Finite Group Databasebeta

← C3×#10592707 #13163046

Subgroups — C3×#10592707 #13163046

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal
4C3×#63635C212yesmaximal, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10592707C311yesmaximal, characteristic, direct product factor
7C3×#5008—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C33×C4—21no—
10C2×C3×#634—21no—
11C33×C4⋊C4C2211yescharacteristic
12C33×C22⋊C4—21no—
13C33×C42C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4×D4—24no—
18C32×C4×D4D314yes—
19C32×C4×D4C2×C311yescharacteristic
20#64078C2×C311yescharacteristic, complemented
21C3×#869—34no—
22#63659C2×C312yes—
23#63635C2×C312yescomplemented
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
25C3×#832—38no—
26C3×#808—38no—
27C3×C4⋊(C3⋊C8)—34no—
28C23×C33—21no—
29C33×D4—21no—
30C33×D4Q812yescomplemented
31C2×C33×C4—21no—
32C2×C33×C4—42no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C3×#634—42no—
38C3×C3⋊(C3⋊C8)—22no—
39C3×#476—91no—
40#5008—34no—
41C2×C32×D4—24no—
42C2×C32×D4C2×D314yes—
43C2×C32×D4C22×C311yescharacteristic
44C22×C32×C4—28no—
45C22×C32×C4—24no—
46C22×C32×C4—21no—
47C2×#634—21no—
48C2×C3×Dic6—64no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4C2×D314yes—
51C32×C4⋊C4C22×C311yescharacteristic
52C32×C22⋊C4—28no—
53C32×C22⋊C4—24no—
54C32×C22⋊C4—21no—
55C32×C42—24no—
56C32×C42C2×D314yes—
57C32×C42C22×C311yescharacteristic
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C312yes—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×C4⋊Dic3—38no—
62C3×C4⋊Dic3—64no—
63C2×C3×C3⋊C8—38no—
64C22×C33—22no—
65C22×C33—41no—
66C22×C33—41no—
67C22×C33C2×D411yescharacteristic
68C33×C4—21no—
69C33×C4—41no—
70C33×C4SD1612yes—
71C33×C4C2×Q811yescharacteristic
72C33×C4#4111yescharacteristic
73C3×C3⋊Dic3—42no—
74C3×C4⋊Q8—91no—
75C3×C4×D4—24no—
76C3×C4×D4C3×D314yes—
77C3×C4×D4C3⋊D311yescharacteristic
78#869—34no—
79C3×#106—92no—
80C3×C4⋊C8—91no—
81C3×#102—92no—
82#832—38no—
83#808—38no—
84C4⋊(C3⋊C8)—34no—
85C23×C32—28no—
86C23×C32—24no—
87C23×C32—21no—
88C32×D4—28no—
89C32×D4—24no—
90C32×D4—21no—
91C32×D4—28no—
92C32×D4Dic618yescomplemented
93C32×D4C3×Q812yescomplemented
94C2×C32×C4—28no—
95C2×C32×C4—24no—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—416no—
101C2×C32×C4—48no—
102C2×C32×C4—42no—
103C2×C32×C4#6714yes—
104C2×C32×C4#6714yes—
105C2×C32×C4C3×D411yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C22×D314yes—
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3—22no—
110#634—42no—
111C2×C3×Dic3—68no—
112C3×Dic6—128no—
113C3⋊(C3⋊C8)—22no—
114C3×C3⋊C8—68no—
115#476—91noSylow
116C2×C33—42no—
117C2×C33C22⋊Q811yescharacteristic
118C2×C33C2×SD1611yescharacteristic
119C2×C33AGL(1,ℤ/8)11yescharacteristic
120C2×C3×Q8—181no—
121C2×C3×D4—24no—
122C2×C3×D4C2×C3×D314yes—
123C2×C3×D4C2×C3⋊D311yescharacteristic
124C22×C3×C4—28no—
125C22×C3×C4—24no—
126C22×C3×C4—21no—
127C2×Dic6—64no—
128C2×C3×C8—92no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—92no—
131C3×C4⋊C4—181no—
132C3×C4⋊C4C2×C3×D314yes—
133C3×C4⋊C4C2×C3⋊D311yescharacteristic
134C3×C22⋊C4—28no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—21no—
137C3×C42—24no—
138C3×C42C2×C3×D314yes—
139C3×C42C2×C3⋊D311yescharacteristic
140C4⋊Dic3—38no—
141C4⋊Dic3—64no—
142C2×C3⋊C8—38no—
143C22×C32—216no—
144C22×C32—28no—
145C22×C32—22no—
146C22×C32—24no—
147C22×C32—48no—
148C22×C32—48no—
149C22×C32—44no—
150C22×C32—44no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32C2×#6714yes—
154C22×C32C2×C3×D411yescharacteristic
155C32×C4—21no—
156C32×C4—28no—
157C32×C4—24no—
158C32×C4—24no—
159C32×C4—24no—
160C32×C4—44no—
161C32×C4—48no—
162C32×C4—44no—
163C32×C4—41no—
164C32×C4#21418yes—
165C32×C4C3×SD1612yes—
166C32×C4C2×Dic614yes—
167C32×C4#23514yes—
168C32×C4C2×C3×Q811yescharacteristic
169C32×C4C3×#4111yescharacteristic
170C3⋊Dic3—42no—
171C3×Dic3—128no—
172C4⋊Q8—91no—
173C4×D4C3×C3⋊D311yescharacteristic
174#106—92no—
175C4⋊C8—91no—
176#102—92no—
177C33#47611yescharacteristic, complemented, Sylow
178C23×C3—28no—
179C23×C3—24no—
180C23×C3—21no—
181C3×Q8—362no—
182C3×D4—28no—
183C3×D4—24no—
184C3×D4—21no—
185C3×D4—28no—
186C3×D4C3×Dic618yescomplemented
187C3×D4#63412yescomplemented
188C2×C3×C4—28no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—416no—
195C2×C3×C4—48no—
196C2×C3×C4—42no—
197C2×C3×C4—182no—
198C2×C3×C4C3×#6714yes—
199C2×C3×C4C3×#6714yes—
200C2×C3×C4#63811yescharacteristic
201C2×C3×C4#63811yescharacteristic
202C2×C3×C4C22×C3×D314yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204C2×Dic3—68no—
205Dic6—128no—
206C3×C8—182no—
207C3⋊C8—68no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—24no—
211C2×C32—416no—
212C2×C32—48no—
213C2×C32—42no—
214C2×C32#92414yes—
215C2×C32C2×#21414yes—
216C2×C32#94914yes—
217C2×C32C3×C22⋊Q811yescharacteristic
218C2×C32C2×C3×SD1611yescharacteristic
219C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
220C2×Q8—181no—
221C2×D4C2×C3×C3⋊D311yescharacteristic
222C22×C4—21no—
223C2×C8—92no—
224C4⋊C4—92no—
225C4⋊C4—181no—
226C4⋊C4C2×C3×C3⋊D311yescharacteristic
227C22⋊C4—21no—
228C42C2×C3×C3⋊D311yescharacteristic
229C22×C3—216no—
230C22×C3—28no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—48no—
234C22×C3—48no—
235C22×C3—44no—
236C22×C3—44no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3C2×C3×#6714yes—
240C22×C3C2×#63811yescharacteristic
241C3×C4—24no—
242C3×C4—21no—
243C3×C4—28no—
244C3×C4—24no—
245C3×C4—24no—
246C3×C4—44no—
247C3×C4—48no—
248C3×C4—44no—
249C3×C4—41no—
250C3×C4—362no—
251C3×C4C3×#21418yes—
252C3×C4#375412yes—
253C3×C4C2×C3×Dic614yes—
254C3×C4C3×#23514yes—
255C3×C4C2×#63411yescharacteristic
256C3×C4#382811yescharacteristic
257Dic3—128no—
258C32—24no—
259C32#500814yescomplemented
260C32C3×#47611yescharacteristic, complemented
261C23—21no—
262Q8—362no—
263D4—21no—
264D4C3×#63412yescomplemented
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—182no—
268C2×C4C3×#63811yescharacteristic
269C2×C4C3×#63811yescharacteristic
270C2×C4C22×C3×C3⋊D311yescharacteristic
271C8—182no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—416no—
276C2×C3—48no—
277C2×C3—42no—
278C2×C3C3×#92414yes—
279C2×C3C2×C3×#21414yes—
280C2×C3C3×#94914yes—
281C2×C3#6413311yescharacteristic
282C2×C3C2×#375411yescharacteristic
283C2×C3#6415811yescharacteristic
284C22—22no—
285C22—41no—
286C22—41no—
287C22C2×C3×#63811yescharacteristic
288C4—21no—
289C4—41no—
290C4—362no—
291C4C3×#375412yes—
292C4C2×C3×#63411yescharacteristic
293C4C3×#382811yescharacteristic
294C3—24no—
295C3C3×#500814yescomplemented
296C3#1059270711yescharacteristic, direct product factor
297C2—42no—
298C2C3×#6413311yescharacteristic
299C2C2×C3×#375411yescharacteristic
300C2C3×#6415811yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#476
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64078C2semidirect
4. C3×#63635C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10592707C3direct (N × H)
20. #64078C2×C3semidirect
23. #63635C2×C3semidirect
24. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
30. C33×D4Q8semidirect
92. C32×D4Dic6semidirect
93. C32×D4C3×Q8semidirect
177. C33#476semidirect
186. C3×D4C3×Dic6semidirect
187. C3×D4#634semidirect
259. C32#5008semidirect
260. C32C3×#476semidirect
264. D4C3×#634semidirect
295. C3C3×#5008semidirect
296. C3#10592707direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C3×#640782
C3×#636592
C3×#636352
C3×C4⋊(C3⋊(C3⋊C8))2
#105927073
C3×#50083