Finite Group Databasebeta

← C3×#10590314 #13157329

Subgroups — C3×#10590314 #13157329

Order 1728.

530 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C32×#870C212yesmaximal
2C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3C3×#63576C212yesmaximal
4C3×#63574C212yesmaximal
5C3×#5065—32nomaximal
6#10590314C311yesmaximal, characteristic, direct product factor
7C33×C42C2211yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C32×C4⋊Dic3C2214yes—
11C32×Dic3⋊C4—24no—
12C32×#125—32no—
13C3×#870—22no—
14C3×#870D312yes—
15C3×#870C2×C312yes—
16C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
17C3×#808—34no—
18C3×#807—34no—
19C3×C4×C3⋊C8—32no—
20C3×C4×C3⋊C8—61no—
21#63576C2×C312yes—
22#63574C2×C312yes—
23C2×C33×C4D412yes—
24C2×C33×C4C2311yescharacteristic
25C2×C32×Dic3—24no—
26C3×C3⋊(C3⋊C8)Q812yes—
27C3×C3⋊(C3⋊C8)Q812yes—
28C3×#499—91no—
29#5065—32no—
30C32×C4⋊C4—34no—
31C32×C4⋊C4—64no—
32C32×C42—21no—
33C32×C42—22no—
34C32×C42—41no—
35C32×C42C2×D312yes—
36C32×C42C22×C311yescharacteristic
37C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
38C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
39C3×C4⋊Dic3—24no—
40C3×C4⋊Dic3C2×D314yes—
41C3×C4⋊Dic3C22×C314yes—
42C3×Dic3⋊C4—28no—
43C3×Dic3⋊C4—24no—
44C3×Dic3⋊C4—24no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—32no—
47C2×C3×C3⋊C8—61no—
48C2×C3×C3⋊C8—61no—
49C22×C33C2×D411yescharacteristic
50C33×C4—22no—
51C33×C4C2×Q811yescharacteristic
52C33×C4C2×Q811yescharacteristic
53C32×Dic3—44no—
54C3×#125—32no—
55C3×#125—32no—
56C3×#125—62no—
57#870C3×D312yes—
58C3×#107—92no—
59C3×#106—92no—
60C3×C4×C8—91no—
61#808—34no—
62#807—34no—
63C4×C3⋊C8—32no—
64C4×C3⋊C8—61no—
65C2×C32×C4—22no—
66C2×C32×C4—24no—
67C2×C32×C4—21no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4—41no—
71C2×C32×C4—64no—
72C2×C32×C4#6714yes—
73C2×C32×C4C3×D412yes—
74C2×C32×C4C22×D312yes—
75C2×C32×C4C23×C311yescharacteristic
76C2×C3×Dic3—28no—
77C2×C3×Dic3—24no—
78C2×C3×Dic3—24no—
79C3⋊(C3⋊C8)C3×Q812yes—
80C3⋊(C3⋊C8)C3×Q812yes—
81C3×C3⋊C8—34no—
82C3×C3⋊C8—34no—
83C3×C3⋊C8—62no—
84C3×C3⋊C8—62no—
85#499—91noSylow
86C2×C33C4⋊Q811yescharacteristic
87C2×C33#13512yes—
88C2×C3×C8—91no—
89C2×C3×C8—91no—
90C3×C4⋊C4—34no—
91C3×C4⋊C4—34no—
92C3×C4⋊C4—64no—
93C3×C4⋊C4—68no—
94C3×C4⋊C4—64no—
95C3×C4⋊C4—64no—
96C3×C42—21no—
97C3×C42—22no—
98C3×C42—41no—
99C3×C42D3211yescharacteristic
100C3×C42C2×C3×D312yes—
101C4⋊Dic3C2×C3×D314yes—
102Dic3⋊C4—24no—
103C2×C3⋊C8—32no—
104C2×C3⋊C8—32no—
105C2×C3⋊C8—61no—
106C2×C3⋊C8—61no—
107C22×C32—21no—
108C22×C32—22no—
109C22×C32—41no—
110C22×C32C2×#6712yes—
111C22×C32C2×C3×D411yescharacteristic
112C32×C4—22no—
113C32×C4—24no—
114C32×C4—24no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—41no—
120C32×C4—41no—
121C32×C4—44no—
122C32×C4—44no—
123C32×C4—124no—
124C32×C4D3×Q812yes—
125C32×C4D3×Q812yes—
126C32×C4C2×C3×Q811yescharacteristic
127C32×C4C2×C3×Q811yescharacteristic
128C3×Dic3—48no—
129C3×Dic3—44no—
130C3×Dic3—44no—
131#125—32no—
132#107—92no—
133#106—92no—
134C4×C8—91no—
135C33#49911yescharacteristic, complemented, Sylow
136C2×C3×C4—22no—
137C2×C3×C4—24no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4—68no—
143C2×C3×C4—64no—
144C2×C3×C4—64no—
145C2×C3×C4#62512yes—
146C2×C3×C4C3×#6714yes—
147C2×C3×C4C2×D3211yescharacteristic
148C2×C3×C4C22×C3×D312yes—
149C2×Dic3—24no—
150C3×C8—92no—
151C3×C8—92no—
152C3⋊C8—34no—
153C3⋊C8—34no—
154C3⋊C8—62no—
155C3⋊C8—62no—
156C2×C32—22no—
157C2×C32—24no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—41no—
161C2×C32—42no—
162C2×C32#94412yes—
163C2×C32#95014yes—
164C2×C32C3×C4⋊Q811yescharacteristic
165C2×C32C3×#13512yes—
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—34no—
169C4⋊C4—64no—
170C42C3×D3211yescharacteristic
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3C2×#62511yescharacteristic
175C22×C3C2×C3×#6712yes—
176C3×C4—24no—
177C3×C4—24no—
178C3×C4—22no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—44no—
184C3×C4—44no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4—128no—
188C3×C4—124no—
189C3×C4—124no—
190C3×C4#379711yescharacteristic
191C3×C4#379711yescharacteristic
192C3×C4C3×D3×Q812yes—
193C3×C4C3×D3×Q812yes—
194Dic3—44no—
195C32—21no—
196C32—22no—
197C32—41no—
198C32#506512yescomplemented
199C32C3×#49911yescharacteristic, complemented
200C2×C4—64no—
201C2×C4C3×#62512yes—
202C2×C4C2×C3×D3211yescharacteristic
203C8—92no—
204C8—92no—
205C2×C3—22no—
206C2×C3—24no—
207C2×C3—22no—
208C2×C3—21no—
209C2×C3—42no—
210C2×C3—41no—
211C2×C3#6382212yes—
212C2×C3#6387311yescharacteristic
213C2×C3C3×#94412yes—
214C2×C3C3×#95014yes—
215C22C2×C3×#62511yescharacteristic
216C4—22no—
217C4—124no—
218C4C3×#379711yescharacteristic
219C4C3×#379711yescharacteristic
220C3—22no—
221C3—21no—
222C3—41no—
223C3#1059031411yescharacteristic, direct product factor
224C3C3×#506512yescomplemented
225C2C3×#6382212yes—
226C2C3×#6387311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
6. #10590314C3direct (N × H)
135. C33#499semidirect
198. C32#5065semidirect
199. C32C3×#499semidirect
223. C3#10590314direct (N × H)
224. C3C3×#5065semidirect

Maximal subgroups

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

subgroupindex
C32×#8702
C3×C4×C3⋊(C3⋊C8)2
C3×#635762
C3×#635742
C3×#50653
#105903143