Finite Group Databasebeta

← C2×#10620070 #13049302

Subgroups — C2×#10620070 #13049302

Order 1296.

376 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 49 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×#66436C211yesmaximal, characteristic, complemented
2#10620070C212yesmaximal, direct product factor
3C23×C3⋊D9C311yesmaximal, characteristic, complemented
4C2×(C3⋊D3)⋊A4—31nomaximal
5C2×A4×D9—31nomaximal
6C2×D9⋊A4—32nomaximal
7C2×#4207—41nomaximal
8#66436C2211yescharacteristic, complemented
9C23×C3×C9C2×C311yescharacteristic, complemented
10C22×C3⋊D9—31no—
11C22×C3⋊D9—31no—
12C22×C3⋊D9—32no—
13C22×C3⋊D9C2×C312yescomplemented
14C2×C22⋊H(3,𝔽3)D311yescharacteristic
15C2×#1094—21no—
16C2×C9⋊A4—22no—
17C2×C9⋊A4D312yescomplemented
18C2×C9×A4—21no—
19C2×C9×A4D311yescharacteristic, complemented
20(C3⋊D3)⋊A4—32no—
21A4×D9—32no—
22D9⋊A4—34no—
23C2×#737—41no—
24#4207—42no—
25C23×C3⋊D3—31no—
26C2×D3×A4—91no—
27C23×D9—32no—
28C23×D9—31no—
29C22×C3×C9—31no—
30C22×C3×C9—31no—
31C22×C3×C9C22×C311yescharacteristic, complemented
32C2×C3⋊D9—32no—
33C2×C3⋊D9—31no—
34C2×C3⋊D9—32no—
35C2×C3⋊D9—32no—
36C2×C3⋊D9—32no—
37C2×C3⋊D9A411yescharacteristic, complemented
38C2×AGL(1,ℤ/9)—122no—
39C2×(C3⋊D3)⋊C3—121no—
40C2×C3×D9—121no—
41C22⋊H(3,𝔽3)C2×D311yescharacteristic
42#1094—21no—
43C9⋊A4—22no—
44C9⋊A4C2×D312yescomplemented
45C9×A4—21no—
46C9×A4C2×D311yescharacteristic, complemented
47#737—41noSylow
48C23×C32C3×D311yescharacteristic
49C22×C3⋊D3—32no—
50C22×C3⋊D3—91no—
51C22×C3⋊D3—92no—
52C22×C3⋊D3—91no—
53C2×C3×A4—21no—
54C2×C3×A4C3⋊D311yescharacteristic
55D3×A4—92no—
56C23×C9C3×D311yescharacteristic
57C23×C9C3×D312yes—
58C22×D9—34no—
59C22×D9—32no—
60C22×D9—91no—
61C22×D9—92no—
62C22×D9—94no—
63C22×D9—92no—
64C22×D9—92no—
65C22×D9—91no—
66C2×C22⋊C9—22no—
67C2×C22⋊C9—21no—
68C2×3−1+2—42no—
69C2×3−1+2—81no—
70C2×3−1+2—82no—
71C2×H(3,𝔽3)—41no—
72C2×C3×C9—31no—
73C2×C3×C9—31no—
74C2×C3×C9—41no—
75C2×C3×C9—81no—
76C2×C3×C9C2×A411yescharacteristic, complemented
77C3⋊D9—32no—
78C3⋊D9C2×A412yescomplemented
79AGL(1,ℤ/9)—124no—
80(C3⋊D3)⋊C3—122no—
81C3×D9—122no—
82C23×D3—91no—
83C23×D3—271no—
84C22×A4—271no—
85C22×C32—31no—
86C22×C32—31no—
87C22×C32C2×C3×D311yescharacteristic
88C2×C3⋊D3—31no—
89C2×C3⋊D3—92no—
90C2×C3⋊D3—92no—
91C2×C3⋊D3—92no—
92C2×C3⋊D3—91no—
93C2×C3⋊D3—92no—
94C2×C3×D3—361no—
95C3×A4—21no—
96C3×A4C2×C3⋊D311yescharacteristic
97C22×C9—31no—
98C22×C9—32no—
99C22×C9—31no—
100C22×C9—32no—
101C22×C9C2×C3×D311yescharacteristic
102C22×C9C2×C3×D312yes—
103C2×D9—31no—
104C2×D9—32no—
105C2×D9—92no—
106C2×D9—94no—
107C2×D9—92no—
108C2×D9—94no—
109C2×D9—92no—
110C2×D9—94no—
111C2×D9—91no—
112C2×D9—92no—
113C2×D9—94no—
114C2×D9—92no—
115C22⋊C9—22no—
116C22⋊C9—21no—
1173−1+2—42no—
1183−1+2—81no—
1193−1+2—82no—
120H(3,𝔽3)—41no—
121C3×C9—41no—
122C3×C9—81no—
123C3×C9C22×A411yescharacteristic, complemented
124C23×C3—31no—
125C23×C3C3×C3⋊D311yescharacteristic
126C22×D3—92no—
127C22×D3—272no—
128C22×D3—271no—
129C22×D3—273no—
130C22×D3—276no—
131C22×D3—273no—
132C22×D3—272no—
133C22×D3—271no—
134C2×A4—31no—
135C2×A4—61no—
136C2×A4—272no—
137C2×C32—31no—
138C2×C32—31no—
139C2×C32—41no—
140C2×C32—81no—
141C2×C32D3×A411yescharacteristic
142C3⋊D3—32no—
143C3⋊D3—92no—
144C3×D3—362no—
145C2×C9—31no—
146C2×C9—32no—
147C2×C9—31no—
148C2×C9—32no—
149C2×C9—81no—
150C2×C9—82no—
151C2×C9D3×A411yescharacteristic
152C2×C9D3×A412yes—
153D9—32no—
154D9—34no—
155D9—92no—
156D9—94no—
157C24—271noSylow
158C22×C3—31no—
159C22×C3—33no—
160C22×C3—33no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—1081no—
164C22×C3C2×C3×C3⋊D311yescharacteristic
165C2×D3—91no—
166C2×D3—271no—
167C2×D3—272no—
168C2×D3—272no—
169C2×D3—272no—
170C2×D3—276no—
171C2×D3—276no—
172C2×D3—276no—
173C2×D3—276no—
174C2×D3—273no—
175C2×D3—271no—
176C2×D3—272no—
177A4—31no—
178A4—61no—
179C32—41no—
180C32—81no—
181C32C2×D3×A411yescharacteristic
182C9—81no—
183C9—82no—
184C9C2×D3×A411yescharacteristic
185C9C2×D3×A412yes—
186C23—272no—
187C23—811no—
188C23—812no—
189C23—811no—
190C23#420711yescharacteristic, complemented
191C2×C3—33no—
192C2×C3—33no—
193C2×C3—31no—
194C2×C3—31no—
195C2×C3—31no—
196C2×C3—121no—
197C2×C3—241no—
198C2×C3—1082no—
199C2×C3A4×C3⋊D311yescharacteristic
200D3—92no—
201D3—272no—
202D3—276no—
203D3—272no—
204C22—31no—
205C22—31no—
206C22—271no—
207C22—812no—
208C22—812no—
209C22—812no—
210C22—812no—
211C22—811no—
212C22C2×#420711yescharacteristic, complemented
213C3—31no—
214C3—121no—
215C3—241no—
216C3C2×A4×C3⋊D311yescharacteristic
217C2—31no—
218C2—31no—
219C2—272no—
220C2—812no—
221C2#1062007011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#10620070
commutator, cosocle, hypocenterC22×C3×C9C22×C3
FrattiniC3C2×A4×C3⋊D3
FittingC23×C3×C9C2×C3
socleC23×C3C3×C3⋊D3
LCSG ▷ C22×C3×C9
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#66436C2semidirect
2. #10620070C2direct (N × H)
3. C23×C3⋊D9C3semidirect
8. #66436C22semidirect
9. C23×C3×C9C2×C3semidirect
13. C22×C3⋊D9C2×C3semidirect
17. C2×C9⋊A4D3semidirect
19. C2×C9×A4D3semidirect
31. C22×C3×C9C22×C3semidirect
37. C2×C3⋊D9A4semidirect
44. C9⋊A4C2×D3semidirect
46. C9×A4C2×D3semidirect
76. C2×C3×C9C2×A4semidirect
78. C3⋊D9C2×A4semidirect
123. C3×C9C22×A4semidirect
190. C23#4207semidirect
212. C22C2×#4207semidirect
221. C2#10620070direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#664362
#106200702
C23×C3⋊D93
C2×(C3⋊D3)⋊A43
C2×A4×D93
C2×D9⋊A43
C2×#42074