Finite Group Databasebeta

← #10588218⋊C3 #13135618

Subgroups — #10588218⋊C3 #13135618

Order 1728.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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
13−1+2×#124C211yesmaximal, characteristic
2#63393⋊C3C212yesmaximal
3C3×#4541—31nomaximal
4#10588218C311yesmaximal, characteristic, complemented
5C2×Q8×3−1+2C411yescharacteristic, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C22⋊C4×3−1+2—22no—
8C42×3−1+2C411yescharacteristic, complemented
9#3676⋊C3—22no—
10C32×#124D311yescharacteristic
11C3×#834—32no—
12C9×#124—21no—
13C9×#124C2×C311yescharacteristic
14#63393C2×C312yes—
15C23×3−1+2D412yes—
16Q8×3−1+2—41no—
17D4×3−1+2—41no—
18C2×C4×3−1+2—21no—
19C2×C4×3−1+2—21no—
20C2×C4×3−1+2C2×C411yescharacteristic
21C2×3−1+2⋊C4—42no—
22C3×#354—91no—
23#4541—31no—
24C2×C32×Q8Dic311yescharacteristic
25C2×C32×D4C2×D311yescharacteristic
26C32×C22⋊C4—22no—
27C32×C42Dic311yescharacteristic
28C3×#217—62no—
29C2×Q8×C9—21no—
30C2×Q8×C9C3×C411yescharacteristic, complemented
31C2×D4×C9—21no—
32C2×D4×C9C22×C311yescharacteristic
33C9×C22⋊C4—24no—
34C9×C22⋊C4—22no—
35C42×C9—21no—
36C42×C9C3×C411yescharacteristic, complemented
37#3676—22no—
38C22×3−1+2—22no—
39C22×3−1+2—42no—
40C22×3−1+2C22⋊C411yescharacteristic
41C4×3−1+2—21no—
42C4×3−1+2—41no—
43C4×3−1+2—41no—
443−1+2⋊C4—82no—
45C3×#124—31no—
46C3×#124C3×D311yescharacteristic
47C3×#99—92no—
48#834—32no—
49C23×C32#6712yes—
50C32×Q8—41no—
51C32×D4—41no—
52C2×C32×C4—21no—
53C2×C32×C4—21no—
54C2×C32×C4C2×Dic311yescharacteristic
55C2×C3×Dic3—122no—
56C23×C9—22no—
57C23×C9C3×D412yes—
58Q8×C9—41no—
59Q8×C9—42no—
60D4×C9—41no—
61D4×C9—42no—
62C2×C4×C9—22no—
63C2×C4×C9—22no—
64C2×C4×C9—21no—
65C2×C4×C9—21no—
66C2×C4×C9—21no—
67C2×C4×C9C2×C3×C411yescharacteristic
68C2×Dic9—42no—
69#354—91noSylow
70C2×3−1+2—21no—
71C2×3−1+2—42no—
72C2×3−1+2#9911yescharacteristic
73C2×C3×Q8—31no—
74C2×C3×Q8C3×Dic311yescharacteristic
75C2×C3×D4—31no—
76C2×C3×D4C2×C3×D311yescharacteristic
77C3×C22⋊C4—22no—
78C3×C22⋊C4—62no—
79C3×C22⋊C4—182no—
80C3×C42—31no—
81C3×C42C3×Dic311yescharacteristic
82#217—62no—
83C22×C32—22no—
84C22×C32—42no—
85C22×C32#21711yescharacteristic
86C32×C4—21no—
87C32×C4—41no—
88C32×C4—41no—
89C3×Dic3—242no—
90C22×C9—22no—
91C22×C9—21no—
92C22×C9—42no—
93C22×C9—42no—
94C22×C9—44no—
95C22×C9C3×C22⋊C411yescharacteristic
96C4×C9—22no—
97C4×C9—21no—
98C4×C9—42no—
99C4×C9—41no—
100C4×C9—42no—
101C4×C9—41no—
102Dic9—82no—
103#124AGL(1,ℤ/9)11yescharacteristic
104#99—92no—
1053−1+2#35411yescharacteristic, complemented, Sylow
106C23×C3—32no—
107C23×C3C3×#6712yes—
108C3×Q8—41no—
109C3×Q8—121no—
110C3×D4—41no—
111C3×D4—121no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—31no—
115C2×C3×C4—61no—
116C2×C3×C4—61no—
117C2×C3×C4—362no—
118C2×C3×C4C2×C3×Dic311yescharacteristic
119C2×Dic3—122no—
120C2×C32—21no—
121C2×C32—42no—
122C2×C32#83411yescharacteristic
123C2×C9—22no—
124C2×C9—21no—
125C2×C9—21no—
126C2×C9—42no—
127C2×C9—82no—
128C2×C9C3×#9911yescharacteristic
129C2×Q83−1+2⋊C411yescharacteristic, complemented
130C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
131C22⋊C4—22no—
132C22⋊C4—182no—
133C423−1+2⋊C411yescharacteristic, complemented
134C22×C3—22no—
135C22×C3—31no—
136C22×C3—42no—
137C22×C3—62no—
138C22×C3—122no—
139C22×C3C3×#21711yescharacteristic
140C3×C4—21no—
141C3×C4—41no—
142C3×C4—41no—
143C3×C4—61no—
144C3×C4—121no—
145C3×C4—121no—
146C3×C4—722no—
147Dic3—242no—
148C32#454111yescharacteristic
149C9—21no—
150C9C3×#35411yescharacteristic, complemented
151C23#611⋊C312yes—
152Q8—41no—
153D4—41no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—362no—
157C2×C4C2×3−1+2⋊C411yescharacteristic
158C2×C3—21no—
159C2×C3—31no—
160C2×C3—42no—
161C2×C3—61no—
162C2×C3—122no—
163C2×C3C3×#83411yescharacteristic
164C22—22no—
165C22—42no—
166C22#3676⋊C311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—722no—
171C3—31no—
172C3C3×#454111yescharacteristic
173C2—21no—
174C2—42no—
175C2#63393⋊C311yescharacteristic
subgroupG/H
centerC2#63393⋊C3
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#124C2
socleC2×C3C3×#834
prime radicalC2×C32×D4C2×D3
cosocleC2×D4×C9C22×C3
hypercenter#124AGL(1,ℤ/9)
hypocenterC9C3×#354
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
4. #10588218C3semidirect
5. C2×Q8×3−1+2C4semidirect
8. C42×3−1+2C4semidirect
30. C2×Q8×C9C3×C4semidirect
36. C42×C9C3×C4semidirect
105. 3−1+2#354semidirect
129. C2×Q83−1+2⋊C4semidirect
133. C423−1+2⋊C4semidirect
150. C9C3×#354semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1242
#63393⋊C32
C3×#45413
#105882183