Finite Group Databasebeta

← C3×#10589341 #13145899

Subgroups — C3×#10589341 #13145899

Order 1728.

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

normal subgroups only

This group has 748 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×#64146C211yesmaximal, characteristic
2C33×#100C211yesmaximal, characteristic, complemented
3C3×#63661C211yesmaximal, characteristic, complemented
4#10589341C311yesmaximal, characteristic, direct product factor
5C3×#4475—34nomaximal
6C23×C3×C3⋊D3C411yescharacteristic, complemented
7C2×C33×D4C2211yescharacteristic
8C2×C3×#638—21no—
9C33×M16—21no—
10C3×#3757—21no—
11C3×#3757C411yescharacteristic, complemented
12C3×#3752—21no—
13#64146C2×C311yescharacteristic
14C3×#937—34no—
15C32×#100—24no—
16C32×#100D314yescomplemented
17C32×#100C2×C311yescharacteristic, complemented
18#63661C2×C311yescharacteristic, complemented
19C3×#834—34no—
20C23×C33C2×C411yescharacteristic
21C23×C33D411yescharacteristic
22C22×C3×C3⋊D3—21no—
23C22×C3×C3⋊D3—42no—
24C22×C3×C3⋊D3—41no—
25C33×D4—41no—
26C2×C33×C4D411yescharacteristic
27C3×#638—42no—
28C2×C3×C3⋊Dic3—21no—
29C2×C3×C3⋊Dic3—41no—
30C33×C8—41no—
31C3×AΣL(1,𝔽16)—91no—
32#4475—34no—
33C23×C3⋊D3C3×C411yescharacteristic, complemented
34C23×C3×D3—34no—
35C2×C32×D4—24no—
36C2×C32×D4C2×D314yes—
37C2×C32×D4C22×C311yescharacteristic
38C2×#638—21no—
39C2×C3×#67—64no—
40C32×M16—28no—
41C32×M16—24no—
42C32×M16—21no—
43#3757—21no—
44#3757C3×C411yescharacteristic, complemented
45#3752—21no—
46C3×#217—34no—
47C3×#217—64no—
48C3×(C2×D3)⋊C4—64no—
49C22×C33—21no—
50C22×C33—21no—
51C22×C33—41no—
52C22×C33—41no—
53C22×C33C22⋊C411yescharacteristic
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3—41no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3—42no—
59C33×C4—21no—
60C3×C3⋊Dic3—41no—
61C3×C3⋊Dic3—81no—
62C3×#120—91no—
63#937—34no—
64C3×#100—24no—
65C3×#100C3×D314yescomplemented
66C3×#100C3⋊D311yescharacteristic, complemented
67C3×#99—91no—
68#834—34no—
69C23×C32—24no—
70C23×C32—24no—
71C23×C32C4×D314yes—
72C23×C32D1214yes—
73C23×C32C2×C3×C411yescharacteristic
74C23×C32C3×D411yescharacteristic
75C22×C3⋊D3—21no—
76C22×C3⋊D3—42no—
77C22×C3⋊D3—41no—
78C22×C3×D3—64no—
79C22×C3×D3—128no—
80C22×C3×D3—124no—
81C32×D4—44no—
82C32×D4—41no—
83C32×D4—48no—
84C2×C32×C4—24no—
85C2×C32×C4#6714yes—
86C2×C32×C4C3×D411yescharacteristic
87#638—42no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—41no—
90C3×#67—128no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—124no—
93C32×C8—48no—
94C32×C8—44no—
95C32×C8—41no—
96AΣL(1,𝔽16)—91noSylow
97C2×C33—21no—
98C2×C33—41no—
99C2×C33—41no—
100C2×C33#9911yescharacteristic
101C3×C3⋊D3—42no—
102C24×C3—91no—
103C23×D3—34no—
104C2×C3×D4—24no—
105C2×C3×D4—181no—
106C2×C3×D4C2×C3×D314yes—
107C2×C3×D4C2×C3⋊D311yescharacteristic
108C2×#67—64no—
109C3×M16—28no—
110C3×M16—24no—
111C3×M16—21no—
112C3×C22⋊C4—91no—
113C3×C22⋊C4—181no—
114C3×C22⋊C4—181no—
115#217—34no—
116#217—64no—
117(C2×D3)⋊C4—64no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—24no—
121C22×C32—24no—
122C22×C32—24no—
123C22×C32—44no—
124C22×C32—44no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—44no—
128C22×C32—44no—
129C22×C32—84no—
130C22×C32—84no—
131C22×C32(C2×D3)⋊C414yes—
132C22×C32C3×C22⋊C411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3—41no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—41no—
137C2×C3⋊D3—42no—
138C2×C3×D3—68no—
139C2×C3×D3—124no—
140C2×C3×D3—128no—
141C2×C3×D3—124no—
142C2×C3×D3—128no—
143C32×C4—28no—
144C32×C4—24no—
145C32×C4—21no—
146C3⋊Dic3—41no—
147C3⋊Dic3—81no—
148C3×Dic3—124no—
149C3×Dic3—244no—
150#120—91no—
151#100C3×C3⋊D311yescharacteristic, complemented
152#99—91no—
153C33AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
154C23×C3—24no—
155C23×C3—24no—
156C23×C3—181no—
157C23×C3—361no—
158C23×C3—362no—
159C23×C3C3×C4×D314yes—
160C23×C3C3×D1214yes—
161C23×C3C4×C3⋊D311yescharacteristic
162C23×C3C4⋊(C3⋊D3)11yescharacteristic
163C22×D3—64no—
164C22×D3—128no—
165C22×D3—124no—
166C3×D4—44no—
167C3×D4—41no—
168C3×D4—48no—
169C3×D4—362no—
170C2×C3×C4—24no—
171C2×C3×C4—181no—
172C2×C3×C4—361no—
173C2×C3×C4C3×#6714yes—
174C2×C3×C4#63811yescharacteristic
175#67—128no—
176C2×Dic3—64no—
177C2×Dic3—124no—
178C3×C8—48no—
179C3×C8—44no—
180C3×C8—41no—
181C2×C32—21no—
182C2×C32—24no—
183C2×C32—24no—
184C2×C32—44no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32—44no—
188C2×C32—44no—
189C2×C32—84no—
190C2×C32—84no—
191C2×C32#80614yes—
192C2×C32C3×#9911yescharacteristic
193C3⋊D3—42no—
194C3×D3—128no—
195C24—91no—
196C2×D4—181no—
197C2×D4C2×C3×C3⋊D311yescharacteristic
198M16—21no—
199C22⋊C4—91no—
200C22⋊C4—181no—
201C22⋊C4—181no—
202C22×C3—24no—
203C22×C3—24no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—24no—
207C22×C3—44no—
208C22×C3—44no—
209C22×C3—44no—
210C22×C3—41no—
211C22×C3—44no—
212C22×C3—41no—
213C22×C3—84no—
214C22×C3—84no—
215C22×C3—182no—
216C22×C3—361no—
217C22×C3—362no—
218C22×C3—362no—
219C22×C3—361no—
220C22×C3C3×(C2×D3)⋊C414yes—
221C22×C3#375211yescharacteristic
222C2×D3—68no—
223C2×D3—124no—
224C2×D3—128no—
225C2×D3—124no—
226C2×D3—128no—
227C3×C4—28no—
228C3×C4—24no—
229C3×C4—21no—
230C3×C4—361no—
231C3×C4—721no—
232Dic3—124no—
233Dic3—244no—
234C32—24no—
235C32#447514yescomplemented
236C32C3×AΣL(1,𝔽16)11yescharacteristic, complemented
237C23—181no—
238C23—361no—
239C23—362no—
240C23C3×C4×C3⋊D311yescharacteristic
241C23C3×C4⋊(C3⋊D3)11yescharacteristic
242D4—41no—
243D4—362no—
244C2×C4—181no—
245C2×C4—361no—
246C2×C4C3×#63811yescharacteristic
247C8—41no—
248C2×C3—24no—
249C2×C3—21no—
250C2×C3—24no—
251C2×C3—44no—
252C2×C3—44no—
253C2×C3—44no—
254C2×C3—41no—
255C2×C3—41no—
256C2×C3—84no—
257C2×C3—84no—
258C2×C3—362no—
259C2×C3C3×#80614yes—
260C2×C3#6363311yescharacteristic
261D3—128no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—182no—
267C22—361no—
268C22—362no—
269C22—362no—
270C22—361no—
271C22C3×#375211yescharacteristic
272C4—21no—
273C4—361no—
274C4—721no—
275C3—24no—
276C3C3×#447514yescomplemented
277C3#1058934111yescharacteristic, direct product factor
278C2—21no—
279C2—41no—
280C2—41no—
281C2—362no—
282C2C3×#6363311yescharacteristic
subgroupG/H
centerC2×C3#63633
commutatorC23×C32C2×C3×C4
FrattiniC2×D4C2×C3×C3⋊D3
FittingC33×#100C2
socleC2×C33#99
prime radicalC3×#64146C2
cosocleC2×C32×D4C22×C3
hypercenterC3×#100C3⋊D3
hypocenterC32C3×AΣL(1,𝔽16)
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
2. C33×#100C2semidirect
3. C3×#63661C2semidirect
4. #10589341C3direct (N × H)
6. C23×C3×C3⋊D3C4semidirect
11. C3×#3757C4semidirect
16. C32×#100D3semidirect
17. C32×#100C2×C3semidirect
18. #63661C2×C3semidirect
33. C23×C3⋊D3C3×C4semidirect
44. #3757C3×C4semidirect
65. C3×#100C3×D3semidirect
66. C3×#100C3⋊D3semidirect
151. #100C3×C3⋊D3semidirect
153. C33AΣL(1,𝔽16)semidirect
235. C32#4475semidirect
236. C32C3×AΣL(1,𝔽16)semidirect
276. C3C3×#4475semidirect
277. C3#10589341direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641462
C33×#1002
C3×#636612
#105893413
C3×#44753