Finite Group Databasebeta

← #10588803⋊C3 #13143228

Subgroups — #10588803⋊C3 #13143228

Order 1728.

376 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C2×M16×3−1+2C211yesmaximal, characteristic
2#63488⋊C3C211yesmaximal, characteristic
3C22×3−1+2⋊C8C211yesmaximal, characteristic
4(3−1+2⋊C4)⋊C8C214yesmaximal, complemented
5C3×#5119—31nomaximal
6#10588803C311yesmaximal, characteristic, complemented
7C22×C4×3−1+2C2211yescharacteristic
8M16×3−1+2—22no—
9C2×C8×3−1+2C2212yes—
10#3676⋊C3C412yes—
11C4⋊(3−1+2⋊C4)C412yes—
12C4×3−1+2⋊C4C2212yes—
13C2×3−1+2⋊C8—22no—
14C2×3−1+2⋊C8C2212yes—
15C2×C32×M16D311yescharacteristic
16C3×#926—31no—
17C22×C3×C3⋊C8—31no—
18C3×Dic3⋊C8—34no—
19C2×C9×M16—21no—
20C2×C9×M16C2×C311yescharacteristic
21#63488C2×C311yescharacteristic
22C22×C9⋊C8C2×C311yescharacteristic
23Dic9⋊C8C2×C314yescomplemented
24C23×3−1+2C2×C411yescharacteristic
25C2×C4×3−1+2C2×C411yescharacteristic
26C2×C4×3−1+2D412yes—
27C2×C4×3−1+2Q812yes—
28C2×C4×3−1+2C2311yescharacteristic
29C2×3−1+2⋊C4C2×C414yes—
30C8×3−1+2—22no—
313−1+2⋊C8—22no—
32C3×#424—91no—
33#5119—31no—
34C22×C32×C4C2×D311yescharacteristic
35C32×M16—22no—
36C2×C32×C8C2×D312yes—
37C3×#217—32no—
38C3×C4⋊Dic3—32no—
39C3×C4×Dic3—32no—
40C2×C3×C3⋊C8—32no—
41C2×C3×C3⋊C8—62no—
42C22×C4×C9—21no—
43C22×C4×C9C22×C311yescharacteristic
44C9×M16—24no—
45C9×M16—22no—
46C2×C8×C9—22no—
47C2×C8×C9C22×C312yes—
48#3676C3×C412yes—
49C4⋊Dic9C3×C412yes—
50C4×Dic9C22×C312yes—
51C2×C9⋊C8—22no—
52C2×C9⋊C8C22×C312yes—
53C22×3−1+2—22no—
54C22×3−1+2C4⋊C412yes—
55C22×3−1+2C22×C411yescharacteristic
56C4×3−1+2C4⋊C412yes—
57C4×3−1+2C2×D411yescharacteristic
58C4×3−1+2C2×Q811yescharacteristic
593−1+2⋊C4—24no—
60C2×C3×M16—31no—
61C2×C3×M16C3×D311yescharacteristic
62C22×C3×C8—91no—
63C3×#117—91no—
64#926—31no—
65C22×C3⋊C8—31no—
66C3×C4⋊C8—94no—
67Dic3⋊C8—34no—
68C23×C32C4×D311yescharacteristic
69C2×C32×C4Dic612yes—
70C2×C32×C4C4×D311yescharacteristic
71C2×C32×C4#6712yes—
72C2×C32×C4C22×D311yescharacteristic
73C2×C3×Dic3—34no—
74C23×C9—21no—
75C23×C9C2×C3×C411yescharacteristic
76C32×C8—22no—
77C3×C3⋊C8—62no—
78C2×C4×C9—22no—
79C2×C4×C9—21no—
80C2×C4×C9—22no—
81C2×C4×C9—21no—
82C2×C4×C9C2×C3×C411yescharacteristic
83C2×C4×C9C3×D412yes—
84C2×C4×C9C3×Q812yes—
85C2×C4×C9C23×C311yescharacteristic
86C2×Dic9C2×C3×C414yes—
87C8×C9—24no—
88C8×C9—22no—
89C9⋊C8—22no—
90#424—91noSylow
91C2×3−1+2—22no—
92C2×3−1+2C2×C4⋊C411yescharacteristic
93C2×3−1+2#13112yes—
94C22×C3×C4—31no—
95C22×C3×C4C2×C3×D311yescharacteristic
96C3×M16—22no—
97C3×M16—62no—
98C2×C3×C8—32no—
99C2×C3×C8—92no—
100C2×C3×C8—182no—
101C2×C3×C8C2×C3×D312yes—
102C3×C4⋊C4—92no—
103C3×C22⋊C4—92no—
104C3×C42—92no—
105#217—32no—
106C4⋊Dic3—32no—
107C4×Dic3—32no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—62no—
110C22×C32—22no—
111C22×C32Dic3⋊C412yes—
112C22×C32C2×C4×D311yescharacteristic
113C32×C4Dic3⋊C412yes—
114C32×C4C2×Dic611yescharacteristic
115C32×C4C2×#6711yescharacteristic
116C3×Dic3—64no—
117C22×C9—22no—
118C22×C9—22no—
119C22×C9—21no—
120C22×C9—42no—
121C22×C9C3×C4⋊C412yes—
122C22×C9C22×C3×C411yescharacteristic
123C4×C9—22no—
124C4×C9—21no—
125C4×C9—21no—
126C4×C9C3×C4⋊C412yes—
127C4×C9C2×C3×D411yescharacteristic
128C4×C9C2×C3×Q811yescharacteristic
129Dic9—24no—
130C2×M16AGL(1,ℤ/9)11yescharacteristic
131C22×C8—91no—
132#117—91no—
133C4⋊C8—94no—
1343−1+2#42411yescharacteristic, complemented, Sylow
135C23×C3—31no—
136C23×C3C3×C4×D311yescharacteristic
137C2×C3×C4—32no—
138C2×C3×C4—31no—
139C2×C3×C4—32no—
140C2×C3×C4—31no—
141C2×C3×C4—94no—
142C2×C3×C4C3×Dic612yes—
143C2×C3×C4C3×C4×D311yescharacteristic
144C2×C3×C4C3×#6712yes—
145C2×C3×C4C22×C3×D311yescharacteristic
146C2×Dic3—34no—
147C3×C8—22no—
148C3×C8—62no—
149C3×C8—182no—
150C3⋊C8—62no—
151C2×C32—22no—
152C2×C32#90712yes—
153C2×C32C2×Dic3⋊C411yescharacteristic
154C2×C9—22no—
155C2×C9—22no—
156C2×C9—21no—
157C2×C9—42no—
158C2×C9C2×C3×C4⋊C411yescharacteristic
159C2×C9C3×#13112yes—
160C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
161M16—22no—
162C2×C8—92no—
163C2×C8—182no—
164C2×C8C2×AGL(1,ℤ/9)12yes—
165C4⋊C4—92no—
166C22⋊C4—92no—
167C42—92no—
168C22×C3—22no—
169C22×C3—31no—
170C22×C3—32no—
171C22×C3—62no—
172C22×C3C3×Dic3⋊C412yes—
173C22×C3C2×C3×C4×D311yescharacteristic
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—32no—
177C3×C4—184no—
178C3×C4C3×Dic3⋊C412yes—
179C3×C4C2×C3×Dic611yescharacteristic
180C3×C4C2×C3×#6711yescharacteristic
181Dic3—64no—
182C32#511911yescharacteristic
183C9—21no—
184C9C3×#42411yescharacteristic, complemented
185C23C4×AGL(1,ℤ/9)11yescharacteristic
186C2×C4—94no—
187C2×C43−1+2⋊Q812yes—
188C2×C4C4×AGL(1,ℤ/9)11yescharacteristic
189C2×C4#611⋊C312yes—
190C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
191C8—22no—
192C8—182no—
193C2×C3—22no—
194C2×C3—32no—
195C2×C3—31no—
196C2×C3—62no—
197C2×C3C3×#90712yes—
198C2×C3C2×C3×Dic3⋊C411yescharacteristic
199C22—22no—
200C22(3−1+2⋊C4)⋊C412yes—
201C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
202C4—184no—
203C4(3−1+2⋊C4)⋊C412yes—
204C4C2×3−1+2⋊Q811yescharacteristic
205C4C2×#611⋊C311yescharacteristic
206C3—31no—
207C3C3×#511911yescharacteristic
208C2—22no—
209C2#63469⋊C312yes—
210C2C2×(3−1+2⋊C4)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×3−1+2C2
socleC22×C3C2×C3×C4×D3
prime radicalC23×C32C4×D3
cosocleC2×C4×C9C23×C3
hypercenterC2×M16AGL(1,ℤ/9)
hypocenterC9C3×#424
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. (3−1+2⋊C4)⋊C8C2semidirect
6. #10588803C3semidirect
23. Dic9⋊C8C2×C3semidirect
134. 3−1+2#424semidirect
184. C9C3×#424semidirect

Maximal subgroups

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

subgroupindex
C2×M16×3−1+22
#63488⋊C32
C22×3−1+2⋊C82
(3−1+2⋊C4)⋊C82
C3×#51193
#105888033