Finite Group Databasebeta

← #13135771

Subgroups — #13135771

Order 1728.

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

normal subgroups only

This group has 350 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#11721836C211yesmaximal, characteristic
2C3×C9×#99C211yesmaximal, characteristic
3#11720577C211yesmaximal, characteristic
4#10589339—31nomaximal
5#10588149—33nomaximal
6C2×C3×D4×C9C2211yescharacteristic
7C2×#6226C411yescharacteristic, complemented
8C3×C9×C22⋊C4—21no—
9#88551—21no—
10#88551—21no—
11#88551—21no—
12C4×C3⋊Dic9C411yescharacteristic, complemented
13#64145—31no—
14C32×#99D311yescharacteristic
15#63661—31no—
16#63498—33no—
17C9×#99D313yes—
18#63393—33no—
19C23×C3×C9D411yescharacteristic
20C23×C3×C9D411yescharacteristic
21C3×D4×C9—41no—
22C2×C3×C4×C9—41no—
23C2×C3×C4×C9C2×C411yescharacteristic
24C2×C3⋊Dic9—21no—
25C2×C3⋊Dic9—21no—
26C2×C3⋊Dic9—41no—
27#6226—41no—
28#4473—93no—
29#4473—91no—
30C2×C32×D4C2×D311yescharacteristic
31C2×#634—31no—
32C32×C22⋊C4—21no—
33#3757—61no—
34#3757—61no—
35#3757—61no—
36C4×C3⋊Dic3—31no—
37C2×D4×C9C2×D313yes—
38C2×Dic18—33no—
39C9×C22⋊C4—23no—
40#3676—63no—
41#3676—63no—
42#3676—63no—
43C4×Dic9—33no—
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×C4×C9—81no—
51C3⋊Dic9—41no—
52C3⋊Dic9—41no—
53C3⋊Dic9—81no—
54#936—93no—
55#936—91no—
56C3×#99D913yes—
57C3×#99C3⋊D311yescharacteristic
58#834—93no—
59#834—91no—
60C23×C32D1211yescharacteristic
61C23×C32#6711yescharacteristic
62C32×D4—41no—
63C2×C32×C4—41no—
64C2×C32×C4C4×D311yescharacteristic
65C2×C3⋊Dic3—61no—
66C2×C3⋊Dic3—61no—
67C2×C3⋊Dic3—121no—
68#634—121no—
69C23×C9D1213yes—
70C23×C9#6713yes—
71D4×C9—43no—
72C2×C4×C9—43no—
73C2×C4×C9C4×D313yes—
74C2×Dic9—63no—
75C2×Dic9—63no—
76C2×Dic9—123no—
77Dic18—123no—
78#354—271noSylow
79C2×C3×C9—21no—
80C2×C3×C9—41no—
81C2×C3×C9—41no—
82C2×C3×C9#9911yescharacteristic
83C2×C3×D4C2×D913yes—
84C2×C3×D4C2×C3⋊D311yescharacteristic
85C2×Dic6—93no—
86C2×Dic6—91no—
87C3×C22⋊C4—23no—
88C3×C22⋊C4—21no—
89#217—183no—
90#217—183no—
91#217—181no—
92#217—181no—
93#217—183no—
94#217—181no—
95C4×Dic3—93no—
96C4×Dic3—91no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—41no—
100C22×C32—41no—
101C22×C32(C2×D3)⋊C411yescharacteristic
102C32×C4—21no—
103C32×C4—81no—
104C3⋊Dic3—121no—
105C3⋊Dic3—121no—
106C3⋊Dic3—241no—
107C22×C9—23no—
108C22×C9—23no—
109C22×C9—43no—
110C22×C9—43no—
111C22×C9(C2×D3)⋊C413yes—
112C4×C9—23no—
113C4×C9—83no—
114Dic9—123no—
115Dic9—123no—
116Dic9—243no—
117#124—271no—
118#99—271no—
119#99C3⋊D911yescharacteristic
120C3×C9#35411yescharacteristic, complemented, Sylow
121C23×C3D3613yes—
122C23×C3#61113yes—
123C23×C3C4⋊(C3⋊D3)11yescharacteristic
124C23×C3#63811yescharacteristic
125C3×D4—41no—
126C3×D4—43no—
127C2×C3×C4—41no—
128C2×C3×C4—43no—
129C2×C3×C4C4×D913yes—
130C2×C3×C4C4×C3⋊D311yescharacteristic
131C2×Dic3—183no—
132C2×Dic3—183no—
133C2×Dic3—181no—
134C2×Dic3—181no—
135C2×Dic3—361no—
136C2×Dic3—363no—
137Dic6—361no—
138Dic6—363no—
139C2×C32—21no—
140C2×C32—41no—
141C2×C32—41no—
142C2×C32#80611yescharacteristic
143C2×C9—23no—
144C2×C9—43no—
145C2×C9—43no—
146C2×C9#80613yes—
147C2×Q8—271no—
148C2×D4C2×C3⋊D911yescharacteristic
149C22⋊C4—21no—
150C22⋊C4—541no—
151C22⋊C4—541no—
152C22⋊C4—541no—
153C42—271no—
154C22×C3—23no—
155C22×C3—23no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—43no—
159C22×C3—43no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3(C2×D9)⋊C413yes—
163C22×C3#375211yescharacteristic
164C3×C4—23no—
165C3×C4—21no—
166C3×C4—81no—
167C3×C4—83no—
168Dic3—361no—
169Dic3—363no—
170Dic3—363no—
171Dic3—361no—
172Dic3—721no—
173Dic3—723no—
174C32#447311yescharacteristic
175C9#447313yescomplemented
176C23C4⋊(C3⋊D9)11yescharacteristic
177C23#623311yescharacteristic
178Q8—1081no—
179D4—41no—
180C2×C4—41no—
181C2×C4—541no—
182C2×C4—541no—
183C2×C4—1081no—
184C2×C4C4×C3⋊D911yescharacteristic
185C2×C3—21no—
186C2×C3—23no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—43no—
190C2×C3—43no—
191C2×C3#6336513yes—
192C2×C3#6363311yescharacteristic
193C22—21no—
194C22—21no—
195C22—41no—
196C22—41no—
197C22#8853611yescharacteristic
198C4—21no—
199C4—81no—
200C4—1081no—
201C4—1081no—
202C4—2161no—
203C3#1058814913yescomplemented
204C3#1058933911yescharacteristic
205C2—21no—
206C2—41no—
207C2—41no—
208C2#1172052311yescharacteristic
subgroupG/H
centerC2#11720523
commutatorC2×C3×C4×C9C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
FittingC3×C9×#99C2
socleC2×C32#806
prime radicalC2×C32×D4C2×D3
cosocleC2×C3×D4×C9C22
hypercenter#99C3⋊D9
hypocenterC3×C9#354
LCSG ▷ C2×C3×C4×C9 ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
7. C2×#6226C4semidirect
12. C4×C3⋊Dic9C4semidirect
120. C3×C9#354semidirect
175. C9#4473semidirect
203. C3#10588149semidirect

Maximal subgroups

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

subgroupindex
#117218362
C3×C9×#992
#117205772
#105893393
#105881493