Finite Group Databasebeta

← #13165109

Subgroups — #13165109

Order 1728.

178 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 39 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#11724335C211yesmaximal, characteristic
2C4⋊(C3⋊AGL(1,𝔽9))C211yesmaximal, characteristic
3C3×#63744C211yesmaximal, characteristic
4#10593156—31nomaximal
5C2×C3⋊AGL(1,𝔽9)—21no—
6#89040—21no—
7C3×C4×#157C2211yescharacteristic
8C3×#3787—21no—
9#88940—21no—
10#88940C412yes—
11#64222—31no—
12C4⋊AGL(1,𝔽9)—31no—
13#63744D311yescharacteristic
14C2×C3×#157C2×C411yescharacteristic
15C2×C3×#157D411yescharacteristic
16C3⋊AGL(1,𝔽9)—41no—
17C3×C4×C3⋊D3D411yescharacteristic
18C3×#634—41no—
19#6296—41no—
20#6294—22no—
21C3×#622—41no—
22#4482—91nomaximal
23C2×AGL(1,𝔽9)—61no—
24#3797—61no—
25D3×Dic6—121no—
26C4×#157C2×D311yescharacteristic
27#3787—21no—
28#3773—32no—
29#3773—61no—
30C2×C3×C3⋊D3C22⋊C411yescharacteristic
31C3×#157—21no—
32C3×#157SD1611yescharacteristic
33C3×#157Q1611yescharacteristic
34C33×C4SD1611yescharacteristic
35C3×C3⋊Dic3—42no—
36C3×C3⋊Dic3D811yescharacteristic
37#869—91no—
38C3×C4⋊C8—91no—
39C4⋊(C3⋊C8)—91no—
40C2×#157C4×D311yescharacteristic
41C2×#157D1211yescharacteristic
42AGL(1,𝔽9)—121no—
43C4×C3⋊D3#6711yescharacteristic
44#634—41no—
45C3×C4×D3—121no—
46C3×Dic6—121no—
47C3×Dic6—121no—
48C3×Dic6—241no—
49#627—122no—
50#627—121no—
51#624—62no—
52D3×Dic3—122no—
53#622—41no—
54#332—271noSylow
55C2×C33#10211yescharacteristic
56C3×C3⋊D3#10111yescharacteristic
57C3×C3⋊D3#10311yescharacteristic
58D3×Q8—361no—
59C2×Dic6—181no—
60C2×C3×C8—181no—
61C3×C42—91no—
62C4⋊Dic3—92no—
63C4⋊Dic3—181no—
64C2×C3⋊C8—181no—
65C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
66C2×C3×D3—121no—
67#157—21no—
68#157#20411yescharacteristic
69#157Dic1211yescharacteristic
70C32×C4—41no—
71C32×C4—81no—
72C32×C4#21511yescharacteristic
73C3⋊Dic3—42no—
74C3⋊Dic3#21311yescharacteristic
75C3×Dic3—121no—
76C3×Dic3—122no—
77C3×Dic3—122no—
78C3×Dic3—242no—
79C4⋊Q8—271no—
80C4⋊C8—91no—
81C4⋊C8—271no—
82C33#33211yescharacteristic, complemented, Sylow
83C3×Q8—361no—
84C2×C3×C4—91no—
85C2×C3×C4—91no—
86C2×C3×C4—91no—
87C2×Dic3—182no—
88C4×D3—121no—
89C4×D3—362no—
90Dic6—121no—
91Dic6—121no—
92Dic6—241no—
93Dic6—361no—
94Dic6—362no—
95C3×C8—361no—
96C3⋊C8—361no—
97C2×C32—41no—
98C2×C32—81no—
99C2×C32#80911yescharacteristic
100C3⋊D3#81611yescharacteristic
101C3⋊D3#82411yescharacteristic
102C3×D3—121no—
103C3×D3—121no—
104C2×Q8—541no—
105C2×C8—181no—
106C2×C8—541no—
107C4⋊C4—272no—
108C4⋊C4—541no—
109C42—91no—
110C22×C3—91no—
111C2×D3—121no—
112C3×C4—41no—
113C3×C4—81no—
114C3×C4—91no—
115C3×C4—91no—
116C3×C4—91no—
117C3×C4—181no—
118C3×C4—362no—
119C3×C4AΓL(1,𝔽9)11yescharacteristic
120Dic3—121no—
121Dic3—122no—
122Dic3—122no—
123Dic3—242no—
124C32—41no—
125C32—81no—
126C32#448211yescharacteristic, complemented
127Q8—361no—
128Q8—1081no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—542no—
133C8—361no—
134C8—1081no—
135C2×C3—41no—
136C2×C3—81no—
137C2×C3—91no—
138C2×C3—91no—
139C2×C3GO+(4,𝔽2)⋊C411yescharacteristic
140D3—121no—
141D3—121no—
142C22—91no—
143C4—91no—
144C4—91no—
145C4—91no—
146C4—181no—
147C4—362no—
148C4#8909211yescharacteristic
149C3—41no—
150C3—81no—
151C3#1059315611yescharacteristic, complemented
152C2—91no—
153C2—91no—
154C2#1172410111yescharacteristic
subgroupG/H
centerC2#11724101
commutatorC2×C3×#157C2×C4
Frattini, hypercenterC4#89092
FittingC33×C4SD16
socleC2×C33#102
prime radicalC2×C3×C3⋊D3C22⋊C4
cosocleC3×C4×#157C22
hypocenterC33#332
LCSG ▷ C2×C3×#157 ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
82. C33#332semidirect
126. C32#4482semidirect
151. C3#10593156semidirect

Maximal subgroups

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

subgroupindex
#117243352
C4⋊(C3⋊AGL(1,𝔽9))2
C3×#637442
#105931563
#44829