Finite Group Databasebeta

← #10588219⋊C3 #13135619

Subgroups — #10588219⋊C3 #13135619

Order 1728.

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

normal subgroups only

This group has 230 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#63394⋊C3C212yesmaximal, complemented
3C3×#4542—31nomaximal
4#10588219C311yesmaximal, characteristic, complemented
5C2×Q8×3−1+2C2211yescharacteristic
6C2×D4×3−1+2C411yescharacteristic
7C22⋊C4×3−1+2—22no—
8C42×3−1+2C411yescharacteristic
9#3667⋊C3—22no—
10C32×#124D311yescharacteristic
11C3×#836—32no—
12C9×#124—21no—
13C9×#124C2×C311yescharacteristic
14#63394C2×C312yescomplemented
15C23×3−1+2—21no—
16Q8×3−1+2—41no—
17D4×3−1+2—41no—
18C2×C4×3−1+2—21no—
19C2×C4×3−1+2C2×C411yescharacteristic
20C2×C4×3−1+2D412yes—
213−1+2⋊C8—42no—
22C3×#355—91no—
23#4542—31no—
24C2×C32×Q8C2×D311yescharacteristic
25C2×C32×D4Dic311yescharacteristic
26C32×C22⋊C4—22no—
27C32×C42Dic311yescharacteristic
28C3×#208—62no—
29C2×Q8×C9—21no—
30C2×Q8×C9C22×C311yescharacteristic
31C2×D4×C9—21no—
32C2×D4×C9C3×C411yescharacteristic
33C9×C22⋊C4—24no—
34C9×C22⋊C4—22no—
35C42×C9—21no—
36C42×C9C3×C411yescharacteristic
37#3667—22no—
38C22×3−1+2—42no—
39C22×3−1+2—41no—
40C22×3−1+2C22⋊C411yescharacteristic
41C4×3−1+2—21no—
42C4×3−1+2—22no—
43C4×3−1+2—41no—
44C3×#124—31no—
45C3×#124C3×D311yescharacteristic
46C3×#101—92no—
47#836—32no—
48C23×C32—21no—
49C32×Q8—41no—
50C32×D4—41no—
51C2×C32×C4—21no—
52C2×C32×C4C2×Dic311yescharacteristic
53C2×C32×C4#6712yes—
54C23×C9—22no—
55C23×C9—21no—
56C3×C3⋊C8—122no—
57Q8×C9—41no—
58Q8×C9—42no—
59D4×C9—41no—
60D4×C9—42no—
61C2×C4×C9—22no—
62C2×C4×C9—21no—
63C2×C4×C9—21no—
64C2×C4×C9—22no—
65C2×C4×C9C2×C3×C411yescharacteristic
66C2×C4×C9C3×D412yes—
67C9⋊C8—42no—
68#355—91noSylow
69C2×3−1+2—21no—
70C2×3−1+2—81no—
71C2×3−1+2#9911yescharacteristic
72C2×C3×Q8—31no—
73C2×C3×Q8C2×C3×D311yescharacteristic
74C2×C3×D4—31no—
75C2×C3×D4C3×Dic311yescharacteristic
76C3×M16—182no—
77C3×C22⋊C4—22no—
78C3×C22⋊C4—62no—
79C3×C42—31no—
80C3×C42C3×Dic311yescharacteristic
81#208—62no—
82C22×C32—42no—
83C22×C32—41no—
84C22×C32#21711yescharacteristic
85C32×C4—22no—
86C32×C4—21no—
87C32×C4—41no—
88C22×C9—21no—
89C22×C9—42no—
90C22×C9—44no—
91C22×C9—42no—
92C22×C9—41no—
93C22×C9C3×C22⋊C411yescharacteristic
94C4×C9—22no—
95C4×C9—22no—
96C4×C9—21no—
97C4×C9—42no—
98C4×C9—42no—
99C4×C9—41no—
100#124AGL(1,ℤ/9)11yescharacteristic
101#101—92no—
1023−1+2#35511yescharacteristic, complemented, Sylow
103C23×C3—21no—
104C23×C3—61no—
105C3×Q8—41no—
106C3×Q8—121no—
107C3×D4—41no—
108C3×D4—121no—
109C2×C3×C4—21no—
110C2×C3×C4—31no—
111C2×C3×C4—32no—
112C2×C3×C4—61no—
113C2×C3×C4C2×C3×Dic311yescharacteristic
114C2×C3×C4C3×#6712yes—
115C3×C8—362no—
116C3⋊C8—122no—
117C2×C32—21no—
118C2×C32—81no—
119C2×C32#83411yescharacteristic
120C2×C9—22no—
121C2×C9—21no—
122C2×C9—21no—
123C2×C9—82no—
124C2×C9—81no—
125C2×C9C3×#9911yescharacteristic
126C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
127C2×D43−1+2⋊C411yescharacteristic
128M16—182no—
129C22⋊C4—22no—
130C423−1+2⋊C411yescharacteristic
131C22×C3—31no—
132C22×C3—42no—
133C22×C3—41no—
134C22×C3—121no—
135C22×C3—122no—
136C22×C3C3×#21711yescharacteristic
137C3×C4—22no—
138C3×C4—21no—
139C3×C4—41no—
140C3×C4—62no—
141C3×C4—61no—
142C3×C4—121no—
143C32#454211yescharacteristic
144C9—21no—
145C9C3×#35511yescharacteristic, complemented
146C23—21no—
147Q8—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4C2×3−1+2⋊C411yescharacteristic
151C2×C4#611⋊C312yes—
152C8—362no—
153C2×C3—21no—
154C2×C3—31no—
155C2×C3—61no—
156C2×C3—81no—
157C2×C3—241no—
158C2×C3C3×#83411yescharacteristic
159C22—42no—
160C22—41no—
161C22#3676⋊C311yescharacteristic
162C4—22no—
163C4—21no—
164C4—41no—
165C3—31no—
166C3C3×#454211yescharacteristic
167C2—21no—
168C2—81no—
169C2#63393⋊C311yescharacteristic
subgroupG/H
centerC2#63393⋊C3
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×Q8C2×C3×D3
Fitting3−1+2×#124C2
socleC2×C3C3×#834
prime radicalC2×C32×D4Dic3
cosocleC2×Q8×C9C22×C3
hypercenter#124AGL(1,ℤ/9)
hypocenterC9C3×#355
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #63394⋊C3C2semidirect
4. #10588219C3semidirect
14. #63394C2×C3semidirect
102. 3−1+2#355semidirect
145. C9C3×#355semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1242
#63394⋊C32
C3×#45423
#105882193