Finite Group Databasebeta

← C3×C3⋊#5931 #13181422

Subgroups — C3×C3⋊#5931 #13181422

Order 1728.

455 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C32×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊#5931C311yesmaximal, characteristic, direct product factor
3C3×#5931—33nomaximal
4C3×#5227—31nomaximal
5C2×C3×C3⋊S4—41nomaximal
6C3×C3⋊GL(2,𝔽3)—42nomaximal
7C22×C32×Q8D311yescharacteristic, complemented
8C3×C22⋊SL(2,𝔽3)—21no—
9C3×C22⋊SL(2,𝔽3)—23no—
10C3×C22⋊SL(2,𝔽3)D313yescomplemented
11C3×C22⋊SL(2,𝔽3)C2×C311yescharacteristic, complemented
12C2×C3×#215—32no—
13C3×#930—31no—
14C3×#835—32no—
15C3×#830—31no—
16C2×C32×A4—41no—
17C3×C3⋊S4—42no—
18C32×SL(2,𝔽3)—42no—
19C32×SL(2,𝔽3)—81no—
20#5931—33no—
21C3×#448—91no—
22#5227—31no—
23C2×C3⋊S4—41no—
24C2×C3×S4—123no—
25C2×C32×Q8—32no—
26C2×C32×Q8—61no—
27C22×C32×C4—31no—
28C2×C3×#67—61no—
29C2×C3×D12—31no—
30C3⋊GL(2,𝔽3)—42no—
31C3×GL(2,𝔽3)—126no—
32C3×#215—62no—
33C3×#215—62no—
34C3×(C2×D3)⋊C4—61no—
35C3×C4⋊Dic3—31no—
36C2×C3×C3⋊C8—32no—
37C2×C3×C3⋊D3—161no—
38C32×A4—41no—
39C22×C3×Q8—21no—
40C22×C3×Q8C3×D311yescharacteristic, complemented
41C22×C3×Q8C3⋊D311yescharacteristic, complemented
42C22⋊SL(2,𝔽3)—23no—
43C22⋊SL(2,𝔽3)C3×D313yescomplemented
44C2×C3×SD16—92no—
45C3×#121—91no—
46C2×#215—32no—
47#930—31no—
48C3×#103—92no—
49C3×C22⋊C8—91no—
50#835—32no—
51#830—31no—
52C23×C32S411yescharacteristic
53C22×C3×D3—61no—
54C2×C3×A4—41no—
55C2×C3×A4—43no—
56C2×C3×A4—83no—
57C2×C3×A4—81no—
58C3⋊S4—42no—
59C3×S4—126no—
60C32×Q8—21no—
61C32×Q8—32no—
62C32×Q8—61no—
63C32×Q8S412yescomplemented
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—62no—
67C3×#67—121no—
68C3×#67—121no—
69C2×C3×Dic3—61no—
70C3×D12—62no—
71C3×SL(2,𝔽3)—42no—
72C3×SL(2,𝔽3)—46no—
73C3×SL(2,𝔽3)—83no—
74C3×SL(2,𝔽3)—81no—
75C3×SL(2,𝔽3)—86no—
76C3×SL(2,𝔽3)—86no—
77C3×SL(2,𝔽3)—82no—
78C3×SL(2,𝔽3)—82no—
79C3×C3⋊C8—62no—
80#448—91noSylow
81C2×C33—161no—
82C3×C3⋊D3—162no—
83C2×S4—123no—
84C2×C3×Q8—32no—
85C2×C3×Q8—32no—
86C2×C3×Q8—62no—
87C2×C3×Q8—61no—
88C2×C3×Q8—61no—
89C2×C3×Q8—62no—
90C2×C3×D4—91no—
91C2×C3×D4—181no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—61no—
95C2×#67—61no—
96C2×D12—31no—
97GL(2,𝔽3)—126no—
98C3×SD16—182no—
99C3×SD16—182no—
100C2×C3×C8—92no—
101C3×C4⋊C4—91no—
102C3×C22⋊C4—181no—
103#215—62no—
104#215—62no—
105(C2×D3)⋊C4—61no—
106C4⋊Dic3—31no—
107C2×C3⋊C8—32no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32GL(2,𝔽3)11yescharacteristic, complemented
111C2×C3⋊D3—161no—
112C2×C3×D3—121no—
113C2×C3×D3—121no—
114C2×C3×D3—121no—
115C2×C3×D3—483no—
116C3×A4—41no—
117C3×A4—43no—
118C3×A4—83no—
119C3×A4—81no—
120C32×C4—32no—
121C32×C4—61no—
122C3×Dic3—121no—
123C22×Q8C3×C3⋊D311yescharacteristic, complemented
124C2×SD16—92no—
125#121—91no—
126#103—92no—
127C22⋊C8—91no—
128C33—161noSylow
129C23×C3—21no—
130C23×C3—181no—
131C23×C3C3×S411yescharacteristic
132C23×C3C3⋊S411yescharacteristic
133C22×D3—61no—
134C2×A4—43no—
135C2×A4—83no—
136S4—126no—
137C3×Q8—22no—
138C3×Q8—21no—
139C3×Q8—21no—
140C3×Q8—22no—
141C3×Q8—32no—
142C3×Q8—32no—
143C3×Q8—62no—
144C3×Q8—61no—
145C3×Q8—61no—
146C3×Q8—62no—
147C3×Q8C3×S412yescomplemented
148C3×Q8C3⋊S412yescomplemented
149C3×D4—182no—
150C3×D4—361no—
151C3×D4—361no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×C3×C4—62no—
159C2×C3×C4—62no—
160C2×C3×C4—64no—
161C2×C3×C4—181no—
162#67—121no—
163#67—121no—
164C2×Dic3—61no—
165D12—62no—
166SL(2,𝔽3)—46no—
167SL(2,𝔽3)—83no—
168SL(2,𝔽3)—86no—
169SL(2,𝔽3)—86no—
170C3×C8—182no—
171C3⋊C8—62no—
172C2×C32—31no—
173C2×C32—31no—
174C2×C32—161no—
175C2×C32—163no—
176C2×C32—323no—
177C2×C32—321no—
178C2×C32#102011yescharacteristic
179C3⋊D3—162no—
180C3×D3—241no—
181C3×D3—486no—
182C2×Q8—32no—
183C2×Q8—61no—
184C2×D4—91no—
185C2×D4—181no—
186C22×C4—31no—
187SD16—182no—
188SD16—182no—
189C2×C8—92no—
190C4⋊C4—91no—
191C22⋊C4—181no—
192C22×C3—21no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—61no—
198C22×C3—61no—
199C22×C3—361no—
200C22×C3—361no—
201C22×C3—361no—
202C22×C3C3×GL(2,𝔽3)11yescharacteristic, complemented
203C22×C3C3⋊GL(2,𝔽3)11yescharacteristic, complemented
204C2×D3—121no—
205C2×D3—121no—
206C2×D3—121no—
207C2×D3—483no—
208A4—43no—
209A4—83no—
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—62no—
213C3×C4—62no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—361no—
217Dic3—121no—
218C32—161no—
219C32—163no—
220C32—323no—
221C32—321no—
222C32#593111yescharacteristic, complemented
223C23—181no—
224C23C3×C3⋊S411yescharacteristic
225Q8—21no—
226Q8—32no—
227Q8—61no—
228Q8C3×C3⋊S412yescomplemented
229D4—182no—
230D4—361no—
231D4—361no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—62no—
235C2×C4—181no—
236C8—182no—
237C2×C3—21no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—31no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—163no—
245C2×C3—323no—
246C2×C3—721no—
247C2×C3C3×#102011yescharacteristic
248C2×C3C3⋊#102011yescharacteristic
249D3—241no—
250D3—486no—
251C22—31no—
252C22—31no—
253C22—361no—
254C22—361no—
255C22—361no—
256C22C3×C3⋊GL(2,𝔽3)11yescharacteristic, complemented
257C4—32no—
258C4—61no—
259C4—361no—
260C3—21no—
261C3—163no—
262C3—323no—
263C3C3×#593111yescharacteristic, complemented
264C3C3⋊#593111yescharacteristic, direct product factor
265C2—31no—
266C2—31no—
267C2—721no—
268C2C3×C3⋊#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C3C3⋊#1020
commutator, cosocle, hypocenterC3×C22⋊SL(2,𝔽3)C2×C3
FrattiniC2C3×C3⋊#1020
FittingC22×C32×Q8D3
socleC23×C32S4
LCSG ▷ C3×C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊SL(2,𝔽3)C2semidirect
2. C3⋊#5931C3direct (N × H)
7. C22×C32×Q8D3semidirect
10. C3×C22⋊SL(2,𝔽3)D3semidirect
11. C3×C22⋊SL(2,𝔽3)C2×C3semidirect
40. C22×C3×Q8C3×D3semidirect
41. C22×C3×Q8C3⋊D3semidirect
43. C22⋊SL(2,𝔽3)C3×D3semidirect
63. C32×Q8S4semidirect
110. C22×C32GL(2,𝔽3)semidirect
123. C22×Q8C3×C3⋊D3semidirect
147. C3×Q8C3×S4semidirect
148. C3×Q8C3⋊S4semidirect
202. C22×C3C3×GL(2,𝔽3)semidirect
203. C22×C3C3⋊GL(2,𝔽3)semidirect
222. C32#5931semidirect
228. Q8C3×C3⋊S4semidirect
256. C22C3×C3⋊GL(2,𝔽3)semidirect
263. C3C3×#5931semidirect
264. C3C3⋊#5931direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C22⋊SL(2,𝔽3)2
C3⋊#59313
C3×#59313
C3×#52273
C2×C3×C3⋊S44
C3×C3⋊GL(2,𝔽3)4