Finite Group Databasebeta

← #13173185

Subgroups — #13173185

Order 1728.

608 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 117 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×C32×#41⋊C3C211yesmaximal, characteristic
2SL(2,𝔽3)×C3⋊Dic3C212yesmaximal, complemented
3#10596298C311yesmaximal, characteristic, complemented
4#5827⋊C3—34nomaximal
5C32×#41⋊C3C412yescomplemented
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3×C4⋊(C3⋊Dic3)—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C2×C3×#41⋊C3—24no—
10C2×C3×#41⋊C3D314yes—
11Q8×C3⋊Dic3C2×C312yes—
12D4×C3⋊Dic3—32no—
13#64135—31no—
14C2×C4⋊(C3⋊Dic3)—31no—
15Dic3×SL(2,𝔽3)—38no—
16C2×C33×C4—41no—
17C2×C3×C3⋊Dic3—42no—
18C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
19#5827—34no—
20#520⋊C3—91no—
21C32×#41—32no—
22C32×#41C3×C412yescomplemented
23C2×C32×Q8C22×C311yescharacteristic
24C2×C32×D4—31no—
25C22×C32×C4—31no—
26C22×C3⋊Dic3—32no—
27C3×#41⋊C3—28no—
28C3×#41⋊C3Dic318yescomplemented
29C2×C3×SL(2,𝔽3)—24no—
30C2×C3×SL(2,𝔽3)C2×D314yes—
31#3757—32no—
32C4⋊(C3⋊Dic3)—31no—
33C4⋊(C3⋊Dic3)—32no—
34C4⋊(C3⋊Dic3)A411yescharacteristic
35C4×C3⋊Dic3—32no—
36C3×C4⋊Dic3—124no—
37C22×C33—41no—
38C33×C4—42no—
39C3×C3⋊Dic3—82no—
40C2×C3×#41C3×D314yes—
41C2×#41⋊C3—24no—
42C2×#41⋊C3C3⋊D311yescharacteristic
43Q8×Dic3—38no—
44D4×Dic3—98no—
45#926—94no—
46C2×C4⋊Dic3—94no—
47C4×SL(2,𝔽3)—92no—
48C23×C32—31no—
49C32×Q8—31no—
50C32×Q8C2×C3×C411yescharacteristic
51C32×D4—32no—
52C32×D4—32no—
53C2×C32×C4—31no—
54C2×C32×C4—32no—
55C2×C32×C4—32no—
56C2×C32×C4—44no—
57C2×C32×C4—84no—
58C2×C32×C4C2×A411yescharacteristic
59C2×C3⋊Dic3—32no—
60C2×C3⋊Dic3—122no—
61C2×C3⋊Dic3C2×A412yes—
62C2×C3×Dic3—128no—
63C3×SL(2,𝔽3)—24no—
64C3×SL(2,𝔽3)C2×Dic314yescomplemented
65#520—91noSylow
66C2×C33—41no—
67C2×C33—41no—
68C2×C33—41no—
69C3×#41—38no—
70C3×#41C3×Dic318yescomplemented
71C2×C3×Q8C2×C3×D314yes—
72C2×C3×D4—34no—
73C22×C3×C4—34no—
74C22×Dic3—98no—
75#41⋊C3—28no—
76#41⋊C3C3⋊Dic312yescomplemented
77C2×SL(2,𝔽3)—24no—
78C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
79C3×C4⋊C4—361no—
80#217—98no—
81C4⋊Dic3—34no—
82C4⋊Dic3—94no—
83C4⋊Dic3—98no—
84C4×Dic3—98no—
85C22×C32—32no—
86C22×C32—44no—
87C22×C32—61no—
88C22×C32—61no—
89C22×C32—84no—
90C22×C32C22×A411yescharacteristic
91C32×C4—31no—
92C32×C4—31no—
93C32×C4—48no—
94C32×C4—88no—
95C32×C4C4×A412yes—
96C3⋊Dic3—22no—
97C3⋊Dic3—62no—
98C3×Dic3—248no—
99C2×#41C3×C3⋊D311yescharacteristic
100C4×Q8—92no—
101C4×D4—272no—
102#117—271no—
103C2×C4⋊C4—271no—
104C33—41noSylow
105C23×C3—34no—
106C3×Q8—34no—
107C3×Q8C2×C3×Dic314yes—
108C3×D4—38no—
109C3×D4—38no—
110C2×C3×C4—34no—
111C2×C3×C4—38no—
112C2×C3×C4—38no—
113C2×C3×C4—41no—
114C2×C3×C4—84no—
115C2×C3×C4—362no—
116C2×C3×C4D3×A414yes—
117C2×Dic3—38no—
118C2×Dic3—98no—
119C2×Dic3—368no—
120SL(2,𝔽3)—24no—
121SL(2,𝔽3)C2×C3⋊Dic311yescharacteristic, complemented
122C2×C32—44no—
123C2×C32—44no—
124C2×C32—44no—
125C2×C32—62no—
126C2×C32—84no—
127C2×C32—84no—
128C2×C32—84no—
129C2×C32C2×C4×A411yescharacteristic
130C2×C32#99411yescharacteristic
131C2×C322−1+4⋊C311yescharacteristic
132#41—32no—
133#41C3×C3⋊Dic312yescomplemented
134C2×Q8C2×C3×C3⋊D311yescharacteristic
135C2×D4—31no—
136C22×C4—31no—
137C22×C4—272no—
138C4⋊C4—91no—
139C4⋊C4—272no—
140C4⋊C4—271no—
141C22⋊C4—272no—
142C42—272no—
143C22×C3—38no—
144C22×C3—41no—
145C22×C3—64no—
146C22×C3—64no—
147C22×C3—84no—
148C22×C3C2×D3×A414yes—
149C3×C4—34no—
150C3×C4—34no—
151C3×C4—42no—
152C3×C4—88no—
153C3×C4—722no—
154C3×C4Dic3×A418yes—
155Dic3—68no—
156Dic3—188no—
157C32—44no—
158C32—84no—
159C32#520⋊C311yescharacteristic, complemented
160C23—31no—
161Q8—31no—
162Q8C2×C3×C3⋊Dic311yescharacteristic
163D4—32no—
164D4—32no—
165C2×C4—31no—
166C2×C4—32no—
167C2×C4—32no—
168C2×C4—92no—
169C2×C4—272no—
170C2×C4—1082no—
171C2×C4A4×C3⋊D311yescharacteristic
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—68no—
176C2×C3—84no—
177C2×C3—84no—
178C2×C3—84no—
179C2×C3C3⋊#99414yes—
180C2×C3#1010⋊C314yes—
181C2×C3C2×Dic3×A414yes—
182C22—32no—
183C22—61no—
184C22—61no—
185C22C2×A4×C3⋊D311yescharacteristic
186C4—31no—
187C4—31no—
188C4—182no—
189C4—542no—
190C4A4×C3⋊Dic312yes—
191C3—41no—
192C3—84no—
193C3#5827⋊C314yescomplemented
194C2—62no—
195C2#1172449111yescharacteristic
196C2#1172449311yescharacteristic
197C2C2×A4×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, hypocenterC32×Q8C2×C3×C4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
prime radicalC2×C32×#41⋊C3C2
cosocleC2×C32×Q8C22×C3
hypercenterC2×C4A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. #10596298C3semidirect
5. C32×#41⋊C3C4semidirect
18. C32×SL(2,𝔽3)C2×C4semidirect
22. C32×#41C3×C4semidirect
28. C3×#41⋊C3Dic3semidirect
64. C3×SL(2,𝔽3)C2×Dic3semidirect
70. C3×#41C3×Dic3semidirect
76. #41⋊C3C3⋊Dic3semidirect
121. SL(2,𝔽3)C2×C3⋊Dic3semidirect
133. #41C3×C3⋊Dic3semidirect
159. C32#520⋊C3semidirect
193. C3#5827⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
SL(2,𝔽3)×C3⋊Dic32
#105962983
#5827⋊C33
C3×C4⋊(C3⋊Dic3)4