Finite Group Databasebeta

← C3×#10593395 #13166748

Subgroups — C3×#10593395 #13166748

Order 1728.

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

normal subgroups only

This group has 384 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
1C3×#63919C212yesmaximal
2C3×C4×#622C211yesmaximal, characteristic
3C3×#63743C214yesmaximal
4#10593395C311yesmaximal, characteristic, direct product factor
5C2×C3×#627—22no—
6C2×C3×#622C2212yes—
7C3×C4×C3⋊Dic3C2211yescharacteristic
8C32×C4⋊Dic3—22no—
9C3×#3724C2214yes—
10C3×#888—62no—
11#63919C2×C312yes—
12C4×#622C2×C311yescharacteristic
13#63743C2×C314yes—
14C2×C33×C4D411yescharacteristic
15C2×C3×C3⋊Dic3D411yescharacteristic
16C2×C3×C3⋊Dic3C2311yescharacteristic
17C2×C32×Dic3—24no—
18C3×#627—44no—
19C3×#622Q814yescomplemented
20C3×#500—91nomaximal
21C2×C3×Dic6—64no—
22C2×#627—22no—
23C2×#622C22×C312yes—
24C32×C4⋊C4—62no—
25C4×C3⋊Dic3C22×C311yescharacteristic
26C3×C4⋊Dic3—22no—
27C3×C4⋊Dic3—22no—
28C3×C4⋊Dic3—42no—
29C3×Dic3⋊C4—64no—
30C3×C4×Dic3—62no—
31#3724C22×C314yes—
32C22×C33C2×D411yescharacteristic
33C33×C4Q1612yes—
34C3×C3⋊Dic3D812yes—
35C3×C3⋊Dic3C2×Q812yes—
36C32×Dic3—44no—
37C3×C4⋊Q8—92no—
38#888—62no—
39C3×#107—94no—
40C3×C4×C8—91no—
41C2×C32×C4—22no—
42C2×C32×C4—42no—
43C2×C32×C4—64no—
44C2×C32×C4C3×D411yescharacteristic
45C2×C3⋊Dic3C3×D411yescharacteristic
46C2×C3⋊Dic3C23×C311yescharacteristic
47C2×C3×Dic3—24no—
48C2×C3×Dic3—24no—
49C2×C3×Dic3—44no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3—62no—
52C3×Dic6—128no—
53#627—44no—
54#622C3×Q814yescomplemented
55#500—91noSylow
56C2×C33C4⋊Q811yescharacteristic
57C2×C33C2×D811yescharacteristic
58C2×C33C2×Q1611yescharacteristic
59C2×C3×Q8—182no—
60C2×Dic6—64no—
61C2×C3×C8—92no—
62C3×C4⋊C4—62no—
63C3×C4⋊C4—62no—
64C3×C4⋊C4—94no—
65C3×C4⋊C4—122no—
66C3×C42—91no—
67C4⋊Dic3—22no—
68Dic3⋊C4—64no—
69C4×Dic3—62no—
70C22×C32—22no—
71C22×C32—42no—
72C22×C32C2×C3×D411yescharacteristic
73C32×C4—24no—
74C32×C4—44no—
75C32×C4—124no—
76C32×C4C3×Q1612yes—
77C3⋊Dic3C3×D812yes—
78C3⋊Dic3C2×C3×Q812yes—
79C3×Dic3—44no—
80C3×Dic3—44no—
81C3×Dic3—64no—
82C3×Dic3—64no—
83C3×Dic3—84no—
84C4⋊Q8—92no—
85#107—94no—
86C4×C8—91no—
87C33#50011yescharacteristic, complemented, Sylow
88C3×Q8—364no—
89C2×C3×C4—22no—
90C2×C3×C4—42no—
91C2×C3×C4—64no—
92C2×C3×C4—64no—
93C2×C3×C4—91no—
94C2×C3×C4—91no—
95C2×C3×C4—124no—
96C2×C3×C4GO+(4,𝔽2)11yescharacteristic
97C2×Dic3—24no—
98C2×Dic3—62no—
99C2×Dic3—62no—
100Dic6—128no—
101C3×C8—94no—
102C2×C32—22no—
103C2×C32—22no—
104C2×C32—22no—
105C2×C32—42no—
106C2×C32—42no—
107C2×C32—42no—
108C2×C32C3×C4⋊Q811yescharacteristic
109C2×C32C2×C3×D811yescharacteristic
110C2×C32C2×C3×Q1611yescharacteristic
111C2×Q8—182no—
112C2×C8—92no—
113C4⋊C4—62no—
114C4⋊C4—94no—
115C42—91no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3C2×GO+(4,𝔽2)11yescharacteristic
119C3×C4—24no—
120C3×C4—44no—
121C3×C4—92no—
122C3×C4—92no—
123C3×C4—124no—
124C3×C4—124no—
125C3×C4—244no—
126C3×C4#377612yes—
127Dic3—44no—
128Dic3—64no—
129Dic3—64no—
130C32—22no—
131C32—42no—
132C32C3×#50011yescharacteristic, complemented
133Q8—364no—
134C2×C4—64no—
135C2×C4—91no—
136C2×C4—91no—
137C2×C4C3×GO+(4,𝔽2)11yescharacteristic
138C8—94no—
139C2×C3—22no—
140C2×C3—22no—
141C2×C3—22no—
142C2×C3—42no—
143C2×C3—42no—
144C2×C3—42no—
145C2×C3#6422211yescharacteristic
146C2×C3#6422511yescharacteristic
147C2×C3C2×#377611yescharacteristic
148C22C2×C3×GO+(4,𝔽2)11yescharacteristic
149C4—92no—
150C4—92no—
151C4—124no—
152C4C3×#377612yes—
153C3—22no—
154C3—42no—
155C3#1059339511yescharacteristic, direct product factor
156C2C3×#6422211yescharacteristic
157C2C3×#6422511yescharacteristic
158C2C2×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×#500
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. #10593395C3direct (N × H)
19. C3×#622Q8semidirect
54. #622C3×Q8semidirect
87. C33#500semidirect
132. C32C3×#500semidirect
155. C3#10593395direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639192
C3×C4×#6222
C3×#637432
#105933953
C3×#5009