Finite Group Databasebeta

← #13173369

Subgroups — #13173369

Order 1728.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#64116C211yesmaximal, characteristic
2#11723098C211yesmaximal, characteristic, complemented
3C2×C3⋊#3708C211yesmaximal, characteristic, complemented
4#11723090C212yesmaximal, complemented
5#11723089C212yesmaximal, complemented
6#10593772—31nomaximal
7C3×#3828C411yescharacteristic
8C33×M16C411yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×#3743C412yes—
11C3×C8×C3⋊D3C2212yes—
12C3⋊#3708C2211yescharacteristic
13C3⋊#3708C2212yes—
14C3⋊#3708C2211yescharacteristic
15#64116D311yescharacteristic
16C3×#907—62no—
17#63773—31no—
18C2×#3708—31no—
19#63765—32no—
20#63764—32no—
21C2×C33×C4C2×C411yescharacteristic
22C3×#638C812yes—
23C3×C4⋊(C3⋊D3)C811yescharacteristic
24C3×C4×C3⋊D3C2×C412yes—
25C3×#634C811yescharacteristic
26C33×C8C2×C412yes—
27C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29#5140—91nomaximal
30#3828Dic311yescharacteristic
31C3×#235—62no—
32C32×M16—22no—
33C32×M16—42no—
34C32×M16Dic311yescharacteristic
35C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
36#3743Dic312yes—
37C8×C3⋊D3C2×D312yes—
38C2×C3×C3⋊C8—62no—
39C3×ΓL(1,𝔽25)—64no—
40C3×D3×C8—64no—
41#3708—31no—
42#3708—32no—
43#3708—31no—
44C22×C33C2×C811yescharacteristic
45C2×C3×C3⋊D3C2×C812yes—
46C33×C4C2×C811yescharacteristic
47C33×C4C22×C411yescharacteristic
48C3×C3⋊Dic3C2×C812yes—
49C3×#131—91no—
50#907—62no—
51#812—92no—
52#812—91no—
53C2×C3⋊C16—92no—
54C2×C3⋊C16—91no—
55C2×C32×C4—22no—
56C2×C32×C4—42no—
57C2×C32×C4C2×Dic311yescharacteristic
58#638C3⋊C812yes—
59C4⋊(C3⋊D3)C3⋊C811yescharacteristic
60C4×C3⋊D3C2×Dic312yes—
61#634C3⋊C811yescharacteristic
62C3×#67—64no—
63C3×D12—62no—
64C3×C4×D3—64no—
65C3×Dic6—62no—
66C32×C8—24no—
67C32×C8—44no—
68C32×C8C2×Dic312yes—
69C3⋊(C3⋊C8)C2×Dic311yescharacteristic
70C3⋊(C3⋊C8)C22×D311yescharacteristic
71C3×C3⋊C8—62no—
72C3×C3⋊C8—62no—
73#504—271noSylow
74C2×C33—21no—
75C2×C33C22×C811yescharacteristic
76C3×C3⋊D3—22no—
77C3×#41—91no—
78#235—62no—
79C3×M16—22no—
80C3×M16—42no—
81C3×M16—92no—
82C3×M16#15711yescharacteristic
83C2×C3×C8—91no—
84C2×C3×C8—92no—
85C2×C3⋊C8—62no—
86ΓL(1,𝔽25)—64no—
87D3×C8—64no—
88C3⋊C16—91no—
89C3⋊C16—92no—
90C3⋊C16—91no—
91C22×C32—22no—
92C22×C32—42no—
93C22×C32C2×C3⋊C811yescharacteristic
94C2×C3⋊D3C2×C3⋊C812yes—
95C2×C3×D3—64no—
96C32×C4—22no—
97C32×C4—22no—
98C32×C4—42no—
99C32×C4—42no—
100C32×C4C2×C3⋊C811yescharacteristic
101C32×C4C22×Dic311yescharacteristic
102C3⋊Dic3C2×C3⋊C812yes—
103C3×Dic3—64no—
104#131—91no—
105M32—272no—
106M32—271no—
107C2×C16—272no—
108C2×C16—271no—
109C33#50411yescharacteristic, complemented, Sylow
110C3×Q8—91no—
111C3×D4—92no—
112C3×D4—91no—
113C2×C3×C4—22no—
114C2×C3×C4—42no—
115C2×C3×C4—92no—
116C2×C3×C4C2×#15711yescharacteristic
117#67—64no—
118D12—62no—
119C4×D3—64no—
120Dic6—62no—
121C3×C8—24no—
122C3×C8—44no—
123C3×C8—91no—
124C3×C8—91no—
125C3×C8C2×#15712yes—
126C3⋊C8—62no—
127C3⋊C8—62no—
128C2×C32—22no—
129C2×C32—21no—
130C2×C32—42no—
131C2×C32—42no—
132C2×C32—82no—
133C2×C32C22×C3⋊C811yescharacteristic
134C3⋊D3—22no—
135C3×D3—124no—
136#41—91no—
137M16—92no—
138M16C3⋊#15711yescharacteristic
139C2×C8—91no—
140C2×C8—92no—
141C16—271no—
142C16—272no—
143C16—271no—
144C22×C3—22no—
145C22×C3—42no—
146C22×C3—92no—
147C22×C3#378711yescharacteristic
148C2×D3—64no—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—42no—
152C3×C4—42no—
153C3×C4—92no—
154C3×C4#378711yescharacteristic
155C3×C4C22×#15711yescharacteristic
156Dic3—64no—
157C32—22no—
158C32—42no—
159C32#514011yescharacteristic, complemented
160Q8—91no—
161D4—92no—
162D4—91no—
163C2×C4—92no—
164C2×C4C2×C3⋊#15711yescharacteristic
165C8—91no—
166C8—91no—
167C8C2×C3⋊#15712yes—
168C2×C3—21no—
169C2×C3—22no—
170C2×C3—42no—
171C2×C3—42no—
172C2×C3—82no—
173C2×C3—182no—
174C2×C3C2×#378711yescharacteristic
175D3—124no—
176C22—92no—
177C22#8898811yescharacteristic
178C4—92no—
179C4#8898811yescharacteristic
180C4C22×C3⋊#15711yescharacteristic
181C3—22no—
182C3—42no—
183C3#1059377211yescharacteristic, complemented
184C2—21no—
185C2—182no—
186C2C2×#8898811yescharacteristic
subgroupG/H
center, FrattiniC4C22×C3⋊#157
commutator, socleC2×C33C22×C8
FittingC33×M16C4
prime radicalC3×#3828C4
cosocleC3×C3⋊(C3⋊C8)C23
hypercenterM16C3⋊#157
hypocenterC33#504
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. #11723098C2semidirect
3. C2×C3⋊#3708C2semidirect
4. #11723090C2semidirect
5. #11723089C2semidirect
109. C33#504semidirect
159. C32#5140semidirect
183. C3#10593772semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
#117230982
C2×C3⋊#37082
#117230902
#117230892
#105937723
#51409