Finite Group Databasebeta

← C3×#10589338 #13145896

Subgroups — C3×#10589338 #13145896

Order 1728.

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

normal subgroups only

This group has 616 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×#64144C211yesmaximal, characteristic
2C33×#99C211yesmaximal, characteristic
3C3×#63660C211yesmaximal, characteristic
4#10589338C311yesmaximal, characteristic, direct product factor
5C3×#4472—34nomaximal
6C2×C33×D4C2211yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic, complemented
8C33×C22⋊C4—21no—
9C3×#3757—21no—
10C3×#3757C411yescharacteristic, complemented
11C3×#3750—21no—
12C3×#3748—21no—
13#64144C2×C311yescharacteristic
14C3×#935—34no—
15C32×#99—24no—
16C32×#99D314yes—
17C32×#99C2×C311yescharacteristic
18#63660C2×C311yescharacteristic
19C3×#833—34no—
20C23×C33C2×C411yescharacteristic
21C23×C33D411yescharacteristic
22C33×D4—41no—
23C2×C33×C4—41no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—41no—
28C3×C3⋊(C3⋊C8)—41no—
29C3×#352—91no—
30#4472—34no—
31C2×C32×D4—24no—
32C2×C32×D4C2×D314yes—
33C2×C32×D4C22×C311yescharacteristic
34C22×C3⋊Dic3C3×C411yescharacteristic, complemented
35C22×C3×Dic3—34no—
36C32×C22⋊C4—28no—
37C32×C22⋊C4—24no—
38C32×C22⋊C4—21no—
39#3757—21no—
40#3757C3×C411yescharacteristic, complemented
41#3750—21no—
42#3748—21no—
43C3×#217—34no—
44C3×#217—64no—
45C3×Dic3⋊C4—64no—
46C3×#208—64no—
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×#123—91no—
57#935—34no—
58C3×#100—91no—
59C3×#99—24no—
60C3×#99C3×D314yes—
61C3×#99C3⋊D311yescharacteristic
62#833—34no—
63C23×C32—24no—
64C23×C32—24no—
65C23×C32C4×D314yes—
66C23×C32#6714yes—
67C23×C32C2×C3×C411yescharacteristic
68C23×C32C3×D411yescharacteristic
69C32×D4—44no—
70C32×D4—41no—
71C32×D4—48no—
72C2×C32×C4—24no—
73C2×C32×C4—48no—
74C2×C32×C4—44no—
75C2×C32×C4—41no—
76C2×C32×C4D1214yes—
77C2×C32×C4C3×D411yescharacteristic
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—41no—
81C2×C3×Dic3—64no—
82C2×C3×Dic3—64no—
83C2×C3×Dic3—124no—
84C3⋊(C3⋊C8)—41no—
85C3×C3⋊C8—124no—
86#352—91noSylow
87C2×C33—21no—
88C2×C33—41no—
89C2×C33—41no—
90C2×C33#9911yescharacteristic
91C2×C3×D4—24no—
92C2×C3×D4C2×C3×D314yes—
93C2×C3×D4C2×C3⋊D311yescharacteristic
94C22×C3×C4—91no—
95C22×Dic3—34no—
96C3×M16—181no—
97C3×C4⋊C4—181no—
98C3×C22⋊C4—28no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—91no—
102C3×C22⋊C4—181no—
103#217—34no—
104#217—64no—
105Dic3⋊C4—64no—
106#208—64no—
107C22×C32—28no—
108C22×C32—24no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—44no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32—44no—
117C22×C32—44no—
118C22×C32—84no—
119C22×C32—84no—
120C22×C32(C2×D3)⋊C414yes—
121C22×C32C3×C22⋊C411yescharacteristic
122C32×C4—21no—
123C32×C4—24no—
124C32×C4—44no—
125C32×C4—88no—
126C32×C4—84no—
127C32×C4—81no—
128C3⋊Dic3—41no—
129C3⋊Dic3—41no—
130C3×Dic3—124no—
131C3×Dic3—124no—
132#123—91no—
133#100—91no—
134#99C3×C3⋊D311yescharacteristic
135C33#35211yescharacteristic, complemented, Sylow
136C23×C3—24no—
137C23×C3—24no—
138C23×C3C3×C4×D314yes—
139C23×C3C3×#6714yes—
140C23×C3C4×C3⋊D311yescharacteristic
141C23×C3#63811yescharacteristic
142C3×D4—44no—
143C3×D4—41no—
144C3×D4—48no—
145C2×C3×C4—24no—
146C2×C3×C4—48no—
147C2×C3×C4—44no—
148C2×C3×C4—41no—
149C2×C3×C4—181no—
150C2×C3×C4—181no—
151C2×C3×C4—361no—
152C2×C3×C4C3×D1214yes—
153C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
154C2×Dic3—64no—
155C2×Dic3—64no—
156C2×Dic3—124no—
157C3×C8—361no—
158C3⋊C8—124no—
159C2×C32—21no—
160C2×C32—24no—
161C2×C32—24no—
162C2×C32—44no—
163C2×C32—48no—
164C2×C32—44no—
165C2×C32—41no—
166C2×C32—41no—
167C2×C32—44no—
168C2×C32—84no—
169C2×C32#80614yes—
170C2×C32C3×#9911yescharacteristic
171C2×D4C2×C3×C3⋊D311yescharacteristic
172C22×C4—91no—
173M16—181no—
174C4⋊C4—181no—
175C22⋊C4—21no—
176C22⋊C4—91no—
177C22⋊C4—181no—
178C22×C3—28no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—24no—
184C22×C3—44no—
185C22×C3—44no—
186C22×C3—44no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—84no—
190C22×C3—84no—
191C22×C3C3×(C2×D3)⋊C414yes—
192C22×C3#375211yescharacteristic
193C3×C4—24no—
194C3×C4—21no—
195C3×C4—44no—
196C3×C4—88no—
197C3×C4—84no—
198C3×C4—81no—
199C3×C4—361no—
200C3×C4—361no—
201Dic3—124no—
202Dic3—124no—
203C32—24no—
204C32#447214yescomplemented
205C32C3×#35211yescharacteristic, complemented
206C23C3×C4×C3⋊D311yescharacteristic
207C23C3×#63811yescharacteristic
208D4—41no—
209C2×C4—41no—
210C2×C4—181no—
211C2×C4—181no—
212C2×C4—361no—
213C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
214C8—361no—
215C2×C3—24no—
216C2×C3—21no—
217C2×C3—24no—
218C2×C3—44no—
219C2×C3—48no—
220C2×C3—44no—
221C2×C3—44no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3—84no—
225C2×C3C3×#80614yes—
226C2×C3#6363311yescharacteristic
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22C3×#375211yescharacteristic
232C4—21no—
233C4—81no—
234C4—361no—
235C4—361no—
236C3—24no—
237C3C3×#447214yescomplemented
238C3#1058933811yescharacteristic, direct product factor
239C2—21no—
240C2—41no—
241C2—41no—
242C2C3×#6363311yescharacteristic
subgroupG/H
centerC2×C3#63633
commutatorC23×C32C2×C3×C4
FrattiniC2×D4C2×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33#99
prime radicalC2×C33×D4C22
cosocleC2×C32×D4C22×C3
hypercenterC3×#99C3⋊D3
hypocenterC32C3×#352
LCSG ▷ C23×C32 ▷ 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. #10589338C3direct (N × H)
7. C22×C3×C3⋊Dic3C4semidirect
10. C3×#3757C4semidirect
34. C22×C3⋊Dic3C3×C4semidirect
40. #3757C3×C4semidirect
135. C33#352semidirect
204. C32#4472semidirect
205. C32C3×#352semidirect
237. C3C3×#4472semidirect
238. C3#10589338direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641442
C33×#992
C3×#636602
#105893383
C3×#44723