Finite Group Databasebeta

← #13138273

Subgroups — #13138273

Order 1728.

294 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 97 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#11720935C212yesmaximal
2#11720540C211yesmaximal, characteristic
3C9×#817C211yesmaximal, characteristic
4C3×C8×Dic9C211yesmaximal, characteristic
5#11720262C212yesmaximal
6#10590871—31nomaximal
7#10588538—31nomaximal
8C2×C3×C8×C9C2211yescharacteristic
9C4⋊(C3⋊Dic9)C2212yes—
10C9×C4⋊Dic3C2212yes—
11C3×C4×Dic9C2211yescharacteristic
12C2×C3×C9⋊C8C2211yescharacteristic
13Dic3⋊Dic9—22no—
14#88442—22no—
15#63869—32no—
16#63644—31no—
17C3×#817D311yescharacteristic
18C3×C8×Dic3—31no—
19#63575—32no—
20#63452—32no—
21C9×#106—31no—
22#63377—31no—
23#63377—61no—
24C8×Dic9D311yescharacteristic
25#63366—32no—
26C2×C3×C4×C9C2311yescharacteristic
27C2×C3⋊Dic9—22no—
28C2×C9×Dic3—22no—
29C2×C3×Dic9D412yes—
30C3×C8×C9Q812yes—
31C3×C9⋊C8Q812yes—
32#4858—91no—
33#4683—91no—
34C2×C32×C8C2×D311yescharacteristic
35C4⋊(C3⋊Dic3)—32no—
36C3×C4⋊Dic3C2×D312yes—
37C3×C4×Dic3—31no—
38C2×C3×C3⋊C8—31no—
39#3724—62no—
40Dic3⋊Dic3—62no—
41C2×C8×C9—21no—
42C2×C8×C9C2×D311yescharacteristic
43C9×C4⋊C4—32no—
44C4⋊Dic9—32no—
45C4⋊Dic9—62no—
46C4⋊Dic9—62no—
47Dic9⋊C4—62no—
48C4×Dic9C2×D311yescharacteristic
49C2×C9⋊C8C2×D311yescharacteristic
50C22×C3×C9C2×D411yescharacteristic
51C3×C4×C9C2×Q811yescharacteristic
52C3×C4×C9C2×Q811yescharacteristic
53C3⋊Dic9—42no—
54C9×Dic3—42no—
55C3×Dic9—22no—
56#890—92no—
57#870—92no—
58C3×#106—31no—
59C3×C4×C8—91no—
60#818—92no—
61#817—91no—
62#817—91no—
63#817—181no—
64#817D911yescharacteristic
65C8×Dic3—31no—
66#807—92no—
67C2×C32×C4C22×D311yescharacteristic
68C2×C3⋊Dic3—62no—
69C2×C3×Dic3—22no—
70C2×C3×Dic3—32no—
71C32×C8Dic612yes—
72C3×C3⋊C8—32no—
73C2×C4×C9—21no—
74C2×C4×C9—62no—
75C2×C4×C9C22×D311yescharacteristic
76C2×Dic9—62no—
77C2×Dic9—122no—
78C2×Dic9D1212yes—
79C8×C9—22no—
80C8×C9Dic612yes—
81C9⋊C8Dic612yes—
82#499—271noSylow
83C2×C3×C9C4⋊Q811yescharacteristic
84C2×C3×C9#13512yes—
85C2×C3×C8—21no—
86C2×C3×C8—91no—
87C2×C3×C8C2×D911yescharacteristic
88C2×C3×C8D3211yescharacteristic
89C3×C4⋊C4—32no—
90C3×C42—91no—
91C4⋊Dic3—92no—
92C4⋊Dic3—92no—
93C4⋊Dic3—182no—
94C4⋊Dic3—182no—
95C4⋊Dic3C2×D912yes—
96Dic3⋊C4—182no—
97Dic3⋊C4—182no—
98Dic3⋊C4—182no—
99C4×Dic3—31no—
100C2×C3⋊C8—31no—
101C22×C32D3×D411yescharacteristic
102C32×C4C2×Dic611yescharacteristic
103C32×C4D3×Q811yescharacteristic
104C3⋊Dic3—122no—
105C3×Dic3—42no—
106C3×Dic3—62no—
107C22×C9—21no—
108C22×C9C2×D1211yescharacteristic
109C4×C9—21no—
110C4×C9—21no—
111C4×C9—122no—
112C4×C9C2×Dic611yescharacteristic
113C4×C9C2×Dic611yescharacteristic
114Dic9—22no—
115Dic9—122no—
116Dic9—242no—
117#125—272no—
118#107—272no—
119#106—31no—
120#106—271no—
121C4×C8—91no—
122C3×C9#49911yescharacteristic, complemented, Sylow
123C2×C3×C4—21no—
124C2×C3×C4—62no—
125C2×C3×C4—92no—
126C2×C3×C4C22×D911yescharacteristic
127C2×C3×C4C2×D3211yescharacteristic
128C2×Dic3—22no—
129C2×Dic3—32no—
130C2×Dic3—182no—
131C2×Dic3—182no—
132C2×Dic3—362no—
133C3×C8—22no—
134C3×C8—92no—
135C3×C8Dic1812yes—
136C3×C8#62712yes—
137C3⋊C8—32no—
138C2×C32#88811yescharacteristic
139C2×C32#91612yes—
140C2×C9—22no—
141C2×C9—21no—
142C2×C9#86911yescharacteristic
143C2×C9#90412yes—
144C2×C8—91no—
145C2×C8D3×D911yescharacteristic
146C4⋊C4—32no—
147C4⋊C4—272no—
148C4⋊C4—542no—
149C4⋊C4—542no—
150C42—91no—
151C22×C3—21no—
152C22×C3D4×D911yescharacteristic
153C22×C3D3×D1211yescharacteristic
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—122no—
157C3×C4—182no—
158C3×C4C2×Dic1811yescharacteristic
159C3×C4Q8×D911yescharacteristic
160C3×C4D3×Dic611yescharacteristic
161C3×C4C2×#62711yescharacteristic
162Dic3—42no—
163Dic3—62no—
164Dic3—362no—
165Dic3—362no—
166Dic3—722no—
167C32#485811yescharacteristic
168C9—21no—
169C9#468311yescharacteristic, complemented
170C2×C4—62no—
171C2×C4—92no—
172C2×C4—542no—
173C2×C4C2×D3×D911yescharacteristic
174C8—92no—
175C8#618912yes—
176C2×C3—21no—
177C2×C3—22no—
178C2×C3#6345011yescharacteristic
179C2×C3#6347812yes—
180C2×C3#6380612yes—
181C2×C3#6391811yescharacteristic
182C22D9×D1211yescharacteristic
183C4—122no—
184C4—182no—
185C4—1082no—
186C4D9×Dic611yescharacteristic
187C4C2×#618911yescharacteristic
188C3—21no—
189C3#1058853811yescharacteristic, complemented
190C3#1059087111yescharacteristic
191C2#1172082112yes—
192C2#1172096911yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#499
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
122. C3×C9#499semidirect
169. C9#4683semidirect
189. C3#10588538semidirect

Maximal subgroups

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

subgroupindex
#117209352
#117205402
C9×#8172
C3×C8×Dic92
#117202622
#105908713
#105885383