Finite Group Databasebeta

← #13146445

Subgroups — #13146445

Order 1728.

231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C22×#6249C212yesmaximal
2C2×C4×#1088C211yesmaximal, characteristic
3C22×#1088C411yescharacteristic
4C22×#1088C2211yescharacteristic
5C2×#6249—22no—
6C2×#6249—22no—
7C2×#6249C414yescomplemented
8C2×C4×#272C411yescharacteristic
9C4×#1088—22no—
10C22×#272C2×C411yescharacteristic
11C2×#1088—21no—
12C2×#1088—21no—
13C2×#1088C812yescomplemented
14C2×#1088C2×C412yes—
15C2×#1088D411yescharacteristic
16C2×#1088D412yes—
17C2×#1088Q811yescharacteristic
18#6249—24no—
19C2×C4×H(3,𝔽3)C811yescharacteristic, complemented
20C2×#1084C811yescharacteristic, complemented
21C4×#272—21no—
22C4×#272—21no—
23#4524—91nomaximal
24C2×C3×C4×D3—121no—
25C22×H(3,𝔽3)C2×C811yescharacteristic
26C2×#272C4211yescharacteristic
27C2×#272C22⋊C411yescharacteristic
28C2×#272C4⋊C411yescharacteristic
29C2×#272C2×C811yescharacteristic
30C2×#272M1611yescharacteristic
31C2×#272M1611yescharacteristic
32#1088—22no—
33#1088C22⋊C412yes—
34#1088C4⋊C412yes—
35C4×H(3,𝔽3)—21no—
36#1084—21no—
37C2×C3×C42—91no—
38C22×C3⋊C8—92no—
39C22×C3×D3—121no—
40C2×C32×C4—41no—
41C2×C3×Dic3—121no—
42C3×C4×D3—122no—
43C3×C4×D3—122no—
44#336—271noSylow
45C2×H(3,𝔽3)C4×C811yescharacteristic
46C2×H(3,𝔽3)C22⋊C811yescharacteristic
47C2×H(3,𝔽3)C4⋊C811yescharacteristic
48#272#9511yescharacteristic
49#272#9711yescharacteristic
50#272C22⋊C811yescharacteristic
51#272C4⋊C811yescharacteristic
52C22×C3×C4—91no—
53C22×C3×C4—91no—
54C22×C3×C4—91no—
55C2×C4×D3—121no—
56C3×C42—182no—
57C2×C3⋊C8—94no—
58C2×C3⋊C8—182no—
59C2×C3⋊C8—182no—
60C22×C32—41no—
61C2×C3×D3—121no—
62C2×C3×D3—121no—
63C2×C3×D3—121no—
64C2×C3×D3—121no—
65C2×C3×D3—121no—
66C2×C3×D3—121no—
67C32×C4—42no—
68C3×Dic3—122no—
69C22×C8—272no—
70C2×C42—91no—
71H(3,𝔽3)#33611yescharacteristic, complemented, Sylow
72C23×C3—91no—
73C22×D3—121no—
74C2×C3×C4—92no—
75C2×C3×C4—91no—
76C2×C3×C4—92no—
77C2×C3×C4—91no—
78C2×C3×C4—92no—
79C2×C3×C4—91no—
80C2×C3×C4—121no—
81C2×C3×C4—181no—
82C2×C3×C4—181no—
83C2×C3×C4—181no—
84C2×C3×C4—181no—
85C2×C3×C4AGL(1,𝔽9)11yescharacteristic
86C2×Dic3—121no—
87C4×D3—122no—
88C4×D3—122no—
89C3⋊C8—184no—
90C2×C32—41no—
91C2×C32—41no—
92C2×C32—41no—
93C3×D3—121no—
94C3×D3—121no—
95C3×D3—121no—
96C3×D3—121no—
97C22×C4—91no—
98C22×C4—91no—
99C22×C4—91no—
100C2×C8—274no—
101C2×C8—542no—
102C2×C8—542no—
103C42—182no—
104C22×C3—91no—
105C22×C3—91no—
106C22×C3—91no—
107C22×C3—91no—
108C22×C3—91no—
109C22×C3—91no—
110C22×C3—121no—
111C22×C3C2×AGL(1,𝔽9)11yescharacteristic
112C2×D3—121no—
113C2×D3—121no—
114C2×D3—121no—
115C2×D3—121no—
116C2×D3—121no—
117C2×D3—121no—
118C3×C4—21no—
119C3×C4—92no—
120C3×C4—92no—
121C3×C4—122no—
122C3×C4—182no—
123C3×C4—181no—
124Dic3—122no—
125C32—41no—
126C23—91no—
127C2×C4—92no—
128C2×C4—92no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—92no—
132C2×C4—91no—
133C2×C4—181no—
134C2×C4—181no—
135C2×C4—181no—
136C2×C4—181no—
137C2×C4#624911yescharacteristic, complemented
138C8—544no—
139C2×C3—91no—
140C2×C3—91no—
141C2×C3—91no—
142C2×C3—91no—
143C2×C3—121no—
144C2×C3—121no—
145C2×C3—121no—
146C2×C3C4×AGL(1,𝔽9)11yescharacteristic
147C2×C3C4⋊AGL(1,𝔽9)11yescharacteristic
148C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
149D3—121no—
150D3—121no—
151D3—121no—
152D3—121no—
153C22—91no—
154C22—91no—
155C22—91no—
156C22—91no—
157C22—91no—
158C22—91no—
159C22C2×#624911yescharacteristic
160C4—21no—
161C4—92no—
162C4—92no—
163C4—182no—
164C4—181no—
165C3—121no—
166C3#1059326711yescharacteristic
167C2—91no—
168C2—91no—
169C2—91no—
170C2—91no—
171C2C4×#624911yescharacteristic
172C2C4⋊#624911yescharacteristic
173C2C22⋊#624911yescharacteristic
subgroupG/H
centerC22C2×#6249
commutatorC2×H(3,𝔽3)C4×C8
Frattini, socleC22×C3C2×AGL(1,𝔽9)
FittingC2×C4×H(3,𝔽3)C8
prime radicalC22×#272C2×C4
cosocleC22×#1088C22
hypercenterC2×C4#6249
hypocenterH(3,𝔽3)#336
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. C2×#6249C4semidirect
13. C2×#1088C8semidirect
19. C2×C4×H(3,𝔽3)C8semidirect
20. C2×#1084C8semidirect
71. H(3,𝔽3)#336semidirect
137. C2×C4#6249semidirect

Maximal subgroups

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

subgroupindex
C22×#62492
C2×C4×#10882
#45249