Finite Group Databasebeta

← C3×#10589411 #13145968

Subgroups — C3×#10589411 #13145968

Order 1728.

948 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 126 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×#127C211yesmaximal, characteristic
2C3×#63661C212yesmaximal, complemented
3#10589411C311yesmaximal, characteristic, direct product factor
4C3×#4545—34nomaximal
5C2×C33×D4—22no—
6C2×C33×D4C411yescharacteristic, complemented
7C2×C33×D4C2211yescharacteristic
8C33×C42C411yescharacteristic, complemented
9C3×#3757—22no—
10C32×#127—24no—
11C32×#127D314yes—
12C32×#127C2×C311yescharacteristic
13#63661C2×C312yescomplemented
14C3×#834—38no—
15C23×C33—21no—
16C23×C33D412yes—
17C33×D4—41no—
18C33×D4—44no—
19C33×D4—41no—
20C2×C33×C4—21no—
21C2×C33×C4C2×C411yescharacteristic
22C2×C3×C3⋊Dic3—42no—
23C3×#353—91no—
24#4545—34no—
25C2×C32×D4—216no—
26C2×C32×D4—24no—
27C2×C32×D4—24no—
28C2×C32×D4—28no—
29C2×C32×D4—22no—
30C2×C32×D4Dic314yescomplemented
31C2×C32×D4C3×C411yescharacteristic, complemented
32C2×C32×D4C2×D314yes—
33C2×C32×D4C22×C311yescharacteristic
34C32×C42—24no—
35C32×C42Dic314yescomplemented
36C32×C42C3×C411yescharacteristic, complemented
37#3757—22no—
38C3×#217—68no—
39C22×C33—22no—
40C22×C33—42no—
41C22×C33—42no—
42C22×C33—41no—
43C22×C33C22⋊C411yescharacteristic
44C33×C4—22no—
45C33×C4—21no—
46C3×C3⋊Dic3—82no—
47C3×#127—24no—
48C3×#127C3×D314yes—
49C3×#127C3⋊D311yescharacteristic
50C3×#99—92no—
51#834—38no—
52C23×C32—28no—
53C23×C32—24no—
54C23×C32—21no—
55C23×C32—28no—
56C23×C32#6718yes—
57C23×C32C3×D412yes—
58C32×D4—44no—
59C32×D4—41no—
60C32×D4—44no—
61C32×D4—41no—
62C32×D4—48no—
63C32×D4—48no—
64C32×D4—416no—
65C32×D4—44no—
66C32×D4—432no—
67C2×C32×C4—28no—
68C2×C32×C4—24no—
69C2×C32×C4—21no—
70C2×C32×C4—24no—
71C2×C32×C4C2×Dic314yes—
72C2×C32×C4C2×C3×C411yescharacteristic
73C2×C3⋊Dic3—42no—
74C2×C3×Dic3—128no—
75#353—91noSylow
76C2×C33—21no—
77C2×C33—42no—
78C2×C33—81no—
79C2×C33#9911yescharacteristic
80C2×C3×D4—216no—
81C2×C3×D4—28no—
82C2×C3×D4—22no—
83C2×C3×D4—24no—
84C2×C3×D4—24no—
85C2×C3×D4C3×Dic314yescomplemented
86C2×C3×D4C3⋊Dic311yescharacteristic, complemented
87C2×C3×D4C2×C3×D314yes—
88C2×C3×D4C2×C3⋊D311yescharacteristic
89C3×C22⋊C4—182no—
90C3×C42—24no—
91C3×C42C3×Dic314yescomplemented
92C3×C42C3⋊Dic311yescharacteristic, complemented
93#217—68no—
94C22×C32—22no—
95C22×C32—24no—
96C22×C32—28no—
97C22×C32—48no—
98C22×C32—48no—
99C22×C32—42no—
100C22×C32—416no—
101C22×C32—416no—
102C22×C32—48no—
103C22×C32—42no—
104C22×C32—48no—
105C22×C32—44no—
106C22×C32—41no—
107C22×C32#21714yes—
108C22×C32C3×C22⋊C411yescharacteristic
109C32×C4—216no—
110C32×C4—28no—
111C32×C4—22no—
112C32×C4—28no—
113C32×C4—24no—
114C32×C4—21no—
115C3⋊Dic3—82no—
116C3×Dic3—248no—
117#127C3×C3⋊D311yescharacteristic
118#99—92no—
119C33#35311yescharacteristic, complemented, Sylow
120C23×C3—28no—
121C23×C3—24no—
122C23×C3—21no—
123C23×C3—28no—
124C23×C3C3×#6718yes—
125C23×C3#63812yes—
126C3×D4—44no—
127C3×D4—41no—
128C3×D4—44no—
129C3×D4—41no—
130C3×D4—48no—
131C3×D4—48no—
132C3×D4—416no—
133C3×D4—44no—
134C3×D4—432no—
135C2×C3×C4—28no—
136C2×C3×C4—24no—
137C2×C3×C4—21no—
138C2×C3×C4—24no—
139C2×C3×C4—362no—
140C2×C3×C4C2×C3×Dic314yes—
141C2×C3×C4C2×C3⋊Dic311yescharacteristic
142C2×Dic3—128no—
143C2×C32—28no—
144C2×C32—24no—
145C2×C32—21no—
146C2×C32—24no—
147C2×C32—42no—
148C2×C32—48no—
149C2×C32—88no—
150C2×C32—88no—
151C2×C32—84no—
152C2×C32—81no—
153C2×C32#83414yes—
154C2×C32C3×#9911yescharacteristic
155C2×D4—22no—
156C2×D4C3×C3⋊Dic311yescharacteristic, complemented
157C2×D4C2×C3×C3⋊D311yescharacteristic
158C22⋊C4—182no—
159C42C3×C3⋊Dic311yescharacteristic, complemented
160C22×C3—28no—
161C22×C3—22no—
162C22×C3—24no—
163C22×C3—48no—
164C22×C3—48no—
165C22×C3—42no—
166C22×C3—416no—
167C22×C3—416no—
168C22×C3—48no—
169C22×C3—42no—
170C22×C3—48no—
171C22×C3—44no—
172C22×C3—41no—
173C22×C3C3×#21714yes—
174C22×C3#375711yescharacteristic
175C3×C4—216no—
176C3×C4—28no—
177C3×C4—22no—
178C3×C4—28no—
179C3×C4—24no—
180C3×C4—21no—
181C3×C4—722no—
182Dic3—248no—
183C32—24no—
184C32#454514yescomplemented
185C32C3×#35311yescharacteristic, complemented
186C23—21no—
187C23C3×#63812yes—
188D4—41no—
189D4—41no—
190D4—44no—
191C2×C4—21no—
192C2×C4—362no—
193C2×C4C2×C3×C3⋊Dic311yescharacteristic
194C2×C3—28no—
195C2×C3—24no—
196C2×C3—21no—
197C2×C3—24no—
198C2×C3—48no—
199C2×C3—42no—
200C2×C3—88no—
201C2×C3—88no—
202C2×C3—84no—
203C2×C3—81no—
204C2×C3C3×#83414yes—
205C2×C3#6366111yescharacteristic
206C22—22no—
207C22—42no—
208C22—42no—
209C22—41no—
210C22C3×#375711yescharacteristic
211C4—22no—
212C4—21no—
213C4—722no—
214C3—24no—
215C3C3×#454514yescomplemented
216C3#1058941111yescharacteristic, direct product factor
217C2—21no—
218C2—42no—
219C2—81no—
220C2C3×#6366111yescharacteristic
subgroupG/H
centerC2×C3#63661
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×D4C2×C3×C3⋊D3
Fitting, prime radicalC33×#127C2
socleC2×C33#99
cosocleC2×C32×D4C22×C3
hypercenterC3×#127C3⋊D3
hypocenterC32C3×#353
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
2. C3×#63661C2semidirect
3. #10589411C3direct (N × H)
6. C2×C33×D4C4semidirect
8. C33×C42C4semidirect
13. #63661C2×C3semidirect
30. C2×C32×D4Dic3semidirect
31. C2×C32×D4C3×C4semidirect
35. C32×C42Dic3semidirect
36. C32×C42C3×C4semidirect
85. C2×C3×D4C3×Dic3semidirect
86. C2×C3×D4C3⋊Dic3semidirect
91. C3×C42C3×Dic3semidirect
92. C3×C42C3⋊Dic3semidirect
119. C33#353semidirect
156. C2×D4C3×C3⋊Dic3semidirect
159. C42C3×C3⋊Dic3semidirect
184. C32#4545semidirect
185. C32C3×#353semidirect
215. C3C3×#4545semidirect
216. C3#10589411direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1272
C3×#636612
#105894113
C3×#45453