Finite Group Databasebeta

← #13135068

Subgroups — #13135068

Order 1728.

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

normal subgroups only

This group has 239 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#11720550C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589125—31nomaximal
5#10589123—31nomaximal
6#3828⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4⋊#1084—21no—
9#3745⋊C3C411yescharacteristic, complemented
10#3744⋊C3—21no—
11#3746⋊C3C411yescharacteristic, complemented
12(C3⋊(C3⋊C16))⋊C3—21no—
13#64113D311yescharacteristic
14C3×#904—31no—
15#63639D311yescharacteristic, complemented
16C3×#817—31no—
17C3×#817—31no—
18C3×#817—61no—
19C3×#812—31no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C2×#1084—41no—
22#638⋊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
36#3746Dic311yescharacteristic, complemented
37#3745Dic311yescharacteristic, complemented
38#3744—21no—
39C3×C4⋊Dic3—62no—
40C3×C4⋊Dic3—61no—
41C3×C4⋊Dic3—61no—
42C3×Dic12—31no—
43C3×D24—31no—
44C3×#204—61no—
45C3⋊(C3⋊C16)—21no—
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#1084—81no—
52(C3⋊Dic3)⋊C3—41no—
53C3×#135—91no—
54#904—31no—
55#904—91no—
56C3×M32—91no—
57C3×#106—91no—
58#817—31no—
59#817—31no—
60#817—61no—
61#812—31no—
62#812—91no—
63C2×C32×C4—21no—
64C2×C32×C4D1211yescharacteristic
65C2×C32×C4#6711yescharacteristic
66#638—41no—
67C4⋊(C3⋊D3)—21no—
68C4×C3⋊D3—41no—
69#634—21no—
70C3×#67—121no—
71C2×C3×Dic3—122no—
72C2×C3×Dic3—121no—
73C2×C3×Dic3—121no—
74C3×D12—61no—
75C3×C4×D3—121no—
76C3×Dic6—61no—
77C32×C8—21no—
78C32×C8—21no—
79C32×C8C4×D311yescharacteristic
80C32×C8C2×Dic311yescharacteristic
81C32×C8#6711yescharacteristic
82C32×C8#6711yescharacteristic
83#360—271noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)#10211yescharacteristic
86(C3⋊D3)⋊C3—81no—
87C3×#41—181no—
88#235—61no—
89#235—181no—
90C3×Q16—91no—
91C3×SD16—181no—
92C3×D8—91no—
93C2×C3×C8—31no—
94C2×C3×C8—31no—
95C2×C3×C8—61no—
96C2×C3×C8D3211yescharacteristic
97C3×C4⋊C4—181no—
98C4⋊Dic3—61no—
99C4⋊Dic3—61no—
100C4⋊Dic3—62no—
101Dic12—31no—
102Dic12—91no—
103D24—31no—
104D24—91no—
105#204—61no—
106#204—181no—
107C3×C16—181no—
108C3⋊C16—61no—
109C3⋊C16—181no—
110C22×C32—21no—
111C22×C32D2411yescharacteristic
112C22×C32#21311yescharacteristic
113C2×C3⋊D3—41no—
114C2×C3×D3—121no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4#20411yescharacteristic
118C32×C4(C2×D3)⋊C411yescharacteristic
119C32×C4#21411yescharacteristic
120C32×C4#21711yescharacteristic
121C3⋊Dic3—41no—
122C3×Dic3—121no—
123C3×Dic3—242no—
124C3×Dic3—241no—
125C3×Dic3—241no—
126#135—91no—
127M32—91no—
128#106—91no—
129H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
130C3×Q8—181no—
131C3×D4—181no—
132C3×D4—361no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—61no—
136C2×C3×C4—361no—
137C2×C3×C4—361no—
138C2×C3×C4#62611yescharacteristic
139#67—121no—
140#67—361no—
141C2×Dic3—121no—
142C2×Dic3—121no—
143C2×Dic3—122no—
144D12—61no—
145D12—181no—
146C4×D3—121no—
147C4×D3—361no—
148Dic6—61no—
149Dic6—181no—
150C3×C8—31no—
151C3×C8—31no—
152C3×C8—31no—
153C3×C8—31no—
154C3×C8—61no—
155C3×C8—61no—
156C3×C8D3×Dic311yescharacteristic
157C3×C8#62511yescharacteristic
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—21no—
162C2×C32#82111yescharacteristic
163C2×C32#83211yescharacteristic
164C3⋊D3—81no—
165C3×D3—241no—
166#41—181no—
167Q16—91no—
168SD16—181no—
169D8—91no—
170C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
171C4⋊C4—181no—
172C16—181no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—61no—
176C22×C3—361no—
177C22×C3#371411yescharacteristic
178C2×D3—121no—
179C2×D3—361no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—361no—
187C3×C4—721no—
188C3×C4#371611yescharacteristic
189C3×C4(C2×D3)⋊Dic311yescharacteristic
190Dic3—121no—
191Dic3—241no—
192Dic3—241no—
193Dic3—242no—
194Dic3—361no—
195C32—21no—
196C32#451811yescharacteristic, complemented
197C32#456611yescharacteristic, complemented
198Q8—181no—
199D4—181no—
200D4—361no—
201C2×C4—361no—
202C2×C4—361no—
203C2×C4#620011yescharacteristic
204C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
205C8#619811yescharacteristic
206C2×C3—21no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—62no—
212C2×C3—61no—
213C2×C3—721no—
214C2×C3#6356311yescharacteristic
215D3—241no—
216D3—721no—
217C22—361no—
218C22#8843611yescharacteristic
219C4—361no—
220C4—721no—
221C4#8843811yescharacteristic
222C4#8845011yescharacteristic
223C3—31no—
224C3—31no—
225C3—61no—
226C3#1058912411yescharacteristic
227C2—21no—
228C2—721no—
229C2#1172030811yescharacteristic
subgroupG/H
centerC2#11720308
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63563
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. #11720550C2semidirect
3. #63639⋊C3C2semidirect
9. #3745⋊C3C4semidirect
11. #3746⋊C3C4semidirect
15. #63639D3semidirect
36. #3746Dic3semidirect
37. #3745Dic3semidirect
129. H(3,𝔽3)#360semidirect
196. C32#4518semidirect
197. C32#4566semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#117205502
#63639⋊C32
#105891253
#105891233