Finite Group Databasebeta

← #13135778

Subgroups — #13135778

Order 1728.

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

normal subgroups only

This group has 340 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
1#11721835C211yesmaximal, characteristic
2C3×C9×#100C211yesmaximal, characteristic
3#11720577C211yesmaximal, characteristic
4#10589342—31nomaximal
5#10588152—33nomaximal
6C2×C3×D4×C9C2211yescharacteristic
7C22×C3⋊Dic9C411yescharacteristic, complemented
8C3×C9×M16—21no—
9#88551—21no—
10#88551—21no—
11#88551C411yescharacteristic, complemented
12#88534—21no—
13#64144—31no—
14C32×#100D311yescharacteristic
15#63661—31no—
16#63497—33no—
17C9×#100D313yes—
18#63393—33no—
19C23×C3×C9C2×C411yescharacteristic
20C23×C3×C9D411yescharacteristic
21C3×D4×C9—41no—
22C2×C3×C4×C9D411yescharacteristic
23C2×C3⋊Dic9—21no—
24C2×C3⋊Dic9—21no—
25C2×C3⋊Dic9—41no—
26C2×C3⋊Dic9—41no—
27C3×C8×C9—41no—
28#4476—93no—
29#4476—91no—
30C2×C32×D4C2×D311yescharacteristic
31C22×C3⋊Dic3—31no—
32C32×M16—21no—
33#3757—31no—
34#3757—61no—
35#3757—61no—
36#3750—61no—
37C2×D4×C9C2×D313yes—
38C22×Dic9—33no—
39C9×M16—23no—
40#3676—33no—
41#3676—63no—
42#3676—63no—
43Dic9⋊C4—63no—
44C22×C3×C9—21no—
45C22×C3×C9—21no—
46C22×C3×C9—41no—
47C22×C3×C9—41no—
48C22×C3×C9C22⋊C411yescharacteristic
49C3×C4×C9—21no—
50C3⋊Dic9—41no—
51C3⋊Dic9—41no—
52C3⋊Dic9—81no—
53#935—93no—
54#935—91no—
55C3×#100D913yes—
56C3×#100C3⋊D311yescharacteristic
57#834—93no—
58#834—91no—
59C23×C32C4×D311yescharacteristic
60C23×C32D1211yescharacteristic
61C32×D4—41no—
62C2×C32×C4#6711yescharacteristic
63C2×C3⋊Dic3—61no—
64C2×C3⋊Dic3—61no—
65C2×C3⋊Dic3—121no—
66C2×C3⋊Dic3—121no—
67C23×C9C4×D313yes—
68C23×C9D1213yes—
69C32×C8—41no—
70D4×C9—43no—
71C2×C4×C9#6713yes—
72C2×Dic9—63no—
73C2×Dic9—63no—
74C2×Dic9—123no—
75C2×Dic9—123no—
76C8×C9—43no—
77#352—271noSylow
78C2×C3×C9—21no—
79C2×C3×C9—41no—
80C2×C3×C9—41no—
81C2×C3×C9#9911yescharacteristic
82C2×C3×D4C2×D913yes—
83C2×C3×D4C2×C3⋊D311yescharacteristic
84C22×Dic3—93no—
85C22×Dic3—91no—
86C3×M16—23no—
87C3×M16—21no—
88#217—93no—
89#217—91no—
90#217—183no—
91#217—181no—
92#217—183no—
93#217—181no—
94Dic3⋊C4—183no—
95Dic3⋊C4—181no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32—41no—
99C22×C32—41no—
100C22×C32(C2×D3)⋊C411yescharacteristic
101C32×C4—21no—
102C3⋊Dic3—121no—
103C3⋊Dic3—121no—
104C3⋊Dic3—241no—
105C22×C9—23no—
106C22×C9—23no—
107C22×C9—43no—
108C22×C9—43no—
109C22×C9(C2×D3)⋊C413yes—
110C4×C9—23no—
111Dic9—123no—
112Dic9—123no—
113Dic9—243no—
114#123—271no—
115#100C3⋊D911yescharacteristic
116#99—271no—
117C3×C9#35211yescharacteristic, complemented, Sylow
118C23×C3C4×D913yes—
119C23×C3D3613yes—
120C23×C3C4×C3⋊D311yescharacteristic
121C23×C3C4⋊(C3⋊D3)11yescharacteristic
122C3×D4—41no—
123C3×D4—43no—
124C2×C3×C4#61113yes—
125C2×C3×C4#63811yescharacteristic
126C2×Dic3—183no—
127C2×Dic3—183no—
128C2×Dic3—181no—
129C2×Dic3—181no—
130C2×Dic3—361no—
131C2×Dic3—363no—
132C2×Dic3—361no—
133C2×Dic3—363no—
134C3×C8—41no—
135C3×C8—43no—
136C2×C32—21no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32#80611yescharacteristic
140C2×C9—23no—
141C2×C9—43no—
142C2×C9—43no—
143C2×C9#80613yes—
144C2×D4C2×C3⋊D911yescharacteristic
145C22×C4—271no—
146M16—21no—
147C4⋊C4—541no—
148C22⋊C4—271no—
149C22⋊C4—541no—
150C22⋊C4—541no—
151C22×C3—23no—
152C22×C3—23no—
153C22×C3—21no—
154C22×C3—21no—
155C22×C3—43no—
156C22×C3—43no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3(C2×D9)⋊C413yes—
160C22×C3#375211yescharacteristic
161C3×C4—23no—
162C3×C4—21no—
163Dic3—361no—
164Dic3—363no—
165Dic3—363no—
166Dic3—361no—
167Dic3—721no—
168Dic3—723no—
169C32#447611yescharacteristic
170C9#447613yescomplemented
171C23C4×C3⋊D911yescharacteristic
172C23C4⋊(C3⋊D9)11yescharacteristic
173D4—41no—
174C2×C4—541no—
175C2×C4—541no—
176C2×C4—1081no—
177C2×C4—1081no—
178C2×C4#623311yescharacteristic
179C8—41no—
180C2×C3—21no—
181C2×C3—23no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—43no—
185C2×C3—43no—
186C2×C3#6336513yes—
187C2×C3#6363311yescharacteristic
188C22—21no—
189C22—21no—
190C22—41no—
191C22—41no—
192C22#8853611yescharacteristic
193C4—21no—
194C4—1081no—
195C4—1081no—
196C4—2161no—
197C3#1058815213yescomplemented
198C3#1058934211yescharacteristic
199C2—21no—
200C2—41no—
201C2—41no—
202C2#1172052311yescharacteristic
subgroupG/H
centerC2#11720523
commutatorC23×C3×C9C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
FittingC3×C9×#100C2
socleC2×C32#806
prime radicalC2×C32×D4C2×D3
cosocleC2×C3×D4×C9C22
hypercenter#100C3⋊D9
hypocenterC3×C9#352
LCSG ▷ C23×C3×C9 ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
7. C22×C3⋊Dic9C4semidirect
11. #88551C4semidirect
117. C3×C9#352semidirect
170. C9#4476semidirect
197. C3#10588152semidirect

Maximal subgroups

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

subgroupindex
#117218352
C3×C9×#1002
#117205772
#105893423
#105881523