Finite Group Databasebeta

← C3×#10593450 #13166819

Subgroups — C3×#10593450 #13166819

Order 1728.

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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×C3×#3776C214yesmaximal
2C3×#63919C212yesmaximal
3C3×C4×#622C211yesmaximal, characteristic
4#10593450C311yesmaximal, characteristic, direct product factor
5C2×C3×#627C2214yes—
6C2×C3×#622C2212yes—
7C3×#3776—28no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9C32×C4⋊Dic3—22no—
10C3×#3724—22no—
11C2×#3776C2×C314yes—
12C3×#888—62no—
13#63919C2×C312yes—
14C4×#622C2×C311yescharacteristic
15C2×C33×C4D411yescharacteristic
16C2×C3×C3⋊Dic3D411yescharacteristic
17C2×C3×C3⋊Dic3C2311yescharacteristic
18C2×C32×Dic3—24no—
19C3×#627—28no—
20C3×#622D414yes—
21C3×#494—91nomaximal
22C2×C3×Dic6—64no—
23C2×#627C22×C314yes—
24C2×#622C22×C312yes—
25#3776—28no—
26C32×C4⋊C4—62no—
27C4×C3⋊Dic3C22×C311yescharacteristic
28C3×C4⋊Dic3—22no—
29C3×C4⋊Dic3—22no—
30C3×C4⋊Dic3—42no—
31C3×Dic3⋊C4—64no—
32C3×C4×Dic3—62no—
33#3724—22no—
34C22×C33C2×D411yescharacteristic
35C33×C4Q1612yescomplemented
36C3×C3⋊Dic3Q1612yescomplemented
37C3×C3⋊Dic3C2×D412yes—
38C32×Dic3—44no—
39C2×C3×Q16—94no—
40C3×C4⋊Q8—92no—
41#888—62no—
42C3×C4×C8—91no—
43C2×C32×C4—22no—
44C2×C32×C4—42no—
45C2×C32×C4—64no—
46C2×C32×C4C3×D411yescharacteristic
47C2×C3⋊Dic3C3×D411yescharacteristic
48C2×C3⋊Dic3C23×C311yescharacteristic
49C2×C3×Dic3—24no—
50C2×C3×Dic3—24no—
51C2×C3×Dic3—44no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—62no—
54C3×Dic6—128no—
55#627—28no—
56#622C3×D414yes—
57#494—91noSylow
58C2×C33#12711yescharacteristic
59C2×C33C2×Q1611yescharacteristic
60C2×C33C2×Q1611yescharacteristic
61C2×C3×Q8—94no—
62C2×Dic6—64no—
63C3×Q16—188no—
64C2×C3×C8—92no—
65C3×C4⋊C4—62no—
66C3×C4⋊C4—62no—
67C3×C4⋊C4—122no—
68C3×C4⋊C4—182no—
69C3×C42—91no—
70C4⋊Dic3—22no—
71Dic3⋊C4—64no—
72C4×Dic3—62no—
73C22×C32—22no—
74C22×C32—42no—
75C22×C32C2×C3×D411yescharacteristic
76C32×C4—24no—
77C32×C4—44no—
78C32×C4—124no—
79C32×C4C3×Q1612yescomplemented
80C3⋊Dic3C3×Q1612yescomplemented
81C3⋊Dic3C2×C3×D412yes—
82C3×Dic3—44no—
83C3×Dic3—44no—
84C3×Dic3—64no—
85C3×Dic3—64no—
86C3×Dic3—84no—
87C2×Q16—94no—
88C4⋊Q8—92no—
89C4×C8—91no—
90C33#49411yescharacteristic, complemented, Sylow
91C3×Q8—188no—
92C2×C3×C4—22no—
93C2×C3×C4—42no—
94C2×C3×C4—64no—
95C2×C3×C4—64no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×C3×C4—124no—
99C2×C3×C4GO+(4,𝔽2)11yescharacteristic
100C2×Dic3—24no—
101C2×Dic3—62no—
102C2×Dic3—62no—
103Dic6—128no—
104C3×C8—94no—
105C2×C32—22no—
106C2×C32—22no—
107C2×C32—22no—
108C2×C32—42no—
109C2×C32—42no—
110C2×C32—42no—
111C2×C32C3×#12711yescharacteristic
112C2×C32C2×C3×Q1611yescharacteristic
113C2×C32C2×C3×Q1611yescharacteristic
114C2×Q8—94no—
115Q16—188no—
116C2×C8—92no—
117C4⋊C4—62no—
118C4⋊C4—182no—
119C42—91no—
120C22×C3—22no—
121C22×C3—42no—
122C22×C3C2×GO+(4,𝔽2)11yescharacteristic
123C3×C4—24no—
124C3×C4—44no—
125C3×C4—92no—
126C3×C4—92no—
127C3×C4—124no—
128C3×C4—124no—
129C3×C4—244no—
130C3×C4#377612yescomplemented
131Dic3—44no—
132Dic3—64no—
133Dic3—64no—
134C32—22no—
135C32—42no—
136C32C3×#49411yescharacteristic, complemented
137Q8—188no—
138C2×C4—64no—
139C2×C4—91no—
140C2×C4—91no—
141C2×C4C3×GO+(4,𝔽2)11yescharacteristic
142C8—94no—
143C2×C3—22no—
144C2×C3—22no—
145C2×C3—22no—
146C2×C3—42no—
147C2×C3—42no—
148C2×C3—42no—
149C2×C3#6422811yescharacteristic
150C2×C3#6423111yescharacteristic
151C2×C3C2×#377611yescharacteristic
152C22C2×C3×GO+(4,𝔽2)11yescharacteristic
153C4—92no—
154C4—92no—
155C4—124no—
156C4C3×#377612yescomplemented
157C3—22no—
158C3—42no—
159C3#1059345011yescharacteristic, direct product factor
160C2C3×#6422811yescharacteristic
161C2C3×#6423111yescharacteristic
162C2C2×C3×#377611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#494
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10593450C3direct (N × H)
35. C33×C4Q16semidirect
36. C3×C3⋊Dic3Q16semidirect
79. C32×C4C3×Q16semidirect
80. C3⋊Dic3C3×Q16semidirect
90. C33#494semidirect
130. C3×C4#3776semidirect
136. C32C3×#494semidirect
156. C4C3×#3776semidirect
159. C3#10593450direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C3×#639192
C3×C4×#6222
#105934503
C3×#4949