Finite Group Databasebeta

← #13180357

Subgroups — #13180357

Order 1728.

250 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 62 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
1D3×#3787C211yesmaximal, characteristic
2#11724238C211yesmaximal, characteristic, complemented
3C3×#64214C211yesmaximal, characteristic, complemented
4#11724093C212yesmaximal, complemented
5#11724092C212yesmaximal, complemented
6#10596481—31nomaximal
7C4×D3×C3⋊D3C411yescharacteristic
8#88988C411yescharacteristic
9C3×#3787C2211yescharacteristic
10#88924C411yescharacteristic
11#88924C2211yescharacteristic
12D3×#622C411yescharacteristic
13D3×#622C2211yescharacteristic
14C3⋊#3771C2212yescomplemented
15C3×#3771C2212yescomplemented
16C2×#3787—31no—
17#64218—32no—
18#64214—31no—
19#64214D311yescharacteristic, complemented
20#64213—32no—
21C2×D3×C3⋊D3C811yescharacteristic
22C4×C3⋊(C3⋊D3)C811yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C32×C4×D3C811yescharacteristic
25#6289C2×C411yescharacteristic
26Dic3×C3⋊D3C811yescharacteristic
27D3×C3⋊Dic3C2×C411yescharacteristic
28C3⋊#622C2×C411yescharacteristic
29C3⋊#622C2×C411yescharacteristic
30C3×#622C2×C411yescharacteristic
31C3×#622C2311yescharacteristic
32D3×M32—91nomaximal
33C2×C4×C3⋊D3—31no—
34C4×D32—121no—
35C2×#622—31no—
36C2×#622—31no—
37#3787—31no—
38#3787—31no—
39#3787—31no—
40#3787C2×D311yescharacteristic
41#3771—32no—
42#3771C2×D312yescomplemented
43C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
44C2×C3×C3⋊D3C2×C811yescharacteristic
45C2×C32×D3C2×C811yescharacteristic
46D3×C3⋊D3—22no—
47C33×C4C2×C811yescharacteristic
48C3⋊(C3⋊Dic3)C2×C811yescharacteristic
49C3×C3⋊Dic3C22×C411yescharacteristic
50C32×Dic3C2×C811yescharacteristic
51C2×D3×C8—91no—
52C3×M32—91no—
53#812—91no—
54#798—92no—
55D3×C16—92no—
56C22×C3⋊D3—31no—
57C2×D32—121no—
58C2×C32×C4—31no—
59C2×C3⋊Dic3—31no—
60C4×C3⋊D3—31no—
61C4×C3⋊D3—31no—
62C4×C3⋊D3—31no—
63C4×C3⋊D3—121no—
64C4×C3⋊D3—241no—
65C4×C3⋊D3C4×D311yescharacteristic
66C3×C4×D3—41no—
67C3×C4×D3—121no—
68#624—121no—
69D3×Dic3—121no—
70D3×Dic3—121no—
71#622—31no—
72#622—31no—
73#622C4×D311yescharacteristic
74#622C22×D311yescharacteristic
75C2×M32—271noSylow
76C2×C33C22×C811yescharacteristic
77C3⋊(C3⋊D3)M3212yescomplemented
78C3×C3⋊D3—21no—
79C32×D3—21no—
80C2×C4×D3—91no—
81C2×C4×D3—361no—
82C2×C3×C8—91no—
83C2×C3⋊C8—91no—
84D3×C8—91no—
85D3×C8—91no—
86D3×C8—91no—
87D3×C8—91no—
88C3×C16—92no—
89C3⋊C16—92no—
90C22×C32—31no—
91C2×C3⋊D3—31no—
92C2×C3⋊D3—62no—
93C2×C3⋊D3—121no—
94C2×C3⋊D3—241no—
95C2×C3⋊D3D3×C811yescharacteristic
96C2×C3×D3—41no—
97C2×C3×D3—121no—
98D32—124no—
99C32×C4—31no—
100C32×C4—41no—
101C32×C4—81no—
102C32×C4D3×C811yescharacteristic
103C3⋊Dic3—31no—
104C3⋊Dic3—121no—
105C3⋊Dic3—241no—
106C3⋊Dic3C2×C4×D311yescharacteristic
107C3×Dic3—41no—
108C3×Dic3—121no—
109C22×C8—271no—
110M32—91no—
111M32—272no—
112M32—271no—
113C2×C16—272no—
114C33C2×M3211yescharacteristic, complemented, Sylow
115C22×D3—91no—
116C22×D3—361no—
117C2×C3×C4—91no—
118C2×C3×C4—121no—
119C2×Dic3—91no—
120C2×Dic3—361no—
121C4×D3—91no—
122C4×D3—91no—
123C4×D3—91no—
124C4×D3—121no—
125C4×D3—361no—
126C4×D3—361no—
127C4×D3—361no—
128C4×D3—721no—
129C4×D3AGL(1,𝔽9)11yescharacteristic
130C3×C8—91no—
131C3×C8—91no—
132C3⋊C8—91no—
133C3⋊C8—91no—
134C2×C32—41no—
135C2×C32—61no—
136C2×C32—81no—
137C2×C32C2×D3×C811yescharacteristic
138C3⋊D3—21no—
139C3⋊D3—32no—
140C3⋊D3—122no—
141C3⋊D3—242no—
142C3×D3—42no—
143C3×D3—122no—
144C22×C4—271no—
145C2×C8—91no—
146C2×C8—271no—
147C2×C8—271no—
148C2×C8—271no—
149C2×C8—271no—
150C2×C8—271no—
151C16—92no—
152C16—272no—
153C22×C3—91no—
154C22×C3—121no—
155C2×D3—91no—
156C2×D3—121no—
157C2×D3—182no—
158C2×D3—361no—
159C2×D3—364no—
160C2×D3—721no—
161C2×D3C2×AGL(1,𝔽9)11yescharacteristic
162C3×C4—41no—
163C3×C4—81no—
164C3×C4—91no—
165C3×C4—121no—
166C3×C4C2×AGL(1,𝔽9)11yescharacteristic
167Dic3—91no—
168Dic3—121no—
169Dic3—361no—
170Dic3—721no—
171Dic3C2×AGL(1,𝔽9)11yescharacteristic
172C32—41no—
173C32—81no—
174C32D3×M3211yescharacteristic, complemented
175C23—271no—
176C2×C4—31no—
177C2×C4—91no—
178C2×C4—271no—
179C2×C4—271no—
180C2×C4—271no—
181C2×C4—271no—
182C8—91no—
183C8—91no—
184C8—271no—
185C8—271no—
186C2×C3—41no—
187C2×C3—81no—
188C2×C3—122no—
189C2×C3—181no—
190C2×C3C22×AGL(1,𝔽9)11yescharacteristic
191D3—21no—
192D3—92no—
193D3—122no—
194D3—362no—
195D3—722no—
196C22—31no—
197C22—91no—
198C22—271no—
199C22—542no—
200C4—31no—
201C4—91no—
202C4—271no—
203C4D3×AGL(1,𝔽9)11yescharacteristic
204C3—41no—
205C3—81no—
206C3#1059648111yescharacteristic, complemented
207C2—61no—
208C2—181no—
209C2—272no—
210C2C2×D3×AGL(1,𝔽9)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×AGL(1,𝔽9)
commutator, socleC2×C33C22×C8
FittingC33×C4C2×C8
prime radicalC2×D3×C3⋊D3C8
cosocleC3×#622C23
hypercenterC4D3×AGL(1,𝔽9)
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724238C2semidirect
3. C3×#64214C2semidirect
4. #11724093C2semidirect
5. #11724092C2semidirect
14. C3⋊#3771C22semidirect
15. C3×#3771C22semidirect
19. #64214D3semidirect
42. #3771C2×D3semidirect
77. C3⋊(C3⋊D3)M32semidirect
114. C33C2×M32semidirect
174. C32D3×M32semidirect
206. C3#10596481semidirect

Maximal subgroups

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

subgroupindex
D3×#37872
#117242382
C3×#642142
#117240932
#117240922
#105964813
D3×M329