Finite Group Databasebeta

← #13147360

Subgroups — #13147360

Order 1728.

331 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 75 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#11722448C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4#11720773C212yesmaximal, complemented
5C8×#1088C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C22⋊#1088C412yes—
8C2×#6188C412yescomplemented
9C4⋊#1088C412yes—
10C4×#1088C2212yes—
11#88626C412yescomplemented
12#88626C2211yescharacteristic
13#88626C2211yescharacteristic
14M16×H(3,𝔽3)C411yescharacteristic, complemented
15#88538C411yescharacteristic, complemented
16#88527C412yescomplemented
17C8×#272C2212yes—
18C3×D3×M16—62no—
19C22×#272C2×C411yescharacteristic
20C2×#1088C2×C414yes—
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×#1084C2×C411yescharacteristic
23C4×#272C2×C411yescharacteristic
24C4×#272C2×C411yescharacteristic
25C4×#272C2×C411yescharacteristic
26C4×#272C2311yescharacteristic
27#6188C2×C412yes—
28#6188C2×C412yes—
29C8×H(3,𝔽3)C2×C412yes—
30#6180C2×C412yes—
31C3×#405—91nomaximal
32C2×C3×C4×D3—62no—
33C32×M16—22no—
34C3×#208—62no—
35C3×ΓL(1,𝔽25)—64no—
36C3×D3×C8—64no—
37C22×H(3,𝔽3)C4211yescharacteristic
38C2×#272—22no—
39C2×#272C4211yescharacteristic
40C2×#272C22×C411yescharacteristic
41#1088—24no—
42C4×H(3,𝔽3)C4211yescharacteristic
43C4×H(3,𝔽3)C22×C411yescharacteristic
44#1084C4211yescharacteristic
45#1084C22×C411yescharacteristic
46C2×C3×M16—91no—
47C22×C3×C8—91no—
48C3×#117—91no—
49D3×M16—62no—
50C3×#97—92no—
51C3×C4×C8—92no—
52C22×C3×D3—62no—
53C2×C32×C4—22no—
54C2×C3×Dic3—62no—
55C3×C4×D3—62no—
56C3×C4×D3—62no—
57C3×C4×D3—62no—
58C3×C4×D3—62no—
59C32×C8—24no—
60C3×C3⋊C8—64no—
61#405—91noSylow
62C2×H(3,𝔽3)—21no—
63C2×H(3,𝔽3)C2×C4211yescharacteristic
64#272—21no—
65#272#13112yes—
66C22×C3×C4—91no—
67C2×C4×D3—62no—
68C3×M16—62no—
69C3×M16—91no—
70C3×M16—92no—
71C3×M16#15711yescharacteristic
72C2×C3×C8—92no—
73C2×C3×C8—91no—
74C2×C3×C8—91no—
75C2×C3×C8—92no—
76C2×C3×C8—92no—
77C3×C4⋊C4—92no—
78C3×C22⋊C4—92no—
79C3×C42—92no—
80#208—62no—
81ΓL(1,𝔽25)—64no—
82D3×C8—64no—
83C22×C32—22no—
84C2×C3×D3—62no—
85C2×C3×D3—62no—
86C2×C3×D3—124no—
87C32×C4—22no—
88C32×C4—22no—
89C3×Dic3—62no—
90C3×Dic3—62no—
91C2×M16—91no—
92C22×C8—91no—
93#117—91no—
94#97—92no—
95C4×C8—92no—
96H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—62no—
99C2×C3×C4—62no—
100C2×C3×C4—91no—
101C2×C3×C4—91no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4—91no—
105C2×C3×C4—94no—
106C2×C3×C4C2×#15711yescharacteristic
107C2×Dic3—62no—
108C4×D3—62no—
109C4×D3—62no—
110C4×D3—62no—
111C4×D3—62no—
112C3×C8—64no—
113C3×C8—92no—
114C3×C8—92no—
115C3×C8—92no—
116C3×C8C2×#15712yes—
117C3⋊C8—64no—
118C2×C32—22no—
119C2×C32—42no—
120C3×D3—64no—
121C3×D3—122no—
122C22×C4—91no—
123M16—91no—
124M16—92no—
125M16#108811yescharacteristic, complemented
126C2×C8—91no—
127C2×C8—91no—
128C2×C8—92no—
129C2×C8—92no—
130C2×C8—92no—
131C4⋊C4—92no—
132C22⋊C4—92no—
133C42—92no—
134C22×C3—62no—
135C22×C3—91no—
136C22×C3—91no—
137C22×C3—182no—
138C22×C3C4×#15711yescharacteristic
139C2×D3—62no—
140C2×D3—62no—
141C2×D3—124no—
142C3×C4—62no—
143C3×C4—62no—
144C3×C4—91no—
145C3×C4—91no—
146C3×C4—184no—
147C3×C4C4×#15711yescharacteristic
148C3×C4C22×#15711yescharacteristic
149Dic3—62no—
150Dic3—62no—
151C32—22no—
152C23—91no—
153C2×C4—91no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4—91no—
157C2×C4—91no—
158C2×C4—94no—
159C2×C4C2×#108811yescharacteristic
160C8—92no—
161C8—92no—
162C8—92no—
163C8C2×#108812yes—
164C2×C3—21no—
165C2×C3—62no—
166C2×C3—92no—
167C2×C3—122no—
168C2×C3—181no—
169C2×C3C2×C4×#15711yescharacteristic
170D3—64no—
171D3—122no—
172C22—91no—
173C22—91no—
174C22—182no—
175C22C4×#108811yescharacteristic
176C4—91no—
177C4—91no—
178C4—184no—
179C4C4×#108811yescharacteristic
180C4C22×#108811yescharacteristic
181C3—62no—
182C3#1059377511yescharacteristic
183C2—21no—
184C2—92no—
185C2—181no—
186C2C2×C4×#108811yescharacteristic
subgroupG/H
center, FrattiniC3×C4C22×#157
commutatorC2×H(3,𝔽3)C2×C42
FittingM16×H(3,𝔽3)C4
socleC2×C3C2×C4×#157
prime radicalC22×#272C2×C4
cosocleC4×#272C23
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. #11720773C2semidirect
5. C8×#1088C2semidirect
8. C2×#6188C4semidirect
11. #88626C4semidirect
14. M16×H(3,𝔽3)C4semidirect
15. #88538C4semidirect
16. #88527C4semidirect
96. H(3,𝔽3)#405semidirect
125. M16#1088semidirect

Maximal subgroups

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

subgroupindex
#117224482
C2×#886262
M16×#2722
#117207732
C8×#10882
C3×#4059