Finite Group Databasebeta

← #13180937

Subgroups — #13180937

Order 1728.

250 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 83 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×C3×#3787C211yesmaximal, characteristic
2#11724242C212yesmaximal, complemented
3#11724238C212yesmaximal, complemented
4#11724237C212yesmaximal, complemented
5#10596481—31nomaximal
6C2×C3×C4×C3⋊D3C411yescharacteristic
7C2×C3×#622C411yescharacteristic
8C2×C3×#622C2211yescharacteristic
9C3×#3787C411yescharacteristic
10C3×#3787C411yescharacteristic
11C3×#3787C2211yescharacteristic
12C3×#3787C2211yescharacteristic
13C3⋊#3771C2214yescomplemented
14C2×#3787D311yescharacteristic
15#64218—32no—
16#64214—32no—
17#64213—32no—
18C22×C3×C3⋊D3C811yescharacteristic
19C2×C33×C4C811yescharacteristic
20C2×C3×C3⋊Dic3C2×C411yescharacteristic
21C3×C4×C3⋊D3C811yescharacteristic
22C3×C4×C3⋊D3C811yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×#622C2×C411yescharacteristic
26C3×#622C2×C411yescharacteristic
27C3×#622C2×C411yescharacteristic
28C3×#622C2311yescharacteristic
29C2×#812—91nomaximal
30C2×C4×C3⋊D3Dic311yescharacteristic
31C2×C3×C4×D3—121no—
32C2×#622Dic311yescharacteristic
33C2×#622C2×D311yescharacteristic
34#3787Dic311yescharacteristic
35#3787Dic311yescharacteristic
36#3787C2×D311yescharacteristic
37#3787C2×D311yescharacteristic
38#3771—34no—
39C22×C33C2×C811yescharacteristic
40C2×C3×C3⋊D3—22no—
41C2×C3×C3⋊D3C2×C811yescharacteristic
42C2×C3×C3⋊D3C2×C811yescharacteristic
43C33×C4C2×C811yescharacteristic
44C33×C4C2×C811yescharacteristic
45C3×C3⋊Dic3C2×C811yescharacteristic
46C3×C3⋊Dic3C22×C411yescharacteristic
47C22×C3×C8—91no—
48#812—92no—
49#812—92no—
50C2×C3⋊C16—92no—
51C22×C3⋊D3C3⋊C811yescharacteristic
52C22×C3×D3—121no—
53C2×C32×C4—41no—
54C2×C32×C4—81no—
55C2×C32×C4C3⋊C811yescharacteristic
56C2×C3⋊Dic3C2×Dic311yescharacteristic
57C4×C3⋊D3C3⋊C811yescharacteristic
58C4×C3⋊D3C3⋊C811yescharacteristic
59C4×C3⋊D3C2×Dic311yescharacteristic
60C4×C3⋊D3C2×Dic311yescharacteristic
61C2×C3×Dic3—121no—
62C3×C4×D3—121no—
63C3×C4×D3—121no—
64C3×C4×D3—121no—
65C3×C4×D3—121no—
66#622C2×Dic311yescharacteristic
67#622C2×Dic311yescharacteristic
68#622C2×Dic311yescharacteristic
69#622C22×D311yescharacteristic
70C2×M32—271noSylow
71C2×C33—21no—
72C2×C33C22×C811yescharacteristic
73C3×C3⋊D3—21no—
74C3×C3⋊D3M3212yescomplemented
75C22×C3×C4—91no—
76C2×C4×D3—121no—
77C2×C3×C8—91no—
78C2×C3×C8—91no—
79C2×C3×C8—91no—
80C2×C3×C8—91no—
81C2×C3×C8—91no—
82C2×C3×C8—91no—
83C3⋊C16—94no—
84C22×C32—41no—
85C22×C32—81no—
86C22×C32C2×C3⋊C811yescharacteristic
87C2×C3⋊D3—22no—
88C2×C3⋊D3C2×C3⋊C811yescharacteristic
89C2×C3⋊D3C2×C3⋊C811yescharacteristic
90C2×C3×D3—121no—
91C2×C3×D3—121no—
92C2×C3×D3—124no—
93C32×C4—41no—
94C32×C4—41no—
95C32×C4—81no—
96C32×C4—81no—
97C32×C4C2×C3⋊C811yescharacteristic
98C32×C4C2×C3⋊C811yescharacteristic
99C3⋊Dic3C2×C3⋊C811yescharacteristic
100C3⋊Dic3C22×Dic311yescharacteristic
101C3×Dic3—121no—
102C3×Dic3—121no—
103C22×C8—91no—
104M32—272no—
105M32—272no—
106C2×C16—272no—
107C33C2×M3211yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—121no—
110C2×C3×C4—41no—
111C2×C3×C4—81no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4AGL(1,𝔽9)11yescharacteristic
118C2×Dic3—121no—
119C4×D3—121no—
120C4×D3—121no—
121C4×D3—121no—
122C4×D3—121no—
123C3×C8—91no—
124C3×C8—91no—
125C3×C8—91no—
126C3×C8—91no—
127C2×C32—21no—
128C2×C32—42no—
129C2×C32—41no—
130C2×C32—82no—
131C2×C32—81no—
132C2×C32C22×C3⋊C811yescharacteristic
133C3⋊D3—21no—
134C3⋊D3#81212yescomplemented
135C3×D3—122no—
136C3×D3—122no—
137C22×C4—91no—
138C2×C8—91no—
139C2×C8—91no—
140C2×C8—91no—
141C2×C8—91no—
142C2×C8—91no—
143C2×C8—91no—
144C16—274no—
145C22×C3—41no—
146C22×C3—81no—
147C22×C3—91no—
148C22×C3—91no—
149C22×C3—182no—
150C22×C3C2×AGL(1,𝔽9)11yescharacteristic
151C2×D3—121no—
152C2×D3—121no—
153C2×D3—124no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—81no—
157C3×C4—81no—
158C3×C4—91no—
159C3×C4—91no—
160C3×C4C2×AGL(1,𝔽9)11yescharacteristic
161C3×C4C2×AGL(1,𝔽9)11yescharacteristic
162Dic3—121no—
163Dic3—121no—
164C32—41no—
165C32—81no—
166C32C2×#81211yescharacteristic, complemented
167C23—91no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4C3⋊AGL(1,𝔽9)11yescharacteristic
174C8—91no—
175C8—91no—
176C8—91no—
177C8—91no—
178C2×C3—21no—
179C2×C3—42no—
180C2×C3—41no—
181C2×C3—82no—
182C2×C3—81no—
183C2×C3—92no—
184C2×C3—181no—
185C2×C3C22×AGL(1,𝔽9)11yescharacteristic
186D3—122no—
187D3—122no—
188C22—91no—
189C22—91no—
190C22—182no—
191C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
192C4—91no—
193C4—91no—
194C4C2×C3⋊AGL(1,𝔽9)11yescharacteristic
195C4C2×C3⋊AGL(1,𝔽9)11yescharacteristic
196C3—41no—
197C3—81no—
198C3#1059648111yescharacteristic, complemented
199C2—21no—
200C2—92no—
201C2—181no—
202C2C22×C3⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊AGL(1,𝔽9)
commutator, socleC2×C33C22×C8
FrattiniC2C22×C3⋊AGL(1,𝔽9)
FittingC2×C33×C4C8
prime radicalC22×C3×C3⋊D3C8
cosocleC3×#622C23
hypercenterC2×C4C3⋊AGL(1,𝔽9)
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724242C2semidirect
3. #11724238C2semidirect
4. #11724237C2semidirect
13. C3⋊#3771C22semidirect
74. C3×C3⋊D3M32semidirect
107. C33C2×M32semidirect
134. C3⋊D3#812semidirect
166. C32C2×#812semidirect
198. C3#10596481semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
#117242422
#117242382
#117242372
#105964813
C2×#8129