Finite Group Databasebeta

← #63459⋊C3 #11721483

Subgroups — #63459⋊C3 #11721483

Order 864.

235 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
3C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
5C4⋊(3−1+2⋊C4)C212yesmaximal, complemented
6C3×#897—31nomaximal
7#63459C311yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/9)C2211yescharacteristic
9C2×C4×3−1+2C2212yes—
10C2×C4×3−1+2C2211yescharacteristic
11C2×3−1+2⋊C4C2212yes—
12C2×3−1+2⋊C4C2211yescharacteristic
13C4×AGL(1,ℤ/9)—22no—
143−1+2⋊Q8—22no—
15C2×C3×C4×D3—31no—
16C2×C3×Dic6—31no—
17C32×C4⋊C4D311yescharacteristic
18C3×(C2×D3)⋊C4—32no—
19C3×C4⋊Dic3—32no—
20C2×C4×D9C2×C311yescharacteristic
21C2×Dic18C2×C311yescharacteristic, complemented
22C9×C4⋊C4—21no—
23C9×C4⋊C4C2×C311yescharacteristic, complemented
24(C2×D9)⋊C4C2×C312yes—
25C4⋊Dic9C2×C312yescomplemented
26C22×3−1+2C2311yescharacteristic
27C2×AGL(1,ℤ/9)—21no—
28C2×AGL(1,ℤ/9)—21no—
29C2×AGL(1,ℤ/9)Q812yescomplemented
30C4×3−1+2—22no—
31C4×3−1+2D412yes—
323−1+2⋊C4—22no—
333−1+2⋊C4—21no—
34C3×C22⋊Q8—91no—
35#897—31no—
36C22×C3×D3—31no—
37C2×C32×C4C2×D312yes—
38C2×C32×C4C2×D311yescharacteristic
39C2×C3×Dic3—31no—
40C2×C3×Dic3—32no—
41C3×C4×D3—62no—
42C3×Dic6—62no—
43C22×D9C22×C311yescharacteristic
44C2×C4×C9—21no—
45C2×C4×C9—22no—
46C2×C4×C9C22×C312yes—
47C2×C4×C9C22×C311yescharacteristic
48C2×Dic9C22×C311yescharacteristic
49C2×Dic9C22×C312yes—
50C4×D9—22no—
51Dic18—22no—
52C2×3−1+2C2×D411yescharacteristic
53C2×3−1+2C2×Q811yescharacteristic
54C2×3−1+2#4111yescharacteristic
55AGL(1,ℤ/9)—22no—
56C2×C3×Q8—91no—
57C22×C3×C4—91no—
58C2×C4×D3—31no—
59C2×Dic6—31no—
60C3×C4⋊C4—31no—
61C3×C4⋊C4—92no—
62C3×C4⋊C4C3×D311yescharacteristic
63C3×C22⋊C4—92no—
64(C2×D3)⋊C4—32no—
65C4⋊Dic3—32no—
66C22×C32C22×D311yescharacteristic
67C2×C3×D3—32no—
68C2×C3×D3—61no—
69C2×C3×D3—61no—
70C32×C4—22no—
71C32×C4D1212yes—
72C3×Dic3—62no—
73C3×Dic3—61no—
74C22×C9—21no—
75C22×C9C23×C311yescharacteristic
76C2×D9—21no—
77C2×D9—21no—
78C2×D9C3×Q812yescomplemented
79C4×C9—22no—
80C4×C9—22no—
81C4×C9—42no—
82C4×C9C3×D412yes—
83Dic9—22no—
84Dic9—21no—
85C22⋊Q8—91noSylow
863−1+2C22⋊Q811yescharacteristic, complemented, Sylow
87C23×C3—91no—
88C22×D3—31no—
89C3×Q8—182no—
90C2×C3×C4—31no—
91C2×C3×C4—32no—
92C2×C3×C4—92no—
93C2×C3×C4—91no—
94C2×C3×C4—182no—
95C2×C3×C4C2×C3×D312yes—
96C2×C3×C4C2×C3×D311yescharacteristic
97C2×Dic3—31no—
98C2×Dic3—32no—
99C4×D3—62no—
100Dic6—62no—
101C2×C32C2×D1211yescharacteristic
102C2×C32#23711yescharacteristic
103C2×C32D3×Q811yescharacteristic
104C3×D3—62no—
105C2×C9—21no—
106C2×C9—21no—
107C2×C9—21no—
108C2×C9C2×C3×D411yescharacteristic
109C2×C9C2×C3×Q811yescharacteristic
110C2×C9C3×#4111yescharacteristic
111D9—22no—
112C2×Q8—91no—
113C22×C4—91no—
114C4⋊C4—92no—
115C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
116C22⋊C4—92no—
117C22×C3—31no—
118C22×C3—92no—
119C22×C3—181no—
120C22×C3—181no—
121C22×C3C22×C3×D311yescharacteristic
122C2×D3—32no—
123C2×D3—61no—
124C2×D3—61no—
125C3×C4—22no—
126C3×C4—32no—
127C3×C4—62no—
128C3×C4—181no—
129C3×C4—182no—
130C3×C4C3×D1212yes—
131Dic3—62no—
132Dic3—61no—
133C32#89711yescharacteristic
134C9—21no—
135C9C3×C22⋊Q811yescharacteristic, complemented
136C23—91no—
137Q8—182no—
138C2×C4—92no—
139C2×C4—91no—
140C2×C4—182no—
141C2×C4C2×AGL(1,ℤ/9)12yes—
142C2×C4C2×AGL(1,ℤ/9)11yescharacteristic
143C2×C3—31no—
144C2×C3—31no—
145C2×C3—31no—
146C2×C3—182no—
147C2×C3C2×C3×D1211yescharacteristic
148C2×C3C3×#23711yescharacteristic
149C2×C3C3×D3×Q811yescharacteristic
150D3—62no—
151C22—92no—
152C22—181no—
153C22—181no—
154C22C22×AGL(1,ℤ/9)11yescharacteristic
155C4—22no—
156C4—181no—
157C4—182no—
158C4C4⋊AGL(1,ℤ/9)12yes—
159C3—31no—
160C3C3×#89711yescharacteristic
161C2—182no—
162C2C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
163C2#3699⋊C311yescharacteristic
164C2Q8×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC4⋊C4×3−1+2C2
prime radicalC22×AGL(1,ℤ/9)C22
hypercenterC4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×3−1+2⋊Q8C2semidirect
3. C4⋊C4×3−1+2C2semidirect
5. C4⋊(3−1+2⋊C4)C2semidirect
7. #63459C3semidirect
21. C2×Dic18C2×C3semidirect
23. C9×C4⋊C4C2×C3semidirect
25. C4⋊Dic9C2×C3semidirect
29. C2×AGL(1,ℤ/9)Q8semidirect
78. C2×D9C3×Q8semidirect
86. 3−1+2C22⋊Q8semidirect
115. C4⋊C4AGL(1,ℤ/9)semidirect
135. C9C3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×AGL(1,ℤ/9)2
C2×3−1+2⋊Q82
C4⋊C4×3−1+22
(C2×AGL(1,ℤ/9))⋊C42
C4⋊(3−1+2⋊C4)2
C3×#8973
#634593