Finite Group Databasebeta

← #13048656

Subgroups — #13048656

Order 1296.

614 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 91 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×D9×C3⋊D3C212yesmaximal, complemented
2C4×C3⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C9×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×D36C211yesmaximal, characteristic, complemented
5#10619877C212yesmaximal, complemented
6#89026—31nomaximal
7#88646—34nomaximal
8C2×C3⋊(C3⋊D9)C2211yescharacteristic
9C2×C9×C3⋊D3C2212yescomplemented
10C2×C32×D9C2212yescomplemented
11D9×C3⋊D3—24no—
12C32×C4×C9C2211yescharacteristic, complemented
13C3⋊(C3⋊Dic9)C2211yescharacteristic, complemented
14C2×D3×C3⋊D3—32no—
15C4×C3⋊(C3⋊D3)—31no—
16C3×C4⋊(C3⋊D3)D311yescharacteristic
17C32×D12—31no—
18#6290—32no—
19C2×D3×D9—38no—
20C4×C3⋊D9—34no—
21C4×C3⋊D9—64no—
22C9×D12—34no—
23C3×D36D314yescomplemented
24#6194—38no—
25C2×C32×C9C2311yescharacteristic
26C3⋊(C3⋊D9)D412yescomplemented
27C9×C3⋊D3—22no—
28C32×D9—22no—
29D4×C3⋊D3—91no—
30#3802—94no—
31D4×D9—91no—
32C2×C3⋊(C3⋊D3)—31no—
33C2×C3×C3⋊D3C2×D312yes—
34C2×C32×D3—32no—
35D3×C3⋊D3—64no—
36C33×C4C2×D311yescharacteristic
37C3⋊(C3⋊Dic3)—31no—
38C2×C3⋊D9—34no—
39C2×C3⋊D9—64no—
40C2×D3×C9—38no—
41C2×C3×D9C2×D318yescomplemented
42D3×D9—616no—
43C3×C4×C9—24no—
44C3×C4×C9C2×D314yescomplemented
45C3⋊Dic9—34no—
46C3⋊Dic9—64no—
47C32×C9C2×D411yescharacteristic, complemented, Sylow
48C22×C3⋊D3—92no—
49C2×D32—98no—
50C32×D4—91no—
51#638—92no—
52C4⋊(C3⋊D3)D911yescharacteristic, complemented
53C4×C3⋊D3—91no—
54C4×C3⋊D3—94no—
55C4×C3⋊D3—184no—
56C3×D12—34no—
57C3×D12—34no—
58#625—98no—
59C22×D9—92no—
60D4×C9—91no—
61#611—92no—
62D36C3⋊D311yescharacteristic, complemented
63C4×D9—91no—
64C4×D9—184no—
65C2×C33C22×D311yescharacteristic
66C3⋊(C3⋊D3)—32no—
67C3×C3⋊D3—22no—
68C32×D3—62no—
69C2×C3×C9—24no—
70C2×C3×C9C22×D314yes—
71C3⋊D9—38no—
72C3⋊D9—68no—
73D3×C9—68no—
74C3×D9—28no—
75D3×D4—271no—
76D3×D4—274no—
77C22×C32—92no—
78C2×C3⋊D3—91no—
79C2×C3⋊D3—94no—
80C2×C3⋊D3—184no—
81C2×C3⋊D3—184no—
82C2×C3⋊D3C2×D912yescomplemented
83C2×C3×D3—38no—
84C2×C3×D3—38no—
85D32—1816no—
86C32×C4—24no—
87C32×C4C2×D911yescharacteristic, complemented
88C32×C4D3214yes—
89C3⋊Dic3—91no—
90C3⋊Dic3—94no—
91C3⋊Dic3—184no—
92C22×C9—92no—
93C2×D9—91no—
94C2×D9—184no—
95C2×D9—184no—
96C2×D9C2×C3⋊D312yescomplemented
97C4×C9—24no—
98C4×C9C2×C3⋊D311yescharacteristic, complemented
99Dic9—91no—
100Dic9—184no—
101C33D3×D411yescharacteristic
102C3×C9—24no—
103C3×C9D3×D414yescomplemented
104C22×D3—272no—
105C22×D3—278no—
106C3×D4—94no—
107C3×D4—91no—
108#67—272no—
109#67—278no—
110D12—34no—
111D12—31no—
112C4×D3—274no—
113C4×D3—271no—
114C4×D3—544no—
115C2×C32—24no—
116C2×C32—182no—
117C2×C32C22×D911yescharacteristic
118C2×C32C2×D3214yes—
119C3⋊D3—22no—
120C3⋊D3—92no—
121C3⋊D3—98no—
122C3⋊D3—188no—
123C3×D3—68no—
124C3×D3—68no—
125C2×C9—24no—
126C2×C9—182no—
127C2×C9C22×C3⋊D311yescharacteristic
128D9—22no—
129D9—92no—
130D9—188no—
131C2×D4—811noSylow
132C22×C3—98no—
133C22×C3—92no—
134C2×D3—38no—
135C2×D3—32no—
136C2×D3—274no—
137C2×D3—271no—
138C2×D3—544no—
139C2×D3—544no—
140C2×D3—5416no—
141C3×C4—24no—
142C3×C4D3×D914yescomplemented
143C3×C4D3×C3⋊D311yescharacteristic
144Dic3—274no—
145Dic3—271no—
146Dic3—544no—
147C32—24no—
148C32D4×D911yescharacteristic, complemented
149C32#380214yes—
150C9—24no—
151C9D4×C3⋊D311yescharacteristic, complemented
152C23—812no—
153D4—91no—
154D4—91no—
155D4—812no—
156C2×C4—811no—
157C2×C3—24no—
158C2×C3—188no—
159C2×C3—182no—
160C2×C3C2×D3×D914yes—
161C2×C3C2×D3×C3⋊D311yescharacteristic
162D3—68no—
163D3—62no—
164D3—278no—
165D3—272no—
166D3—548no—
167C22—92no—
168C22—92no—
169C22—811no—
170C22—1624no—
171C4—811no—
172C4D9×C3⋊D311yescharacteristic, complemented
173C3—24no—
174C3#8864614yescomplemented
175C3#8902611yescharacteristic
176C2—182no—
177C2—182no—
178C2—812no—
179C2C2×D9×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D9×C3⋊D3
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC32×C4×C9C22
socleC2×C33C22×D3
hypercenterC4D9×C3⋊D3
hypocenterC32×C9C2×D4
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D9×C3⋊D3C2semidirect
2. C4×C3⋊(C3⋊D9)C2semidirect
3. C9×C4⋊(C3⋊D3)C2semidirect
4. C32×D36C2semidirect
5. #10619877C2semidirect
9. C2×C9×C3⋊D3C22semidirect
10. C2×C32×D9C22semidirect
12. C32×C4×C9C22semidirect
13. C3⋊(C3⋊Dic9)C22semidirect
23. C3×D36D3semidirect
26. C3⋊(C3⋊D9)D4semidirect
41. C2×C3×D9C2×D3semidirect
44. C3×C4×C9C2×D3semidirect
47. C32×C9C2×D4semidirect
52. C4⋊(C3⋊D3)D9semidirect
62. D36C3⋊D3semidirect
82. C2×C3⋊D3C2×D9semidirect
87. C32×C4C2×D9semidirect
96. C2×D9C2×C3⋊D3semidirect
98. C4×C9C2×C3⋊D3semidirect
103. C3×C9D3×D4semidirect
142. C3×C4D3×D9semidirect
148. C32D4×D9semidirect
151. C9D4×C3⋊D3semidirect
172. C4D9×C3⋊D3semidirect
174. C3#88646semidirect

Maximal subgroups

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

subgroupindex
C2×D9×C3⋊D32
C4×C3⋊(C3⋊D9)2
C9×C4⋊(C3⋊D3)2
C32×D362
#106198772
#890263
#886463