Finite Group Databasebeta

← #10595339⋊C3 #13166141

Subgroups — #10595339⋊C3 #13166141

Order 1728.

463 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 60 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
3#10595339C311yesmaximal, characteristic, complemented
4#5572⋊C3—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3⋊D3×SL(2,𝔽3)C412yescomplemented
7C3×#3722—41nomaximal
8C2×Q8×C3⋊D3C2×C311yescharacteristic
9C2×D3×SL(2,𝔽3)—32no—
10C2×D3×SL(2,𝔽3)—61no—
11C3×Q8×Dic3C2×C312yescomplemented
12C3×C4×SL(2,𝔽3)—32no—
13#63874—31no—
14#63849—31no—
15#63844—32no—
16Dic3×SL(2,𝔽3)—22no—
17Dic3×SL(2,𝔽3)D312yescomplemented
18C22×C3×C3⋊D3—41no—
19C2×C32×Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic
21#5572—32no—
22#519⋊C3—91no—
23C2×C32×Q8C22×C311yescharacteristic
24Q8×C3⋊D3—32no—
25Q8×C3⋊D3C3×C412yescomplemented
26C2×C4×C3⋊D3—31no—
27C2×#634—31no—
28C2×C3×SL(2,𝔽3)—22no—
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)—41no—
31C2×C3×SL(2,𝔽3)C2×D312yes—
32D3×SL(2,𝔽3)—34no—
33D3×SL(2,𝔽3)—62no—
34C3×(C2×D3)⋊C4—122no—
35C3×C4⋊Dic3—32no—
36C3×C4×Dic3—32no—
37#3724—32no—
38#3722—31no—
39#3722A411yescharacteristic
40Dic32—32no—
41C22×C33—41no—
42C2×C3×C3⋊D3—42no—
43C2×C3×C3⋊D3—82no—
44C32×Dic3—82no—
45C2×D3×Q8—32no—
46C2×D3×Q8—61no—
47C22×SL(2,𝔽3)—91no—
48C3×C4×Q8—32no—
49Q8×Dic3C3×D312yescomplemented
50#892—92no—
51#887—92no—
52#872—92no—
53C4×Dic6—92no—
54C4×SL(2,𝔽3)—32no—
55C4×SL(2,𝔽3)—62no—
56C22×C3⋊D3C2×A411yescharacteristic
57C22×C3×D3—122no—
58C22×C3×D3—241no—
59C32×Q8—31no—
60C32×Q8C2×C3×C411yescharacteristic
61C2×C32×C4—31no—
62C2×C32×C4—122no—
63C2×C3⋊Dic3—31no—
64C4×C3⋊D3—32no—
65C4×C3⋊D3—32no—
66#634—32no—
67#634—32no—
68C2×C3×Dic3—32no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3—82no—
71C2×C3×Dic3C2×A412yes—
72C3×SL(2,𝔽3)—22no—
73C3×SL(2,𝔽3)—21no—
74C3×SL(2,𝔽3)—41no—
75C3×SL(2,𝔽3)C4×D312yes—
76#519—91noSylow
77C2×C33—41no—
78C2×C33—42no—
79C3×C3⋊D3—82no—
80C2×C3×Q8—21no—
81C2×C3×Q8C2×C3×D312yes—
82D3×Q8—34no—
83D3×Q8—62no—
84D3×Q8—94no—
85D3×Q8—182no—
86C2×C4×D3—92no—
87C2×C4×D3—181no—
88C2×Dic6—92no—
89C2×Dic6—181no—
90C2×SL(2,𝔽3)—22no—
91C2×SL(2,𝔽3)—41no—
92C2×SL(2,𝔽3)—92no—
93C2×SL(2,𝔽3)D3211yescharacteristic
94C3×C4⋊C4—92no—
95C3×C22⋊C4—361no—
96C3×C42—92no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4—92no—
99C4⋊Dic3—32no—
100Dic3⋊C4—92no—
101Dic3⋊C4—92no—
102C4×Dic3—32no—
103C4×Dic3—94no—
104C22×C32—42no—
105C22×C32—81no—
106C22×C32—82no—
107C22×C32—161no—
108C22×C32C22×A411yescharacteristic
109C2×C3⋊D3—22no—
110C2×C3⋊D3C4×A412yes—
111C2×C3×D3—124no—
112C2×C3×D3—242no—
113C2×C3×D3—242no—
114C2×C3×D3—242no—
115C2×C3×D3—244no—
116C32×C4—31no—
117C32×C4—31no—
118C32×C4—242no—
119C3⋊Dic3—32no—
120C3×Dic3—22no—
121C3×Dic3—62no—
122C3×Dic3—82no—
123C3×Dic3—84no—
124C22×Q8—91no—
125C4×Q8—32no—
126C4×Q8—272no—
127#117—271no—
128#117—271no—
129C33—41noSylow
130C23×C3—361no—
131C22×D3—32no—
132C22×D3—61no—
133C3×Q8—21no—
134C3×Q8—32no—
135C3×Q8—61no—
136C3×Q8C3×C4×D312yes—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—61no—
140C2×C3×C4—92no—
141C2×C3×C4—122no—
142C2×C3×C4—242no—
143C2×Dic3—32no—
144C2×Dic3—92no—
145C2×Dic3—181no—
146C2×Dic3D3×A412yes—
147C4×D3—94no—
148C4×D3—94no—
149C4×D3—182no—
150C4×D3—182no—
151Dic6—94no—
152Dic6—94no—
153Dic6—182no—
154Dic6—182no—
155SL(2,𝔽3)—22no—
156SL(2,𝔽3)—41no—
157SL(2,𝔽3)#62411yescharacteristic
158C2×C32—42no—
159C2×C32—42no—
160C2×C32—42no—
161C2×C32—81no—
162C2×C32—82no—
163C2×C32—82no—
164C2×C32—82no—
165C2×C32—82no—
166C2×C32—161no—
167C2×C32—162no—
168C2×C32C2×C4×A411yescharacteristic
169C2×C322−1+4⋊C312yes—
170C3⋊D3—22no—
171C3×D3—244no—
172C3×D3—244no—
173C2×Q8—92no—
174C2×Q8—271no—
175C2×Q8—272no—
176C2×Q8C3×D3211yescharacteristic
177C22×C4—271no—
178C4⋊C4—92no—
179C4⋊C4—272no—
180C22⋊C4—91no—
181C22⋊C4—271no—
182C42—92no—
183C42—272no—
184C22×C3—21no—
185C22×C3—41no—
186C22×C3—82no—
187C22×C3—161no—
188C22×C3—362no—
189C22×C3—722no—
190C22×C3C2×D3×A412yes—
191C2×D3—34no—
192C2×D3—62no—
193C2×D3—64no—
194C2×D3—62no—
195C2×D3—62no—
196C3×C4—32no—
197C3×C4—32no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—62no—
201C3×C4—182no—
202C3×C4—242no—
203C3×C4—244no—
204Dic3—22no—
205Dic3—62no—
206Dic3—94no—
207Dic3—182no—
208C32—42no—
209C32—81no—
210C32—82no—
211C32—161no—
212C32#519⋊C311yescharacteristic, complemented
213C23—91no—
214Q8—31no—
215Q8—272no—
216Q8—272no—
217Q8C3×#62411yescharacteristic
218C2×C4—31no—
219C2×C4—32no—
220C2×C4—92no—
221C2×C4—272no—
222C2×C4—272no—
223C2×C4—271no—
224C2×C3—22no—
225C2×C3—21no—
226C2×C3—42no—
227C2×C3—41no—
228C2×C3—82no—
229C2×C3—82no—
230C2×C3—82no—
231C2×C3—162no—
232C2×C3—161no—
233C2×C3—722no—
234C2×C3C4×D3×A412yes—
235C2×C3#1007⋊C312yes—
236C2×C3#1010⋊C312yes—
237D3—64no—
238D3—64no—
239C22—92no—
240C22—182no—
241C22D32×A411yescharacteristic
242C4—31no—
243C4—31no—
244C4—62no—
245C4—182no—
246C4—272no—
247C3—21no—
248C3—41no—
249C3—82no—
250C3—161no—
251C3#5572⋊C312yescomplemented
252C2—182no—
253C2#64315⋊C312yes—
254C2A4×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. #10595339C3semidirect
6. C3⋊D3×SL(2,𝔽3)C4semidirect
11. C3×Q8×Dic3C2×C3semidirect
17. Dic3×SL(2,𝔽3)D3semidirect
25. Q8×C3⋊D3C3×C4semidirect
49. Q8×Dic3C3×D3semidirect
212. C32#519⋊C3semidirect
251. C3#5572⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
#105953393
#5572⋊C33
C3×#37224