Finite Group Databasebeta

← C3×#10592782 #13163125

Subgroups — C3×#10592782 #13163125

Order 1728.

992 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3755C212yesmaximal
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
4C3×#63662C212yesmaximal, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10592782C311yesmaximal, characteristic, direct product factor
7C3×#5083—34nomaximal
8C2×C33×Q8C2212yes—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4×C3⋊D3—21no—
11C33×C4⋊C4—21no—
12C33×C4⋊C4—21no—
13C33×C42C2211yescharacteristic
14C3×#3755—24no—
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C32×C4⋊Q8—24no—
19C32×C4⋊Q8D314yescomplemented
20C32×C4⋊Q8C2×C311yescharacteristic, complemented
21C2×#3755C2×C312yes—
22C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
23C2×C3×#215—38no—
24C3×C4×D12—34no—
25#63662C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#835—38no—
28C3×C4⋊(C3⋊C8)—34no—
29C22×C3×C3⋊D3—21no—
30C33×Q8—22no—
31C33×Q8—22no—
32C2×C33×C4—22no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C3×C4⋊(C3⋊D3)—21no—
38C3×C4⋊(C3⋊D3)D412yes—
39C3×C4×C3⋊D3—42no—
40C3×C3⋊(C3⋊C8)—22no—
41C3×#461—91no—
42#5083—34no—
43C2×C32×Q8—28no—
44C2×C32×Q8C2×D318yes—
45C2×C32×Q8C22×C312yes—
46C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
47C2×C4×C3⋊D3—21no—
48C2×C3×D12—34no—
49C2×C3×C4×D3—64no—
50C32×C4⋊C4—28no—
51C32×C4⋊C4—28no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4—24no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C42—24no—
57C32×C42C2×D314yes—
58C32×C42C22×C311yescharacteristic
59#3755—24no—
60#3752—21no—
61C4⋊(C3⋊Dic3)C22×C311yescharacteristic
62C2×C3⋊(C3⋊C8)C22×C312yes—
63C3×#215—616no—
64C3×(C2×D3)⋊C4—64no—
65C3×C4⋊Dic3—34no—
66C2×C3×C3⋊C8—38no—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—22no—
69C2×C3×C3⋊D3—41no—
70C2×C3×C3⋊D3—41no—
71C33×C4—21no—
72C33×C4—42no—
73C33×C4SD1612yescomplemented
74C33×C4C2×D411yescharacteristic
75C33×C4#4111yescharacteristic
76C3×C3⋊Dic3—41no—
77C2×C3×SD16—92no—
78C3×C4⋊Q8—24no—
79C3×C4⋊Q8C3×D314yescomplemented
80C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
81C3×C4×D4—91no—
82C2×#215—38no—
83C4×D12—34no—
84C3×C4⋊C8—91no—
85C3×#103—92no—
86#835—38no—
87C4⋊(C3⋊C8)—34no—
88C22×C3⋊D3—21no—
89C22×C3×D3—64no—
90C32×Q8—22no—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—28no—
94C32×Q8—48no—
95C32×Q8—48no—
96C2×C32×C4—216no—
97C2×C32×C4—28no—
98C2×C32×C4—22no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4#6714yes—
103C2×C32×C4#6714yes—
104C2×C32×C4C3×D411yescharacteristic
105C2×C32×C4C3×D411yescharacteristic
106C2×C32×C4C22×D314yes—
107C2×C32×C4C23×C311yescharacteristic
108C2×C3⋊Dic3—21no—
109C4⋊(C3⋊D3)—21no—
110C4⋊(C3⋊D3)C3×D412yes—
111C4×C3⋊D3—42no—
112C2×C3×Dic3—64no—
113C3×D12—38no—
114C3×D12—64no—
115C3×C4×D3—128no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—68no—
118#461—91noSylow
119C2×C33#12111yescharacteristic
120C2×C33C2×SD1611yescharacteristic
121C2×C33#13711yescharacteristic
122C3×C3⋊D3—42no—
123C2×C3×Q8—28no—
124C2×C3×Q8C2×C3×D318yes—
125C2×C3×Q8C2×C3⋊D312yes—
126C2×C3×D4—91no—
127C22×C3×C4—181no—
128C2×D12—34no—
129C2×C4×D3—64no—
130C3×SD16—184no—
131C2×C3×C8—92no—
132C3×C4⋊C4—28no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—91no—
139C3×C22⋊C4—181no—
140C3×C42—24no—
141C3×C42C2×C3×D314yes—
142C3×C42C2×C3⋊D311yescharacteristic
143#215—616no—
144(C2×D3)⋊C4—64no—
145C4⋊Dic3—34no—
146C2×C3⋊C8—38no—
147C22×C32—24no—
148C22×C32C2×#6714yes—
149C22×C32C2×C3×D411yescharacteristic
150C2×C3⋊D3—22no—
151C2×C3⋊D3—41no—
152C2×C3⋊D3—41no—
153C2×C3×D3—68no—
154C2×C3×D3—124no—
155C2×C3×D3—124no—
156C32×C4—28no—
157C32×C4—24no—
158C32×C4—21no—
159C32×C4—28no—
160C32×C4—24no—
161C32×C4—24no—
162C32×C4—416no—
163C32×C4—48no—
164C32×C4—42no—
165C32×C4#21518yescomplemented
166C32×C4C3×SD1612yescomplemented
167C32×C4D3×D414yes—
168C32×C4#23714yes—
169C32×C4C2×C3×D411yescharacteristic
170C32×C4C3×#4111yescharacteristic
171C3⋊Dic3—41no—
172C3×Dic3—124no—
173C2×SD16—92no—
174C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
175C4×D4—91no—
176C4⋊C8—91no—
177#103—92no—
178C33#46111yescharacteristic, complemented, Sylow
179C23×C3—181no—
180C22×D3—64no—
181C3×Q8—28no—
182C3×Q8—28no—
183C3×Q8—22no—
184C3×Q8—22no—
185C3×Q8—48no—
186C3×Q8—48no—
187C3×D4—92no—
188C3×D4—181no—
189C2×C3×C4—216no—
190C2×C3×C4—28no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—181no—
196C2×C3×C4—362no—
197C2×C3×C4C3×#6714yes—
198C2×C3×C4C3×#6714yes—
199C2×C3×C4#63811yescharacteristic
200C2×C3×C4#63811yescharacteristic
201C2×C3×C4C22×C3×D314yes—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203C2×Dic3—64no—
204D12—38no—
205D12—64no—
206C4×D3—128no—
207C3×C8—182no—
208C3⋊C8—68no—
209C2×C32—24no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32#93814yes—
213C2×C32C2×#21514yes—
214C2×C32#94214yes—
215C2×C32C3×#12111yescharacteristic
216C2×C32C2×C3×SD1611yescharacteristic
217C2×C32C3×#13711yescharacteristic
218C3⋊D3—42no—
219C3×D3—128no—
220C2×Q8C2×C3×C3⋊D312yes—
221C2×D4—91no—
222C22×C4—181no—
223SD16—184no—
224C2×C8—92no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—91no—
228C22⋊C4—181no—
229C42C2×C3×C3⋊D311yescharacteristic
230C22×C3—24no—
231C22×C3—182no—
232C22×C3—361no—
233C22×C3—361no—
234C22×C3C2×C3×#6714yes—
235C22×C3C2×#63811yescharacteristic
236C2×D3—68no—
237C2×D3—124no—
238C2×D3—124no—
239C3×C4—28no—
240C3×C4—24no—
241C3×C4—21no—
242C3×C4—28no—
243C3×C4—24no—
244C3×C4—24no—
245C3×C4—416no—
246C3×C4—48no—
247C3×C4—42no—
248C3×C4—361no—
249C3×C4C3×#21518yescomplemented
250C3×C4#375512yescomplemented
251C3×C4C3×D3×D414yes—
252C3×C4C3×#23714yes—
253C3×C4D4×C3⋊D311yescharacteristic
254C3×C4#383011yescharacteristic
255Dic3—124no—
256C32—24no—
257C32#508314yescomplemented
258C32C3×#46111yescharacteristic, complemented
259C23—181no—
260Q8—22no—
261Q8—22no—
262D4—92no—
263D4—181no—
264C2×C4—22no—
265C2×C4—181no—
266C2×C4—362no—
267C2×C4C3×#63811yescharacteristic
268C2×C4C3×#63811yescharacteristic
269C2×C4C22×C3×C3⋊D311yescharacteristic
270C8—182no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3—362no—
275C2×C3C3×#93814yes—
276C2×C3C2×C3×#21514yes—
277C2×C3C3×#94214yes—
278C2×C3#6414711yescharacteristic
279C2×C3C2×#375511yescharacteristic
280C2×C3#6415111yescharacteristic
281D3—128no—
282C22—182no—
283C22—361no—
284C22—361no—
285C22C2×C3×#63811yescharacteristic
286C4—21no—
287C4—42no—
288C4—361no—
289C4C3×#375512yescomplemented
290C4C3×D4×C3⋊D311yescharacteristic
291C4C3×#383011yescharacteristic
292C3—24no—
293C3C3×#508314yescomplemented
294C3#1059278211yescharacteristic, direct product factor
295C2—362no—
296C2C3×#6414711yescharacteristic
297C2C2×C3×#375511yescharacteristic
298C2C3×#6415111yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
4. C3×#63662C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10592782C3direct (N × H)
19. C32×C4⋊Q8D3semidirect
20. C32×C4⋊Q8C2×C3semidirect
25. #63662C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
73. C33×C4SD16semidirect
79. C3×C4⋊Q8C3×D3semidirect
80. C3×C4⋊Q8C3⋊D3semidirect
165. C32×C4#215semidirect
166. C32×C4C3×SD16semidirect
174. C4⋊Q8C3×C3⋊D3semidirect
178. C33#461semidirect
249. C3×C4C3×#215semidirect
250. C3×C4#3755semidirect
257. C32#5083semidirect
258. C32C3×#461semidirect
289. C4C3×#3755semidirect
293. C3C3×#5083semidirect
294. C3#10592782direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C2×C3×#37552
C3×C4×C4⋊(C3⋊D3)2
C3×#636622
C3×C4⋊(C3⋊(C3⋊C8))2
#105927823
C3×#50833