Finite Group Databasebeta

← #13048766

Subgroups — #13048766

Order 1296.

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

normal subgroups only

This group has 348 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
1C3×#6233C212yesmaximal, complemented
2C2×C9×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#10619897C211yesmaximal, characteristic, complemented
4#10619896C212yesmaximal, complemented
5#10619895C211yesmaximal, characteristic, complemented
6#89046—31nomaximal
7#88658—32nomaximal
8C22×C32×C9C2211yescharacteristic
9C2×C3×C3⋊D9C2212yes—
10C9×C3⋊Dic3C2211yescharacteristic
11C9×C3⋊Dic3C2211yescharacteristic
12C3×C3⋊Dic9C2212yes—
13C3×#638—32no—
14C2×C3×C3⋊Dic3D311yescharacteristic
15#6296—31no—
16#6295—31no—
17#6294—32no—
18#6233D312yescomplemented
19C3×#611—32no—
20C3×#611—62no—
21C2×C9×Dic3—32no—
22C2×C9×Dic3—61no—
23#6192—32no—
24(C3⋊D9)⋊C4—32no—
25Dic3×D9—32no—
26#6189—32no—
27C2×C32×C9—21no—
28C2×C32×C9C2311yescharacteristic
29C3×C3⋊D9—22no—
30#3805—91no—
31#3804—92no—
32#3697—91no—
33C22×C33C2×D311yescharacteristic
34C2×C3×C3⋊D3—32no—
35C3×C3⋊Dic3—32no—
36C3×C3⋊Dic3C2×D311yescharacteristic
37C3×C3⋊Dic3C2×D311yescharacteristic
38C22×C3×C9—21no—
39C22×C3×C9—22no—
40C22×C3×C9—41no—
41C22×C3×C9C2×D312yes—
42C2×C3⋊D9C2×D312yes—
43C2×C3×D9—32no—
44C2×C3×D9—62no—
45C3⋊Dic9C2×D312yes—
46C9×Dic3—32no—
47C9×Dic3—32no—
48C9×Dic3—61no—
49C9×Dic3—61no—
50C3×Dic9—32no—
51C3×Dic9—62no—
52C32×C9#4111yescharacteristic, complemented, Sylow
53#638—32no—
54C2×C3⋊Dic3D911yescharacteristic, complemented
55C3×#67—92no—
56C3×#67—92no—
57C3×#67—182no—
58C2×C3×Dic3—32no—
59C2×C3×Dic3—61no—
60#627—92no—
61#627—91no—
62#626—92no—
63#625—91no—
64#624—92no—
65D3×Dic3—92no—
66D3×Dic3—92no—
67C2×C4×C9—91no—
68#611—32no—
69#611—62no—
70D36—91no—
71C4×D9—92no—
72Dic18—91no—
73C2×C33—21no—
74C2×C33C22×D311yescharacteristic
75C3×C3⋊D3—62no—
76C2×C3×C9—22no—
77C2×C3×C9—22no—
78C2×C3×C9—22no—
79C2×C3×C9—21no—
80C2×C3×C9—41no—
81C2×C3×C9—42no—
82C2×C3×C9—42no—
83C2×C3×C9C22×D312yes—
84C3⋊D9—22no—
85C3×D9—62no—
86C3×D9—122no—
87#237—272no—
88#235—271no—
89C22×C32—22no—
90C22×C32—21no—
91C22×C32—41no—
92C22×C32C2×D911yescharacteristic
93C22×C32D3212yes—
94C2×C3⋊D3—32no—
95C2×C3×D3—92no—
96C2×C3×D3—92no—
97C2×C3×D3—182no—
98C3⋊Dic3—32no—
99C3⋊Dic3C2×D911yescharacteristic
100C3⋊Dic3C2×D911yescharacteristic
101C3×Dic3—32no—
102C3×Dic3—32no—
103C3×Dic3—61no—
104C3×Dic3—61no—
105C3×Dic3—92no—
106C3×Dic3—92no—
107C3×Dic3—182no—
108C22×C9—22no—
109C22×C9—41no—
110C22×C9D3211yescharacteristic
111C2×D9—32no—
112C2×D9—62no—
113C4×C9—91no—
114C4×C9—91no—
115Dic9—32no—
116Dic9—62no—
117C33#23511yescharacteristic
118C3×C9—22no—
119C3×C9—21no—
120C3×C9—41no—
121C3×C9#23712yescomplemented
122C3×D4—272no—
123C2×C3×C4—91no—
124#67—92no—
125#67—92no—
126#67—182no—
127#67—272no—
128C2×Dic3—32no—
129C2×Dic3—61no—
130C2×Dic3—272no—
131D12—271no—
132C4×D3—272no—
133C4×D3—272no—
134Dic6—272no—
135Dic6—271no—
136C2×C32—21no—
137C2×C32—22no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—21no—
141C2×C32—41no—
142C2×C32—42no—
143C2×C32—42no—
144C2×C32C22×D911yescharacteristic
145C2×C32C2×D3212yes—
146C3⋊D3—62no—
147C3×D3—182no—
148C3×D3—182no—
149C3×D3—362no—
150C2×C9—21no—
151C2×C9—22no—
152C2×C9—42no—
153C2×C9—42no—
154C2×C9—41no—
155C2×C9C2×D3211yescharacteristic
156D9—62no—
157D9—122no—
158#41—811noSylow
159C22×C3—21no—
160C22×C3—22no—
161C22×C3—41no—
162C22×C3—272no—
163C22×C3D3×D912yes—
164C22×C3#111311yescharacteristic
165C2×D3—92no—
166C2×D3—92no—
167C2×D3—182no—
168C3×C4—91no—
169C3×C4—91no—
170C3×C4—272no—
171Dic3—32no—
172Dic3—32no—
173Dic3—61no—
174Dic3—61no—
175Dic3—92no—
176Dic3—92no—
177Dic3—182no—
178C32—22no—
179C32—21no—
180C32—41no—
181C32#369711yescharacteristic, complemented
182C32#380412yes—
183C9—22no—
184C9—41no—
185C9#380511yescharacteristic, complemented
186Q8—811no—
187D4—272no—
188D4—811no—
189C2×C4—91no—
190C2×C4—812no—
191C2×C3—22no—
192C2×C3—22no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—22no—
196C2×C3—42no—
197C2×C3—42no—
198C2×C3—41no—
199C2×C3—542no—
200C2×C3C2×D3×D912yes—
201C2×C3C2×#111311yescharacteristic
202D3—182no—
203D3—182no—
204D3—362no—
205C22—272no—
206C22#6643111yescharacteristic
207C4—91no—
208C4—91no—
209C4—272no—
210C3—22no—
211C3—21no—
212C3—41no—
213C3#8865812yescomplemented
214C3#8904611yescharacteristic
215C2—21no—
216C2—542no—
217C2C2×#6643111yescharacteristic
subgroupG/H
centerC2C2×#66431
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×#1113
FittingC22×C32×C9C22
socleC2×C33C22×D3
hypercenterC22#66431
hypocenterC32×C9#41
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C3×#6233C2semidirect
2. C2×C9×C3⋊Dic3C2semidirect
3. #10619897C2semidirect
4. #10619896C2semidirect
5. #10619895C2semidirect
18. #6233D3semidirect
52. C32×C9#41semidirect
54. C2×C3⋊Dic3D9semidirect
121. C3×C9#237semidirect
181. C32#3697semidirect
185. C9#3805semidirect
213. C3#88658semidirect

Maximal subgroups

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

subgroupindex
C3×#62332
C2×C9×C3⋊Dic32
#106198972
#106198962
#106198952
#890463
#886583