Finite Group Databasebeta

← #13135901

Subgroups — #13135901

Order 1728.

517 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 73 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
1H(3,𝔽3)×#120C211yesmaximal, characteristic
2#11720580C211yesmaximal, characteristic, complemented
3#11720579C211yesmaximal, characteristic, complemented
4C3×#4536—34nomaximal
5C24×H(3,𝔽3)C411yescharacteristic, complemented
6C2×D4×H(3,𝔽3)—21no—
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C22⋊C4×H(3,𝔽3)—21no—
9C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
10#88552—21no—
11#88538—21no—
12C32×#120D314yes—
13C3×#834—34no—
14C3×#833—34no—
15C23×H(3,𝔽3)—21no—
16C23×H(3,𝔽3)—42no—
17C23×H(3,𝔽3)—41no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C23×H(3,𝔽3)D411yescharacteristic
20D4×H(3,𝔽3)—41no—
21D4×H(3,𝔽3)—42no—
22C2×C4×H(3,𝔽3)—21no—
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×#1084—41no—
25#6180—41no—
26C3×AΣL(1,𝔽16)—91no—
27#4536—34no—
28C24×C32Dic314yescomplemented
29C2×C32×D4—24no—
30C2×C32×D4C2×D314yes—
31C32×C22⋊C4—24no—
32C32×C22⋊C4Dic314yescomplemented
33C3×#217—64no—
34C3×#208—64no—
35C22×H(3,𝔽3)—22no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)—41no—
39C22×H(3,𝔽3)—41no—
40C22×H(3,𝔽3)—41no—
41C22×H(3,𝔽3)—42no—
42C22×H(3,𝔽3)—41no—
43C22×H(3,𝔽3)—42no—
44C22×H(3,𝔽3)C22⋊C411yescharacteristic
45C4×H(3,𝔽3)—21no—
46C4×H(3,𝔽3)—41no—
47#1084—81no—
48C3×#120—34no—
49C3×#120C3⋊D311yescharacteristic
50C3×#100—91no—
51C3×#99—91no—
52#834—34no—
53#833—34no—
54C23×C32—24no—
55C23×C32—48no—
56C23×C32—44no—
57C23×C32C2×Dic314yes—
58C23×C32#6714yes—
59C32×D4—44no—
60C32×D4—48no—
61C2×C32×C4—24no—
62C2×C32×C4#6714yes—
63C2×C3×Dic3—124no—
64C3×C3⋊C8—124no—
65AΣL(1,𝔽16)—91noSylow
66C2×H(3,𝔽3)—21no—
67C2×H(3,𝔽3)—41no—
68C2×H(3,𝔽3)—42no—
69C2×H(3,𝔽3)—41no—
70C2×H(3,𝔽3)#9911yescharacteristic
71C24×C3—34no—
72C24×C3C3⋊Dic311yescharacteristic
73C2×C3×D4—21no—
74C2×C3×D4—34no—
75C2×C3×D4—64no—
76C2×C3×D4C2×C3⋊D311yescharacteristic
77C3×M16—181no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—34no—
80C3×C22⋊C4—64no—
81C3×C22⋊C4—181no—
82C3×C22⋊C4C3⋊Dic311yescharacteristic
83#217—64no—
84#208—64no—
85C22×C32—28no—
86C22×C32—24no—
87C22×C32—24no—
88C22×C32—44no—
89C22×C32—44no—
90C22×C32—44no—
91C22×C32—48no—
92C22×C32—48no—
93C22×C32—44no—
94C22×C32#21714yes—
95C32×C4—24no—
96C32×C4—44no—
97C3×Dic3—244no—
98#120#27211yescharacteristic
99#100—91no—
100#99—91no—
101H(3,𝔽3)AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
102C23×C3—21no—
103C23×C3—34no—
104C23×C3—34no—
105C23×C3—42no—
106C23×C3—41no—
107C23×C3—64no—
108C23×C3—124no—
109C23×C3—128no—
110C23×C3C2×C3⋊Dic311yescharacteristic
111C23×C3#63811yescharacteristic
112C3×D4—41no—
113C3×D4—42no—
114C3×D4—124no—
115C3×D4—128no—
116C2×C3×C4—21no—
117C2×C3×C4—34no—
118C2×C3×C4—64no—
119C2×C3×C4—361no—
120C2×C3×C4#63811yescharacteristic
121C2×Dic3—124no—
122C3×C8—361no—
123C3⋊C8—124no—
124C2×C32—24no—
125C2×C32—44no—
126C2×C32—48no—
127C2×C32—44no—
128C2×C32#83414yes—
129C24#108411yescharacteristic, complemented
130C2×D4—21no—
131C2×D4C2×#27211yescharacteristic
132M16—181no—
133C22⋊C4—21no—
134C22⋊C4—181no—
135C22⋊C4#108411yescharacteristic, complemented
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—34no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3—41no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—41no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3—68no—
149C22×C3—124no—
150C22×C3—124no—
151C22×C3—124no—
152C22×C3—124no—
153C22×C3—128no—
154C22×C3—128no—
155C22×C3#375711yescharacteristic
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—64no—
159C3×C4—124no—
160C3×C4—721no—
161Dic3—244no—
162C32#453614yescomplemented
163C23—21no—
164C23—42no—
165C23—41no—
166C23C2×#108411yescharacteristic
167C23#623511yescharacteristic
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—361no—
172C2×C4#623511yescharacteristic
173C8—361no—
174C2×C3—21no—
175C2×C3—34no—
176C2×C3—41no—
177C2×C3—42no—
178C2×C3—41no—
179C2×C3—64no—
180C2×C3—124no—
181C2×C3—124no—
182C2×C3—128no—
183C2×C3#6366111yescharacteristic
184C22—22no—
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22#8855211yescharacteristic
194C4—21no—
195C4—41no—
196C4—721no—
197C3—34no—
198C3#1058940211yescharacteristic
199C2—21no—
200C2—41no—
201C2—42no—
202C2—41no—
203C2#1172058011yescharacteristic
subgroupG/H
center, socleC2×C3#63661
commutatorC23×H(3,𝔽3)C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×#120C2
cosocleC2×D4×H(3,𝔽3)C22
hypercenterC3×#120C3⋊D3
hypocenterH(3,𝔽3)AΣL(1,𝔽16)
LCSG ▷ C23×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
2. #11720580C2semidirect
3. #11720579C2semidirect
5. C24×H(3,𝔽3)C4semidirect
9. C22⋊C4×H(3,𝔽3)C4semidirect
28. C24×C32Dic3semidirect
32. C32×C22⋊C4Dic3semidirect
101. H(3,𝔽3)AΣL(1,𝔽16)semidirect
129. C24#1084semidirect
135. C22⋊C4#1084semidirect
162. C32#4536semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1202
#117205802
#117205792
C3×#45363