Finite Group Databasebeta

← C3×C4×C3⋊GL(2,𝔽3) #13172994

Subgroups — C3×C4×C3⋊GL(2,𝔽3) #13172994

Order 1728.

588 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 78 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
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)C311yesmaximal, characteristic, direct product factor
5C3×C4×GL(2,𝔽3)—33nomaximal
6C3×C4×#215—31nomaximal
7C2×C3×C4×C3⋊D3—41nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3×C3⋊GL(2,𝔽3)C412yesdirect product factor
10C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
11C2×C3×GL(2,𝔽3)—33no—
12C32×C4×Q8D311yescharacteristic, complemented
13C2×C3×#215—31no—
14C3×C4×D12—31no—
15C3×C4×SL(2,𝔽3)—21no—
16C3×C4×SL(2,𝔽3)—23no—
17C3×C4×SL(2,𝔽3)D313yescomplemented
18C3×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C3⋊#859C2×C311yescharacteristic, complemented
20C3×#859—33no—
21C3×#835—31no—
22C3×#809—31no—
23C3×#808—31no—
24C3×C4×C3⋊C8—31no—
25C22×C3×C3⋊D3—41no—
26C2×C33×C4—41no—
27C2×C3×C3⋊Dic3—41no—
28C3×C4×C3⋊D3—44no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30C4×GL(2,𝔽3)—33no—
31C3×C4×SD16—91no—
32C4×#215—31no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4×C3⋊D3—41no—
35C2×C3×D12—31no—
36C2×C3×C4×D3—61no—
37C2×C3×C4×D3—123no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—23no—
40C2×C3×SL(2,𝔽3)C2×D313yes—
41C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
42C3⋊GL(2,𝔽3)C3×C412yesdirect product factor
43C3×GL(2,𝔽3)—36no—
44C32×C4⋊C4—31no—
45C32×C42—31no—
46C3×#215—32no—
47C3×#215—32no—
48C3×(C2×D3)⋊C4—61no—
49C3×C4⋊Dic3—31no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52C22×C33—41no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—42no—
56C33×C4—42no—
57C3×C3⋊Dic3—42no—
58C2×GL(2,𝔽3)—33no—
59C2×C3×SD16—91no—
60C3×C4×Q8—21no—
61C3×C4×Q8C3×D311yescharacteristic, complemented
62C3×C4×Q8C3⋊D311yescharacteristic, complemented
63C3×C4×D4—91no—
64C2×#215—31no—
65C4×D12—31no—
66C4×SL(2,𝔽3)—23no—
67C4×SL(2,𝔽3)C3×D313yescomplemented
68#859—33no—
69C3×#106—91no—
70C3×#103—91no—
71C3×#102—91no—
72C3×C4×C8—91no—
73#835—31no—
74#809—31no—
75#808—31no—
76C4×C3⋊C8—31no—
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—31no—
84C2×C32×C4—41no—
85C2×C32×C4—43no—
86C2×C32×C4—83no—
87C2×C32×C4—81no—
88C2×C32×C4S411yescharacteristic
89C2×C3⋊Dic3—41no—
90C4×C3⋊D3—44no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—123no—
93C3×D12—32no—
94C3×D12—61no—
95C3×C4×D3—1212no—
96C3×C4×D3—122no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—23no—
99C3×SL(2,𝔽3)C4×D313yescomplemented
100C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
101C3×C3⋊C8—32no—
102C3×C3⋊C8—61no—
103C4×SD16—91noSylow
104C2×C33—41no—
105C2×C33—41no—
106C2×C33—41no—
107C3×C3⋊D3—44no—
108C2×C3×Q8—21no—
109C2×C3×Q8C2×C3×D311yescharacteristic
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—91no—
112C22×C3×C4—181no—
113C2×D12—31no—
114C2×C4×D3—61no—
115C2×C4×D3—123no—
116C2×SL(2,𝔽3)—23no—
117C2×SL(2,𝔽3)C2×C3×D313yes—
118GL(2,𝔽3)—36no—
119C3×SD16—92no—
120C3×SD16—92no—
121C2×C3×C8—91no—
122C2×C3×C8—91no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—61no—
126C3×C4⋊C4—91no—
127C3×C22⋊C4—181no—
128C3×C42—31no—
129C3×C42—31no—
130C3×C42—61no—
131#215—32no—
132#215—32no—
133(C2×D3)⋊C4—61no—
134C4⋊Dic3—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
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—42no—
154C32×C4—46no—
155C32×C4—61no—
156C32×C4—86no—
157C32×C4—82no—
158C32×C4GL(2,𝔽3)12yescomplemented
159C3⋊Dic3—42no—
160C3×Dic3—126no—
161C3×Dic3—121no—
162C2×SD16—91no—
163C4×Q8C3×C3⋊D311yescharacteristic, complemented
164C4×D4—91no—
165#106—91no—
166#103—91no—
167#102—91no—
168C4×C8—91no—
169C33—41noSylow
170C23×C3—181no—
171C22×D3—61no—
172C22×D3—123no—
173C3×Q8—21no—
174C3×Q8—31no—
175C3×Q8—31no—
176C3×Q8—61no—
177C3×Q8C3×C4×D311yescharacteristic, complemented
178C3×Q8C4×C3⋊D311yescharacteristic, complemented
179C3×D4—92no—
180C3×D4—181no—
181C2×C3×C4—21no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—43no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—83no—
190C2×C3×C4—181no—
191C2×C3×C4—362no—
192C2×C3×C4C3×S411yescharacteristic
193C2×C3×C4C3⋊S411yescharacteristic
194C2×Dic3—61no—
195C2×Dic3—123no—
196D12—32no—
197D12—61no—
198C4×D3—1212no—
199C4×D3—122no—
200SL(2,𝔽3)—23no—
201SL(2,𝔽3)C3×C4×D313yescomplemented
202C3×C8—92no—
203C3×C8—181no—
204C3⋊C8—32no—
205C3⋊C8—61no—
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×Q8C2×C3×C3⋊D311yescharacteristic
225C2×D4—91no—
226C22×C4—181no—
227SD16—92no—
228SD16—92no—
229C2×C8—91no—
230C2×C8—91no—
231C4⋊C4—31no—
232C4⋊C4—91no—
233C22⋊C4—181no—
234C42—31no—
235C22×C3—21no—
236C22×C3—43no—
237C22×C3—83no—
238C22×C3—182no—
239C22×C3—361no—
240C22×C3—361no—
241C22×C3C2×C3×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—22no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—46no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—86no—
260C3×C4—121no—
261C3×C4—361no—
262C3×C4C3×GL(2,𝔽3)12yescomplemented
263C3×C4C3⋊GL(2,𝔽3)12yesdirect product factor
264Dic3—126no—
265Dic3—121no—
266C32—41no—
267C32—43no—
268C32—83no—
269C32—81no—
270C32C4×GL(2,𝔽3)11yescharacteristic, complemented
271C23—181no—
272Q8—31no—
273Q8C3×C4×C3⋊D311yescharacteristic, complemented
274D4—92no—
275D4—181no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—181no—
279C2×C4—362no—
280C2×C4C3×C3⋊S411yescharacteristic
281C8—92no—
282C8—181no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—43no—
287C2×C3—43no—
288C2×C3—43no—
289C2×C3—83no—
290C2×C3—83no—
291C2×C3—83no—
292C2×C3—362no—
293C2×C3C3×C4×S411yescharacteristic
294C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
295C2×C3C3×#98511yescharacteristic
296C2×C3C4×C3⋊S411yescharacteristic
297C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
298C2×C3#6426611yescharacteristic
299D3—1212no—
300D3—122no—
301C22—182no—
302C22—361no—
303C22—361no—
304C22C2×C3×C3⋊S411yescharacteristic
305C4—31no—
306C4—31no—
307C4—61no—
308C4—361no—
309C4C3×C3⋊GL(2,𝔽3)12yesdirect product factor
310C3—21no—
311C3—43no—
312C3—83no—
313C3C3×C4×GL(2,𝔽3)11yescharacteristic, complemented
314C3C4×C3⋊GL(2,𝔽3)11yescharacteristic, direct product factor
315C2—362no—
316C2C3×C4×C3⋊S411yescharacteristic
317C2C2×C3×C3⋊GL(2,𝔽3)11yescharacteristic
318C2C3×#6426611yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×SL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
4. C4×C3⋊GL(2,𝔽3)C3direct (N × H)
9. C3×C3⋊GL(2,𝔽3)C4direct (N × H)
12. C32×C4×Q8D3semidirect
17. C3×C4×SL(2,𝔽3)D3semidirect
18. C3×C4×SL(2,𝔽3)C2×C3semidirect
19. C3⋊#859C2×C3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
42. C3⋊GL(2,𝔽3)C3×C4direct (N × H)
61. C3×C4×Q8C3×D3semidirect
62. C3×C4×Q8C3⋊D3semidirect
67. C4×SL(2,𝔽3)C3×D3semidirect
81. C32×Q8C4×D3semidirect
99. C3×SL(2,𝔽3)C4×D3semidirect
100. C3×SL(2,𝔽3)C2×C3×C4semidirect
158. C32×C4GL(2,𝔽3)semidirect
163. C4×Q8C3×C3⋊D3semidirect
177. C3×Q8C3×C4×D3semidirect
178. C3×Q8C4×C3⋊D3semidirect
201. SL(2,𝔽3)C3×C4×D3semidirect
262. C3×C4C3×GL(2,𝔽3)semidirect
263. C3×C4C3⋊GL(2,𝔽3)direct (N × H)
270. C32C4×GL(2,𝔽3)semidirect
273. Q8C3×C4×C3⋊D3semidirect
309. C4C3×C3⋊GL(2,𝔽3)direct (N × H)
313. C3C3×C4×GL(2,𝔽3)semidirect
314. 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
C3×C3⋊#8592
C4×C3⋊GL(2,𝔽3)3
C3×C4×GL(2,𝔽3)3
C3×C4×#2153
C2×C3×C4×C3⋊D34