Finite Group Databasebeta

← #13136957

Subgroups — #13136957

Order 1728.

401 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 23 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
1#11721876C211yesmaximal, characteristic
2#11720794C211yesmaximal, characteristic, complemented
3#11720792C211yesmaximal, characteristic, complemented
4C24×H(3,𝔽3)C411yescharacteristic, complemented
5C2×#6235—21no—
6C2×#6235C2211yescharacteristic
7C22⋊#1088—21no—
8#88631—21no—
9#88552—21no—
10#88552C411yescharacteristic, complemented
11C3×#953—62no—
12C23×H(3,𝔽3)—21no—
13C23×H(3,𝔽3)—42no—
14C23×H(3,𝔽3)—41no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C22×#272D411yescharacteristic
17C2×#1088—41no—
18#6235—41no—
19#6235—42no—
20C2×#1084—21no—
21C2×#1084D411yescharacteristic
22#6188—41no—
23C3×AΣL(1,𝔽16)—91nomaximal
24C24×C32—22no—
25C2×C3×#67—62no—
26C2×C3×#67—64no—
27C3×#217—62no—
28C3×#217—64no—
29C22×H(3,𝔽3)—22no—
30C22×H(3,𝔽3)—21no—
31C22×H(3,𝔽3)—41no—
32C22×H(3,𝔽3)—41no—
33C22×H(3,𝔽3)—42no—
34C22×H(3,𝔽3)—41no—
35C22×H(3,𝔽3)—42no—
36C22×H(3,𝔽3)C22⋊C411yescharacteristic
37C2×#272—21no—
38C2×#272—41no—
39#1088—81no—
40#1084—21no—
41#1084—41no—
42C3×#120—91no—
43#953—62no—
44C3×#100—91no—
45C3×#99—91no—
46C23×C32—24no—
47C23×C32—22no—
48C23×C32—48no—
49C23×C32—44no—
50C22×C3×D3—62no—
51C3×#67—124no—
52C3×#67—128no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—64no—
55AΣL(1,𝔽16)—91noSylow
56C2×H(3,𝔽3)—21no—
57C2×H(3,𝔽3)—42no—
58C2×H(3,𝔽3)—41no—
59C2×H(3,𝔽3)#9911yescharacteristic
60#272—41no—
61C24×C3—62no—
62C24×C3#15711yescharacteristic
63C2×C3×D4—91no—
64C2×C3×D4—181no—
65C2×#67—62no—
66C2×#67—64no—
67C3×M16—181no—
68C3×C22⋊C4—91no—
69C3×C22⋊C4—181no—
70C3×C22⋊C4—181no—
71#217—62no—
72#217—64no—
73C22×C32—28no—
74C22×C32—24no—
75C22×C32—22no—
76C22×C32—44no—
77C22×C32—44no—
78C22×C32—44no—
79C22×C32—48no—
80C22×C32—48no—
81C2×C3×D3—122no—
82C2×C3×D3—124no—
83C3×Dic3—122no—
84C3×Dic3—124no—
85#120—91no—
86#100—91no—
87#99—91no—
88H(3,𝔽3)AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
89C23×C3—21no—
90C23×C3—42no—
91C23×C3—41no—
92C23×C3—64no—
93C23×C3—62no—
94C23×C3—91no—
95C23×C3—128no—
96C23×C3—124no—
97C23×C3C2×#15711yescharacteristic
98C22×D3—62no—
99C3×D4—361no—
100C3×D4—362no—
101C2×C3×C4—91no—
102C2×C3×C4—181no—
103C2×C3×C4—361no—
104#67—124no—
105#67—128no—
106C2×Dic3—62no—
107C2×Dic3—64no—
108C3×C8—361no—
109C2×C32—24no—
110C2×C32—22no—
111C2×C32—48no—
112C2×C32—44no—
113C3×D3—242no—
114C24#108811yescharacteristic, complemented
115C2×D4—91no—
116C2×D4—181no—
117M16—181no—
118C22⋊C4—91no—
119C22⋊C4—181no—
120C22⋊C4—181no—
121C22×C3—22no—
122C22×C3—21no—
123C22×C3—41no—
124C22×C3—41no—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—41no—
128C22×C3—68no—
129C22×C3—64no—
130C22×C3—62no—
131C22×C3—124no—
132C22×C3—124no—
133C22×C3—124no—
134C22×C3—128no—
135C22×C3—128no—
136C22×C3—181no—
137C22×C3—361no—
138C22×C3C22⋊#15711yescharacteristic
139C2×D3—122no—
140C2×D3—124no—
141C3×C4—181no—
142C3×C4—361no—
143C3×C4—721no—
144Dic3—122no—
145Dic3—124no—
146C32—22no—
147C23—21no—
148C23—42no—
149C23—41no—
150C23—91no—
151C23C2×#108811yescharacteristic
152D4—361no—
153D4—362no—
154C2×C4—91no—
155C2×C4—181no—
156C2×C4—361no—
157C8—361no—
158C2×C3—21no—
159C2×C3—42no—
160C2×C3—41no—
161C2×C3—64no—
162C2×C3—62no—
163C2×C3—128no—
164C2×C3—124no—
165C2×C3—361no—
166C2×C3#6378611yescharacteristic
167D3—242no—
168C22—22no—
169C22—21no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22—42no—
174C22—41no—
175C22—181no—
176C22—361no—
177C22C22⋊#108811yescharacteristic
178C4—181no—
179C4—361no—
180C4—721no—
181C3—62no—
182C3#1059022311yescharacteristic
183C2—21no—
184C2—42no—
185C2—41no—
186C2—361no—
187C2#1172079211yescharacteristic
subgroupG/H
center, socleC2×C3#63786
commutatorC23×H(3,𝔽3)C2×C4
FrattiniC23×C3C2×#157
FittingC24×H(3,𝔽3)C4
prime radical#11721876C2
cosocleC2×#6235C22
hypercenterC24×C3#157
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 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. #11720794C2semidirect
3. #11720792C2semidirect
4. C24×H(3,𝔽3)C4semidirect
10. #88552C4semidirect
88. H(3,𝔽3)AΣL(1,𝔽16)semidirect
114. C24#1088semidirect

Maximal subgroups

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

subgroupindex
#117218762
#117207942
#117207922
C3×AΣL(1,𝔽16)9