Finite Group Databasebeta

← #13138846

Subgroups — #13138846

Order 1728.

498 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 95 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×#3745⋊C3C211yesmaximal, characteristic
2#11721132C212yesmaximal, complemented
3#11720550C211yesmaximal, characteristic, complemented
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590781—31nomaximal
7#10590768—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3745⋊C3C414yescomplemented
15#88450—22no—
16C2×#3745D311yescharacteristic
17C2×C3×D24—31no—
18#63898—32no—
19#63894—32no—
20#63642D311yescharacteristic, complemented
21C3×#817—31no—
22C3×#817—31no—
23C3×#817—61no—
24C3×#815—31no—
25#63564—32no—
26#63562—32no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3D412yes—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
33((C3⋊D3)⋊C3)⋊C4—44no—
34C8×H(3,𝔽3)C2×C412yescomplemented
35(C3⋊(C3⋊C8))⋊C3—21no—
36#5152—91no—
37#4869—91no—
38#4711—91no—
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C3×D12—32no—
41C2×#624—62no—
42C2×D3×Dic3—62no—
43C2×C32×C8—21no—
44C2×C32×C8C2×D311yescharacteristic
45C2×C32×C8C2×D311yescharacteristic
46C4×C3⋊Dic3C2×D311yescharacteristic
47C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
48#3745Dic314yescomplemented
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—62no—
52C3×C4×Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C3×D24—34no—
55#3722—62no—
56(C2×D3)⋊Dic3—62no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×(C3⋊D3)⋊C3—24no—
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C4×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)#4111yescharacteristic
63#1084—42no—
64(C3⋊Dic3)⋊C3—22no—
65C2×C3×D8—91no—
66D4×Dic3—92no—
67C2×D24—31no—
68C2×D24—91no—
69#893—92no—
70C4×D12—92no—
71C3×#106—91no—
72C3×#97—91no—
73#832—92no—
74#821—92no—
75#817—31no—
76#817—31no—
77#817—61no—
78#815—31no—
79#815—91no—
80#809—92no—
81C22×C3⋊D3—22no—
82C22×C3×D3—62no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D311yescharacteristic
85C2×C32×C4C22×D311yescharacteristic
86C2×C3⋊Dic3D1212yes—
87C4⋊(C3⋊D3)—22no—
88C4⋊(C3⋊D3)C2×Dic314yescomplemented
89C2×C3×Dic3—32no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C3×D12—34no—
94C3×D12—62no—
95#624—124no—
96D3×Dic3—124no—
97C32×C8—22no—
98C32×C8C4×D312yescomplemented
99C32×C8C2×Dic312yescomplemented
100C3⋊(C3⋊C8)—21no—
101C3×C3⋊C8—61no—
102#442—271noSylow
103C2×H(3,𝔽3)C4×D411yescharacteristic
104C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
105(C3⋊D3)⋊C3—44no—
106C2×C3×D4—92no—
107C22×Dic3—182no—
108C2×D12—32no—
109C2×D12—92no—
110C2×C4×D3—182no—
111C2×C4×D3—182no—
112C3×D8—94no—
113C2×C3×C8—31no—
114C2×C3×C8—31no—
115C2×C3×C8—61no—
116C2×C3×C8—91no—
117C2×C3×C8D3211yescharacteristic
118C3×C4⋊C4—92no—
119C3×C42—91no—
120#217—182no—
121(C2×D3)⋊C4—182no—
122(C2×D3)⋊C4—182no—
123C4⋊Dic3—32no—
124C4⋊Dic3—32no—
125C4⋊Dic3—62no—
126C4×Dic3—31no—
127C4×Dic3—91no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—91no—
130D24—34no—
131D24—94no—
132C22×C32—21no—
133C22×C32C2×D1211yescharacteristic
134C22×C32D3×D411yescharacteristic
135C2×C3⋊D3—24no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3—42no—
138C2×C3×D3—64no—
139C2×C3×D3—122no—
140C2×C3×D3—122no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4C2×C4×D311yescharacteristic
144C32×C4#23511yescharacteristic
145C32×C4#23711yescharacteristic
146C32×C4C22×Dic311yescharacteristic
147C3⋊Dic3—22no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C2×D8—91no—
153C4×D4—272no—
154#106—91no—
155#102—272no—
156#97—91no—
157H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
158C23×C3—182no—
159C22×D3—62no—
160C22×D3—182no—
161C3×D4—94no—
162C3×D4—182no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—61no—
166C2×C3×C4—92no—
167C2×C3×C4—182no—
168C2×C3×C4C2×D3211yescharacteristic
169C2×Dic3—32no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—92no—
173C2×Dic3—122no—
174C2×Dic3—364no—
175D12—34no—
176D12—62no—
177D12—94no—
178D12—182no—
179C4×D3—364no—
180C4×D3—364no—
181C3×C8—32no—
182C3×C8—32no—
183C3×C8—62no—
184C3×C8—181no—
185C3×C8D3×Dic312yes—
186C3⋊C8—61no—
187C3⋊C8—181no—
188C2×C32—22no—
189C2×C32—21no—
190C2×C32C4×D1211yescharacteristic
191C2×C32#90812yes—
192C2×C32#91112yes—
193C2×C32D4×Dic311yescharacteristic
194C3⋊D3—44no—
195C3×D3—124no—
196C2×D4—92no—
197C22×C4—542no—
198D8—94no—
199C2×C8—91no—
200C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
201C4⋊C4—92no—
202C22⋊C4—542no—
203C42—91no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—184no—
208C22×C3—362no—
209C22×C3—362no—
210C22×C3D3×D1211yescharacteristic
211C2×D3—64no—
212C2×D3—122no—
213C2×D3—122no—
214C2×D3—184no—
215C2×D3—362no—
216C2×D3—362no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—31no—
220C3×C4—31no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—182no—
224C3×C4—362no—
225C3×C4#379511yescharacteristic
226C3×C4C2×D3×Dic311yescharacteristic
227Dic3—62no—
228Dic3—122no—
229Dic3—122no—
230Dic3—182no—
231Dic3—242no—
232C32—21no—
233C32#471111yescharacteristic, complemented
234C32#515211yescharacteristic, complemented
235C23—182no—
236D4—94no—
237D4—182no—
238C2×C4—92no—
239C2×C4—182no—
240C2×C4—1084no—
241C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C8—181no—
243C8((C3⋊D3)⋊C3)⋊C412yescomplemented
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—62no—
249C2×C3—61no—
250C2×C3—364no—
251C2×C3#6379512yes—
252C2×C3Dic3×D1211yescharacteristic
253D3—124no—
254D3—364no—
255C22—184no—
256C22—362no—
257C22—362no—
258C22#8865411yescharacteristic
259C4—182no—
260C4—362no—
261C4#8864811yescharacteristic
262C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
263C3—31no—
264C3—31no—
265C3—61no—
266C3#1059077111yescharacteristic
267C2—364no—
268C2#1172084912yes—
269C2#1172113211yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#3745⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721132C2semidirect
3. #11720550C2semidirect
4. #63642⋊C3C2semidirect
5. #11720308C2semidirect
14. #3745⋊C3C4semidirect
20. #63642D3semidirect
32. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
34. C8×H(3,𝔽3)C2×C4semidirect
48. #3745Dic3semidirect
88. C4⋊(C3⋊D3)C2×Dic3semidirect
98. C32×C8C4×D3semidirect
99. C32×C8C2×Dic3semidirect
157. H(3,𝔽3)#442semidirect
233. C32#4711semidirect
234. C32#5152semidirect
243. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211322
#117205502
#63642⋊C32
#117203082
#105907813
#105907683