Finite Group Databasebeta

← #13180371

Subgroups — #13180371

Order 1728.

224 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 60 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#11724266C211yesmaximal, characteristic
2C2×C3⋊#3771C211yesmaximal, characteristic, complemented
3C3×#64218C211yesmaximal, characteristic, complemented
4#11724094C211yesmaximal, characteristic, complemented
5#11724093C211yesmaximal, characteristic, complemented
6#11724092C211yesmaximal, characteristic, complemented
7D3×#3771C211yesmaximal, characteristic, complemented
8#10596485—31nomaximal
9#89028C411yescharacteristic
10#88993C411yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88926C411yescharacteristic
13#88925C411yescharacteristic
14#88924C2211yescharacteristic
15D3×#622C2211yescharacteristic
16C3⋊#3771C2211yescharacteristic
17C3⋊#3771C2211yescharacteristic
18C3×#3771C2211yescharacteristic
19C3×#3771C2211yescharacteristic
20#64287—31no—
21#64218—31no—
22#64218D311yescharacteristic, complemented
23C2×#3771—31no—
24C2×#3771—31no—
25#64214—31no—
26#64213—31no—
27#6312C811yescharacteristic
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C32×#67C811yescharacteristic
30#6293C811yescharacteristic
31#6291C811yescharacteristic
32#6289C2×C411yescharacteristic
33D3×C3⋊Dic3C2×C411yescharacteristic
34C3⋊#622C2×C411yescharacteristic
35C3⋊#622C2×C411yescharacteristic
36C3×#622C2×C411yescharacteristic
37C3×#622C2311yescharacteristic
38#4907—91nomaximal
39#3830—31no—
40#3804—121no—
41#3792—31no—
42#3792—31no—
43C2×#622—31no—
44C2×#622C2×D311yescharacteristic
45#3788—31no—
46#3787—31no—
47#3771—31no—
48#3771—31no—
49#3771C2×D311yescharacteristic
50#3771C2×D311yescharacteristic
51C22×C33C2×C811yescharacteristic
52C2×C3⋊(C3⋊D3)C2×C811yescharacteristic
53C2×C32×D3C2×C811yescharacteristic
54C3⋊(C3⋊Dic3)C2×C811yescharacteristic
55C3×C3⋊Dic3C2×C811yescharacteristic
56C3×C3⋊Dic3C22×C411yescharacteristic
57C32×Dic3C2×C811yescharacteristic
58#901—91no—
59C3×M32—91no—
60C2×C3⋊C16—91no—
61#798—91no—
62#798—91no—
63D3×C16—91no—
64D3×C16—91no—
65C32×D4—31no—
66#638—31no—
67#638—31no—
68#638—121no—
69#638—241no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3C4×D311yescharacteristic
72C4×C3⋊D3—31no—
73#634—31no—
74C3×#67—41no—
75C2×C3×Dic3—121no—
76#627—121no—
77#626—121no—
78#624—121no—
79D3×Dic3—121no—
80#622—31no—
81#622—31no—
82#622C4×D311yescharacteristic
83#622C22×D311yescharacteristic
84#504—271noSylow
85C2×C33—21no—
86C2×C33C22×C811yescharacteristic
87C3⋊(C3⋊D3)—21no—
88C32×D3—21no—
89#237—361no—
90#235—91no—
91C2×C3×C8—91no—
92#208—91no—
93ΓL(1,𝔽25)—91no—
94ΓL(1,𝔽25)—91no—
95D3×C8—91no—
96D3×C8—91no—
97C3×C16—91no—
98C3×C16—91no—
99C3⋊C16—91no—
100C3⋊C16—91no—
101C22×C32—31no—
102C22×C32—41no—
103C22×C32—81no—
104C22×C32D3×C811yescharacteristic
105C2×C3⋊D3—31no—
106C2×C3⋊D3—121no—
107C2×C3⋊D3—241no—
108C2×C3×D3—41no—
109C32×C4—31no—
110C3⋊Dic3—31no—
111C3⋊Dic3—121no—
112C3⋊Dic3—241no—
113C3⋊Dic3D3×C811yescharacteristic
114C3⋊Dic3C2×C4×D311yescharacteristic
115C3×Dic3—41no—
116C3×Dic3—121no—
117C3×Dic3—121no—
118#131—271no—
119M32—91no—
120M32—271no—
121M32—271no—
122C2×C16—271no—
123C2×C16—271no—
124C2×C16—271no—
125C33#50411yescharacteristic, complemented, Sylow
126C3×D4—121no—
127C2×C3×C4—91no—
128#67—91no—
129#67—361no—
130#67—361no—
131#67—721no—
132#67AGL(1,𝔽9)11yescharacteristic
133C2×Dic3—121no—
134C2×Dic3—361no—
135D12—91no—
136C4×D3—91no—
137C4×D3—91no—
138C4×D3—361no—
139Dic6—91no—
140Dic6—361no—
141C3×C8—91no—
142C3×C8—91no—
143C3⋊C8—91no—
144C3⋊C8—91no—
145C2×C32—21no—
146C2×C32—41no—
147C2×C32—61no—
148C2×C32—81no—
149C2×C32—81no—
150C2×C32—161no—
151C2×C32C2×D3×C811yescharacteristic
152C3⋊D3—61no—
153C3⋊D3—241no—
154C3⋊D3—481no—
155C3×D3—81no—
156#41—271no—
157M16—271no—
158M16—271no—
159M16—271no—
160C2×C8—91no—
161C2×C8—271no—
162C2×C8—271no—
163C16—91no—
164C16—91no—
165C16—271no—
166C16—271no—
167C22×C3—41no—
168C22×C3—81no—
169C22×C3—121no—
170C22×C3C2×AGL(1,𝔽9)11yescharacteristic
171C2×D3—91no—
172C2×D3—361no—
173C2×D3—721no—
174C2×D3C2×AGL(1,𝔽9)11yescharacteristic
175C3×C4—91no—
176C3×C4—91no—
177C3×C4—121no—
178Dic3—91no—
179Dic3—121no—
180Dic3—121no—
181Dic3—361no—
182Dic3—721no—
183Dic3C2×AGL(1,𝔽9)11yescharacteristic
184C32—41no—
185C32—81no—
186C32#490711yescharacteristic, complemented
187Q8—271no—
188D4—31no—
189D4—271no—
190D4—271no—
191C2×C4—91no—
192C2×C4—271no—
193C2×C4—271no—
194C8—91no—
195C8—91no—
196C8—271no—
197C8—271no—
198C2×C3—21no—
199C2×C3—41no—
200C2×C3—81no—
201C2×C3—81no—
202C2×C3—161no—
203C2×C3—241no—
204C2×C3C22×AGL(1,𝔽9)11yescharacteristic
205D3—21no—
206D3—181no—
207D3—721no—
208D3—1441no—
209C22—31no—
210C22—271no—
211C22D3×AGL(1,𝔽9)11yescharacteristic
212C4—31no—
213C4—91no—
214C4—91no—
215C4—271no—
216C3—41no—
217C3—81no—
218C3#1059648511yescharacteristic, complemented
219C2—21no—
220C2—61no—
221C2—541no—
222C2C2×D3×AGL(1,𝔽9)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×AGL(1,𝔽9)
commutator, socleC2×C33C22×C8
FittingC22×C33C2×C8
prime radical#89028C4
cosocleC3×#622C23
hypercenterC22D3×AGL(1,𝔽9)
hypocenterC33#504
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3771C2semidirect
3. C3×#64218C2semidirect
4. #11724094C2semidirect
5. #11724093C2semidirect
6. #11724092C2semidirect
7. D3×#3771C2semidirect
22. #64218D3semidirect
125. C33#504semidirect
186. C32#4907semidirect
218. C3#10596485semidirect

Maximal subgroups

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

subgroupindex
#117242662
C2×C3⋊#37712
C3×#642182
#117240942
#117240932
#117240922
D3×#37712
#105964853
#49079