Finite Group Databasebeta

← #13163802

Subgroups — #13163802

Order 1728.

1680 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1680 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
2#11723985C211yesmaximal, characteristic, complemented
3D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
4C2×#88861C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6#11722186C211yesmaximal, characteristic, complemented
7#11722184C211yesmaximal, characteristic, complemented
8#10592861—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12C2×#6308C2211yescharacteristic
13C33×SD16—22no—
14C2×C33×C8C2211yescharacteristic
15#88864—21no—
16#88861—22no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18#88857—21no—
19#88857—21no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C2×C32×SD16D3113yes—
23#64153—313no—
24D4×C3⋊Dic3—313no—
25C2×#3754—313no—
26#63662—313no—
27#63643—313no—
28#63641—313no—
29C23×C33—21no—
30C33×Q8—21no—
31C33×Q8—21no—
32C33×D4—21no—
33C33×D4D412yes—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3⋊(C3⋊Dic3)—21no—
38C2×C3⋊(C3⋊Dic3)—42no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41#6308—21no—
42#6308—21no—
43C33×C8—21no—
44C3⋊(C3⋊(C3⋊C8))—21no—
45#5162—913no—
46C2×C32×Q8C2×D3113yes—
47C2×C32×D4C2×D3113yes—
48C22×C3⋊Dic3—613no—
49C2×#634—313no—
50C32×SD16—226no—
51C2×C32×C8C2×D3113yes—
52#3757—613no—
53#3754—626no—
54C4⋊(C3⋊Dic3)—313no—
55#3750—613no—
56#3750—613no—
57C4×C3⋊Dic3—313no—
58C2×C3⋊(C3⋊C8)—313no—
59C22×C33—22no—
60C22×C33—41no—
61C22×C33—41no—
62C22×C33C2×D411yescharacteristic
63C33×C4—41no—
64C33×C4C2×D411yescharacteristic
65C33×C4#4111yescharacteristic
66C3⋊(C3⋊Dic3)—21no—
67C3⋊(C3⋊Dic3)—41no—
68C3⋊(C3⋊Dic3)—41no—
69C3⋊(C3⋊Dic3)SD1612yescomplemented
70C2×C3×SD16C3⋊D3113yes—
71#944—913no—
72D4×Dic3—913no—
73C2×#214—913no—
74#835—913no—
75#816—913no—
76Dic3⋊C8—913no—
77C23×C32—213no—
78C32×Q8—213no—
79C32×Q8—213no—
80C32×D4—213no—
81C32×D4#67126yes—
82C2×C32×C4—213no—
83C2×C32×C4C22×D3113yes—
84C2×C3⋊Dic3—313no—
85C2×C3⋊Dic3—313no—
86C2×C3⋊Dic3—613no—
87C2×C3⋊Dic3—613no—
88C2×C3⋊Dic3—1226no—
89#634—613no—
90#634—613no—
91C32×C8—213no—
92C3⋊(C3⋊C8)—613no—
93#461—271noSylow
94C2×C33—42no—
95C2×C33#12111yescharacteristic
96C2×C33C2×SD1611yescharacteristic
97C2×C33#13711yescharacteristic
98C2×C3×Q8C2×C3⋊D3113yes—
99C2×C3×D4C2×C3⋊D3113yes—
100C22×Dic3—1813no—
101C2×Dic6—913no—
102C3×SD16—226no—
103C2×C3×C8C2×C3⋊D3113yes—
104#217—1813no—
105#214—1826no—
106C4⋊Dic3—913no—
107Dic3⋊C4—1813no—
108Dic3⋊C4—1813no—
109C4×Dic3—913no—
110C2×C3⋊C8—913no—
111C22×C32—226no—
112C22×C32—413no—
113C22×C32—413no—
114C22×C32D3×D4113yes—
115C32×C4—413no—
116C32×C4#237113yes—
117C32×C4C2×#67113yes—
118C3⋊Dic3—326no—
119C3⋊Dic3—613no—
120C3⋊Dic3—1213no—
121C3⋊Dic3—1213no—
122C2×SD16—271no—
123C2×SD16C3⋊(C3⋊D3)11yescharacteristic
124C4⋊Q8—271no—
125C4×D4—271no—
126C4⋊C8—271no—
127#103—271no—
128#103—271no—
129C33#46111yescharacteristic, complemented, Sylow
130C23×C3—213no—
131C3×Q8—213no—
132C3×Q8—213no—
133C3×D4—213no—
134C3×D4#638126yes—
135C2×C3×C4—213no—
136C2×C3×C4C22×C3⋊D3113yes—
137C2×Dic3—913no—
138C2×Dic3—913no—
139C2×Dic3—1813no—
140C2×Dic3—1813no—
141C2×Dic3—3626no—
142Dic6—1813no—
143Dic6—1813no—
144C3×C8—213no—
145C3⋊C8—1813no—
146C2×C32—426no—
147C2×C32D3×SD16113yes—
148C2×C32#915113yes—
149C2×C32#939113yes—
150C2×Q8—271no—
151C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
152C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
153C22×C4—541no—
154SD16—22no—
155SD16—542no—
156C2×C8—271no—
157C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
158C4⋊C4—271no—
159C4⋊C4—541no—
160C4⋊C4—541no—
161C22⋊C4—541no—
162C42—271no—
163C22×C3—226no—
164C22×C3—413no—
165C22×C3—413no—
166C22×C3D4×C3⋊D3113yes—
167C3×C4—413no—
168C3×C4#3830113yes—
169C3×C4C2×#638113yes—
170Dic3—926no—
171Dic3—1813no—
172Dic3—3613no—
173Dic3—3613no—
174C32#5162113yescomplemented
175C23—21no—
176Q8—21no—
177Q8—21no—
178Q8—541no—
179Q8—541no—
180D4—21no—
181D4#631212yes—
182C2×C4—21no—
183C2×C4—271no—
184C2×C4—271no—
185C2×C4—541no—
186C2×C4—541no—
187C2×C4—1082no—
188C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
189C8—21no—
190C8—541no—
191C2×C3—426no—
192C2×C3SD16×C3⋊D3113yes—
193C2×C3#64124113yes—
194C2×C3#64148113yes—
195C22—22no—
196C22—41no—
197C22—41no—
198C22D4×C3⋊(C3⋊D3)11yescharacteristic
199C4—41no—
200C4—272no—
201C4—541no—
202C4—1081no—
203C4—1081no—
204C4#8907811yescharacteristic
205C4C2×#631211yescharacteristic
206C3#10592861113yescomplemented
207C2—42no—
208C2SD16×C3⋊(C3⋊D3)11yescharacteristic
209C2#1172395611yescharacteristic
210C2#1172398011yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33#461
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
2. #11723985C2semidirect
5. #11722205C2semidirect
6. #11722186C2semidirect
7. #11722184C2semidirect
69. C3⋊(C3⋊Dic3)SD16semidirect
129. C33#461semidirect
174. C32#5162semidirect
206. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
C2×C33×SD162
#117239852
D4×C3⋊(C3⋊Dic3)2
C2×#888612
#117222052
#117221862
#117221842
#105928613