Finite Group Databasebeta

← C3×#10592764 #13163103

Subgroups — C3×#10592764 #13163103

Order 1728.

740 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C33×#125C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic
3C3×#63635C212yesmaximal
4C3×#63634C212yesmaximal
5C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
6#10592764C311yesmaximal, characteristic, direct product factor
7C3×#5065—34nomaximal
8C33×C4⋊C4—22no—
9C33×C4⋊C4C2212yes—
10C33×C42C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×#3750—22no—
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×#125—24no—
16C32×#125D314yes—
17C32×#125C2×C311yescharacteristic
18#64079C2×C311yescharacteristic
19C3×#870—34no—
20#63635C2×C312yes—
21#63634C2×C312yes—
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
23C3×#808—38no—
24C3×#807—38no—
25C3×C4×C3⋊C8—34no—
26C2×C33×C4—22no—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C3×C3⋊(C3⋊C8)Q812yes—
31C3×C3⋊(C3⋊C8)Q812yes—
32C3×#499—91no—
33#5065—34no—
34C32×C4⋊C4—216no—
35C32×C4⋊C4—28no—
36C32×C4⋊C4—28no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4C2×D318yes—
39C32×C4⋊C4C22×C312yes—
40C32×C42—24no—
41C32×C42C2×D314yes—
42C32×C42C22×C311yescharacteristic
43C4⋊(C3⋊Dic3)C22×C312yes—
44#3750—22no—
45C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
46C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
47C3×C4⋊Dic3—38no—
48C3×Dic3⋊C4—68no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—34no—
51C22×C33C2×D411yescharacteristic
52C33×C4—22no—
53C33×C4—42no—
54C33×C4C2×Q811yescharacteristic
55C33×C4C2×Q811yescharacteristic
56C3×C3⋊Dic3—42no—
57C3×#125—24no—
58C3×#125—91no—
59C3×#125C3×D314yes—
60C3×#125C3⋊D311yescharacteristic
61#870—34no—
62C3×#107—92no—
63C3×#106—92no—
64C3×C4×C8—91no—
65#808—38no—
66#807—38no—
67C4×C3⋊C8—34no—
68C2×C32×C4—216no—
69C2×C32×C4—28no—
70C2×C32×C4—22no—
71C2×C32×C4—28no—
72C2×C32×C4—24no—
73C2×C32×C4#6718yes—
74C2×C32×C4C3×D412yes—
75C2×C32×C4C22×D314yes—
76C2×C32×C4C23×C311yescharacteristic
77C2×C3⋊Dic3—22no—
78C2×C3×Dic3—68no—
79C3⋊(C3⋊C8)C3×Q812yes—
80C3⋊(C3⋊C8)C3×Q812yes—
81C3×C3⋊C8—38no—
82C3×C3⋊C8—38no—
83#499—91noSylow
84C2×C33C4⋊Q811yescharacteristic
85C2×C33#13512yes—
86C2×C3×C8—91no—
87C2×C3×C8—91no—
88C3×C4⋊C4—216no—
89C3×C4⋊C4—28no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—28no—
92C3×C4⋊C4—92no—
93C3×C4⋊C4—182no—
94C3×C4⋊C4C2×C3×D318yes—
95C3×C4⋊C4C2×C3⋊D312yes—
96C3×C42—24no—
97C3×C42C2×C3×D314yes—
98C3×C42C2×C3⋊D311yescharacteristic
99C4⋊Dic3—38no—
100Dic3⋊C4—68no—
101C2×C3⋊C8—34no—
102C2×C3⋊C8—34no—
103C22×C32—24no—
104C22×C32C2×#6714yes—
105C22×C32C2×C3×D411yescharacteristic
106C32×C4—22no—
107C32×C4—24no—
108C32×C4—24no—
109C32×C4—28no—
110C32×C4—48no—
111C32×C4—416no—
112C32×C4—48no—
113C32×C4—42no—
114C32×C4D3×Q814yes—
115C32×C4D3×Q814yes—
116C32×C4C2×C3×Q811yescharacteristic
117C32×C4C2×C3×Q811yescharacteristic
118C3⋊Dic3—42no—
119C3×Dic3—128no—
120#125—91no—
121#125C3×C3⋊D311yescharacteristic
122#107—92no—
123#106—92no—
124C4×C8—91no—
125C33#49911yescharacteristic, complemented, Sylow
126C2×C3×C4—216no—
127C2×C3×C4—28no—
128C2×C3×C4—22no—
129C2×C3×C4—28no—
130C2×C3×C4—24no—
131C2×C3×C4—182no—
132C2×C3×C4C3×#6718yes—
133C2×C3×C4#63812yes—
134C2×C3×C4C22×C3×D314yes—
135C2×C3×C4C22×C3⋊D311yescharacteristic
136C2×Dic3—68no—
137C3×C8—92no—
138C3×C8—92no—
139C3⋊C8—38no—
140C3⋊C8—38no—
141C2×C32—24no—
142C2×C32—28no—
143C2×C32#94414yes—
144C2×C32#95018yes—
145C2×C32C3×C4⋊Q811yescharacteristic
146C2×C32C3×#13512yes—
147C2×C8—91no—
148C2×C8—91no—
149C4⋊C4—22no—
150C4⋊C4—92no—
151C4⋊C4—182no—
152C4⋊C4C2×C3×C3⋊D312yes—
153C42C2×C3×C3⋊D311yescharacteristic
154C22×C3—24no—
155C22×C3C2×C3×#6714yes—
156C22×C3C2×#63811yescharacteristic
157C3×C4—28no—
158C3×C4—22no—
159C3×C4—24no—
160C3×C4—24no—
161C3×C4—48no—
162C3×C4—416no—
163C3×C4—48no—
164C3×C4—42no—
165C3×C4—362no—
166C3×C4C3×D3×Q814yes—
167C3×C4C3×D3×Q814yes—
168C3×C4Q8×C3⋊D311yescharacteristic
169C3×C4Q8×C3⋊D311yescharacteristic
170Dic3—128no—
171C32—24no—
172C32#506514yescomplemented
173C32C3×#49911yescharacteristic, complemented
174C2×C4—22no—
175C2×C4—182no—
176C2×C4C3×#63812yes—
177C2×C4C22×C3×C3⋊D311yescharacteristic
178C8—92no—
179C8—92no—
180C2×C3—24no—
181C2×C3—28no—
182C2×C3C3×#94414yes—
183C2×C3C3×#95018yes—
184C2×C3#6415311yescharacteristic
185C2×C3#6415912yes—
186C22C2×C3×#63811yescharacteristic
187C4—22no—
188C4—42no—
189C4—362no—
190C4C3×Q8×C3⋊D311yescharacteristic
191C4C3×Q8×C3⋊D311yescharacteristic
192C3—24no—
193C3C3×#506514yescomplemented
194C3#1059276411yescharacteristic, direct product factor
195C2C3×#6415311yescharacteristic
196C2C3×#6415912yes—
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#125C2
socle, prime radicalC22×C33C2×D4
hypercenterC3×#125C3⋊D3
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
6. #10592764C3direct (N × H)
125. C33#499semidirect
172. C32#5065semidirect
173. C32C3×#499semidirect
193. C3C3×#5065semidirect
194. C3#10592764direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1252
C3×#640792
C3×#636352
C3×#636342
C3×C4×C3⋊(C3⋊C8)2
#105927643
C3×#50653