Finite Group Databasebeta

← C4×(C3⋊GL(2,𝔽3))⋊C3 #13146825

Subgroups — C4×(C3⋊GL(2,𝔽3))⋊C3 #13146825

Order 1728.

347 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 49 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⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3(C3⋊#859)⋊C3C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)C311yesmaximal, characteristic, complemented
5C3×C4×#215—31nomaximal
6C2×C4×(C3⋊D3)⋊C3—41nomaximal
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊GL(2,𝔽3))⋊C3C412yesdirect product factor
9C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
10C32×C4×Q8D311yescharacteristic, complemented
11C2×C3×#215—31no—
12C3×C4×D12—31no—
13C3×C4×SL(2,𝔽3)—21no—
14C3×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
15C3⋊#859C2×C311yescharacteristic, complemented
16C3×#835—31no—
17C3×#809—31no—
18C3×#808—31no—
19C3×C4×C3⋊C8—31no—
20C22×(C3⋊D3)⋊C3—41no—
21C2×C4×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C4×(C3⋊D3)⋊C3—44no—
24Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
25C4×GL(2,𝔽3)—91no—
26C3×C4×SD16—91no—
27C4×#215—31no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C4×C3⋊D3—41no—
30C2×C3×D12—31no—
31C2×C3×C4×D3—61no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
34C3⋊GL(2,𝔽3)C3×C412yescomplemented
35C32×C4⋊C4—31no—
36C32×C42—31no—
37C3×#215—32no—
38C3×#215—32no—
39C3×(C2×D3)⋊C4—61no—
40C3×C4⋊Dic3—31no—
41C2×C3×C3⋊C8—31no—
42C2×C3×C3⋊C8—31no—
43C22×H(3,𝔽3)—41no—
44C2×(C3⋊D3)⋊C3—42no—
45C2×(C3⋊D3)⋊C3—42no—
46C2×(C3⋊D3)⋊C3—42no—
47C4×H(3,𝔽3)—42no—
48(C3⋊Dic3)⋊C3—42no—
49C2×GL(2,𝔽3)—91no—
50C2×C3×SD16—91no—
51C3×C4×Q8—31no—
52C3×C4×Q8C3×D311yescharacteristic
53C3×C4×D4—91no—
54C2×#215—31no—
55C4×D12—31no—
56C4×SL(2,𝔽3)—31no—
57C4×SL(2,𝔽3)—61no—
58#859—91no—
59C3×#106—91no—
60C3×#103—91no—
61C3×#102—91no—
62C3×C4×C8—91no—
63#835—31no—
64#809—31no—
65#808—31no—
66C4×C3⋊C8—31no—
67C22×C3⋊D3—41no—
68C22×C3×D3—61no—
69C32×Q8—31no—
70C32×Q8C4×D311yescharacteristic, complemented
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—41no—
74C2×C32×C4—81no—
75C2×C32×C4S411yescharacteristic
76C2×C3⋊Dic3—41no—
77C4×C3⋊D3—44no—
78C2×C3×Dic3—61no—
79C3×D12—32no—
80C3×D12—61no—
81C3×C4×D3—122no—
82C3×SL(2,𝔽3)—21no—
83C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
84C3×C3⋊C8—32no—
85C3×C3⋊C8—61no—
86C4×SD16—91noSylow
87C2×H(3,𝔽3)—41no—
88C2×H(3,𝔽3)—41no—
89C2×H(3,𝔽3)—41no—
90(C3⋊D3)⋊C3—44no—
91C2×C3×Q8—31no—
92C2×C3×Q8C2×C3×D311yescharacteristic
93C2×C3×D4—91no—
94C22×C3×C4—181no—
95C2×D12—31no—
96C2×C4×D3—61no—
97C2×C4×D3—361no—
98C2×SL(2,𝔽3)—31no—
99C2×SL(2,𝔽3)—61no—
100GL(2,𝔽3)—92no—
101C3×SD16—92no—
102C3×SD16—92no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—61no—
108C3×C4⋊C4—91no—
109C3×C22⋊C4—181no—
110C3×C42—31no—
111C3×C42—31no—
112C3×C42—61no—
113#215—32no—
114#215—32no—
115(C2×D3)⋊C4—61no—
116C4⋊Dic3—31no—
117C2×C3⋊C8—31no—
118C2×C3⋊C8—31no—
119C22×C32—41no—
120C22×C32—81no—
121C22×C32C2×S411yescharacteristic
122C2×C3⋊D3—42no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—42no—
125C2×C3×D3—62no—
126C2×C3×D3—121no—
127C2×C3×D3—121no—
128C32×C4—31no—
129C32×C4—31no—
130C32×C4—42no—
131C32×C4—61no—
132C32×C4—82no—
133C32×C4GL(2,𝔽3)12yescomplemented
134C3⋊Dic3—42no—
135C3×Dic3—121no—
136C2×SD16—91no—
137C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
138C4×D4—91no—
139#106—91no—
140#103—91no—
141#102—91no—
142C4×C8—91no—
143H(3,𝔽3)—41noSylow
144C23×C3—181no—
145C22×D3—61no—
146C22×D3—361no—
147C3×Q8—31no—
148C3×Q8—31no—
149C3×Q8—31no—
150C3×Q8—61no—
151C3×Q8C3×C4×D311yescharacteristic
152C3×D4—92no—
153C3×D4—181no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—121no—
162C2×C3×C4—181no—
163C2×C3×C4—241no—
164C2×C3×C4—362no—
165C2×C3×C4C3×S411yescharacteristic
166C2×Dic3—61no—
167C2×Dic3—361no—
168D12—32no—
169D12—61no—
170C4×D3—122no—
171C4×D3—364no—
172SL(2,𝔽3)—31no—
173SL(2,𝔽3)—61no—
174C3×C8—92no—
175C3×C8—181no—
176C3⋊C8—32no—
177C3⋊C8—61no—
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—81no—
182C2×C32—81no—
183C2×C32—81no—
184C2×C32C4×S411yescharacteristic
185C2×C32C2×GL(2,𝔽3)11yescharacteristic
186C2×C32#98511yescharacteristic
187C3⋊D3—44no—
188C3×D3—122no—
189C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
190C2×D4—91no—
191C22×C4—181no—
192SD16—92no—
193SD16—92no—
194C2×C8—91no—
195C2×C8—91no—
196C4⋊C4—31no—
197C4⋊C4—91no—
198C22⋊C4—181no—
199C42—31no—
200C22×C3—31no—
201C22×C3—121no—
202C22×C3—182no—
203C22×C3—241no—
204C22×C3—361no—
205C22×C3—361no—
206C22×C3C2×C3×S411yescharacteristic
207C2×D3—62no—
208C2×D3—121no—
209C2×D3—121no—
210C2×D3—362no—
211C2×D3—362no—
212C2×D3—362no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—32no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—121no—
223C3×C4—122no—
224C3×C4—242no—
225C3×C4—361no—
226C3×C4C3×GL(2,𝔽3)12yes—
227Dic3—121no—
228Dic3—362no—
229C32—41no—
230C32—81no—
231C32C4×GL(2,𝔽3)11yescharacteristic, complemented
232C23—181no—
233Q8—31no—
234Q8C4×(C3⋊D3)⋊C311yescharacteristic, complemented
235D4—92no—
236D4—181no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—181no—
240C2×C4—362no—
241C2×C4(C3⋊S4)⋊C311yescharacteristic
242C8—92no—
243C8—181no—
244C2×C3—31no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—121no—
248C2×C3—121no—
249C2×C3—121no—
250C2×C3—241no—
251C2×C3—241no—
252C2×C3—241no—
253C2×C3—362no—
254C2×C3C3×C4×S411yescharacteristic
255C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
256C2×C3C3×#98511yescharacteristic
257D3—122no—
258D3—364no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22C2×(C3⋊S4)⋊C311yescharacteristic
263C4—31no—
264C4—31no—
265C4—61no—
266C4—361no—
267C4(C3⋊GL(2,𝔽3))⋊C312yesdirect product factor
268C3—31no—
269C3—121no—
270C3—241no—
271C3C3×C4×GL(2,𝔽3)11yescharacteristic
272C2—362no—
273C2C4×(C3⋊S4)⋊C311yescharacteristic
274C2C2×(C3⋊GL(2,𝔽3))⋊C311yescharacteristic
275C2#64266⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC32×C4×Q8D3
prime radicalC2×(C3⋊GL(2,𝔽3))⋊C3C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×Q8⋊H(3,𝔽3)C2semidirect
3. (C3⋊#859)⋊C3C2semidirect
4. C4×C3⋊GL(2,𝔽3)C3semidirect
8. (C3⋊GL(2,𝔽3))⋊C3C4direct (N × H)
10. C32×C4×Q8D3semidirect
14. C3×C4×SL(2,𝔽3)C2×C3semidirect
15. C3⋊#859C2×C3semidirect
24. Q8⋊H(3,𝔽3)C2×C4semidirect
34. C3⋊GL(2,𝔽3)C3×C4semidirect
70. C32×Q8C4×D3semidirect
83. C3×SL(2,𝔽3)C2×C3×C4semidirect
133. C32×C4GL(2,𝔽3)semidirect
137. C4×Q8(C3⋊D3)⋊C3semidirect
231. C32C4×GL(2,𝔽3)semidirect
234. Q8C4×(C3⋊D3)⋊C3semidirect
267. C4(C3⋊GL(2,𝔽3))⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×(C3⋊GL(2,𝔽3))⋊C32
C4×Q8⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
C4×C3⋊GL(2,𝔽3)3
C3×C4×#2153
C2×C4×(C3⋊D3)⋊C34