Finite Group Databasebeta

← #13163882

Subgroups — #13163882

Order 1728.

2352 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 2352 conjugacy classes of subgroups — too many to draw. Only its 459 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#11723990C212yesmaximal, complemented
3#11723981C211yesmaximal, characteristic, complemented
4#11722207C212yesmaximal, complemented
5#11722203C211yesmaximal, characteristic, complemented
6#10592940—313nomaximal
7C33×#41—22no—
8C33×#41C2212yes—
9C2×C33×D4—22no—
10C2×C33×D4—22no—
11C2×C33×D4C2211yescharacteristic
12C2×#6312—22no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
14#88862—22no—
15#88860—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17#88855C2212yescomplemented
18C32×H(5,𝔽2)D3113yescomplemented
19#64158—326no—
20#64149—313no—
21#63664—326no—
22#63660—313no—
23C23×C33—22no—
24C23×C33D412yes—
25C22×C3⋊(C3⋊D3)—21no—
26C33×Q8D412yescomplemented
27C33×D4—24no—
28C33×D4—22no—
29C33×D4—22no—
30C33×D4D412yescomplemented
31C2×C33×C4—22no—
32C2×C33×C4—22no—
33C2×C33×C4C2311yescharacteristic
34#6312—44no—
35C2×C3⋊(C3⋊Dic3)—21no—
36C4⋊(C3⋊(C3⋊D3))—22no—
37C3⋊(C3⋊(C3⋊C8))—22no—
38#5241—913no—
39C32×#41—226no—
40C32×#41C2×D3126yes—
41C2×C32×D4—226no—
42C2×C32×D4—226no—
43C2×C32×D4C2×D3113yes—
44C2×#638—626no—
45C2×C4⋊(C3⋊D3)—313no—
46#3755—626no—
47#3753—626no—
48C4×C3⋊Dic3—313no—
49#3748—326no—
50C22×C33—22no—
51C22×C33—22no—
52C22×C33—42no—
53C22×C33—82no—
54C22×C33C2×D411yescharacteristic
55C2×C3⋊(C3⋊D3)—42no—
56C2×C3⋊(C3⋊D3)—41no—
57C33×C4—22no—
58C33×C4C2×D412yes—
59C3⋊(C3⋊Dic3)—22no—
60C3×H(5,𝔽2)C3⋊D3113yescomplemented
61#949—926no—
62#940—913no—
63#837—926no—
64#833—913no—
65C23×C32—226no—
66C23×C32#67126yes—
67C22×C3⋊D3—613no—
68C32×Q8#67126yescomplemented
69C32×D4—252no—
70C32×D4—226no—
71C32×D4—226no—
72C32×D4#67126yescomplemented
73C2×C32×C4—226no—
74C2×C32×C4—226no—
75C2×C32×C4C22×D3113yes—
76#638—1252no—
77C2×C3⋊Dic3—613no—
78C4⋊(C3⋊D3)—626no—
79C3⋊(C3⋊C8)—626no—
80#453—271noSylow
81C2×C33—21no—
82C2×C33—42no—
83C2×C33—42no—
84C2×C33#12011yescharacteristic
85C3⋊(C3⋊D3)—81no—
86C3×#41—226no—
87C3×#41C2×C3⋊D3126yes—
88C2×C3×D4—226no—
89C2×C3×D4—226no—
90C2×C3×D4C2×C3⋊D3113yes—
91C2×#67—1826no—
92C2×D12—913no—
93#215—1826no—
94#213—1826no—
95C4×Dic3—913no—
96#208—926no—
97C22×C32—226no—
98C22×C32—226no—
99C22×C32—426no—
100C22×C32—826no—
101C22×C32C2×#67113yes—
102C2×C3⋊D3—1226no—
103C2×C3⋊D3—1213no—
104C32×C4—226no—
105C32×C4C2×#67126yes—
106C3⋊Dic3—626no—
107H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic, complemented
108AGL(1,ℤ/8)—272no—
109#127—271no—
110#104—272no—
111#100—271no—
112C33#45311yescharacteristic, complemented, Sylow
113C23×C3—226no—
114C23×C3#638126yes—
115C22×D3—1813no—
116C3×Q8#638126yescomplemented
117C3×D4—252no—
118C3×D4—226no—
119C3×D4—226no—
120C3×D4#638126yescomplemented
121C2×C3×C4—226no—
122C2×C3×C4—226no—
123C2×C3×C4C22×C3⋊D3113yes—
124#67—3652no—
125C2×Dic3—1813no—
126D12—1826no—
127C3⋊C8—1826no—
128C2×C32—213no—
129C2×C32—426no—
130C2×C32—426no—
131C2×C32#953113yes—
132C3⋊D3—2413no—
133#41—22no—
134#41C2×C3⋊(C3⋊D3)12yes—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—271no—
138C2×D4—542no—
139C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
140SD16—542no—
141D8—542no—
142M16—272no—
143C42—271no—
144C22×C3—226no—
145C22×C3—226no—
146C22×C3—426no—
147C22×C3—826no—
148C22×C3C2×#638113yes—
149C2×D3—3626no—
150C2×D3—3613no—
151C3×C4—226no—
152C3×C4C2×#638126yes—
153Dic3—1826no—
154C32#5241113yescomplemented
155C23—22no—
156C23—541no—
157C23#631212yes—
158Q8#631212yescomplemented
159D4—24no—
160D4—22no—
161D4—22no—
162D4—542no—
163D4—1084no—
164D4#631212yescomplemented
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—541no—
168C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
169C8—542no—
170C2×C3—213no—
171C2×C3—426no—
172C2×C3—426no—
173C2×C3#64162113yes—
174D3—7213no—
175C22—22no—
176C22—22no—
177C22—42no—
178C22—82no—
179C22—1082no—
180C22—1081no—
181C22C2×#631211yescharacteristic
182C4—22no—
183C4—542no—
184C4C2×#631212yes—
185C3#10592940113yescomplemented
186C2—21no—
187C2—42no—
188C2—42no—
189C2—2161no—
190C2#1172399411yescharacteristic
subgroupG/H
centerC2#11723994
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterH(5,𝔽2)C3⋊(C3⋊D3)
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. #11723990C2semidirect
3. #11723981C2semidirect
4. #11722207C2semidirect
5. #11722203C2semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C4×C3⋊(C3⋊Dic3)C22semidirect
17. #88855C22semidirect
18. C32×H(5,𝔽2)D3semidirect
26. C33×Q8D4semidirect
30. C33×D4D4semidirect
60. C3×H(5,𝔽2)C3⋊D3semidirect
68. C32×Q8#67semidirect
72. C32×D4#67semidirect
107. H(5,𝔽2)C3⋊(C3⋊D3)semidirect
112. C33#453semidirect
116. C3×Q8#638semidirect
120. C3×D4#638semidirect
154. C32#5241semidirect
158. Q8#6312semidirect
164. D4#6312semidirect
185. C3#10592940semidirect

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
#117239902
#117239812
#117222072
#117222032
#105929403