Finite Group Databasebeta

← #63508⋊C3 #11721679

Subgroups — #63508⋊C3 #11721679

Order 864.

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

normal subgroups only

This group has 241 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×Q8×3−1+2C211yesmaximal, characteristic, complemented
2C2×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic
3(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
4C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
6C3×#946—31nomaximal
7#63508C311yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/9)C2211yescharacteristic
9Q8×3−1+2—22no—
10C2×C4×3−1+2C2212yes—
11C2×C4×3−1+2C2211yescharacteristic
12C2×3−1+2⋊C4C2211yescharacteristic
13C2×3−1+2⋊C4C2212yes—
14C4×AGL(1,ℤ/9)—22no—
15C2×C32×Q8D311yescharacteristic
16C2×C3×C4×D3—31no—
17C3×(C2×D3)⋊C4—32no—
18C3×C4⋊Dic3—31no—
19C3×Dic3⋊C4—32no—
20C2×Q8×C9—21no—
21C2×Q8×C9C2×C311yescharacteristic, complemented
22C2×C4×D9C2×C311yescharacteristic
23(C2×D9)⋊C4C2×C312yes—
24C4⋊Dic9C2×C311yescharacteristic, complemented
25Dic9⋊C4C2×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—21no—
333−1+2⋊C4—22no—
34C3×C22⋊Q8—91no—
35#946—31no—
36C22×C3×D3—31no—
37C32×Q8—22no—
38C2×C32×C4C2×D312yes—
39C2×C32×C4C2×D311yescharacteristic
40C2×C3×Dic3—31no—
41C2×C3×Dic3—32no—
42C3×C4×D3—62no—
43C22×D9C22×C311yescharacteristic
44Q8×C9—24no—
45Q8×C9—22no—
46C2×C4×C9—22no—
47C2×C4×C9—21no—
48C2×C4×C9C22×C312yes—
49C2×C4×C9C22×C311yescharacteristic
50C2×Dic9C22×C311yescharacteristic
51C2×Dic9C22×C312yes—
52C4×D9—22no—
53C2×3−1+2C2×D411yescharacteristic
54C2×3−1+2C2×Q811yescharacteristic
55C2×3−1+2#4111yescharacteristic
56AGL(1,ℤ/9)—22no—
57C2×C3×Q8—31no—
58C2×C3×Q8C3×D311yescharacteristic
59C22×C3×C4—91no—
60C2×C4×D3—31no—
61C3×C4⋊C4—92no—
62C3×C4⋊C4—91no—
63C3×C22⋊C4—92no—
64(C2×D3)⋊C4—32no—
65C4⋊Dic3—31no—
66Dic3⋊C4—32no—
67C22×C32C22×D311yescharacteristic
68C2×C3×D3—32no—
69C2×C3×D3—61no—
70C2×C3×D3—61no—
71C32×C4—22no—
72C32×C4#6712yes—
73C3×Dic3—62no—
74C3×Dic3—61no—
75C22×C9—21no—
76C22×C9C23×C311yescharacteristic
77C2×D9—21no—
78C2×D9—21no—
79C2×D9C3×Q812yescomplemented
80C4×C9—24no—
81C4×C9—22no—
82C4×C9—22no—
83C4×C9C3×D412yes—
84Dic9—21no—
85Dic9—22no—
86C22⋊Q8—91noSylow
873−1+2C22⋊Q811yescharacteristic, complemented, Sylow
88C23×C3—91no—
89C22×D3—31no—
90C3×Q8—22no—
91C3×Q8—62no—
92C2×C3×C4—31no—
93C2×C3×C4—32no—
94C2×C3×C4—92no—
95C2×C3×C4—91no—
96C2×C3×C4—182no—
97C2×C3×C4C2×C3×D312yes—
98C2×C3×C4C2×C3×D311yescharacteristic
99C2×Dic3—31no—
100C2×Dic3—32no—
101C4×D3—62no—
102C2×C32D3×Q811yescharacteristic
103C2×C32#23911yescharacteristic
104C2×C32C2×#6711yescharacteristic
105C3×D3—62no—
106C2×C9—21no—
107C2×C9—21no—
108C2×C9—21no—
109C2×C9C2×C3×D411yescharacteristic
110C2×C9C2×C3×Q811yescharacteristic
111C2×C9C3×#4111yescharacteristic
112D9—22no—
113C2×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
114C22×C4—91no—
115C4⋊C4—92no—
116C4⋊C4—91no—
117C22⋊C4—92no—
118C22×C3—31no—
119C22×C3—92no—
120C22×C3—181no—
121C22×C3—181no—
122C22×C3C22×C3×D311yescharacteristic
123C2×D3—32no—
124C2×D3—61no—
125C2×D3—61no—
126C3×C4—22no—
127C3×C4—32no—
128C3×C4—62no—
129C3×C4—181no—
130C3×C4—182no—
131C3×C4C3×#6712yes—
132Dic3—62no—
133Dic3—61no—
134C32#94611yescharacteristic
135C9—21no—
136C9C3×C22⋊Q811yescharacteristic, complemented
137C23—91no—
138Q8—22no—
139C2×C4—92no—
140C2×C4—91no—
141C2×C4—182no—
142C2×C4C2×AGL(1,ℤ/9)12yes—
143C2×C4C2×AGL(1,ℤ/9)11yescharacteristic
144C2×C3—31no—
145C2×C3—31no—
146C2×C3—31no—
147C2×C3—182no—
148C2×C3C3×D3×Q811yescharacteristic
149C2×C3C3×#23911yescharacteristic
150C2×C3C2×C3×#6711yescharacteristic
151D3—62no—
152C22—92no—
153C22—181no—
154C22—181no—
155C22C22×AGL(1,ℤ/9)11yescharacteristic
156C4—22no—
157C4—182no—
158C4—181no—
159C4#611⋊C312yes—
160C3—31no—
161C3C3×#94611yescharacteristic
162C2—182no—
163C2Q8×AGL(1,ℤ/9)11yescharacteristic
164C2#8872411yescharacteristic
165C2C2×#611⋊C311yescharacteristic
subgroupG/H
centerC22C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC2×Q8×3−1+2C2
prime radicalC22×AGL(1,ℤ/9)C22
hypercenterC2×Q8AGL(1,ℤ/9)
hypocenterC9C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×3−1+2C2semidirect
4. C4⋊(3−1+2⋊C4)C2semidirect
5. (3−1+2⋊C4)⋊C4C2semidirect
7. #63508C3semidirect
21. C2×Q8×C9C2×C3semidirect
24. C4⋊Dic9C2×C3semidirect
25. Dic9⋊C4C2×C3semidirect
29. C2×AGL(1,ℤ/9)Q8semidirect
79. C2×D9C3×Q8semidirect
87. 3−1+2C22⋊Q8semidirect
113. C2×Q8AGL(1,ℤ/9)semidirect
136. C9C3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×3−1+22
C2×C4×AGL(1,ℤ/9)2
(C2×AGL(1,ℤ/9))⋊C42
C4⋊(3−1+2⋊C4)2
(3−1+2⋊C4)⋊C42
C3×#9463
#635083