Finite Group Databasebeta

← #13163810

Subgroups — #13163810

Order 1728.

1820 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 407 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
1C2×C33×SD16C211yesmaximal, characteristic, complemented
2#11723988C211yesmaximal, characteristic, complemented
3C2×#88863C211yesmaximal, characteristic, complemented
4#11723977C211yesmaximal, characteristic, complemented
5C2×#88860C211yesmaximal, characteristic
6C2×#88851C211yesmaximal, characteristic, complemented
7C8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8#10592867—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13C33×SD16—22no—
14C2×C33×C8C2211yescharacteristic
15#88864—22no—
16#88863—22no—
17#88860—22no—
18#88859—22no—
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88851—22no—
22C2×C32×SD16D3113yescomplemented
23#64156—313no—
24C2×#3756—313no—
25#64145—313no—
26C2×#3753—313no—
27C2×#3744—313no—
28C8×C3⋊Dic3—313no—
29C23×C33—21no—
30C22×C3⋊(C3⋊D3)—21no—
31C33×Q8—21no—
32C33×Q8—21no—
33C33×D4—21no—
34C33×D4—21no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—21no—
38C2×C3⋊(C3⋊Dic3)D412yes—
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))—21no—
41#6308—21no—
42#6308—21no—
43C33×C8D412yes—
44C3⋊(C3⋊(C3⋊C8))D412yes—
45#5168—913no—
46C2×C32×Q8C2×D3113yes—
47C2×C32×D4C2×D3113yes—
48C2×C4⋊(C3⋊D3)—313no—
49C2×#634—313no—
50C32×SD16—226no—
51C2×C32×C8C2×D3113yes—
52#3757—626no—
53#3756—626no—
54#3753—626no—
55#3752—626no—
56C4×C3⋊Dic3—313no—
57C2×C3⋊(C3⋊C8)—313no—
58#3744—626no—
59C22×C33—41no—
60C22×C33—42no—
61C22×C33C2×D411yescharacteristic
62C2×C3⋊(C3⋊D3)—41no—
63C2×C3⋊(C3⋊D3)—42no—
64C33×C4—41no—
65C33×C4C2×D411yescharacteristic
66C33×C4C2×D411yescharacteristic
67C3⋊(C3⋊Dic3)—22no—
68C3⋊(C3⋊Dic3)—41no—
69C2×C3×SD16C3⋊D3113yescomplemented
70#947—913no—
71C2×#216—913no—
72#936—913no—
73C2×#213—913no—
74C2×#204—913no—
75C8×Dic3—913no—
76C23×C32—213no—
77C22×C3⋊D3—613no—
78C32×Q8—213no—
79C32×Q8—213no—
80C32×D4—213no—
81C32×D4—213no—
82C2×C32×C4—213no—
83C2×C32×C4C22×D3113yes—
84C2×C3⋊Dic3—326no—
85C2×C3⋊Dic3—613no—
86C4⋊(C3⋊D3)—613no—
87C4⋊(C3⋊D3)—613no—
88#634—613no—
89#634—613no—
90C32×C8#67126yes—
91C3⋊(C3⋊C8)—326no—
92#495—271noSylow
93C2×C33—81no—
94C2×C33#12711yescharacteristic
95C2×C33#13512yes—
96C3⋊(C3⋊D3)—81no—
97C2×C3×Q8C2×C3⋊D3113yes—
98C2×C3×D4C2×C3⋊D3113yes—
99C2×D12—913no—
100C2×Dic6—913no—
101C3×SD16—226no—
102C2×C3×C8C2×C3⋊D3113yes—
103#217—1826no—
104#216—1826no—
105#213—1826no—
106(C2×D3)⋊C4—1826no—
107C4×Dic3—913no—
108C2×C3⋊C8—913no—
109#204—1826no—
110C22×C32—426no—
111C22×C32—413no—
112C22×C32D3×D4113yes—
113C2×C3⋊D3—1226no—
114C2×C3⋊D3—1213no—
115C32×C4—413no—
116C32×C4D3×D4113yes—
117C32×C4C2×#67113yes—
118C3⋊Dic3—626no—
119C3⋊Dic3—1213no—
120C2×Q16—271no—
121C2×SD16—271no—
122C2×SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
123C2×D8—271no—
124#124—271no—
125#124—271no—
126C4×C8—271no—
127C33#49511yescharacteristic, complemented, Sylow
128C23×C3—213no—
129C22×D3—1813no—
130C3×Q8—213no—
131C3×Q8—213no—
132C3×D4—213no—
133C3×D4—213no—
134C2×C3×C4—213no—
135C2×C3×C4C22×C3⋊D3113yes—
136C2×Dic3—926no—
137C2×Dic3—1813no—
138D12—1813no—
139D12—1813no—
140Dic6—1813no—
141Dic6—1813no—
142C3×C8#638126yes—
143C3⋊C8—926no—
144C2×C32—813no—
145C2×C32#916126yes—
146C2×C32#940113yes—
147C3⋊D3—2413no—
148C2×Q8—271no—
149C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
150C2×D4—271no—
151C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
152Q16—542no—
153SD16—22no—
154SD16—542no—
155D8—542no—
156C2×C8—271no—
157C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
158C22⋊C4—542no—
159C22⋊C4—542no—
160C42—271no—
161C22×C3—426no—
162C22×C3—413no—
163C22×C3D4×C3⋊D3113yes—
164C2×D3—3626no—
165C2×D3—3613no—
166C3×C4—413no—
167C3×C4D4×C3⋊D3113yes—
168C3×C4C2×#638113yes—
169Dic3—1826no—
170Dic3—3613no—
171C32#5168113yescomplemented
172C23—21no—
173C23—541no—
174Q8—21no—
175Q8—21no—
176Q8—541no—
177Q8—541no—
178D4—21no—
179D4—21no—
180D4—541no—
181D4—541no—
182C2×C4—21no—
183C2×C4—272no—
184C2×C4—541no—
185C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
186C8—272no—
187C8#631212yes—
188C2×C3—813no—
189C2×C3#64125126yes—
190C2×C3#64149113yes—
191D3—7213no—
192C22—42no—
193C22—41no—
194C22—1082no—
195C22—1081no—
196C22D4×C3⋊(C3⋊D3)11yescharacteristic
197C4—41no—
198C4—542no—
199C4—1081no—
200C4D4×C3⋊(C3⋊D3)11yescharacteristic
201C4C2×#631211yescharacteristic
202C3#10592867113yescomplemented
203C2—81no—
204C2—2161no—
205C2#1172395712yes—
206C2#1172398111yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×#88860C2
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C33×SD16C2semidirect
2. #11723988C2semidirect
3. C2×#88863C2semidirect
4. #11723977C2semidirect
6. C2×#88851C2semidirect
7. C8×C3⋊(C3⋊Dic3)C2semidirect
22. C2×C32×SD16D3semidirect
69. C2×C3×SD16C3⋊D3semidirect
122. C2×SD16C3⋊(C3⋊D3)semidirect
127. C33#495semidirect
171. C32#5168semidirect
202. C3#10592867semidirect

Maximal subgroups

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

subgroupindex
C2×C33×SD162
#117239882
C2×#888632
#117239772
C2×#888602
C2×#888512
C8×C3⋊(C3⋊Dic3)2
#105928673