Finite Group Databasebeta

← #63448⋊C3 #11721436

Subgroups — #63448⋊C3 #11721436

Order 864.

251 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 77 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×#611⋊C3C211yesmaximal, characteristic
2C22×3−1+2⋊C4C211yesmaximal, characteristic, complemented
3C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
4(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal, complemented
5C4⋊(3−1+2⋊C4)C212yesmaximal, complemented
6C3×#886—31nomaximal
7#63448C311yesmaximal, characteristic, complemented
8C23×3−1+2C2211yescharacteristic
9C22×AGL(1,ℤ/9)C2211yescharacteristic
10C2×C4×3−1+2C2212yes—
11#611⋊C3—22no—
12C2×3−1+2⋊C4—22no—
13C2×3−1+2⋊C4C2212yes—
14C2×3−1+2⋊C4C2211yescharacteristic
15C2×C3×#67—31no—
16C22×C3×Dic3—31no—
17C32×C22⋊C4D311yescharacteristic
18C3×(C2×D3)⋊C4—32no—
19C3×C4⋊Dic3—32no—
20C2×#611C2×C311yescharacteristic
21C22×Dic9C2×C311yescharacteristic, complemented
22C9×C22⋊C4—21no—
23C9×C22⋊C4C2×C311yescharacteristic, complemented
24(C2×D9)⋊C4C2×C312yescomplemented
25C4⋊Dic9C2×C312yescomplemented
26C22×3−1+2—22no—
27C22×3−1+2D412yes—
28C22×3−1+2C2311yescharacteristic
29C2×AGL(1,ℤ/9)—21no—
30C2×AGL(1,ℤ/9)—22no—
31C4×3−1+2—22no—
323−1+2⋊C4—22no—
333−1+2⋊C4—21no—
34C3×#123—91no—
35#886—31no—
36C23×C32C2×D311yescharacteristic
37C22×C3×D3—31no—
38C2×C32×C4C2×D312yes—
39C3×#67—62no—
40C2×C3×Dic3—31no—
41C2×C3×Dic3—32no—
42C2×C3×Dic3—62no—
43C23×C9—21no—
44C23×C9C22×C311yescharacteristic
45C22×D9C22×C311yescharacteristic
46C2×C4×C9—22no—
47C2×C4×C9C22×C312yes—
48#611—22no—
49C2×Dic9—22no—
50C2×Dic9C22×C311yescharacteristic
51C2×Dic9C22×C312yes—
52C2×3−1+2—22no—
53C2×3−1+2C2×D411yescharacteristic
54C2×3−1+2#4112yes—
55AGL(1,ℤ/9)—41no—
56C2×C3×D4—91no—
57C22×C3×C4—91no—
58C2×#67—31no—
59C22×Dic3—31no—
60C3×C4⋊C4—92no—
61C3×C22⋊C4—31no—
62C3×C22⋊C4—92no—
63C3×C22⋊C4C3×D311yescharacteristic
64(C2×D3)⋊C4—32no—
65C4⋊Dic3—32no—
66C22×C32—22no—
67C22×C32D1212yes—
68C22×C32C22×D311yescharacteristic
69C2×C3×D3—61no—
70C2×C3×D3—62no—
71C32×C4—22no—
72C3×Dic3—62no—
73C3×Dic3—61no—
74C22×C9—22no—
75C22×C9—22no—
76C22×C9—21no—
77C22×C9—42no—
78C22×C9C3×D412yes—
79C22×C9C23×C311yescharacteristic
80C2×D9—21no—
81C2×D9—22no—
82C4×C9—22no—
83C4×C9—42no—
84Dic9—22no—
85Dic9—21no—
86#123—91noSylow
873−1+2#12311yescharacteristic, complemented, Sylow
88C23×C3—31no—
89C23×C3—91no—
90C23×C3C2×C3×D311yescharacteristic
91C22×D3—31no—
92C3×D4—182no—
93C2×C3×C4—32no—
94C2×C3×C4—92no—
95C2×C3×C4—91no—
96C2×C3×C4—182no—
97C2×C3×C4C2×C3×D312yes—
98#67—62no—
99C2×Dic3—31no—
100C2×Dic3—32no—
101C2×Dic3—62no—
102C2×C32—22no—
103C2×C32C2×D1211yescharacteristic
104C2×C32#23712yes—
105C3×D3—121no—
106C2×C9—22no—
107C2×C9—21no—
108C2×C9—22no—
109C2×C9—42no—
110C2×C9C2×C3×D411yescharacteristic
111C2×C9C3×#4112yes—
112D9—41no—
113C2×D4—91no—
114C22×C4—91no—
115C4⋊C4—92no—
116C22⋊C4—92no—
117C22⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
118C22×C3—22no—
119C22×C3—32no—
120C22×C3—31no—
121C22×C3—62no—
122C22×C3—181no—
123C22×C3—182no—
124C22×C3C3×D1212yes—
125C22×C3C22×C3×D311yescharacteristic
126C2×D3—61no—
127C2×D3—62no—
128C3×C4—22no—
129C3×C4—62no—
130C3×C4—182no—
131C3×C4—181no—
132Dic3—62no—
133Dic3—61no—
134C32#88611yescharacteristic
135C9—21no—
136C9C3×#12311yescharacteristic, complemented
137C23—91no—
138C23C2×AGL(1,ℤ/9)11yescharacteristic
139D4—182no—
140C2×C4—92no—
141C2×C4—91no—
142C2×C4—182no—
143C2×C4C2×AGL(1,ℤ/9)12yes—
144C2×C3—22no—
145C2×C3—31no—
146C2×C3—32no—
147C2×C3—62no—
148C2×C3—361no—
149C2×C3C2×C3×D1211yescharacteristic
150C2×C3C3×#23712yes—
151D3—121no—
152C22—22no—
153C22—181no—
154C22—182no—
155C22C4⋊AGL(1,ℤ/9)12yes—
156C22C22×AGL(1,ℤ/9)11yescharacteristic
157C4—22no—
158C4—182no—
159C4—181no—
160C3—31no—
161C3C3×#88611yescharacteristic
162C2—22no—
163C2—361no—
164C2C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
165C2#3699⋊C312yes—
subgroupG/H
centerC22C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22⋊C4×3−1+2C2
prime radicalC2×#611⋊C3C2
hypercenterC22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#123
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×3−1+2⋊C4C2semidirect
3. C22⋊C4×3−1+2C2semidirect
4. (C2×AGL(1,ℤ/9))⋊C4C2semidirect
5. C4⋊(3−1+2⋊C4)C2semidirect
7. #63448C3semidirect
21. C22×Dic9C2×C3semidirect
23. C9×C22⋊C4C2×C3semidirect
24. (C2×D9)⋊C4C2×C3semidirect
25. C4⋊Dic9C2×C3semidirect
87. 3−1+2#123semidirect
117. C22⋊C4AGL(1,ℤ/9)semidirect
136. C9C3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#611⋊C32
C22×3−1+2⋊C42
C22⋊C4×3−1+22
(C2×AGL(1,ℤ/9))⋊C42
C4⋊(3−1+2⋊C4)2
C3×#8863
#634483