Finite Group Databasebeta

← C3×#10593390 #13166739

Subgroups — C3×#10593390 #13166739

Order 1728.

336 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 70 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×#63870C212yesmaximal
2C3×C4×#622C211yesmaximal, characteristic
3C3×#63743C211yesmaximal, characteristic
4C3×#63743C211yesmaximal, characteristic
5C3×#63742C212yesmaximal
6#10593390C311yesmaximal, characteristic, direct product factor
7C2×C3×#622C2211yescharacteristic
8C2×C3×#622C2211yescharacteristic
9C3×C4×C3⋊Dic3C2211yescharacteristic
10C32×Dic3⋊C4—22no—
11C3×#3724C2212yes—
12C3×#3724C2212yes—
13C3×Dic3⋊Dic3—22no—
14C3×#889—62no—
15#63870C2×C312yes—
16C4×#622C2×C311yescharacteristic
17#63743C2×C311yescharacteristic
18#63743C2×C311yescharacteristic
19#63742C2×C312yes—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C3×C3⋊Dic3C2311yescharacteristic
23C2×C32×Dic3—22no—
24C2×C32×Dic3—22no—
25C3×#622Q812yes—
26C3×#622Q812yes—
27C3×#499—91nomaximal
28C2×#622C22×C311yescharacteristic
29C2×#622C22×C311yescharacteristic
30C32×C4⋊C4—62no—
31C4×C3⋊Dic3C22×C311yescharacteristic
32C3×C4⋊Dic3—62no—
33C3×Dic3⋊C4—22no—
34C3×Dic3⋊C4—22no—
35C3×Dic3⋊C4—42no—
36C3×Dic3⋊C4—62no—
37C3×Dic3⋊C4—62no—
38C3×Dic3⋊C4—62no—
39C3×C4×Dic3—62no—
40#3724C22×C312yes—
41#3724C22×C312yes—
42Dic3⋊Dic3—22no—
43C22×C33C2×D411yescharacteristic
44C33×C4—21no—
45C3×C3⋊Dic3—21no—
46C3×C3⋊Dic3C2×Q811yescharacteristic
47C3×C3⋊Dic3C2×Q811yescharacteristic
48C32×Dic3—42no—
49C32×Dic3—42no—
50C3×#125—92no—
51#889—62no—
52C3×#107—91no—
53C3×#107—91no—
54C3×#106—92no—
55C3×C4×C8—91no—
56C2×C32×C4—22no—
57C2×C32×C4—42no—
58C2×C32×C4—62no—
59C2×C32×C4—62no—
60C2×C32×C4C3×D411yescharacteristic
61C2×C3⋊Dic3C3×D411yescharacteristic
62C2×C3⋊Dic3C23×C311yescharacteristic
63C2×C3×Dic3—22no—
64C2×C3×Dic3—22no—
65C2×C3×Dic3—22no—
66C2×C3×Dic3—22no—
67C2×C3×Dic3—42no—
68C2×C3×Dic3—42no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71#622C3×Q812yes—
72#622C3×Q812yes—
73#499—91noSylow
74C2×C33C4⋊Q811yescharacteristic
75C2×C33#13511yescharacteristic
76C2×C33#13511yescharacteristic
77C2×C3×C8—91no—
78C2×C3×C8—91no—
79C3×C4⋊C4—62no—
80C3×C4⋊C4—62no—
81C3×C4⋊C4—92no—
82C3×C4⋊C4—92no—
83C3×C4⋊C4—122no—
84C3×C4⋊C4—182no—
85C3×C42—91no—
86C4⋊Dic3—62no—
87Dic3⋊C4—22no—
88Dic3⋊C4—62no—
89Dic3⋊C4—62no—
90Dic3⋊C4—62no—
91C4×Dic3—62no—
92C22×C32—22no—
93C22×C32—42no—
94C22×C32C2×C3×D411yescharacteristic
95C32×C4—21no—
96C32×C4—42no—
97C32×C4—82no—
98C32×C4—122no—
99C32×C4—122no—
100C3⋊Dic3—21no—
101C3⋊Dic3C2×C3×Q811yescharacteristic
102C3⋊Dic3C2×C3×Q811yescharacteristic
103C3×Dic3—42no—
104C3×Dic3—42no—
105C3×Dic3—44no—
106C3×Dic3—42no—
107C3×Dic3—42no—
108C3×Dic3—62no—
109C3×Dic3—62no—
110C3×Dic3—82no—
111C3×Dic3—122no—
112#125—92no—
113#107—91no—
114#107—91no—
115#106—92no—
116C4×C8—91no—
117C33#49911yescharacteristic, complemented, Sylow
118C2×C3×C4—22no—
119C2×C3×C4—42no—
120C2×C3×C4—62no—
121C2×C3×C4—62no—
122C2×C3×C4—62no—
123C2×C3×C4—62no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—122no—
127C2×C3×C4—122no—
128C2×C3×C4GO+(4,𝔽2)11yescharacteristic
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—62no—
132C2×Dic3—62no—
133C3×C8—92no—
134C3×C8—92no—
135C2×C32—22no—
136C2×C32—22no—
137C2×C32—22no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32—42no—
141C2×C32C3×C4⋊Q811yescharacteristic
142C2×C32C3×#13511yescharacteristic
143C2×C32C3×#13511yescharacteristic
144C2×C8—91no—
145C2×C8—91no—
146C4⋊C4—62no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4—182no—
150C42—91no—
151C22×C3—22no—
152C22×C3—42no—
153C22×C3C2×GO+(4,𝔽2)11yescharacteristic
154C3×C4—21no—
155C3×C4—42no—
156C3×C4—82no—
157C3×C4—91no—
158C3×C4—91no—
159C3×C4—122no—
160C3×C4—122no—
161C3×C4—124no—
162C3×C4—122no—
163C3×C4—122no—
164C3×C4—181no—
165C3×C4—242no—
166Dic3—42no—
167Dic3—42no—
168Dic3—62no—
169Dic3—62no—
170Dic3—122no—
171C32—22no—
172C32—42no—
173C32C3×#49911yescharacteristic, complemented
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4C3×GO+(4,𝔽2)11yescharacteristic
179C8—92no—
180C8—92no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3—42no—
187C2×C3#6422211yescharacteristic
188C2×C3#6422311yescharacteristic
189C2×C3#6423711yescharacteristic
190C22C2×C3×GO+(4,𝔽2)11yescharacteristic
191C4—21no—
192C4—91no—
193C4—91no—
194C4—122no—
195C4—122no—
196C4—181no—
197C3—22no—
198C3—42no—
199C3#1059339011yescharacteristic, direct product factor
200C2C3×#6422211yescharacteristic
201C2C3×#6422311yescharacteristic
202C2C3×#6423711yescharacteristic
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×#499
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
6. #10593390C3direct (N × H)
117. C33#499semidirect
173. C32C3×#499semidirect
199. C3#10593390direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638702
C3×C4×#6222
C3×#637432
C3×#637422
#105933903
C3×#4999