Finite Group Databasebeta

← #13137378

Subgroups — #13137378

Order 1728.

537 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88430C212yesmaximal
3#11720520C211yesmaximal, characteristic, complemented
4#63630⋊C3C211yesmaximal, characteristic, complemented
5#11720310C212yesmaximal, complemented
6#10590334—31nomaximal
7#10590247—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C412yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C4×(C3⋊Dic3)⋊C3C412yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88430C418yescomplemented
16C42×C3⋊D3D311yescharacteristic
17C3×C42×D3—31no—
18C2×#3712—32no—
19C2×#3711—32no—
20#63630D311yescharacteristic, complemented
21C3×#803—31no—
22C3×#803—31no—
23C3×#803—31no—
24C3×#803—61no—
25#63553—32no—
26Dic3×C3⋊C8—32no—
27C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32C4×(C3⋊D3)⋊C3—24no—
33C4×(C3⋊D3)⋊C3C2×C414yes—
34#6180C2×C414yescomplemented
35(C3⋊(C3⋊C8))⋊C3C2×C414yescomplemented
36C4×#208—91no—
37#4707—91no—
38#4707—91no—
39C2×C4×C3⋊D3Dic312yes—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—31no—
42C2×C3×C4×D3—32no—
43C32×C42—21no—
44C32×C42C2×D311yescharacteristic
45C32×C42C2×D311yescharacteristic
46C4×C3⋊Dic3Dic312yes—
47C4×C3⋊Dic3C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)C2×D312yes—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—31no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—32no—
53C2×C3×C3⋊C8—32no—
54C2×C3×C3⋊C8—62no—
55#3712—38no—
56#3711—38no—
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C2×(C3⋊D3)⋊C3—22no—
59C2×(C3⋊D3)⋊C3C4212yes—
60C4×H(3,𝔽3)—22no—
61C4×H(3,𝔽3)C22×C412yes—
62(C3⋊Dic3)⋊C3C4212yes—
63(C3⋊Dic3)⋊C3M1614yescomplemented
64C2×C3×C42—91no—
65C2×#208—92no—
66C2×ΓL(1,𝔽25)—92no—
67C2×ΓL(1,𝔽25)—92no—
68C42×D3—31no—
69C42×D3—91no—
70C3×#97—91no—
71C3×#97—91no—
72C8×Dic3—92no—
73C8×Dic3—92no—
74#803—31no—
75#803—31no—
76#803—31no—
77#803—61no—
78#803—91no—
79C4×C3⋊C8—92no—
80C22×C3⋊D3C2×Dic311yescharacteristic
81C22×C3×D3—31no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C4×D312yes—
85C2×C32×C4C2×Dic312yes—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3C2×Dic312yes—
89C2×C3⋊Dic3C2×Dic311yescharacteristic
90C4×C3⋊D3—24no—
91C4×C3⋊D3C4×D314yes—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—32no—
94C3×C4×D3—34no—
95C3×C4×D3—64no—
96C3⋊(C3⋊C8)C4×D314yescomplemented
97C3×C3⋊C8—34no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—64no—
101C4×M16—271noSylow
102C2×H(3,𝔽3)C2×C4211yescharacteristic
103C2×H(3,𝔽3)C2×M1612yes—
104(C3⋊D3)⋊C3—22no—
105C22×C3×C4—91no—
106C22×C3×C4—92no—
107C2×C4×D3—31no—
108C2×C4×D3—32no—
109C2×C4×D3—91no—
110C2×C4×D3—92no—
111C2×C3×C8—92no—
112C2×C3×C8—92no—
113C3×C42—31no—
114C3×C42—31no—
115C3×C42—61no—
116C3×C42—91no—
117C3×C42—92no—
118C3×C42D3211yescharacteristic
119C4×Dic3—32no—
120C4×Dic3—31no—
121C4×Dic3—91no—
122C4×Dic3—92no—
123#208—98no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—62no—
128C2×C3⋊C8—92no—
129ΓL(1,𝔽25)—98no—
130ΓL(1,𝔽25)—98no—
131C22×C32—21no—
132C22×C32C2×C4×D311yescharacteristic
133C22×C32C22×Dic311yescharacteristic
134C2×C3⋊D3—22no—
135C2×C3⋊D3C4×Dic312yes—
136C2×C3×D3—32no—
137C2×C3×D3—62no—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—22no—
141C32×C4—42no—
142C32×C4C2×C4×D312yes—
143C32×C4C2×C4×D312yes—
144C3⋊Dic3#20814yescomplemented
145C3⋊Dic3C4×Dic312yes—
146C3×Dic3—32no—
147C3×Dic3—34no—
148C2×M16—272no—
149C2×C42—91no—
150#97—91no—
151#97—91no—
152C4×C8—272no—
153H(3,𝔽3)C4×M1611yescharacteristic, complemented, Sylow
154C23×C3—91no—
155C22×D3—31no—
156C22×D3—91no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—94no—
164C2×C3×C4—91no—
165C2×C3×C4—92no—
166C2×C3×C4—184no—
167C2×C3×C4D3×Dic312yes—
168C2×C3×C4C2×D3211yescharacteristic
169C2×Dic3—32no—
170C2×Dic3—31no—
171C2×Dic3—91no—
172C2×Dic3—92no—
173C4×D3—34no—
174C4×D3—64no—
175C4×D3—94no—
176C4×D3—184no—
177C3×C8—94no—
178C3×C8—94no—
179C3⋊C8—34no—
180C3⋊C8—34no—
181C3⋊C8—34no—
182C3⋊C8—64no—
183C3⋊C8—94no—
184C2×C32—22no—
185C2×C32—21no—
186C2×C32C42×D311yescharacteristic
187C2×C32D3×M1612yes—
188C2×C32C2×#20812yes—
189C2×C32C2×C4×Dic311yescharacteristic
190C3⋊D3—22no—
191C3×D3—62no—
192C22×C4—91no—
193C22×C4—92no—
194M16—278no—
195C2×C8—92no—
196C2×C8—92no—
197C42—92no—
198C42—91no—
199C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—61no—
203C22×C3—92no—
204C22×C3—182no—
205C22×C3C2×D3×Dic311yescharacteristic
206C2×D3—32no—
207C2×D3—62no—
208C2×D3—92no—
209C2×D3—182no—
210C3×C4—22no—
211C3×C4—32no—
212C3×C4—32no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—62no—
216C3×C4—92no—
217C3×C4—94no—
218C3×C4—122no—
219C3×C4C4×D3212yes—
220Dic3—34no—
221Dic3—32no—
222Dic3—92no—
223Dic3—94no—
224C32—21no—
225C32#470711yescharacteristic, complemented
226C32C4×#20811yescharacteristic, complemented
227C23—91no—
228C2×C4—92no—
229C2×C4—91no—
230C2×C4—94no—
231C2×C4—184no—
232C2×C4((C3⋊D3)⋊C3)⋊C412yes—
233C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
234C8—94no—
235C8—94no—
236C2×C3—32no—
237C2×C3—32no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—62no—
241C2×C3—61no—
242C2×C3—182no—
243C2×C3D3×#20812yes—
244C2×C3C4×D3×Dic311yescharacteristic
245D3—62no—
246D3—182no—
247C22—92no—
248C22—182no—
249C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
250C4—22no—
251C4—94no—
252C4—92no—
253C4C4×((C3⋊D3)⋊C3)⋊C212yes—
254C3—31no—
255C3—31no—
256C3—61no—
257C3#1059033411yescharacteristic
258C2—182no—
259C2#1172105312yes—
260C2C4×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #11720520C2semidirect
4. #63630⋊C3C2semidirect
5. #11720310C2semidirect
15. #88430C4semidirect
20. #63630D3semidirect
34. #6180C2×C4semidirect
35. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
63. (C3⋊Dic3)⋊C3M16semidirect
96. C3⋊(C3⋊C8)C4×D3semidirect
144. C3⋊Dic3#208semidirect
153. H(3,𝔽3)C4×M16semidirect
225. C32#4707semidirect
226. C32C4×#208semidirect

Maximal subgroups

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

subgroupindex
C42×(C3⋊D3)⋊C32
C2×#884302
#117205202
#63630⋊C32
#117203102
#105903343
#105902473