Finite Group Databasebeta

← C3×#10591138 #13158166

Subgroups — C3×#10591138 #13158166

Order 1728.

738 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C32×#926C212yesmaximal
3C3×#63576C212yesmaximal, complemented
4C3×#63574C212yesmaximal, complemented
5C3×#4998—32nomaximal
6#10591138C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C414yescomplemented
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×#217—22no—
12C32×C4⋊Dic3C2214yes—
13C32×C4×Dic3—22no—
14C32×#117—32no—
15C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
16C3×#926—22no—
17C3×#926D312yes—
18C3×#926C2×C312yes—
19C22×C3×C3⋊C8—32no—
20C22×C3×C3⋊C8—61no—
21C3×#808—34no—
22C3×#807—34no—
23#63576C2×C312yescomplemented
24#63574C2×C312yescomplemented
25C23×C33D411yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4D411yescharacteristic
28C2×C33×C4Q812yes—
29C2×C33×C4C2311yescharacteristic
30C2×C32×Dic3—24no—
31C3×C3⋊(C3⋊C8)C2×C412yes—
32C3×C3⋊(C3⋊C8)C2×C412yes—
33C3×#427—91no—
34#4998—32no—
35C22×C32×C4—21no—
36C22×C32×C4—22no—
37C22×C32×C4—41no—
38C22×C32×C4C2×D312yes—
39C22×C32×C4C22×C311yescharacteristic
40C32×C4⋊C4—34no—
41C32×C22⋊C4—62no—
42C32×C42—62no—
43C2×C3⋊(C3⋊C8)C3×C414yescomplemented
44C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
45C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
46C3×#217—24no—
47C3×#217—22no—
48C3×#217—22no—
49C3×C4⋊Dic3—24no—
50C3×C4⋊Dic3C2×D314yes—
51C3×C4⋊Dic3C22×C314yes—
52C3×C4×Dic3—24no—
53C3×C4×Dic3—22no—
54C3×C4×Dic3—22no—
55C2×C3×C3⋊C8—38no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—32no—
58C2×C3×C3⋊C8—64no—
59C2×C3×C3⋊C8—61no—
60C2×C3×C3⋊C8—61no—
61C22×C33—22no—
62C22×C33C4⋊C412yes—
63C22×C33C2×D411yescharacteristic
64C33×C4C4⋊C412yes—
65C33×C4C22×C411yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C32×Dic3—44no—
68C22×C3×C8—91no—
69C3×#117—32no—
70C3×#117—32no—
71C3×#117—62no—
72#926C3×D312yes—
73C22×C3⋊C8—32no—
74C22×C3⋊C8—61no—
75C3×#107—92no—
76C3×#106—92no—
77#808—34no—
78#807—34no—
79C23×C32—21no—
80C23×C32—22no—
81C23×C32—41no—
82C23×C32#6712yes—
83C23×C32C3×D411yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4—42no—
93C2×C32×C4—41no—
94C2×C32×C4—42no—
95C2×C32×C4—41no—
96C2×C32×C4—64no—
97C2×C32×C4Dic614yes—
98C2×C32×C4C4×D314yes—
99C2×C32×C4#6712yes—
100C2×C32×C4C2×C3×C412yes—
101C2×C32×C4C3×D411yescharacteristic
102C2×C32×C4C3×Q812yes—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3×Dic3—28no—
106C2×C3×Dic3—24no—
107C2×C3×Dic3—24no—
108C3⋊(C3⋊C8)C2×C3×C412yes—
109C3⋊(C3⋊C8)C2×C3×C412yes—
110C3×C3⋊C8—34no—
111C3×C3⋊C8—34no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—62no—
114#427—91noSylow
115C2×C33—22no—
116C2×C33C2×C4⋊C411yescharacteristic
117C2×C33#13512yes—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4D3211yescharacteristic
122C22×C3×C4C2×C3×D312yes—
123C2×C3×C8—94no—
124C2×C3×C8—91no—
125C2×C3×C8—91no—
126C3×C4⋊C4—34no—
127C3×C4⋊C4—34no—
128C3×C4⋊C4—64no—
129C3×C22⋊C4—64no—
130C3×C22⋊C4—62no—
131C3×C22⋊C4—62no—
132C3×C42—64no—
133C3×C42—62no—
134C3×C42—62no—
135#217—22no—
136C4⋊Dic3C2×C3×D314yes—
137C4×Dic3—22no—
138C2×C3⋊C8—38no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—64no—
142C2×C3⋊C8—61no—
143C2×C3⋊C8—61no—
144C22×C32—22no—
145C22×C32—24no—
146C22×C32—24no—
147C22×C32—22no—
148C22×C32—24no—
149C22×C32—21no—
150C22×C32—22no—
151C22×C32—42no—
152C22×C32—41no—
153C22×C32—44no—
154C22×C32—44no—
155C22×C32Dic3⋊C414yes—
156C22×C32C3×C4⋊C412yes—
157C22×C32C2×#6712yes—
158C22×C32C2×C3×D411yescharacteristic
159C32×C4—22no—
160C32×C4—24no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C32×C4—41no—
167C32×C4—41no—
168C32×C4—124no—
169C32×C4Dic3⋊C414yes—
170C32×C4C3×C4⋊C412yes—
171C32×C4C2×Dic612yes—
172C32×C4C2×C4×D312yes—
173C32×C4C22×C3×C411yescharacteristic
174C32×C4C2×C3×Q811yescharacteristic
175C3×Dic3—48no—
176C3×Dic3—44no—
177C3×Dic3—44no—
178C22×C8—91no—
179#117—32no—
180#107—92no—
181#106—92no—
182C33#42711yescharacteristic, complemented, Sylow
183C23×C3—21no—
184C23×C3—22no—
185C23×C3—41no—
186C23×C3#62511yescharacteristic
187C23×C3C3×#6712yes—
188C2×C3×C4—24no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—24no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—68no—
201C2×C3×C4—64no—
202C2×C3×C4—64no—
203C2×C3×C4#62412yes—
204C2×C3×C4#62511yescharacteristic
205C2×C3×C4#62712yes—
206C2×C3×C4C3×Dic614yes—
207C2×C3×C4C3×C4×D314yes—
208C2×C3×C4C3×#6712yes—
209C2×C3×C4C2×D3211yescharacteristic
210C2×C3×C4C22×C3×D312yes—
211C2×Dic3—24no—
212C3×C8—92no—
213C3×C8—92no—
214C3⋊C8—34no—
215C3⋊C8—34no—
216C3⋊C8—62no—
217C3⋊C8—62no—
218C2×C32—22no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—22no—
222C2×C32—24no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32—44no—
228C2×C32—44no—
229C2×C32C2×Dic3⋊C412yes—
230C2×C32#95014yes—
231C2×C32C2×C3×C4⋊C411yescharacteristic
232C2×C32C3×#13512yes—
233C22×C4C3×D3211yescharacteristic
234C2×C8—94no—
235C2×C8—91no—
236C2×C8—91no—
237C4⋊C4—34no—
238C22⋊C4—62no—
239C42—62no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—44no—
248C22×C3—44no—
249C22×C3—41no—
250C22×C3—42no—
251C22×C3#372412yes—
252C22×C3C3×Dic3⋊C414yes—
253C22×C3C2×#62511yescharacteristic
254C22×C3C2×C3×#6712yes—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—21no—
259C3×C4—24no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—42no—
264C3×C4—128no—
265C3×C4—124no—
266C3×C4—124no—
267C3×C4#372412yes—
268C3×C4C3×Dic3⋊C414yes—
269C3×C4C2×#62411yescharacteristic
270C3×C4C2×#62711yescharacteristic
271C3×C4C2×C3×Dic612yes—
272C3×C4C2×C3×C4×D312yes—
273Dic3—44no—
274C32—22no—
275C32—21no—
276C32—41no—
277C32#499812yescomplemented
278C32C3×#42711yescharacteristic, complemented
279C23C3×#62511yescharacteristic
280C2×C4—64no—
281C2×C4C3×#62412yes—
282C2×C4C3×#62511yescharacteristic
283C2×C4C3×#62712yes—
284C2×C4C2×C3×D3211yescharacteristic
285C8—92no—
286C8—92no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—24no—
293C2×C3—22no—
294C2×C3—44no—
295C2×C3—44no—
296C2×C3—41no—
297C2×C3—42no—
298C2×C3#6382212yes—
299C2×C3C2×#372411yescharacteristic
300C2×C3C2×C3×Dic3⋊C412yes—
301C2×C3C3×#95014yes—
302C22—22no—
303C22C3×#372412yes—
304C22C2×C3×#62511yescharacteristic
305C4—124no—
306C4C3×#372412yes—
307C4C2×C3×#62411yescharacteristic
308C4C2×C3×#62711yescharacteristic
309C3—21no—
310C3—22no—
311C3—41no—
312C3#1059113811yescharacteristic, direct product factor
313C3C3×#499812yescomplemented
314C2—22no—
315C2C3×#6382212yes—
316C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63576C2semidirect
4. C3×#63574C2semidirect
6. #10591138C3direct (N × H)
8. C2×C3×C3⋊(C3⋊C8)C4semidirect
23. #63576C2×C3semidirect
24. #63574C2×C3semidirect
43. C2×C3⋊(C3⋊C8)C3×C4semidirect
182. C33#427semidirect
277. C32#4998semidirect
278. C32C3×#427semidirect
312. C3#10591138direct (N × H)
313. C3C3×#4998semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C32×#9262
C3×#635762
C3×#635742
C3×#49983
#105911383