Finite Group Databasebeta

← #13146225

Subgroups — #13146225

Order 1728.

190 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 35 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
1SL(2,𝔽3)×Dic9C211yesmaximal, characteristic
2C9⋊#859C211yesmaximal, characteristic
3C9×#859C211yesmaximal, characteristic
4#10593196—31nomaximal
5#10588476—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7#88442—41nomaximal
8Dic3×SL(2,𝔽3)—31no—
9C3⋊#859—31no—
10C3×#859D311yescharacteristic
11Q8×Dic9D311yescharacteristic
12#63452—31no—
13#63421—61no—
14C9×#103—31no—
15#63395—31no—
16#63378—31no—
17Dic9⋊C8—31no—
18#63367—31no—
19C2×C3⋊Dic9—41no—
20C2×C9×Dic3—41no—
21C2×C3×Dic9—41no—
22C9×SL(2,𝔽3)Q811yescharacteristic
23#5389—91no—
24#4796—91no—
25C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
26#3724—121no—
27C2×Q8×C9C2×D311yescharacteristic
28C2×Q8⋊C9—21no—
29C2×C8×C9—31no—
30C9×C4⋊C4—31no—
31C4⋊Dic9—31no—
32C4⋊Dic9—31no—
33C4⋊Dic9—61no—
34Dic9⋊C4—61no—
35C4×Dic9—31no—
36C2×C9⋊C8—31no—
37C22×C3×C9—41no—
38C3⋊Dic9—42no—
39C9×Dic3—81no—
40C3×Dic9—81no—
41Q8×Dic3—31no—
42#890—91no—
43C4×SL(2,𝔽3)—91no—
44#859—91no—
45#859—181no—
46#859D911yescharacteristic
47C3×#103—31no—
48#835—91no—
49#818—91no—
50Dic3⋊C8—91no—
51#808—91no—
52C2×C3⋊Dic3—121no—
53C2×C3×Dic3—41no—
54C2×C3×Dic3—121no—
55C3×SL(2,𝔽3)Dic611yescharacteristic
56Q8×C9—31no—
57Q8×C9Dic611yescharacteristic
58C2×C4×C9—31no—
59C2×C4×C9—61no—
60C2×Dic9—31no—
61C2×Dic9—61no—
62C2×Dic9—241no—
63C2×Dic9S411yescharacteristic
64Q8⋊C9—21no—
65C8×C9—61no—
66C9⋊C8—61no—
67#479—271noSylow
68C2×C3×C9—41no—
69C2×C3×C9—41no—
70C2×C3×C9—41no—
71C2×C3×Q8D3211yescharacteristic
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)C2×D911yescharacteristic
74C2×C3×C8—31no—
75C3×C4⋊C4—31no—
76C4⋊Dic3—91no—
77C4⋊Dic3—91no—
78C4⋊Dic3—181no—
79Dic3⋊C4—181no—
80Dic3⋊C4—361no—
81C4×Dic3—91no—
82C2×C3⋊C8—91no—
83C22×C32—41no—
84C3⋊Dic3—122no—
85C3×Dic3—81no—
86C3×Dic3—241no—
87C22×C9—81no—
88C22×C9C2×S411yescharacteristic
89C4×C9—31no—
90C4×C9—31no—
91C4×C9—121no—
92Dic9—21no—
93Dic9—61no—
94Dic9—121no—
95Dic9—242no—
96#125—271no—
97C4×Q8—91no—
98#107—271no—
99#106—271no—
100C4⋊C8—271no—
101#103—31no—
102#103—271no—
103C3×C9—41noSylow
104C3×Q8—31no—
105C3×Q8#62711yescharacteristic
106C2×C3×C4—31no—
107C2×C3×C4—61no—
108C2×C3×C4—361no—
109C2×Dic3—31no—
110C2×Dic3—41no—
111C2×Dic3—91no—
112C2×Dic3—181no—
113C2×Dic3—361no—
114C2×Dic3—721no—
115SL(2,𝔽3)—21no—
116SL(2,𝔽3)Dic1811yescharacteristic
117C3×C8—61no—
118C3⋊C8—181no—
119C2×C32—41no—
120C2×C32—41no—
121C2×C32—41no—
122C2×C9—81no—
123C2×C9—81no—
124C2×C9—81no—
125C2×C9A4⋊Q811yescharacteristic
126C2×C9#98311yescharacteristic
127C2×C9#98511yescharacteristic
128C2×Q8D3×D911yescharacteristic
129C2×C8—31no—
130C2×C8—271no—
131C4⋊C4—31no—
132C4⋊C4—271no—
133C4⋊C4—271no—
134C4⋊C4—541no—
135C4⋊C4—541no—
136C42—271no—
137C22×C3—41no—
138C22×C3—81no—
139C22×C3D3×S411yescharacteristic
140C3×C4—31no—
141C3×C4—31no—
142C3×C4—121no—
143C3×C4—721no—
144Dic3—61no—
145Dic3—81no—
146Dic3—181no—
147Dic3—362no—
148Dic3—361no—
149Dic3—722no—
150C32—41no—
151C9—81no—
152C9#538911yescharacteristic, complemented
153Q8—31no—
154Q8#618911yescharacteristic
155C2×C4—31no—
156C2×C4—61no—
157C2×C4—91no—
158C2×C4—271no—
159C2×C4—541no—
160C8—61no—
161C8—541no—
162C2×C3—41no—
163C2×C3—41no—
164C2×C3—41no—
165C2×C3—81no—
166C2×C3—81no—
167C2×C3—81no—
168C2×C3#6419811yescharacteristic
169C2×C3#6420111yescharacteristic
170C2×C3#6420411yescharacteristic
171C22D9×S411yescharacteristic
172C4—31no—
173C4—31no—
174C4—121no—
175C4—181no—
176C4—541no—
177C4—1081no—
178C3—41no—
179C3—81no—
180C3#1059319611yescharacteristic
181C2#1172223211yescharacteristic
182C2#1172223311yescharacteristic
183C2#1172224411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, cosocleC2×C9×SL(2,𝔽3)C22
Frattini, socleC22×C3D3×S4
FittingC2×Q8×C9C2×D3
prime radicalC2×C3×SL(2,𝔽3)C2×D3
hypocenterC9×SL(2,𝔽3)Q8
LCSG ▷ C2×C9×SL(2,𝔽3) ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
152. C9#5389semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic92
C9⋊#8592
C9×#8592
#105931963
#105884763
#884424