Finite Group Databasebeta

← #10588225⋊C3 #13135643

Subgroups — #10588225⋊C3 #13135643

Order 1728.

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

normal subgroups only

This group has 214 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×C4⋊Q8C211yesmaximal, characteristic
2#63394⋊C3C212yesmaximal
3C3×#4548—31nomaximal
4#10588225C311yesmaximal, characteristic, complemented
5C2×Q8×3−1+2C411yescharacteristic
6C2×Q8×3−1+2C2211yescharacteristic
7C4⋊C4×3−1+2—22no—
8C42×3−1+2C411yescharacteristic
9#3667⋊C3—22no—
10C32×C4⋊Q8D311yescharacteristic
11C3×#836—32no—
12C9×C4⋊Q8—21no—
13C9×C4⋊Q8C2×C311yescharacteristic
14#63394C2×C312yes—
15Q8×3−1+2—41no—
16Q8×3−1+2—41no—
17C2×C4×3−1+2—21no—
18C2×C4×3−1+2—21no—
19C2×C4×3−1+2C2×C411yescharacteristic
20C2×C4×3−1+2D412yes—
213−1+2⋊C8—42no—
22C3×#356—91no—
23#4548—31no—
24C2×C32×Q8Dic311yescharacteristic
25C2×C32×Q8C2×D311yescharacteristic
26C32×C4⋊C4—22no—
27C32×C42Dic311yescharacteristic
28C3×#208—62no—
29C2×Q8×C9—21no—
30C2×Q8×C9—21no—
31C2×Q8×C9C3×C411yescharacteristic
32C2×Q8×C9C22×C311yescharacteristic
33C9×C4⋊C4—24no—
34C9×C4⋊C4—22no—
35C42×C9—21no—
36C42×C9C3×C411yescharacteristic
37#3667—22no—
38C22×3−1+2C22⋊C411yescharacteristic
39C4×3−1+2—22no—
40C4×3−1+2—21no—
41C4×3−1+2—22no—
42C4×3−1+2—41no—
43C3×C4⋊Q8—31no—
44C3×C4⋊Q8C3×D311yescharacteristic
45C3×#101—92no—
46#836—32no—
47C32×Q8—41no—
48C32×Q8—41no—
49C2×C32×C4—21no—
50C2×C32×C4—21no—
51C2×C32×C4C2×Dic311yescharacteristic
52C2×C32×C4#6712yes—
53C3×C3⋊C8—122no—
54Q8×C9—41no—
55Q8×C9—41no—
56Q8×C9—42no—
57Q8×C9—42no—
58C2×C4×C9—22no—
59C2×C4×C9—21no—
60C2×C4×C9—22no—
61C2×C4×C9—21no—
62C2×C4×C9—21no—
63C2×C4×C9—22no—
64C2×C4×C9C2×C3×C411yescharacteristic
65C2×C4×C9C3×D412yes—
66C9⋊C8—42no—
67#356—91noSylow
68C2×3−1+2—21no—
69C2×3−1+2#9911yescharacteristic
70C2×C3×Q8—31no—
71C2×C3×Q8—31no—
72C2×C3×Q8C3×Dic311yescharacteristic
73C2×C3×Q8C2×C3×D311yescharacteristic
74C3×M16—182no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—62no—
77C3×C42—31no—
78C3×C42C3×Dic311yescharacteristic
79#208—62no—
80C22×C32#21711yescharacteristic
81C32×C4—22no—
82C32×C4—22no—
83C32×C4—21no—
84C32×C4—41no—
85C22×C9—21no—
86C22×C9C3×C22⋊C411yescharacteristic
87C4×C9—24no—
88C4×C9—22no—
89C4×C9—22no—
90C4×C9—22no—
91C4×C9—21no—
92C4×C9—42no—
93C4×C9—42no—
94C4×C9—41no—
95C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
96#101—92no—
973−1+2#35611yescharacteristic, complemented, Sylow
98C3×Q8—41no—
99C3×Q8—41no—
100C3×Q8—121no—
101C3×Q8—121no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—31no—
105C2×C3×C4—32no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4C2×C3×Dic311yescharacteristic
109C2×C3×C4C3×#6712yes—
110C3×C8—362no—
111C3⋊C8—122no—
112C2×C32—21no—
113C2×C32#83411yescharacteristic
114C2×C9—22no—
115C2×C9—21no—
116C2×C9—21no—
117C2×C9C3×#9911yescharacteristic
118C2×Q83−1+2⋊C411yescharacteristic
119C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
120M16—182no—
121C4⋊C4—22no—
122C423−1+2⋊C411yescharacteristic
123C22×C3—31no—
124C22×C3C3×#21711yescharacteristic
125C3×C4—22no—
126C3×C4—22no—
127C3×C4—21no—
128C3×C4—41no—
129C3×C4—62no—
130C3×C4—61no—
131C3×C4—62no—
132C3×C4—121no—
133C32#454811yescharacteristic
134C9—21no—
135C9C3×#35611yescharacteristic, complemented
136Q8—41no—
137Q8—41no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4C2×3−1+2⋊C411yescharacteristic
141C2×C4#611⋊C312yes—
142C8—362no—
143C2×C3—21no—
144C2×C3—31no—
145C2×C3—61no—
146C2×C3C3×#83411yescharacteristic
147C22#3676⋊C311yescharacteristic
148C4—22no—
149C4—22no—
150C4—21no—
151C4—41no—
152C3—31no—
153C3C3×#454811yescharacteristic
154C2—21no—
155C2#63393⋊C311yescharacteristic
subgroupG/H
centerC2#63393⋊C3
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×Q8C2×C3×D3
Fitting3−1+2×C4⋊Q8C2
socleC2×C3C3×#834
prime radicalC22×C32#217
cosocleC2×Q8×C9C22×C3
hypercenterC4⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#356
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #10588225C3semidirect
97. 3−1+2#356semidirect
135. C9C3×#356semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C4⋊Q82
#63394⋊C32
C3×#45483
#105882253