Finite Group Databasebeta

← 2−1+4×C3⋊(C3⋊D3) #13181766

Subgroups — 2−1+4×C3⋊(C3⋊D3) #13181766

Order 1728.

11116 conjugacy classes of subgroups; only the 2213 normal subgroups are shown.

normal subgroups only

This group has 11116 conjugacy classes of subgroups — too many to draw. Only its 2213 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
1C33×2−1+4C211yesmaximal, characteristic, complemented
2#11724685C2110yesmaximal, complemented
3#41×C3⋊(C3⋊D3)C2110yesmaximal, complemented
4#11724682C215yesmaximal, complemented
5C2×Q8×C3⋊(C3⋊D3)C215yesmaximal, complemented
6C3⋊D3×2−1+4—313nomaximal
7C33×#41C22110yescomplemented
8C2×C33×Q8C2215yescomplemented
9#89080C22110yescomplemented
10Q8×C3⋊(C3⋊D3)C22110yes—
11Q8×C3⋊(C3⋊D3)C22120yescomplemented
12#89078C22130yescomplemented
13D4×C3⋊(C3⋊D3)C22110yes—
14#89076C22130yescomplemented
15C2×C4×C3⋊(C3⋊D3)C22115yes—
16C2×#6308C22115yescomplemented
17C32×2−1+4D3113yescomplemented
18#64367—3130no—
19#41×C3⋊D3—3130no—
20#64364—365no—
21C2×Q8×C3⋊D3—365no—
22C22×C3⋊(C3⋊D3)C2315yes—
23C33×Q8C23110yes—
24C33×D4C23110yes—
25C2×C33×C4C23115yes—
26#6312C23120yes—
27C2×C3⋊(C3⋊Dic3)C23115yes—
28C4⋊(C3⋊(C3⋊D3))C23110yes—
29C4×C3⋊(C3⋊D3)C23110yes—
30C4×C3⋊(C3⋊D3)C23130yes—
31#6308C23130yes—
32D3×2−1+4—913no—
33C32×#41C2×D31130yescomplemented
34C2×C32×Q8C2×D3165yescomplemented
35#3832—3130no—
36Q8×C3⋊D3—3130no—
37Q8×C3⋊D3—3260no—
38#3830—3390no—
39D4×C3⋊D3—3130no—
40#3828—3390no—
41C2×C4×C3⋊D3—3195no—
42C2×#634—3195no—
43C22×C33C2415yes—
44C2×C3⋊(C3⋊D3)—210no—
45C2×C3⋊(C3⋊D3)C2415yes—
46C2×C3⋊(C3⋊D3)C2411yescharacteristic
47C33×C4C24110yes—
48C3⋊(C3⋊Dic3)C24110yes—
49C3×2−1+4C3⋊D3113yescomplemented
50#1010—9130no—
51D3×#41—9130no—
52#1007—965no—
53C2×D3×Q8—965no—
54C22×C3⋊D3—365no—
55C32×Q8C22×D31130yes—
56C32×D4C22×D31130yes—
57C2×C32×C4C22×D31195yes—
58#638—3260no—
59C2×C3⋊Dic3—3195no—
60C4⋊(C3⋊D3)—3130no—
61C4×C3⋊D3—3130no—
62C4×C3⋊D3—3390no—
63#634—3390no—
64C2×2−1+4—271noSylow
65C2×C33—25no—
66C2×C33C2511yescharacteristic
67C3⋊(C3⋊D3)—25no—
68C3⋊(C3⋊D3)2−1+412yesdirect product factor
69C3×#41C2×C3⋊D31130yescomplemented
70C2×C3×Q8C2×C3⋊D3165yescomplemented
71#239—9130no—
72D3×Q8—9130no—
73D3×Q8—9260no—
74#237—9390no—
75D3×D4—9130no—
76#235—9390no—
77C2×C4×D3—9195no—
78C2×Dic6—9195no—
79C22×C32C23×D3165yes—
80C2×C3⋊D3—313no—
81C2×C3⋊D3—365no—
82C2×C3⋊D3—6130no—
83C32×C4C23×D31130yes—
84C3⋊Dic3—3130no—
852−1+4—2710no—
862−1+4—275no—
872−1+4C3⋊(C3⋊D3)11yescharacteristic, direct product factor
88C2×#41—2710no—
89C22×Q8—275no—
90C33C2×2−1+411yescharacteristic, complemented, Sylow
91C22×D3—965no—
92C3×Q8C22×C3⋊D31130yes—
93C3×D4C22×C3⋊D31130yes—
94C2×C3×C4C22×C3⋊D31195yes—
95#67—9260no—
96C2×Dic3—9195no—
97D12—9130no—
98C4×D3—9130no—
99C4×D3—9390no—
100Dic6—9390no—
101C2×C32—265no—
102C2×C32C24×D3113yes—
103C3⋊D3—326no—
104C3⋊D3—665no—
105#41—2710no—
106#41—2730no—
107#41—2730no—
108#41C2×C3⋊(C3⋊D3)110yescomplemented
109C2×Q8—2710no—
110C2×Q8—2720no—
111C2×Q8—2715no—
112C2×Q8C2×C3⋊(C3⋊D3)15yescomplemented
113C2×D4—2710no—
114C22×C4—2715no—
115C22×C3C23×C3⋊D3165yes—
116C2×D3—913no—
117C2×D3—965no—
118C2×D3—18130no—
119C3×C4C23×C3⋊D31130yes—
120Dic3—9130no—
121C32D3×2−1+4113yescomplemented
122C23—275no—
123Q8—2730no—
124Q8C22×C3⋊(C3⋊D3)110yes—
125D4—2720no—
126D4—2710no—
127D4C22×C3⋊(C3⋊D3)110yes—
128C2×C4—2715no—
129C2×C4—2710no—
130C2×C4—2730no—
131C2×C4C22×C3⋊(C3⋊D3)115yes—
132C2×C3—265no—
133C2×C3C24×C3⋊D3113yes—
134D3—926no—
135D3—1865no—
136C22—271no—
137C22—275no—
138C22—5410no—
139C22C23×C3⋊(C3⋊D3)15yes—
140C4—2710no—
141C4C23×C3⋊(C3⋊D3)110yes—
142C3C3⋊D3×2−1+4113yescomplemented
143C2—25no—
144C2—272no—
145C2—545no—
146C2C24×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C24×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C33C25
FittingC33×2−1+4C2
hypercenter2−1+4C3⋊(C3⋊D3)
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. C33×2−1+4C2semidirect
2. #11724685C2semidirect
3. #41×C3⋊(C3⋊D3)C2semidirect
4. #11724682C2semidirect
5. C2×Q8×C3⋊(C3⋊D3)C2semidirect
7. C33×#41C22semidirect
8. C2×C33×Q8C22semidirect
9. #89080C22semidirect
11. Q8×C3⋊(C3⋊D3)C22semidirect
12. #89078C22semidirect
14. #89076C22semidirect
16. C2×#6308C22semidirect
17. C32×2−1+4D3semidirect
33. C32×#41C2×D3semidirect
34. C2×C32×Q8C2×D3semidirect
49. C3×2−1+4C3⋊D3semidirect
68. C3⋊(C3⋊D3)2−1+4direct (N × H)
69. C3×#41C2×C3⋊D3semidirect
70. C2×C3×Q8C2×C3⋊D3semidirect
87. 2−1+4C3⋊(C3⋊D3)direct (N × H)
90. C33C2×2−1+4semidirect
108. #41C2×C3⋊(C3⋊D3)semidirect
112. C2×Q8C2×C3⋊(C3⋊D3)semidirect
121. C32D3×2−1+4semidirect
142. C3C3⋊D3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C33×2−1+42
#117246852
#41×C3⋊(C3⋊D3)2
#117246822
C2×Q8×C3⋊(C3⋊D3)2
C3⋊D3×2−1+43