Finite Group Databasebeta

← C3×#10592786 #13163126

Subgroups — C3×#10592786 #13163126

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#63636C212yesmaximal
4C3×#63634C212yesmaximal, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10592786C311yesmaximal, characteristic, direct product factor
7C3×#5087—34nomaximal
8C2×C33×Q8—21no—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4×C3⋊D3—21no—
11C33×C4⋊C4—21no—
12C33×C4⋊C4C2212yes—
13C33×C42C2211yescharacteristic
14C3×#3752—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Q8—24no—
18C32×C4⋊Q8D314yescomplemented
19C32×C4⋊Q8C2×C311yescharacteristic, complemented
20C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
21C3×C4×D12—34no—
22#63636C2×C312yes—
23#63634C2×C312yescomplemented
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
25C3×#809—38no—
26C3×#807—38no—
27C3×C4⋊(C3⋊C8)—34no—
28C22×C3×C3⋊D3—21no—
29C33×Q8—42no—
30C2×C33×C4—22no—
31C2×C33×C4D411yescharacteristic
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C3×C4⋊(C3⋊D3)—21no—
36C3×C4⋊(C3⋊D3)Q812yescomplemented
37C3×C4×C3⋊D3—42no—
38C3×C3⋊(C3⋊C8)—22no—
39C3×#474—91no—
40#5087—34no—
41C2×C32×Q8—28no—
42C2×C32×Q8—24no—
43C2×C32×Q8—21no—
44C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×C3×D12—34no—
47C2×C3×C4×D3—64no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—28no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D318yes—
53C32×C4⋊C4C22×C312yes—
54C32×C42—24no—
55C32×C42C2×D314yes—
56C32×C42C22×C311yescharacteristic
57#3752—21no—
58C4⋊(C3⋊Dic3)C22×C311yescharacteristic
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C3×(C2×D3)⋊C4—64no—
61C3×C4⋊Dic3—34no—
62C2×C3×C3⋊C8—38no—
63C22×C33C2×D411yescharacteristic
64C2×C3×C3⋊D3—22no—
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—41no—
67C33×C4—21no—
68C33×C4—42no—
69C33×C4D812yes—
70C33×C4C2×Q811yescharacteristic
71C33×C4#4111yescharacteristic
72C3×C3⋊Dic3—41no—
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8C3×D314yescomplemented
75C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
76C3×C4×D4—91no—
77C4×D12—34no—
78C3×#107—92no—
79C3×C4⋊C8—91no—
80C3×#102—92no—
81#809—38no—
82#807—38no—
83C4⋊(C3⋊C8)—34no—
84C22×C3⋊D3—21no—
85C22×C3×D3—64no—
86C32×Q8—416no—
87C32×Q8—48no—
88C32×Q8—42no—
89C2×C32×C4—216no—
90C2×C32×C4—28no—
91C2×C32×C4—22no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4#6714yes—
96C2×C32×C4#6714yes—
97C2×C32×C4C3×D411yescharacteristic
98C2×C32×C4C3×D411yescharacteristic
99C2×C32×C4C22×D314yes—
100C2×C32×C4C23×C311yescharacteristic
101C2×C3⋊Dic3—21no—
102C4⋊(C3⋊D3)—21no—
103C4⋊(C3⋊D3)C3×Q812yescomplemented
104C4×C3⋊D3—42no—
105C2×C3×Dic3—64no—
106C3×D12—38no—
107C3×D12—64no—
108C3×C4×D3—128no—
109C3⋊(C3⋊C8)—22no—
110C3×C3⋊C8—68no—
111#474—91noSylow
112C2×C33C22⋊Q811yescharacteristic
113C2×C33C2×D811yescharacteristic
114C2×C33#13711yescharacteristic
115C3×C3⋊D3—42no—
116C2×C3×Q8—28no—
117C2×C3×Q8—24no—
118C2×C3×Q8—21no—
119C2×C3×D4—91no—
120C22×C3×C4—181no—
121C2×D12—34no—
122C2×C4×D3—64no—
123C2×C3×C8—92no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—28no—
128C3×C4⋊C4—91no—
129C3×C4⋊C4C2×C3×D318yes—
130C3×C4⋊C4C2×C3⋊D312yes—
131C3×C22⋊C4—181no—
132C3×C42—24no—
133C3×C42C2×C3×D314yes—
134C3×C42C2×C3⋊D311yescharacteristic
135(C2×D3)⋊C4—64no—
136C4⋊Dic3—34no—
137C2×C3⋊C8—38no—
138C22×C32—24no—
139C22×C32C2×#6714yes—
140C22×C32C2×C3×D411yescharacteristic
141C2×C3⋊D3—22no—
142C2×C3⋊D3—41no—
143C2×C3⋊D3—41no—
144C2×C3×D3—68no—
145C2×C3×D3—124no—
146C2×C3×D3—124no—
147C32×C4—28no—
148C32×C4—24no—
149C32×C4—21no—
150C32×C4—28no—
151C32×C4—24no—
152C32×C4—24no—
153C32×C4—416no—
154C32×C4—48no—
155C32×C4—42no—
156C32×C4#21318yes—
157C32×C4C3×D812yes—
158C32×C4D3×Q814yes—
159C32×C4#23914yes—
160C32×C4C2×C3×Q811yescharacteristic
161C32×C4C3×#4111yescharacteristic
162C3⋊Dic3—41no—
163C3×Dic3—124no—
164C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
165C4×D4—91no—
166#107—92no—
167C4⋊C8—91no—
168#102—92no—
169C33#47411yescharacteristic, complemented, Sylow
170C23×C3—181no—
171C22×D3—64no—
172C3×Q8—416no—
173C3×Q8—48no—
174C3×Q8—42no—
175C3×D4—92no—
176C3×D4—181no—
177C2×C3×C4—216no—
178C2×C3×C4—28no—
179C2×C3×C4—22no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—181no—
184C2×C3×C4—362no—
185C2×C3×C4C3×#6714yes—
186C2×C3×C4C3×#6714yes—
187C2×C3×C4#63811yescharacteristic
188C2×C3×C4#63811yescharacteristic
189C2×C3×C4C22×C3×D314yes—
190C2×C3×C4C22×C3⋊D311yescharacteristic
191C2×Dic3—64no—
192D12—38no—
193D12—64no—
194C4×D3—128no—
195C3×C8—182no—
196C3⋊C8—68no—
197C2×C32—24no—
198C2×C32—24no—
199C2×C32—24no—
200C2×C32C2×#21314yes—
201C2×C32#94214yes—
202C2×C32#94614yes—
203C2×C32C3×C22⋊Q811yescharacteristic
204C2×C32C2×C3×D811yescharacteristic
205C2×C32C3×#13711yescharacteristic
206C3⋊D3—42no—
207C3×D3—128no—
208C2×Q8—21no—
209C2×D4—91no—
210C22×C4—181no—
211C2×C8—92no—
212C4⋊C4—21no—
213C4⋊C4—91no—
214C4⋊C4C2×C3×C3⋊D312yes—
215C22⋊C4—181no—
216C42C2×C3×C3⋊D311yescharacteristic
217C22×C3—24no—
218C22×C3—182no—
219C22×C3—361no—
220C22×C3—361no—
221C22×C3C2×C3×#6714yes—
222C22×C3C2×#63811yescharacteristic
223C2×D3—68no—
224C2×D3—124no—
225C2×D3—124no—
226C3×C4—28no—
227C3×C4—24no—
228C3×C4—21no—
229C3×C4—28no—
230C3×C4—24no—
231C3×C4—24no—
232C3×C4—416no—
233C3×C4—48no—
234C3×C4—42no—
235C3×C4—361no—
236C3×C4C3×#21318yes—
237C3×C4#375312yes—
238C3×C4C3×D3×Q814yes—
239C3×C4C3×#23914yes—
240C3×C4Q8×C3⋊D311yescharacteristic
241C3×C4#383211yescharacteristic
242Dic3—124no—
243C32—24no—
244C32#508714yescomplemented
245C32C3×#47411yescharacteristic, complemented
246C23—181no—
247Q8—42no—
248D4—92no—
249D4—181no—
250C2×C4—22no—
251C2×C4—181no—
252C2×C4—362no—
253C2×C4C3×#63811yescharacteristic
254C2×C4C3×#63811yescharacteristic
255C2×C4C22×C3×C3⋊D311yescharacteristic
256C8—182no—
257C2×C3—24no—
258C2×C3—24no—
259C2×C3—24no—
260C2×C3—362no—
261C2×C3C2×C3×#21314yes—
262C2×C3C3×#94214yes—
263C2×C3C3×#94614yes—
264C2×C3C2×#375311yescharacteristic
265C2×C3#6415111yescharacteristic
266C2×C3#6415511yescharacteristic
267D3—128no—
268C22—182no—
269C22—361no—
270C22—361no—
271C22C2×C3×#63811yescharacteristic
272C4—21no—
273C4—42no—
274C4—361no—
275C4C3×#375312yes—
276C4C3×Q8×C3⋊D311yescharacteristic
277C4C3×#383211yescharacteristic
278C3—24no—
279C3C3×#508714yescomplemented
280C3#1059278611yescharacteristic, direct product factor
281C2—362no—
282C2C2×C3×#375311yescharacteristic
283C2C3×#6415111yescharacteristic
284C2C3×#6415511yescharacteristic
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×#474
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×#63634C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10592786C3direct (N × H)
18. C32×C4⋊Q8D3semidirect
19. C32×C4⋊Q8C2×C3semidirect
23. #63634C2×C3semidirect
24. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
36. C3×C4⋊(C3⋊D3)Q8semidirect
74. C3×C4⋊Q8C3×D3semidirect
75. C3×C4⋊Q8C3⋊D3semidirect
103. C4⋊(C3⋊D3)C3×Q8semidirect
164. C4⋊Q8C3×C3⋊D3semidirect
169. C33#474semidirect
244. C32#5087semidirect
245. C32C3×#474semidirect
279. C3C3×#5087semidirect
280. C3#10592786direct (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
C3×C4×C4⋊(C3⋊D3)2
C3×#636362
C3×#636342
C3×C4⋊(C3⋊(C3⋊C8))2
#105927863
C3×#50873