Finite Group Databasebeta

← C3×#10589339 #13145897

Subgroups — C3×#10589339 #13145897

Order 1728.

628 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 106 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×#64145C211yesmaximal, characteristic
2C33×#99C211yesmaximal, characteristic
3C3×#63661C211yesmaximal, characteristic
4#10589339C311yesmaximal, characteristic, direct product factor
5C3×#4473—34nomaximal
6C2×C33×D4C2211yescharacteristic
7C2×C3×#634C411yescharacteristic, complemented
8C33×C22⋊C4—21no—
9C3×#3757—21no—
10C3×#3757—21no—
11C3×#3757—21no—
12C3×C4×C3⋊Dic3C411yescharacteristic, complemented
13#64145C2×C311yescharacteristic
14C3×#936—34no—
15C32×#99—24no—
16C32×#99D314yes—
17C32×#99C2×C311yescharacteristic
18#63661C2×C311yescharacteristic
19C3×#834—34no—
20C23×C33D411yescharacteristic
21C23×C33D411yescharacteristic
22C33×D4—41no—
23C2×C33×C4—41no—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—41no—
28C3×#634—41no—
29C3×#354—91no—
30#4473—34no—
31C2×C32×D4—24no—
32C2×C32×D4C2×D314yes—
33C2×C32×D4C22×C311yescharacteristic
34C2×#634C3×C411yescharacteristic, complemented
35C2×C3×Dic6—34no—
36C32×C22⋊C4—28no—
37C32×C22⋊C4—24no—
38C32×C22⋊C4—21no—
39#3757—21no—
40#3757—21no—
41#3757—21no—
42C4×C3⋊Dic3C3×C411yescharacteristic, complemented
43C3×#217—64no—
44C3×#217—64no—
45C3×#217—64no—
46C3×C4×Dic3—34no—
47C22×C33—21no—
48C22×C33—21no—
49C22×C33—41no—
50C22×C33—41no—
51C22×C33C22⋊C411yescharacteristic
52C33×C4—21no—
53C33×C4—81no—
54C3×C3⋊Dic3—41no—
55C3×C3⋊Dic3—41no—
56C3×C3⋊Dic3—81no—
57C3×#124—91no—
58#936—34no—
59C3×#99—24no—
60C3×#99—91no—
61C3×#99C3×D314yes—
62C3×#99C3⋊D311yescharacteristic
63#834—34no—
64C23×C32—24no—
65C23×C32—24no—
66C23×C32D1214yes—
67C23×C32#6714yes—
68C23×C32C3×D411yescharacteristic
69C23×C32C3×D411yescharacteristic
70C32×D4—44no—
71C32×D4—41no—
72C32×D4—48no—
73C2×C32×C4—24no—
74C2×C32×C4—48no—
75C2×C32×C4—44no—
76C2×C32×C4—41no—
77C2×C32×C4C4×D314yes—
78C2×C32×C4C2×C3×C411yescharacteristic
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3—41no—
82#634—41no—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—124no—
86C3×Dic6—124no—
87#354—91noSylow
88C2×C33—21no—
89C2×C33—41no—
90C2×C33—41no—
91C2×C33#9911yescharacteristic
92C2×C3×Q8—91no—
93C2×C3×D4—24no—
94C2×C3×D4C2×C3×D314yes—
95C2×C3×D4C2×C3⋊D311yescharacteristic
96C2×Dic6—34no—
97C3×C22⋊C4—28no—
98C3×C22⋊C4—24no—
99C3×C22⋊C4—21no—
100C3×C22⋊C4—181no—
101C3×C22⋊C4—181no—
102C3×C22⋊C4—181no—
103C3×C42—91no—
104#217—64no—
105#217—64no—
106#217—64no—
107C4×Dic3—34no—
108C22×C32—28no—
109C22×C32—24no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—44no—
115C22×C32—41no—
116C22×C32—41no—
117C22×C32—44no—
118C22×C32—44no—
119C22×C32—84no—
120C22×C32—84no—
121C22×C32(C2×D3)⋊C414yes—
122C22×C32C3×C22⋊C411yescharacteristic
123C32×C4—21no—
124C32×C4—24no—
125C32×C4—44no—
126C32×C4—88no—
127C32×C4—84no—
128C32×C4—81no—
129C3⋊Dic3—41no—
130C3⋊Dic3—41no—
131C3⋊Dic3—81no—
132C3×Dic3—124no—
133C3×Dic3—124no—
134C3×Dic3—244no—
135#124—91no—
136#99—91no—
137#99C3×C3⋊D311yescharacteristic
138C33#35411yescharacteristic, complemented, Sylow
139C23×C3—24no—
140C23×C3—24no—
141C23×C3C3×D1214yes—
142C23×C3C3×#6714yes—
143C23×C3C4⋊(C3⋊D3)11yescharacteristic
144C23×C3#63811yescharacteristic
145C3×Q8—361no—
146C3×D4—44no—
147C3×D4—41no—
148C3×D4—48no—
149C2×C3×C4—24no—
150C2×C3×C4—48no—
151C2×C3×C4—44no—
152C2×C3×C4—41no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4—361no—
156C2×C3×C4C3×C4×D314yes—
157C2×C3×C4C4×C3⋊D311yescharacteristic
158C2×Dic3—64no—
159C2×Dic3—64no—
160C2×Dic3—124no—
161Dic6—124no—
162C2×C32—21no—
163C2×C32—24no—
164C2×C32—24no—
165C2×C32—44no—
166C2×C32—48no—
167C2×C32—44no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—44no—
171C2×C32—84no—
172C2×C32#80614yes—
173C2×C32C3×#9911yescharacteristic
174C2×Q8—91no—
175C2×D4C2×C3×C3⋊D311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—181no—
178C22⋊C4—181no—
179C22⋊C4—181no—
180C42—91no—
181C22×C3—28no—
182C22×C3—24no—
183C22×C3—24no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—24no—
187C22×C3—44no—
188C22×C3—44no—
189C22×C3—44no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—84no—
193C22×C3—84no—
194C22×C3C3×(C2×D3)⋊C414yes—
195C22×C3#375211yescharacteristic
196C3×C4—24no—
197C3×C4—21no—
198C3×C4—44no—
199C3×C4—88no—
200C3×C4—84no—
201C3×C4—81no—
202C3×C4—361no—
203C3×C4—361no—
204C3×C4—721no—
205Dic3—124no—
206Dic3—124no—
207Dic3—244no—
208C32—24no—
209C32#447314yescomplemented
210C32C3×#35411yescharacteristic, complemented
211C23C3×C4⋊(C3⋊D3)11yescharacteristic
212C23C3×#63811yescharacteristic
213Q8—361no—
214D4—41no—
215C2×C4—41no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C4—361no—
219C2×C4C3×C4×C3⋊D311yescharacteristic
220C2×C3—24no—
221C2×C3—21no—
222C2×C3—24no—
223C2×C3—44no—
224C2×C3—48no—
225C2×C3—44no—
226C2×C3—44no—
227C2×C3—41no—
228C2×C3—41no—
229C2×C3—84no—
230C2×C3C3×#80614yes—
231C2×C3#6363311yescharacteristic
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22C3×#375211yescharacteristic
237C4—21no—
238C4—81no—
239C4—361no—
240C4—361no—
241C4—721no—
242C3—24no—
243C3C3×#447314yescomplemented
244C3#1058933911yescharacteristic, direct product factor
245C2—21no—
246C2—41no—
247C2—41no—
248C2C3×#6363311yescharacteristic
subgroupG/H
centerC2×C3#63633
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×D4C2×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33#99
prime radicalC2×C33×D4C22
cosocleC2×C32×D4C22×C3
hypercenterC3×#99C3⋊D3
hypocenterC32C3×#354
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
4. #10589339C3direct (N × H)
7. C2×C3×#634C4semidirect
12. C3×C4×C3⋊Dic3C4semidirect
34. C2×#634C3×C4semidirect
42. C4×C3⋊Dic3C3×C4semidirect
138. C33#354semidirect
209. C32#4473semidirect
210. C32C3×#354semidirect
243. C3C3×#4473semidirect
244. C3#10589339direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641452
C33×#992
C3×#636612
#105893393
C3×#44733