Finite Group Databasebeta

← Dic3×C3⋊GL(2,𝔽3) #13166037

Subgroups — Dic3×C3⋊GL(2,𝔽3) #13166037

Order 1728.

586 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 64 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)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)—31nomaximal
5Dic3×GL(2,𝔽3)—33nomaximal
6Dic3×#215—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×Dic3×C3⋊D3—41nomaximal
9C3×C3⋊GL(2,𝔽3)C412yescomplemented
10C2×C3⋊GL(2,𝔽3)D311yescharacteristic
11C2×C3×GL(2,𝔽3)—33no—
12C3×Q8×Dic3D311yescharacteristic, complemented
13C2×C3×#215—31no—
14C3×C4×SL(2,𝔽3)—31no—
15Dic3×D12—31no—
16Dic3×SL(2,𝔽3)D313yescomplemented
17C3⋊#859—31no—
18C3⋊#859—33no—
19C3⋊#859—61no—
20C3⋊#859—63no—
21#63662—31no—
22#63573—31no—
23#63562—31no—
24Dic3×C3⋊C8—31no—
25C22×C3×C3⋊D3—41no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
29Dic3×C3⋊D3—44no—
30C4×GL(2,𝔽3)—93no—
31Dic3×SD16—91no—
32C4×#215—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4×C3⋊D3—121no—
35C2×C3×D12—31no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—23no—
38C2×C3×SL(2,𝔽3)C2×D313yes—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40C2×D3×Dic3—61no—
41C2×D3×Dic3—123no—
42C3⋊GL(2,𝔽3)Dic312yesdirect product factor
43C3×GL(2,𝔽3)—36no—
44C4⋊(C3⋊Dic3)—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C3×#215—32no—
47C3×#215—32no—
48C3×C4⋊Dic3—31no—
49C3×C4×Dic3—31no—
50C2×C3×C3⋊C8—31no—
51(C2×D3)⋊Dic3—61no—
52C22×C33—41no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—42no—
56C3⋊(C3⋊Dic3)—42no—
57C32×Dic3—42no—
58C2×GL(2,𝔽3)—33no—
59C2×C3×SD16—91no—
60C3×C4×Q8—31no—
61Q8×Dic3C3⋊D311yescharacteristic, complemented
62C2×#215—31no—
63D4×Dic3—91no—
64C4×D12—91no—
65C4×SL(2,𝔽3)—33no—
66#859—93no—
67#859—183no—
68#835—91no—
69#835—91no—
70#835—181no—
71#832—91no—
72#817—91no—
73C8×Dic3—91no—
74#809—91no—
75#808—91no—
76C4×C3⋊C8—91no—
77C22×C3⋊D3—41no—
78C22×C3×D3—61no—
79C22×C3×D3—123no—
80C32×Q8—31no—
81C32×Q8C4×D311yescharacteristic, complemented
82C2×C32×C4—31no—
83C2×C32×C4—121no—
84C2×C3⋊Dic3—61no—
85C2×C3⋊Dic3—121no—
86C2×C3⋊Dic3—123no—
87C2×C3⋊Dic3—241no—
88C2×C3⋊Dic3—243no—
89C4×C3⋊D3—124no—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—43no—
92C2×C3×Dic3S411yescharacteristic
93C3×D12—32no—
94C3×D12—61no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—23no—
97C3×SL(2,𝔽3)C4×D313yescomplemented
98C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
99D3×Dic3—1212no—
100D3×Dic3—122no—
101C3⋊(C3⋊C8)—61no—
102C3×C3⋊C8—32no—
103C4×SD16—271noSylow
104C2×C33—41no—
105C2×C33—41no—
106C2×C33—41no—
107C3×C3⋊D3—44no—
108C2×C3×Q8—21no—
109C2×C3×Q8D3211yescharacteristic
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—91no—
112C22×Dic3—181no—
113C2×D12—31no—
114C2×C4×D3—181no—
115C2×C4×D3—363no—
116C2×SL(2,𝔽3)—23no—
117C2×SL(2,𝔽3)D3213yes—
118GL(2,𝔽3)—36no—
119C3×SD16—92no—
120C3×SD16—92no—
121C2×C3×C8—91no—
122C3×C4⋊C4—91no—
123C3×C42—91no—
124#217—181no—
125#215—32no—
126#215—32no—
127(C2×D3)⋊C4—181no—
128C4⋊Dic3—31no—
129C4⋊Dic3—91no—
130C4⋊Dic3—91no—
131C4⋊Dic3—181no—
132C4×Dic3—31no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—91no—
135C2×C3⋊C8—91no—
136C2×C3⋊C8—181no—
137C22×C32—41no—
138C22×C32—43no—
139C22×C32—83no—
140C22×C32—81no—
141C22×C32C2×S411yescharacteristic
142C2×C3⋊D3—42no—
143C2×C3⋊D3—42no—
144C2×C3⋊D3—42no—
145C2×C3×D3—62no—
146C2×C3×D3—126no—
147C2×C3×D3—126no—
148C2×C3×D3—126no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C32×C4—31no—
152C32×C4—31no—
153C32×C4—122no—
154C3⋊Dic3—122no—
155C3⋊Dic3—126no—
156C3⋊Dic3—121no—
157C3⋊Dic3—246no—
158C3⋊Dic3—242no—
159C3×Dic3—46no—
160C3×Dic3—61no—
161C3×Dic3GL(2,𝔽3)12yescomplemented
162C2×SD16—91no—
163C4×Q8—31no—
164C4×D4—271no—
165#106—271no—
166#103—271no—
167#102—271no—
168C4×C8—271no—
169C33—41noSylow
170C23×C3—181no—
171C22×D3—61no—
172C22×D3—123no—
173C3×Q8—21no—
174C3×Q8—31no—
175C3×Q8—31no—
176C3×Q8—61no—
177C3×Q8D3×Dic311yescharacteristic, complemented
178C3×Q8C4×C3⋊D311yescharacteristic, complemented
179C3×D4—92no—
180C3×D4—181no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—123no—
187C2×Dic3—31no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—363no—
191C2×Dic3—361no—
192C2×Dic3—362no—
193C2×Dic3—723no—
194C2×Dic3C3⋊S411yescharacteristic
195D12—32no—
196D12—61no—
197C4×D3—362no—
198C4×D3—3612no—
199SL(2,𝔽3)—23no—
200SL(2,𝔽3)D3×Dic313yescomplemented
201C3×C8—92no—
202C3⋊C8—32no—
203C3⋊C8—181no—
204C3⋊C8—181no—
205C3⋊C8—361no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—43no—
210C2×C32—43no—
211C2×C32—43no—
212C2×C32—83no—
213C2×C32—83no—
214C2×C32—83no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32—81no—
218C2×C32C4×S411yescharacteristic
219C2×C32C2×GL(2,𝔽3)11yescharacteristic
220C2×C32#98511yescharacteristic
221C3⋊D3—44no—
222C3×D3—1212no—
223C3×D3—122no—
224C2×Q8D3×C3⋊D311yescharacteristic
225C2×D4—91no—
226C22×C4—541no—
227SD16—92no—
228SD16—92no—
229C2×C8—91no—
230C2×C8—271no—
231C4⋊C4—91no—
232C4⋊C4—271no—
233C22⋊C4—541no—
234C42—91no—
235C22×C3—21no—
236C22×C3—43no—
237C22×C3—83no—
238C22×C3—182no—
239C22×C3—361no—
240C22×C3—361no—
241C22×C3D3×S411yescharacteristic
242C22×C3C2×C3⋊S411yescharacteristic
243C2×D3—62no—
244C2×D3—126no—
245C2×D3—126no—
246C2×D3—126no—
247C2×D3—121no—
248C2×D3—121no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—126no—
257C3×C4—181no—
258Dic3—61no—
259Dic3—366no—
260Dic3—361no—
261Dic3—361no—
262Dic3—721no—
263Dic3—726no—
264Dic3C3⋊GL(2,𝔽3)12yesdirect product factor
265C32—41no—
266C32—43no—
267C32—83no—
268C32—81no—
269C32C4×GL(2,𝔽3)11yescharacteristic, complemented
270C23—181no—
271Q8—31no—
272Q8Dic3×C3⋊D311yescharacteristic, complemented
273D4—92no—
274D4—181no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—91no—
278C2×C4—541no—
279C2×C4—1082no—
280C8—92no—
281C8—541no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—43no—
286C2×C3—43no—
287C2×C3—43no—
288C2×C3—83no—
289C2×C3—83no—
290C2×C3—83no—
291C2×C3—362no—
292C2×C3#6419711yescharacteristic
293C2×C3D3×GL(2,𝔽3)11yescharacteristic
294C2×C3Dic3×S411yescharacteristic
295C2×C3C4×C3⋊S411yescharacteristic
296C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
297C2×C3#6426611yescharacteristic
298D3—1212no—
299D3—122no—
300C22—182no—
301C22—361no—
302C22—361no—
303C22D3×C3⋊S411yescharacteristic
304C4—32no—
305C4—31no—
306C4—31no—
307C4—181no—
308C4—1081no—
309C3—21no—
310C3—43no—
311C3—83no—
312C3Dic3×GL(2,𝔽3)11yescharacteristic, complemented
313C3C4×C3⋊GL(2,𝔽3)11yescharacteristic, complemented
314C2—362no—
315C2#1172419211yescharacteristic
316C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
317C2Dic3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
9. C3×C3⋊GL(2,𝔽3)C4semidirect
12. C3×Q8×Dic3D3semidirect
16. Dic3×SL(2,𝔽3)D3semidirect
28. C32×SL(2,𝔽3)C2×C4semidirect
42. C3⋊GL(2,𝔽3)Dic3direct (N × H)
61. Q8×Dic3C3⋊D3semidirect
81. C32×Q8C4×D3semidirect
97. C3×SL(2,𝔽3)C4×D3semidirect
98. C3×SL(2,𝔽3)C2×Dic3semidirect
161. C3×Dic3GL(2,𝔽3)semidirect
177. C3×Q8D3×Dic3semidirect
178. C3×Q8C4×C3⋊D3semidirect
200. SL(2,𝔽3)D3×Dic3semidirect
264. Dic3C3⋊GL(2,𝔽3)direct (N × H)
269. C32C4×GL(2,𝔽3)semidirect
272. Q8Dic3×C3⋊D3semidirect
312. C3Dic3×GL(2,𝔽3)semidirect
313. C3C4×C3⋊GL(2,𝔽3)semidirect

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
C3×Dic3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C4×C3⋊GL(2,𝔽3)3
Dic3×GL(2,𝔽3)3
Dic3×#2153
C2×Dic3×C3⋊D34