Finite Group Databasebeta

← #13163843

Subgroups — #13163843

Order 1728.

1456 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 349 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×#137C211yesmaximal, characteristic, complemented
2#11723990C211yesmaximal, characteristic
3#11723989C211yesmaximal, characteristic, complemented
4C2×#88862C211yesmaximal, characteristic, complemented
5#11722206C211yesmaximal, characteristic, complemented
6#11722193C211yesmaximal, characteristic, complemented
7#11722192C211yesmaximal, characteristic, complemented
8#10592902—313nomaximal
9C33×#41C2211yescharacteristic
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C33×Q16—21no—
13C33×SD16—21no—
14C33×M16C2211yescharacteristic
15#88862—21no—
16#88862—21no—
17#88862—21no—
18#88860—21no—
19#88855—21no—
20#88855C2211yescharacteristic
21#88855C2211yescharacteristic
22C2×C3⋊(C3⋊(C3⋊C8))—21no—
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C32×#137D3113yescomplemented
25#64158—313no—
26#64157—313no—
27C2×#3755—313no—
28#63663—313no—
29#63650—313no—
30#63649—313no—
31C22×C3⋊(C3⋊D3)—21no—
32C33×Q8—21no—
33C33×Q8—21no—
34C33×Q8D411yescharacteristic
35C33×D4—21no—
36C33×D4D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C33×C8—21no—
43C3⋊(C3⋊(C3⋊C8))—21no—
44C3⋊(C3⋊(C3⋊C8))—21no—
45C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
46C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
47#5203—913no—
48C32×#41C2×D3113yes—
49C2×C32×Q8C2×D3113yes—
50C2×C4⋊(C3⋊D3)—313no—
51C32×Q16—213no—
52C32×SD16—213no—
53C32×M16C2×D3113yes—
54#3755—613no—
55#3755—613no—
56#3755—613no—
57#3753—613no—
58#3748—313no—
59#3748—313no—
60#3748—613no—
61C2×C3⋊(C3⋊C8)—313no—
62C2×C3⋊(C3⋊C8)—613no—
63C22×C33—21no—
64C22×C33#4111yescharacteristic
65C2×C3⋊(C3⋊D3)—41no—
66C2×C3⋊(C3⋊D3)—81no—
67C33×C4—21no—
68C33×C4—41no—
69C33×C4C2×D411yescharacteristic
70C33×C4C2×D411yescharacteristic
71C3×#137C3⋊D3113yescomplemented
72#949—913no—
73#948—913no—
74C2×#215—913no—
75#836—913no—
76#823—913no—
77#822—913no—
78C22×C3⋊D3—613no—
79C32×Q8—213no—
80C32×Q8—213no—
81C32×Q8#67113yes—
82C32×D4—213no—
83C32×D4#67113yes—
84C2×C32×C4—213no—
85C2×C32×C4—213no—
86C2×C32×C4C22×D3113yes—
87C4⋊(C3⋊D3)—613no—
88C4⋊(C3⋊D3)—613no—
89C32×C8—213no—
90C3⋊(C3⋊C8)—313no—
91C3⋊(C3⋊C8)—313no—
92C3⋊(C3⋊C8)—613no—
93C3⋊(C3⋊C8)—613no—
94#471—271noSylow
95C2×C33—21no—
96C2×C33—41no—
97C2×C33#12111yescharacteristic
98C3⋊(C3⋊D3)—81no—
99C3×#41C2×C3⋊D3113yes—
100C2×C3×Q8C2×C3⋊D3113yes—
101C2×D12—913no—
102C3×Q16—213no—
103C3×SD16—213no—
104C3×M16C2×C3⋊D3113yes—
105#215—1813no—
106#215—1813no—
107#215—1813no—
108#213—1813no—
109#208—913no—
110#208—913no—
111#208—1813no—
112C2×C3⋊C8—913no—
113C2×C3⋊C8—1813no—
114C22×C32—213no—
115C22×C32#237113yes—
116C2×C3⋊D3—1213no—
117C2×C3⋊D3—2413no—
118C32×C4—213no—
119C32×C4—413no—
120C32×C4D3×D4113yes—
121C32×C4C2×#67113yes—
122#137C3⋊(C3⋊D3)11yescharacteristic, complemented
123AGL(1,ℤ/8)—271no—
124C2×SD16—271no—
125#131—271no—
126#108—271no—
127#101—271no—
128#100—271no—
129C33#47111yescharacteristic, complemented, Sylow
130C22×D3—1813no—
131C3×Q8—213no—
132C3×Q8—213no—
133C3×Q8#638113yes—
134C3×D4—213no—
135C3×D4#638113yes—
136C2×C3×C4—213no—
137C2×C3×C4—213no—
138C2×C3×C4C22×C3⋊D3113yes—
139D12—1813no—
140D12—1813no—
141C3×C8—213no—
142C3⋊C8—913no—
143C3⋊C8—913no—
144C3⋊C8—1813no—
145C3⋊C8—1813no—
146C2×C32—213no—
147C2×C32—413no—
148C2×C32#939113yes—
149C3⋊D3—2413no—
150#41C2×C3⋊(C3⋊D3)11yescharacteristic
151C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
152C2×D4—271no—
153Q16—21no—
154SD16—21no—
155SD16—541no—
156SD16—541no—
157SD16—541no—
158D8—541no—
159M16—271no—
160M16—271no—
161M16—541no—
162M16C2×C3⋊(C3⋊D3)11yescharacteristic
163C2×C8—271no—
164C2×C8—541no—
165C22×C3—213no—
166C22×C3#3830113yes—
167C2×D3—3613no—
168C2×D3—7213no—
169C3×C4—213no—
170C3×C4—413no—
171C3×C4D4×C3⋊D3113yes—
172C3×C4C2×#638113yes—
173C32#5203113yescomplemented
174C23—541no—
175Q8—21no—
176Q8—21no—
177Q8#631211yescharacteristic
178D4—21no—
179D4—541no—
180D4—541no—
181D4#631211yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
185C8—21no—
186C8—271no—
187C8—271no—
188C8—541no—
189C8—541no—
190C2×C3—213no—
191C2×C3—413no—
192C2×C3#64148113yes—
193D3—7213no—
194C22—21no—
195C22—1081no—
196C22—2161no—
197C22#8907811yescharacteristic
198C4—21no—
199C4—41no—
200C4D4×C3⋊(C3⋊D3)11yescharacteristic
201C4C2×#631211yescharacteristic
202C3#10592902113yescomplemented
203C2—21no—
204C2—41no—
205C2—2161no—
206C2#1172398011yescharacteristic
subgroupG/H
centerC2#11723980
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#137C2
socleC2×C33#121
prime radical#11723990C2
hypercenter#137C3⋊(C3⋊D3)
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C33×#137C2semidirect
3. #11723989C2semidirect
4. C2×#88862C2semidirect
5. #11722206C2semidirect
6. #11722193C2semidirect
7. #11722192C2semidirect
24. C32×#137D3semidirect
71. C3×#137C3⋊D3semidirect
122. #137C3⋊(C3⋊D3)semidirect
129. C33#471semidirect
173. C32#5203semidirect
202. C3#10592902semidirect

Maximal subgroups

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

subgroupindex
C33×#1372
#117239902
#117239892
C2×#888622
#117222062
#117221932
#117221922
#105929023