Finite Group Databasebeta

← C3×#10589337 #13145895

Subgroups — C3×#10589337 #13145895

Order 1728.

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

normal subgroups only

This group has 772 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×#99C211yesmaximal, characteristic, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4#10589337C311yesmaximal, characteristic, direct product factor
5C3×#4471—34nomaximal
6C23×C3×C3⋊D3C411yescharacteristic, complemented
7C2×C33×D4C2211yescharacteristic
8C2×C3×#638—21no—
9C33×C22⋊C4—21no—
10C3×#3757C411yescharacteristic, complemented
11C3×#3752—21no—
12C3×#3748—21no—
13#64146C2×C311yescharacteristic
14C3×#937—34no—
15C32×#99—24no—
16C32×#99D314yescomplemented
17C32×#99C2×C311yescharacteristic, complemented
18#63660C2×C311yescharacteristic, complemented
19C3×#833—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×C4—41no—
27C2×C33×C4D411yescharacteristic
28C3×#638—42no—
29C2×C3×C3⋊Dic3—21no—
30C3×C3⋊(C3⋊C8)—41no—
31C3×AΣL(1,𝔽16)—91no—
32#4471—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×C22⋊C4—28no—
41C32×C22⋊C4—24no—
42C32×C22⋊C4—21no—
43#3757C3×C411yescharacteristic, complemented
44#3752—21no—
45#3748—21no—
46C3×#217—34no—
47C3×(C2×D3)⋊C4—64no—
48C3×#208—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—
60C33×C4—81no—
61C3×C3⋊Dic3—41no—
62C3×#120—91no—
63#937—34no—
64C3×#100—91no—
65C3×#99—24no—
66C3×#99C3×D314yescomplemented
67C3×#99C3⋊D311yescharacteristic, complemented
68#833—34no—
69C23×C32—24no—
70C23×C32—24no—
71C23×C32C4×D314yes—
72C23×C32#6714yes—
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—48no—
86C2×C32×C4—44no—
87C2×C32×C4—41no—
88C2×C32×C4D1214yes—
89C2×C32×C4C3×D411yescharacteristic
90#638—42no—
91C2×C3⋊Dic3—21no—
92C3×#67—128no—
93C2×C3×Dic3—64no—
94C3⋊(C3⋊C8)—41no—
95C3×C3⋊C8—124no—
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—181no—
110C3×C22⋊C4—28no—
111C3×C22⋊C4—24no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—91no—
114C3×C22⋊C4—181no—
115#217—34no—
116(C2×D3)⋊C4—64no—
117#208—64no—
118C22×C32—28no—
119C22×C32—24no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—24no—
123C22×C32—24no—
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—21no—
144C32×C4—24no—
145C32×C4—44no—
146C32×C4—88no—
147C32×C4—84no—
148C32×C4—81no—
149C3⋊Dic3—41no—
150C3×Dic3—124no—
151#120—91no—
152#100—91no—
153#99C3×C3⋊D311yescharacteristic, complemented
154C33AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
155C23×C3—24no—
156C23×C3—24no—
157C23×C3—181no—
158C23×C3—361no—
159C23×C3—362no—
160C23×C3C3×C4×D314yes—
161C23×C3C3×#6714yes—
162C23×C3C4×C3⋊D311yescharacteristic
163C23×C3#63811yescharacteristic
164C22×D3—64no—
165C22×D3—128no—
166C22×D3—124no—
167C3×D4—44no—
168C3×D4—41no—
169C3×D4—48no—
170C3×D4—362no—
171C2×C3×C4—24no—
172C2×C3×C4—48no—
173C2×C3×C4—44no—
174C2×C3×C4—41no—
175C2×C3×C4—181no—
176C2×C3×C4C3×D1214yes—
177C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
178#67—128no—
179C2×Dic3—64no—
180C3×C8—361no—
181C3⋊C8—124no—
182C2×C32—21no—
183C2×C32—24no—
184C2×C32—24no—
185C2×C32—44no—
186C2×C32—48no—
187C2×C32—44no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—44no—
191C2×C32—84no—
192C2×C32#80614yes—
193C2×C32C3×#9911yescharacteristic
194C3⋊D3—42no—
195C3×D3—128no—
196C24—91no—
197C2×D4—181no—
198C2×D4C2×C3×C3⋊D311yescharacteristic
199M16—181no—
200C22⋊C4—21no—
201C22⋊C4—91no—
202C22⋊C4—181no—
203C22×C3—28no—
204C22×C3—24no—
205C22×C3—24no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—24no—
209C22×C3—44no—
210C22×C3—44no—
211C22×C3—41no—
212C22×C3—44no—
213C22×C3—41no—
214C22×C3—84no—
215C22×C3—84no—
216C22×C3—182no—
217C22×C3—361no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3—361no—
221C22×C3C3×(C2×D3)⋊C414yes—
222C22×C3#375211yescharacteristic
223C2×D3—68no—
224C2×D3—124no—
225C2×D3—128no—
226C2×D3—124no—
227C2×D3—128no—
228C3×C4—24no—
229C3×C4—21no—
230C3×C4—44no—
231C3×C4—88no—
232C3×C4—84no—
233C3×C4—81no—
234C3×C4—361no—
235Dic3—124no—
236C32—24no—
237C32#447114yescomplemented
238C32C3×AΣL(1,𝔽16)11yescharacteristic, complemented
239C23—181no—
240C23—361no—
241C23—362no—
242C23C3×C4×C3⋊D311yescharacteristic
243C23C3×#63811yescharacteristic
244D4—41no—
245D4—362no—
246C2×C4—41no—
247C2×C4—181no—
248C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
249C8—361no—
250C2×C3—24no—
251C2×C3—21no—
252C2×C3—24no—
253C2×C3—44no—
254C2×C3—48no—
255C2×C3—44no—
256C2×C3—44no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—84no—
260C2×C3—362no—
261C2×C3C3×#80614yes—
262C2×C3#6363311yescharacteristic
263D3—128no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—182no—
269C22—361no—
270C22—362no—
271C22—362no—
272C22—361no—
273C22C3×#375211yescharacteristic
274C4—21no—
275C4—81no—
276C4—361no—
277C3—24no—
278C3C3×#447114yescomplemented
279C3#1058933711yescharacteristic, direct product factor
280C2—21no—
281C2—41no—
282C2—41no—
283C2—362no—
284C2C3×#6363311yescharacteristic
subgroupG/H
centerC2×C3#63633
commutatorC23×C32C2×C3×C4
FrattiniC2×D4C2×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33#99
prime radicalC3×#64146C2
cosocleC2×C32×D4C22×C3
hypercenterC3×#99C3⋊D3
hypocenterC32C3×AΣL(1,𝔽16)
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
2. C33×#99C2semidirect
3. C3×#63660C2semidirect
4. #10589337C3direct (N × H)
6. C23×C3×C3⋊D3C4semidirect
10. C3×#3757C4semidirect
16. C32×#99D3semidirect
17. C32×#99C2×C3semidirect
18. #63660C2×C3semidirect
33. C23×C3⋊D3C3×C4semidirect
43. #3757C3×C4semidirect
66. C3×#99C3×D3semidirect
67. C3×#99C3⋊D3semidirect
153. #99C3×C3⋊D3semidirect
154. C33AΣL(1,𝔽16)semidirect
237. C32#4471semidirect
238. C32C3×AΣL(1,𝔽16)semidirect
278. C3C3×#4471semidirect
279. C3#10589337direct (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×#992
C3×#636602
#105893373
C3×#44713