Finite Group Databasebeta

← #10589384⋊C3 #13135514

Subgroups — #10589384⋊C3 #13135514

Order 1728.

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

normal subgroups only

This group has 229 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
1#64113⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×M32C211yesmaximal, characteristic, complemented
3#63644⋊C3C211yesmaximal, characteristic, complemented
4#10589384C311yesmaximal, characteristic, complemented
5C3×#4518—31nomaximal
6#3828⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4⋊((C3⋊Dic3)⋊C3)—21no—
9#3745⋊C3C411yescharacteristic, complemented
10#3744⋊C3—21no—
11#3746⋊C3C411yescharacteristic, complemented
12C16×H(3,𝔽3)—21no—
13#64113C2×C311yescharacteristic
14C3×#904—31no—
15C32×M32—21no—
16C32×M32D311yescharacteristic, complemented
17C32×M32C2×C311yescharacteristic, complemented
18#63644C2×C311yescharacteristic, complemented
19C3×#817—31no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21#638⋊C3—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C4⋊((C3⋊D3)⋊C3)—21no—
24C4×(C3⋊D3)⋊C3—41no—
25#634⋊C3—21no—
26C8×H(3,𝔽3)C2×C411yescharacteristic
27C8×H(3,𝔽3)D411yescharacteristic
28C3×#360—91no—
29#4518—31no—
30#4518—91no—
31#3828—21no—
32C3×#235—61no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C22×C311yescharacteristic
36C4⋊(C3⋊Dic3)—21no—
37#3746C3×C411yescharacteristic, complemented
38#3745C3×C411yescharacteristic, complemented
39#3744—21no—
40C3×C4⋊Dic3—61no—
41C3×Dic12—31no—
42C3×D24—31no—
43C3×#204—61no—
44C32×C16—22no—
45C32×C16—21no—
46C32×C16—21no—
47C22×H(3,𝔽3)D811yescharacteristic
48C2×(C3⋊D3)⋊C3—41no—
49C4×H(3,𝔽3)C22⋊C411yescharacteristic
50C4×H(3,𝔽3)SD1611yescharacteristic
51(C3⋊Dic3)⋊C3—41no—
52(C3⋊Dic3)⋊C3—81no—
53C3×#135—91no—
54#904—31no—
55#904—91no—
56C3×M32—31no—
57C3×M32—31no—
58C3×M32—61no—
59C3×M32C3×D311yescharacteristic
60C3×#106—91no—
61#817—31no—
62#817—91no—
63C2×C32×C4—21no—
64C2×C32×C4D1211yescharacteristic
65C2×C32×C4C3×D411yescharacteristic
66#638—41no—
67C2×C3⋊Dic3—41no—
68C4⋊(C3⋊D3)—21no—
69C4×C3⋊D3—41no—
70#634—21no—
71C3×#67—121no—
72C2×C3×Dic3—121no—
73C3×D12—61no—
74C3×C4×D3—121no—
75C3×Dic6—61no—
76C32×C8—21no—
77C32×C8—21no—
78C32×C8C4×D311yescharacteristic
79C32×C8#6711yescharacteristic
80C32×C8C2×C3×C411yescharacteristic
81C32×C8C3×D411yescharacteristic
82#360—91noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)#10211yescharacteristic
85(C3⋊D3)⋊C3—81no—
86C3×#41—181no—
87#235—61no—
88#235—181no—
89C3×Q16—91no—
90C3×SD16—181no—
91C3×D8—91no—
92C2×C3×C8—31no—
93C2×C3×C8—31no—
94C2×C3×C8—61no—
95C2×C3×C8C2×C3×D311yescharacteristic
96C3×C4⋊C4—181no—
97C4⋊Dic3—61no—
98C4⋊Dic3—181no—
99Dic12—31no—
100Dic12—91no—
101D24—31no—
102D24—91no—
103#204—61no—
104#204—181no—
105C3×C16—21no—
106C3×C16—61no—
107C3×C16—61no—
108C3×C16—62no—
109C22×C32—21no—
110C22×C32D2411yescharacteristic
111C22×C32C3×D811yescharacteristic
112C2×C3⋊D3—41no—
113C2×C3×D3—121no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4#20411yescharacteristic
117C32×C4(C2×D3)⋊C411yescharacteristic
118C32×C4C3×C22⋊C411yescharacteristic
119C32×C4C3×SD1611yescharacteristic
120C3⋊Dic3—41no—
121C3⋊Dic3—81no—
122C3×Dic3—121no—
123C3×Dic3—241no—
124#135—91no—
125M32(C3⋊D3)⋊C311yescharacteristic, complemented
126#106—91no—
127H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
128C3×Q8—181no—
129C3×D4—181no—
130C3×D4—361no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—61no—
134C2×C3×C4—361no—
135C2×C3×C4—361no—
136C2×C3×C4C3×D1211yescharacteristic
137#67—121no—
138#67—361no—
139C2×Dic3—121no—
140C2×Dic3—361no—
141D12—61no—
142D12—181no—
143C4×D3—121no—
144C4×D3—361no—
145Dic6—61no—
146Dic6—181no—
147C3×C8—31no—
148C3×C8—31no—
149C3×C8—31no—
150C3×C8—31no—
151C3×C8—61no—
152C3×C8—61no—
153C3×C8C3×C4×D311yescharacteristic
154C3×C8C3×#6711yescharacteristic
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—41no—
159C2×C32#82111yescharacteristic
160C2×C32C3×#10211yescharacteristic
161C3⋊D3—81no—
162C3×D3—241no—
163#41—181no—
164Q16—91no—
165SD16—181no—
166D8—91no—
167C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
168C4⋊C4—181no—
169C16—21no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—361no—
174C22×C3C3×D2411yescharacteristic
175C2×D3—121no—
176C2×D3—361no—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—61no—
182C3×C4—61no—
183C3×C4—361no—
184C3×C4—721no—
185C3×C4C3×#20411yescharacteristic
186C3×C4C3×(C2×D3)⋊C411yescharacteristic
187Dic3—121no—
188Dic3—241no—
189Dic3—361no—
190Dic3—721no—
191C32—21no—
192C32#451811yescharacteristic, complemented
193C32C3×#36011yescharacteristic, complemented
194Q8—181no—
195D4—181no—
196D4—361no—
197C2×C4—361no—
198C2×C4—361no—
199C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
200C8C4×(C3⋊D3)⋊C311yescharacteristic
201C8#638⋊C311yescharacteristic
202C2×C3—21no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—121no—
209C2×C3—721no—
210C2×C3C3×#82111yescharacteristic
211D3—241no—
212D3—721no—
213C22—361no—
214C22#3745⋊C311yescharacteristic
215C4—361no—
216C4—721no—
217C4#3744⋊C311yescharacteristic
218C4#3752⋊C311yescharacteristic
219C3—31no—
220C3—31no—
221C3—61no—
222C3C3×#451811yescharacteristic
223C2—21no—
224C2—721no—
225C2#63648⋊C311yescharacteristic
subgroupG/H
centerC2#63648⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingH(3,𝔽3)×M32C2
socleC2×C3C3×#821
prime radical#64113⋊C3C2
cosocleC2×C32×C8C22×C3
hypercenterM32(C3⋊D3)⋊C3
hypocenterC32C3×#360
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×M32C2semidirect
3. #63644⋊C3C2semidirect
4. #10589384C3semidirect
9. #3745⋊C3C4semidirect
11. #3746⋊C3C4semidirect
16. C32×M32D3semidirect
17. C32×M32C2×C3semidirect
18. #63644C2×C3semidirect
37. #3746C3×C4semidirect
38. #3745C3×C4semidirect
125. M32(C3⋊D3)⋊C3semidirect
127. H(3,𝔽3)#360semidirect
192. C32#4518semidirect
193. C32C3×#360semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
H(3,𝔽3)×M322
#63644⋊C32
#105893843
C3×#45183