Finite Group Databasebeta

← #13135096

Subgroups — #13135096

Order 1728.

229 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 55 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
2#11720545C211yesmaximal, characteristic, complemented
3#63644⋊C3C211yesmaximal, characteristic, complemented
4#10589125—31nomaximal
5#10589124—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
12#88386—21no—
13#64113D311yescharacteristic
14C3×#904—31no—
15#63644D311yescharacteristic, complemented
16C3×#817—31no—
17C3×#812—31no—
18C3×#812—31no—
19C3×#812—61no—
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
28#4566—91no—
29#4518—91no—
30#4488—91no—
31#3828—21no—
32C3×#235—61no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C2×D311yescharacteristic
36C4⋊(C3⋊Dic3)—21no—
37#3746Dic311yescharacteristic, complemented
38#3745Dic311yescharacteristic, complemented
39#3744—21no—
40C3×C4⋊Dic3—61no—
41C3×Dic12—31no—
42C3×D24—31no—
43C3×#204—61no—
44C3×C3⋊C16—62no—
45C3×C3⋊C16—61no—
46C3×C3⋊C16—61no—
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—91no—
57C3×#106—91no—
58#817—31no—
59#817—91no—
60#812—31no—
61#812—31no—
62#812—61no—
63C2×C32×C4—21no—
64C2×C32×C4#6711yescharacteristic
65C2×C32×C4#6711yescharacteristic
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×C8D1211yescharacteristic
80C32×C8C2×Dic311yescharacteristic
81C32×C8#6711yescharacteristic
82#360—271noSylow
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×C8D3211yescharacteristic
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—181no—
106C3⋊C16—61no—
107C3⋊C16—61no—
108C3⋊C16—62no—
109C22×C32—21no—
110C22×C32#21311yescharacteristic
111C22×C32#21311yescharacteristic
112C2×C3⋊D3—41no—
113C2×C3×D3—121no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4(C2×D3)⋊C411yescharacteristic
117C32×C4#21411yescharacteristic
118C32×C4#21511yescharacteristic
119C32×C4#21711yescharacteristic
120C3⋊Dic3—41no—
121C3⋊Dic3—81no—
122C3×Dic3—121no—
123C3×Dic3—241no—
124#135—91no—
125M32—91no—
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×C4#62511yescharacteristic
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×C8D3×Dic311yescharacteristic
154C3×C8#62611yescharacteristic
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—41no—
159C2×C32#80911yescharacteristic
160C2×C32#83211yescharacteristic
161C3⋊D3—81no—
162C3×D3—241no—
163#41—181no—
164Q16—91no—
165SD16—181no—
166D8—91no—
167C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
168C4⋊C4—181no—
169C16—181no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—361no—
174C22×C3#371311yescharacteristic
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×C4#371511yescharacteristic
186C3×C4(C2×D3)⋊Dic311yescharacteristic
187Dic3—121no—
188Dic3—241no—
189Dic3—361no—
190Dic3—721no—
191C32—21no—
192C32#448811yescharacteristic, complemented
193C32#456611yescharacteristic, complemented
194Q8—181no—
195D4—181no—
196D4—361no—
197C2×C4—361no—
198C2×C4—361no—
199C2×C4#619811yescharacteristic
200C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
201C8#620011yescharacteristic
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×C3#6356211yescharacteristic
211D3—241no—
212D3—721no—
213C22—361no—
214C22#8843211yescharacteristic
215C4—361no—
216C4—721no—
217C4#8843111yescharacteristic
218C4#8845011yescharacteristic
219C3—31no—
220C3—31no—
221C3—61no—
222C3#1058912311yescharacteristic
223C2—21no—
224C2—721no—
225C2#1172031811yescharacteristic
subgroupG/H
centerC2#11720318
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63562
prime radical#64113⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#360
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720545C2semidirect
3. #63644⋊C3C2semidirect
9. #3745⋊C3C4semidirect
11. #3746⋊C3C4semidirect
15. #63644D3semidirect
37. #3746Dic3semidirect
38. #3745Dic3semidirect
127. H(3,𝔽3)#360semidirect
192. C32#4488semidirect
193. C32#4566semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#117205452
#63644⋊C32
#105891253
#105891243