Finite Group Databasebeta

← C3×#10589330 #13145889

Subgroups — C3×#10589330 #13145889

Order 1728.

580 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 130 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×#64084C211yesmaximal, characteristic
2C33×#97C211yesmaximal, characteristic, complemented
3C3×#63630C211yesmaximal, characteristic, complemented
4#10589330C311yesmaximal, characteristic, direct product factor
5C3×#4464—34nomaximal
6C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
7C2×C3×#634C411yescharacteristic
8C2×C33×C8—21no—
9C33×C42C2211yescharacteristic
10C3×#3752—21no—
11C3×#3752—21no—
12C2×C3×C3⋊(C3⋊C8)—21no—
13#64084C2×C311yescharacteristic
14C3×#875—34no—
15C32×#97—24no—
16C32×#97D314yescomplemented
17C32×#97C2×C311yescharacteristic, complemented
18#63630C2×C311yescharacteristic, complemented
19C3×#803—34no—
20C22×C3×C3⋊D3—21no—
21C2×C33×C4C2×C411yescharacteristic
22C2×C33×C4D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C3×C4⋊(C3⋊D3)—41no—
26C3×#634—41no—
27C33×C8—22no—
28C3×C3⋊(C3⋊C8)—22no—
29C3×#329—91no—
30#4464—34no—
31C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
32C2×#634C3×C411yescharacteristic
33C2×C3×D12—34no—
34C2×C3×Dic6—34no—
35C2×C32×C8—28no—
36C2×C32×C8—24no—
37C2×C32×C8—21no—
38C32×C42—24no—
39C32×C42C2×D314yes—
40C32×C42C22×C311yescharacteristic
41#3752—21no—
42#3752—21no—
43C2×C3⋊(C3⋊C8)—21no—
44C3×(C2×D3)⋊C4—64no—
45C3×(C2×D3)⋊C4—64no—
46C2×C3×C3⋊C8—64no—
47C22×C33C22⋊C411yescharacteristic
48C2×C3×C3⋊D3—41no—
49C2×C3×C3⋊D3—41no—
50C2×C3×C3⋊D3—41no—
51C33×C4—21no—
52C33×C4—21no—
53C33×C4—21no—
54C3×C3⋊Dic3—41no—
55C3×#124—91no—
56#875—34no—
57C3×#97—24no—
58C3×#97—91no—
59C3×#97C3×D314yescomplemented
60C3×#97C3⋊D311yescharacteristic, complemented
61#803—34no—
62C22×C3⋊D3—21no—
63C22×C3×D3—64no—
64C2×C32×C4—24no—
65C2×C32×C4—24no—
66C2×C32×C4—24no—
67C2×C32×C4C4×D314yes—
68C2×C32×C4D1214yes—
69C2×C32×C4#6714yes—
70C2×C32×C4C2×C3×C411yescharacteristic
71C2×C32×C4C3×D411yescharacteristic
72C2×C32×C4C3×D411yescharacteristic
73C2×C3⋊Dic3—21no—
74C4⋊(C3⋊D3)—41no—
75#634—41no—
76C2×C3×Dic3—64no—
77C3×D12—124no—
78C3×Dic6—124no—
79C32×C8—216no—
80C32×C8—28no—
81C32×C8—22no—
82C3⋊(C3⋊C8)—22no—
83C3×C3⋊C8—68no—
84#329—91noSylow
85C2×C33#10011yescharacteristic
86C2×C33#10411yescharacteristic
87C2×C33#10411yescharacteristic
88C3×C3⋊D3—81no—
89C2×C3×Q8—91no—
90C2×C3×D4—91no—
91C2×D12—34no—
92C2×Dic6—34no—
93C2×C3×C8—28no—
94C2×C3×C8—24no—
95C2×C3×C8—21no—
96C2×C3×C8—181no—
97C3×C22⋊C4—181no—
98C3×C22⋊C4—181no—
99C3×C42—24no—
100C3×C42C2×C3×D314yes—
101C3×C42C2×C3⋊D311yescharacteristic
102(C2×D3)⋊C4—64no—
103(C2×D3)⋊C4—64no—
104C2×C3⋊C8—64no—
105C22×C32—24no—
106C22×C32(C2×D3)⋊C414yes—
107C22×C32C3×C22⋊C411yescharacteristic
108C2×C3⋊D3—41no—
109C2×C3⋊D3—41no—
110C2×C3⋊D3—41no—
111C2×C3×D3—124no—
112C2×C3×D3—124no—
113C2×C3×D3—124no—
114C32×C4—28no—
115C32×C4—24no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—24no—
120C32×C4—24no—
121C32×C4—44no—
122C32×C4—44no—
123C3⋊Dic3—41no—
124C3×Dic3—124no—
125#124—91no—
126#97—91no—
127#97C3×C3⋊D311yescharacteristic, complemented
128C33#32911yescharacteristic, complemented, Sylow
129C23×C3—181no—
130C22×D3—64no—
131C3×Q8—361no—
132C3×D4—361no—
133C2×C3×C4—24no—
134C2×C3×C4—24no—
135C2×C3×C4—24no—
136C2×C3×C4—181no—
137C2×C3×C4C3×C4×D314yes—
138C2×C3×C4C3×D1214yes—
139C2×C3×C4C3×#6714yes—
140C2×C3×C4C4×C3⋊D311yescharacteristic
141C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
142C2×C3×C4#63811yescharacteristic
143C2×Dic3—64no—
144D12—124no—
145Dic6—124no—
146C3×C8—216no—
147C3×C8—28no—
148C3×C8—22no—
149C3×C8—182no—
150C3⋊C8—68no—
151C2×C32—24no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32#80514yes—
155C2×C32#82314yes—
156C2×C32#82514yes—
157C2×C32C3×#10011yescharacteristic
158C2×C32C3×#10411yescharacteristic
159C2×C32C3×#10411yescharacteristic
160C3⋊D3—81no—
161C3×D3—244no—
162C2×Q8—91no—
163C2×D4—91no—
164C2×C8—21no—
165C2×C8—181no—
166C22⋊C4—181no—
167C22⋊C4—181no—
168C42C2×C3×C3⋊D311yescharacteristic
169C22×C3—24no—
170C22×C3—361no—
171C22×C3—361no—
172C22×C3—361no—
173C22×C3C3×(C2×D3)⋊C414yes—
174C22×C3#375211yescharacteristic
175C2×D3—124no—
176C2×D3—124no—
177C2×D3—124no—
178C3×C4—28no—
179C3×C4—24no—
180C3×C4—24no—
181C3×C4—24no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—44no—
186C3×C4—44no—
187C3×C4—361no—
188Dic3—124no—
189C32—24no—
190C32#446414yescomplemented
191C32C3×#32911yescharacteristic, complemented
192C23—181no—
193Q8—361no—
194D4—361no—
195C2×C4—181no—
196C2×C4C3×C4×C3⋊D311yescharacteristic
197C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
198C2×C4C3×#63811yescharacteristic
199C8—22no—
200C8—182no—
201C2×C3—24no—
202C2×C3—24no—
203C2×C3—24no—
204C2×C3—721no—
205C2×C3C3×#80514yes—
206C2×C3C3×#82314yes—
207C2×C3C3×#82514yes—
208C2×C3#6363211yescharacteristic
209C2×C3#6365011yescharacteristic
210C2×C3#6365211yescharacteristic
211D3—244no—
212C22—361no—
213C22—361no—
214C22—361no—
215C22C3×#375211yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—361no—
220C3—24no—
221C3C3×#446414yescomplemented
222C3#1058933011yescharacteristic, direct product factor
223C2—721no—
224C2C3×#6363211yescharacteristic
225C2C3×#6365011yescharacteristic
226C2C3×#6365211yescharacteristic
subgroupG/H
centerC22×C3#3752
commutatorC2×C32×C4C2×C3×C4
FrattiniC42C2×C3×C3⋊D3
FittingC33×#97C2
socleC22×C33C22⋊C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC32×C42C22×C3
hypercenterC3×#97C3⋊D3
hypocenterC32C3×#329
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
2. C33×#97C2semidirect
3. C3×#63630C2semidirect
4. #10589330C3direct (N × H)
16. C32×#97D3semidirect
17. C32×#97C2×C3semidirect
18. #63630C2×C3semidirect
59. C3×#97C3×D3semidirect
60. C3×#97C3⋊D3semidirect
127. #97C3×C3⋊D3semidirect
128. C33#329semidirect
190. C32#4464semidirect
191. C32C3×#329semidirect
221. C3C3×#4464semidirect
222. C3#10589330direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640842
C33×#972
C3×#636302
#105893303
C3×#44643