Finite Group Databasebeta

← C3×#10589508 #13146856

Subgroups — C3×#10589508 #13146856

Order 1728.

234 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 40 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×#63692C211yesmaximal, characteristic
2C3×#63690C211yesmaximal, characteristic, complemented
3C3×#131⋊C9C211yesmaximal, characteristic, complemented
4C3×#5141—31nomaximal
5#10589508C311yesmaximal, characteristic, direct product factor
6C3×#41⋊C9C2211yescharacteristic
7C3×#3689—21no—
8C3×#3688—21no—
9C3×#3664—41nomaximal
10C32×#131D311yescharacteristic
11C3×#951—31no—
12C3×#949—31no—
13C2×C3×#204—31no—
14#63692C2×C311yescharacteristic
15#63690C2×C311yescharacteristic, complemented
16C3×#824—31no—
17C3×#823—31no—
18C3×#819—31no—
19#131⋊C9—21no—
20#131⋊C9C2×C311yescharacteristic, complemented
21C3×D36—41no—
22C3×Dic18—41no—
23C3×Q8⋊C9D411yescharacteristic
24C3×C8×C9—41no—
25C3×#471—91no—
26#5141—31no—
27C32×#41C2×D311yescharacteristic
28C2×C3×D12—31no—
29C2×C3×Dic6—31no—
30C32×M16—31no—
31C2×C32×C8—31no—
32C3×#216—61no—
33C3×#215—61no—
34C3×#214—61no—
35C3×#213—61no—
36C3×#208—31no—
37C3×#208—31no—
38C3×#204—61no—
39C3×#204—61no—
40#41⋊C9—21no—
41#41⋊C9C22×C311yescharacteristic
42#3689—21no—
43#3688—21no—
44#3664—41no—
45C2×C3×D9—81no—
46C3×C4×C9—41no—
47C3×Dic9—81no—
48C3×#137—91no—
49C3×AGL(1,ℤ/8)—91no—
50C2×C3×SD16—91no—
51C3×#131—21no—
52C3×#131D911yescharacteristic, complemented
53C3×#131C3×D311yescharacteristic
54#951—31no—
55#949—31no—
56C2×#204—31no—
57C3×#108—91no—
58C3×#101—91no—
59C3×#100—91no—
60#824—31no—
61#823—31no—
62#819—31no—
63C22×C3×D3—61no—
64C32×Q8D1211yescharacteristic
65C32×D4—31no—
66C2×C32×C4—31no—
67C2×C3×Dic3—61no—
68C3×D12—61no—
69C3×D12—61no—
70C3×Dic6—61no—
71C3×Dic6—61no—
72C32×C8—31no—
73C32×C8S411yescharacteristic
74C3×C3⋊C8—61no—
75C3×C3⋊C8—61no—
76D36—41no—
77Dic18—41no—
78Q8⋊C9—21no—
79Q8⋊C9C3×D411yescharacteristic
80C8×C9—41no—
81C8×C9—81no—
82#471—91noSylow
83C2×C3×C9—41no—
84C3×D9—161no—
85C3×#41—21no—
86C3×#41C2×D911yescharacteristic
87C3×#41C2×C3×D311yescharacteristic
88C2×C3×Q8—91no—
89C2×C3×D4—91no—
90C2×D12—31no—
91C2×Dic6—31no—
92C3×Q16—181no—
93C3×SD16—181no—
94C3×SD16—181no—
95C3×SD16—181no—
96C3×SD16—181no—
97C3×D8—181no—
98C3×M16—31no—
99C3×M16—31no—
100C3×M16—61no—
101C3×M16—91no—
102C3×M16—91no—
103C2×C3×C8—31no—
104C2×C3×C8—31no—
105C2×C3×C8—61no—
106#216—61no—
107#215—61no—
108#214—61no—
109#213—61no—
110#208—31no—
111#208—31no—
112#204—61no—
113#204—61no—
114C22×C32—31no—
115C2×C3×D3—121no—
116C2×C3×D3—241no—
117C32×C4—31no—
118C32×C4C2×S411yescharacteristic
119C3×Dic3—121no—
120C2×D9—81no—
121C4×C9—41no—
122C4×C9—81no—
123Dic9—81no—
124#137—91no—
125AGL(1,ℤ/8)—91no—
126C2×SD16—91no—
127#131C3×D911yescharacteristic, complemented
128#108—91no—
129#101—91no—
130#100—91no—
131C3×C9—41noSylow
132C23×C3—181no—
133C22×D3—61no—
134C3×Q8—21no—
135C3×Q8—181no—
136C3×Q8—181no—
137C3×Q8D3611yescharacteristic
138C3×Q8C3×D1211yescharacteristic
139C3×D4—31no—
140C3×D4—31no—
141C3×D4—61no—
142C3×D4—181no—
143C3×D4—181no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—181no—
148C2×Dic3—61no—
149D12—61no—
150D12—61no—
151Dic6—61no—
152Dic6—61no—
153C3×C8—21no—
154C3×C8—31no—
155C3×C8—31no—
156C3×C8—61no—
157C3×C8—181no—
158C3×C8—181no—
159C3×C8(C22⋊C9)⋊C211yescharacteristic
160C3×C8C3×S411yescharacteristic
161C3⋊C8—61no—
162C3⋊C8—61no—
163C2×C32—61no—
164C2×C32C4⋊S411yescharacteristic
165C3×D3—241no—
166C2×C9—41no—
167C2×C9—81no—
168D9—161no—
169#41C2×C3×D911yescharacteristic
170C2×Q8—91no—
171C2×D4—91no—
172Q16—181no—
173SD16—181no—
174SD16—181no—
175SD16—181no—
176SD16—181no—
177D8—181no—
178M16—31no—
179M16—91no—
180M16—91no—
181C2×C8—31no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—61no—
185C22×C3—361no—
186C22×C3—721no—
187C2×D3—121no—
188C2×D3—241no—
189C3×C4—21no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—61no—
193C3×C4—361no—
194C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
195C3×C4C2×C3×S411yescharacteristic
196Dic3—121no—
197C32#540611yescharacteristic
198C9—41no—
199C9—81no—
200C23—181no—
201Q8—181no—
202Q8—181no—
203Q8C3×D3611yescharacteristic
204D4—31no—
205D4—181no—
206D4—181no—
207C2×C4—31no—
208C2×C4—181no—
209C8—31no—
210C8—181no—
211C8—181no—
212C8C3×(C22⋊C9)⋊C211yescharacteristic
213C2×C3—21no—
214C2×C3—61no—
215C2×C3—61no—
216C2×C3—121no—
217C2×C3—721no—
218C2×C3C4⋊((C22⋊C9)⋊C2)11yescharacteristic
219C2×C3C3×C4⋊S411yescharacteristic
220D3—241no—
221C22—31no—
222C22—361no—
223C22—721no—
224C4—31no—
225C4—361no—
226C4C2×C3×(C22⋊C9)⋊C211yescharacteristic
227C3—21no—
228C3#1058950811yescharacteristic, direct product factor
229C3C3×#540611yescharacteristic
230C2—61no—
231C2—721no—
232C2C3×C4⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC2×C3C4⋊((C22⋊C9)⋊C2)
commutator, cosocle#41⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C32C4⋊S4
prime radicalC3×#63692C2
hypercenterC3×C8(C22⋊C9)⋊C2
hypocenterQ8⋊C9C3×D4
LCSG ▷ #41⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63690C2semidirect
3. C3×#131⋊C9C2semidirect
5. #10589508C3direct (N × H)
15. #63690C2×C3semidirect
20. #131⋊C9C2×C3semidirect
52. C3×#131D9semidirect
127. #131C3×D9semidirect
228. C3#10589508direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#636922
C3×#636902
C3×#131⋊C92
C3×#51413
#105895083
C3×#36644