Finite Group Databasebeta

← C3×#10592942 #13163279

Subgroups — C3×#10592942 #13163279

Order 1728.

1624 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 214 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×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#64144C213yesmaximal, complemented
3C3×#63661C213yesmaximal, complemented
4#10592942C311yesmaximal, characteristic, direct product factor
5C3×#5243—34nomaximal
6C33×#41—23no—
7C2×C33×D4—23no—
8C2×C33×D4C2213yes—
9C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
10C3×#3757—23no—
11C3×#3757C2213yescomplemented
12C3×#3750—23no—
13C32×H(5,𝔽2)—24no—
14C32×H(5,𝔽2)D314yes—
15C32×H(5,𝔽2)C2×C311yescharacteristic
16#64144C2×C313yescomplemented
17C3×#935—312no—
18#63661C2×C313yescomplemented
19C3×#834—312no—
20C23×C33—21no—
21C23×C33D413yes—
22C23×C33C2311yescharacteristic
23C33×Q8—21no—
24C33×D4—23no—
25C33×D4—26no—
26C2×C33×C4—23no—
27C2×C33×C4D413yes—
28C2×C3×C3⋊Dic3—23no—
29C2×C3×C3⋊Dic3—23no—
30C3×#458—91no—
31#5243—34no—
32C32×#41—224no—
33C32×#41—212no—
34C32×#41—23no—
35C2×C32×D4—224no—
36C2×C32×D4—212no—
37C2×C32×D4—212no—
38C2×C32×D4—23no—
39C2×C32×D4C2×D3112yes—
40C2×C32×D4C22×C313yes—
41C22×C3⋊Dic3C22×C311yescharacteristic, complemented
42C22×C3×Dic3—34no—
43#3757—23no—
44#3757C22×C313yescomplemented
45#3750—23no—
46C3×#217—312no—
47C3×#217—612no—
48C3×Dic3⋊C4—612no—
49C22×C33—23no—
50C22×C33—41no—
51C22×C33—43no—
52C22×C33—81no—
53C22×C33C2×D413yes—
54C33×C4—23no—
55C3×C3⋊Dic3—43no—
56C3×C3⋊Dic3—41no—
57C3×H(5,𝔽2)—24no—
58C3×H(5,𝔽2)C3×D314yes—
59C3×H(5,𝔽2)C3⋊D311yescharacteristic
60C3×#123—93no—
61#935—312no—
62C3×#99—93no—
63#834—312no—
64C23×C32—28no—
65C23×C32—24no—
66C23×C32—21no—
67C23×C32—212no—
68C23×C32—24no—
69C23×C32#67112yes—
70C23×C32C3×D413yes—
71C23×C32C22×D314yes—
72C23×C32C23×C311yescharacteristic
73C32×Q8—28no—
74C32×Q8—24no—
75C32×Q8—21no—
76C32×D4—224no—
77C32×D4—212no—
78C32×D4—23no—
79C32×D4—248no—
80C32×D4—224no—
81C32×D4—26no—
82C2×C32×C4—224no—
83C2×C32×C4—212no—
84C2×C32×C4—23no—
85C2×C32×C4—212no—
86C2×C32×C4#67112yes—
87C2×C32×C4C3×D413yes—
88C2×C3⋊Dic3—23no—
89C2×C3⋊Dic3—23no—
90C2×C3×Dic3—612no—
91C2×C3×Dic3—612no—
92#458—91noSylow
93C2×C33—23no—
94C2×C33—43no—
95C2×C33#12011yescharacteristic
96C3×#41—224no—
97C3×#41—212no—
98C3×#41—23no—
99C2×C3×D4—224no—
100C2×C3×D4—212no—
101C2×C3×D4—23no—
102C2×C3×D4—212no—
103C2×C3×D4C2×C3×D3112yes—
104C2×C3×D4C2×C3⋊D313yes—
105C22×C3×C4—91no—
106C22×Dic3—34no—
107C3×C4⋊C4—183no—
108C3×C22⋊C4—93no—
109C3×C22⋊C4—183no—
110#217—312no—
111#217—612no—
112Dic3⋊C4—612no—
113C22×C32—224no—
114C22×C32—212no—
115C22×C32—23no—
116C22×C32—212no—
117C22×C32—41no—
118C22×C32—43no—
119C22×C32—44no—
120C22×C32—412no—
121C22×C32—84no—
122C22×C32—812no—
123C22×C32—88no—
124C22×C32—84no—
125C22×C32—81no—
126C22×C32C2×#67112yes—
127C22×C32C2×C3×D413yes—
128C32×C4—224no—
129C32×C4—212no—
130C32×C4—23no—
131C3⋊Dic3—43no—
132C3⋊Dic3—41no—
133C3×Dic3—1212no—
134C3×Dic3—124no—
135H(5,𝔽2)C3×C3⋊D311yescharacteristic
136#123—93no—
137#99—93no—
138C33#45811yescharacteristic, complemented, Sylow
139C23×C3—28no—
140C23×C3—24no—
141C23×C3—21no—
142C23×C3—212no—
143C23×C3—24no—
144C23×C3C3×#67112yes—
145C23×C3#63813yes—
146C23×C3C22×C3×D314yes—
147C23×C3C22×C3⋊D311yescharacteristic
148C3×Q8—28no—
149C3×Q8—24no—
150C3×Q8—21no—
151C3×D4—224no—
152C3×D4—212no—
153C3×D4—23no—
154C3×D4—248no—
155C3×D4—224no—
156C3×D4—26no—
157C2×C3×C4—224no—
158C2×C3×C4—212no—
159C2×C3×C4—23no—
160C2×C3×C4—212no—
161C2×C3×C4—183no—
162C2×C3×C4—183no—
163C2×C3×C4C3×#67112yes—
164C2×C3×C4#63813yes—
165C2×Dic3—612no—
166C2×Dic3—612no—
167C2×C32—23no—
168C2×C32—24no—
169C2×C32—212no—
170C2×C32—412no—
171C2×C32—424no—
172C2×C32—412no—
173C2×C32—43no—
174C2×C32#95314yes—
175C2×C32C3×#12011yescharacteristic
176#41—23no—
177C2×D4—23no—
178C2×D4C2×C3×C3⋊D313yes—
179C22×C4—91no—
180C4⋊C4—183no—
181C22⋊C4—93no—
182C22⋊C4—183no—
183C22×C3—224no—
184C22×C3—212no—
185C22×C3—23no—
186C22×C3—212no—
187C22×C3—44no—
188C22×C3—41no—
189C22×C3—412no—
190C22×C3—43no—
191C22×C3—84no—
192C22×C3—812no—
193C22×C3—88no—
194C22×C3—84no—
195C22×C3—81no—
196C22×C3C2×C3×#67112yes—
197C22×C3C2×#63813yes—
198C3×C4—224no—
199C3×C4—212no—
200C3×C4—23no—
201C3×C4—363no—
202C3×C4—361no—
203Dic3—1212no—
204Dic3—124no—
205C32—24no—
206C32#524314yescomplemented
207C32C3×#45811yescharacteristic, complemented
208C23—21no—
209C23C3×#63813yes—
210C23C22×C3×C3⋊D311yescharacteristic
211Q8—21no—
212D4—23no—
213D4—26no—
214C2×C4—23no—
215C2×C4—183no—
216C2×C4—183no—
217C2×C4C3×#63813yes—
218C2×C3—212no—
219C2×C3—23no—
220C2×C3—24no—
221C2×C3—412no—
222C2×C3—424no—
223C2×C3—412no—
224C2×C3—43no—
225C2×C3C3×#95314yes—
226C2×C3#6416211yescharacteristic
227C22—23no—
228C22—41no—
229C22—43no—
230C22—81no—
231C22C2×C3×#63813yes—
232C4—23no—
233C4—363no—
234C4—361no—
235C3—24no—
236C3C3×#524314yescomplemented
237C3#1059294211yescharacteristic, direct product factor
238C2—23no—
239C2—43no—
240C2C3×#6416211yescharacteristic
subgroupG/H
centerC2×C3#64162
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
Fitting, prime radicalC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. C3×#63661C2semidirect
4. #10592942C3direct (N × H)
9. C22×C3×C3⋊Dic3C22semidirect
11. C3×#3757C22semidirect
16. #64144C2×C3semidirect
18. #63661C2×C3semidirect
41. C22×C3⋊Dic3C22×C3semidirect
44. #3757C22×C3semidirect
138. C33#458semidirect
206. C32#5243semidirect
207. C32C3×#458semidirect
236. C3C3×#5243semidirect
237. C3#10592942direct (N × H)

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C3×#641442
C3×#636612
#105929423
C3×#52433