Finite Group Databasebeta

← #13181765

Subgroups — #13181765

Order 1728.

11340 conjugacy classes of subgroups; only the 2211 normal subgroups are shown.

normal subgroups only

This group has 11340 conjugacy classes of subgroups — too many to draw. Only its 2211 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11724685C216yesmaximal, complemented
3#41×C3⋊(C3⋊D3)C216yesmaximal, complemented
4#11724679C219yesmaximal, complemented
5C2×#89078C219yesmaximal, complemented
6#10596758—313nomaximal
7C33×#41C2216yescomplemented
8C2×C33×D4C2219yescomplemented
9C2×#6312C22118yescomplemented
10C22×C3⋊(C3⋊Dic3)C2216yescomplemented
11#89080C2212yescomplemented
12Q8×C3⋊(C3⋊D3)C2216yescomplemented
13#89078C22118yescomplemented
14#89078C22136yescomplemented
15D4×C3⋊(C3⋊D3)C22118yescomplemented
16#89076C22118yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2219yescomplemented
18C2×#6308C2219yescomplemented
19C32×H(5,𝔽2)D3113yescomplemented
20#64367—378no—
21#41×C3⋊D3—378no—
22#64361—3117no—
23C2×#3830—3117no—
24C23×C33C2316yes—
25C22×C3⋊(C3⋊D3)C2319yes—
26C33×Q8C2312yes—
27C33×D4C23118yes—
28C2×C33×C4C2319yes—
29#6312C23136yes—
30C2×C3⋊(C3⋊Dic3)C23118yes—
31C2×C3⋊(C3⋊Dic3)C2319yes—
32C4⋊(C3⋊(C3⋊D3))C2316yes—
33C4×C3⋊(C3⋊D3)C2316yes—
34C4×C3⋊(C3⋊D3)C23118yes—
35#6308C23118yes—
36#5966—913no—
37C32×#41C2×D3178yescomplemented
38C2×C32×D4C2×D31117yescomplemented
39C2×#638—3234no—
40C22×C3⋊Dic3—378no—
41#3832—326no—
42Q8×C3⋊D3—378no—
43#3830—3234no—
44#3830—3468no—
45D4×C3⋊D3—3234no—
46#3828—3234no—
47C2×C4×C3⋊D3—3117no—
48C2×#634—3117no—
49C22×C33—46no—
50C22×C33C2419yes—
51C2×C3⋊(C3⋊D3)—49no—
52C2×C3⋊(C3⋊D3)C2416yes—
53C33×C4C2416yes—
54C3⋊(C3⋊Dic3)C2419yes—
55C3⋊(C3⋊Dic3)C2411yescharacteristic
56C3×H(5,𝔽2)C3⋊D3113yescomplemented
57#1010—978no—
58D3×#41—978no—
59#1004—9117no—
60C2×#237—9117no—
61C23×C32C22×D3178yes—
62C22×C3⋊D3—3117no—
63C32×Q8C22×D3126yes—
64C32×D4C22×D31234yes—
65C2×C32×C4C22×D31117yes—
66#638—3468no—
67C2×C3⋊Dic3—3234no—
68C2×C3⋊Dic3—3117no—
69C4⋊(C3⋊D3)—378no—
70C4×C3⋊D3—378no—
71C4×C3⋊D3—3234no—
72#634—3234no—
73#585—271noSylow
74C2×C33—29no—
75C2×C33C2511yescharacteristic
76C3⋊(C3⋊D3)—26no—
77C3×#41C2×C3⋊D3178yescomplemented
78C2×C3×D4C2×C3⋊D31117yescomplemented
79C2×#67—9234no—
80C22×Dic3—978no—
81#239—926no—
82D3×Q8—978no—
83#237—9234no—
84#237—9468no—
85D3×D4—9234no—
86#235—9234no—
87C2×C4×D3—9117no—
88C2×Dic6—9117no—
89C22×C32—478no—
90C22×C32C23×D31117yes—
91C2×C3⋊D3—378no—
92C2×C3⋊D3—12117no—
93C32×C4C23×D3178yes—
94C3⋊Dic3—3117no—
95C3⋊Dic3—313no—
962−1+4—276no—
97H(5,𝔽2)—279no—
98H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic, complemented
99C2×#41—276no—
100C2×#41—279no—
101C33#58511yescharacteristic, complemented, Sylow
102C23×C3C22×C3⋊D3178yes—
103C22×D3—9117no—
104C3×Q8C22×C3⋊D3126yes—
105C3×D4C22×C3⋊D31234yes—
106C2×C3×C4C22×C3⋊D31117yes—
107#67—9468no—
108C2×Dic3—9234no—
109C2×Dic3—9117no—
110D12—978no—
111C4×D3—978no—
112C4×D3—9234no—
113Dic6—9234no—
114C2×C32—2117no—
115C2×C32C24×D3113yes—
116C3⋊D3—678no—
117#41—272no—
118#41—2718no—
119#41—2736no—
120#41—2718no—
121#41C2×C3⋊(C3⋊D3)16yescomplemented
122C2×Q8—276no—
123C2×Q8—279no—
124C2×D4—2718no—
125C2×D4—2718no—
126C2×D4C2×C3⋊(C3⋊D3)19yescomplemented
127C22×C4—276no—
128C22×C4—279no—
129C22×C3—478no—
130C22×C3C23×C3⋊D31117yes—
131C2×D3—978no—
132C2×D3—36117no—
133C3×C4C23×C3⋊D3178yes—
134Dic3—9117no—
135Dic3—913no—
136C32#5966113yescomplemented
137C23—279no—
138C23C22×C3⋊(C3⋊D3)16yes—
139Q8—2718no—
140Q8C22×C3⋊(C3⋊D3)12yes—
141D4—2736no—
142D4—276no—
143D4C22×C3⋊(C3⋊D3)118yes—
144C2×C4—279no—
145C2×C4—2718no—
146C2×C4—276no—
147C2×C4—2718no—
148C2×C4C22×C3⋊(C3⋊D3)19yes—
149C2×C3—2117no—
150C2×C3C24×C3⋊D3113yes—
151D3—1878no—
152C22—46no—
153C22—276no—
154C22—1089no—
155C22C23×C3⋊(C3⋊D3)19yes—
156C4—271no—
157C4—279no—
158C4C23×C3⋊(C3⋊D3)16yes—
159C3#10596758113yescomplemented
160C2—29no—
161C2—546no—
162C2C24×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C24×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C33C25
FittingC33×H(5,𝔽2)C2
hypercenterH(5,𝔽2)C3⋊(C3⋊D3)
hypocenterC33#585
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. #11724685C2semidirect
3. #41×C3⋊(C3⋊D3)C2semidirect
4. #11724679C2semidirect
5. C2×#89078C2semidirect
7. C33×#41C22semidirect
8. C2×C33×D4C22semidirect
9. C2×#6312C22semidirect
10. C22×C3⋊(C3⋊Dic3)C22semidirect
11. #89080C22semidirect
12. Q8×C3⋊(C3⋊D3)C22semidirect
13. #89078C22semidirect
14. #89078C22semidirect
15. D4×C3⋊(C3⋊D3)C22semidirect
16. #89076C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
18. C2×#6308C22semidirect
19. C32×H(5,𝔽2)D3semidirect
37. C32×#41C2×D3semidirect
38. C2×C32×D4C2×D3semidirect
56. C3×H(5,𝔽2)C3⋊D3semidirect
77. C3×#41C2×C3⋊D3semidirect
78. C2×C3×D4C2×C3⋊D3semidirect
98. H(5,𝔽2)C3⋊(C3⋊D3)semidirect
101. C33#585semidirect
121. #41C2×C3⋊(C3⋊D3)semidirect
126. C2×D4C2×C3⋊(C3⋊D3)semidirect
136. C32#5966semidirect
159. C3#10596758semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
#117246852
#41×C3⋊(C3⋊D3)2
#117246792
C2×#890782
#105967583