirect
174. SL(2,𝔽3)C3×D12semidirect
230. C3×C4C3×GL(2,𝔽3)semidirect
231. C3×C4C3⋊GL(2,𝔽3)semidirect
236. C32C4⋊GL(2,𝔽3)semidirect
239. Q8C3×C4⋊(C3⋊D3)semidirect
273. C4C3×C3⋊GL(2,𝔽3)semidirect
277. C3C3×C4⋊GL(2,𝔽3)semidirect
278. C3C4⋊(C3⋊GL(2,𝔽3))direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C32×C4×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))3
C3×C4⋊GL(2,𝔽3)3
C3×#50273
C2×C3×C4⋊(C3⋊D3)4