Finite Group Databasebeta

← #13163428

Subgroups — #13163428

Order 1728.

1680 conjugacy classes of subgroups; only the 353 normal subgroups are shown.

normal subgroups only

This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 353 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
1#11723985C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic
3C2×#88851C211yesmaximal, characteristic
4#11723934C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6#11722184C211yesmaximal, characteristic, complemented
7#11722180C211yesmaximal, characteristic, complemented
8#10592517—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×#6308C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88862—22no—
16#88859—21no—
17#88857—21no—
18#88857—21no—
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88851—22no—
22#64153—313no—
23C2×#3755—313no—
24C2×#3744—313no—
25#64102—313no—
26C32×#103D3113yescomplemented
27#63641—313no—
28#63637—313no—
29C22×C3⋊(C3⋊D3)—21no—
30C33×Q8—21no—
31C33×Q8—21no—
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3⋊(C3⋊Dic3)D411yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C4⋊(C3⋊(C3⋊D3))—21no—
39C4⋊(C3⋊(C3⋊D3))D412yes—
40C4×C3⋊(C3⋊D3)—42no—
41#6308—21no—
42#6308—21no—
43C33×C8—21no—
44C3⋊(C3⋊(C3⋊C8))—21no—
45#4818—913no—
46C2×C32×Q8C2×D3113yes—
47C2×C4⋊(C3⋊D3)—313no—
48C2×C4×C3⋊D3—613no—
49C2×#634—313no—
50C2×C32×C8C2×D3113yes—
51C32×C4⋊C4C2×D3113yes—
52#3755—626no—
53#3752—613no—
54#3750—613no—
55#3750—613no—
56C4×C3⋊Dic3—313no—
57C2×C3⋊(C3⋊C8)—313no—
58#3744—626no—
59C22×C33C2×D411yescharacteristic
60C2×C3⋊(C3⋊D3)—22no—
61C2×C3⋊(C3⋊D3)—41no—
62C2×C3⋊(C3⋊D3)—41no—
63C33×C4—41no—
64C33×C4—41no—
65C33×C4C2×D411yescharacteristic
66C33×C4#4111yescharacteristic
67C3⋊(C3⋊Dic3)—21no—
68C3⋊(C3⋊Dic3)—41no—
69C3⋊(C3⋊Dic3)SD1612yescomplemented
70#944—913no—
71C2×#215—913no—
72C2×#204—913no—
73#893—913no—
74C3×#103C3⋊D3113yescomplemented
75Dic3⋊C8—913no—
76#810—913no—
77C22×C3⋊D3—613no—
78C32×Q8—213no—
79C32×Q8—213no—
80C2×C32×C4—213no—
81C2×C32×C4—213no—
82C2×C32×C4C22×D3113yes—
83C2×C3⋊Dic3—313no—
84C2×C3⋊Dic3—313no—
85C2×C3⋊Dic3—613no—
86C4⋊(C3⋊D3)—326no—
87C4⋊(C3⋊D3)—613no—
88C4×C3⋊D3—1226no—
89#634—613no—
90#634—613no—
91C32×C8—213no—
92C3⋊(C3⋊C8)—613no—
93#461—271noSylow
94C2×C33#12111yescharacteristic
95C2×C33C2×SD1611yescharacteristic
96C2×C33#13711yescharacteristic
97C3⋊(C3⋊D3)—42no—
98C2×C3×Q8C2×C3⋊D3113yes—
99C2×D12—913no—
100C2×C4×D3—1813no—
101C2×Dic6—913no—
102C2×C3×C8C2×C3⋊D3113yes—
103C3×C4⋊C4C2×C3⋊D3113yes—
104#215—1826no—
105(C2×D3)⋊C4—1813no—
106Dic3⋊C4—1813no—
107Dic3⋊C4—1813no—
108C4×Dic3—913no—
109C2×C3⋊C8—913no—
110#204—1826no—
111C22×C32D3×D4113yes—
112C2×C3⋊D3—626no—
113C2×C3⋊D3—1213no—
114C2×C3⋊D3—1213no—
115C32×C4—413no—
116C32×C4—413no—
117C32×C4#235113yes—
118C32×C4D3×D4113yes—
119C3⋊Dic3—326no—
120C3⋊Dic3—613no—
121C3⋊Dic3—1213no—
122C2×SD16—271no—
123C2×SD16—271no—
124C4⋊Q8—271no—
125C4×D4—271no—
126C4⋊C8—271no—
127#103—271no—
128#103C3⋊(C3⋊D3)11yescharacteristic, complemented
129C33#46111yescharacteristic, complemented, Sylow
130C22×D3—1813no—
131C3×Q8—213no—
132C3×Q8—213no—
133C2×C3×C4—213no—
134C2×C3×C4—213no—
135C2×C3×C4C22×C3⋊D3113yes—
136C2×Dic3—913no—
137C2×Dic3—913no—
138C2×Dic3—1813no—
139D12—926no—
140D12—1813no—
141C4×D3—3626no—
142Dic6—1813no—
143Dic6—1813no—
144C3×C8—213no—
145C3⋊C8—1813no—
146C2×C32#884113yes—
147C2×C32D3×SD16113yes—
148C2×C32#918113yes—
149C3⋊D3—1226no—
150C2×Q8—271no—
151C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
152C2×D4—271no—
153C22×C4—541no—
154SD16—542no—
155SD16—542no—
156C2×C8—271no—
157C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
158C4⋊C4—541no—
159C4⋊C4—541no—
160C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
161C22⋊C4—541no—
162C42—271no—
163C22×C3D4×C3⋊D3113yes—
164C2×D3—1826no—
165C2×D3—3613no—
166C2×D3—3613no—
167C3×C4—413no—
168C3×C4—413no—
169C3×C4#3828113yes—
170C3×C4D4×C3⋊D3113yes—
171Dic3—926no—
172Dic3—1813no—
173Dic3—3613no—
174C32#4818113yescomplemented
175C23—541no—
176Q8—21no—
177Q8—21no—
178Q8—541no—
179Q8—541no—
180D4—272no—
181D4—541no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—271no—
185C2×C4—271no—
186C2×C4—541no—
187C2×C4—1082no—
188C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
189C8—21no—
190C8—541no—
191C2×C3#64093113yes—
192C2×C3SD16×C3⋊D3113yes—
193C2×C3#64127113yes—
194D3—3626no—
195C22—542no—
196C22—1081no—
197C22—1081no—
198C22D4×C3⋊(C3⋊D3)11yescharacteristic
199C4—41no—
200C4—41no—
201C4—272no—
202C4—541no—
203C4—1081no—
204C4#8907611yescharacteristic
205C4D4×C3⋊(C3⋊D3)11yescharacteristic
206C3#10592517113yescomplemented
207C2—1082no—
208C2#1172392511yescharacteristic
209C2SD16×C3⋊(C3⋊D3)11yescharacteristic
210C2#1172395911yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenter#103C3⋊(C3⋊D3)
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #11723985C2semidirect
5. C33×#103C2semidirect
6. #11722184C2semidirect
7. #11722180C2semidirect
26. C32×#103D3semidirect
69. C3⋊(C3⋊Dic3)SD16semidirect
74. C3×#103C3⋊D3semidirect
128. #103C3⋊(C3⋊D3)semidirect
129. C33#461semidirect
174. C32#4818semidirect
206. C3#10592517semidirect

Maximal subgroups

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

subgroupindex
#117239852
C2×#888622
C2×#888512
#117239342
C33×#1032
#117221842
#117221802
#105925173