Finite Group Databasebeta

← C2×#11722079 #13161826

Subgroups — C2×#11722079 #13161826

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C212yesmaximal
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#11722079C214yesmaximal, direct product factor
4C2×#63574—31nomaximal
5C2×#63573—32nomaximal
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3—22no—
8C3×C4⋊(C3⋊Dic3)—22no—
9C3×C4⋊(C3⋊Dic3)C2214yes—
10C2×C3×C3⋊(C3⋊C8)C414yescomplemented
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C2×C4⋊(C3⋊Dic3)D312yes—
13C22×C3⋊(C3⋊C8)D311yescharacteristic
14C2×C3×C4⋊Dic3—32no—
15C2×C3×C4⋊Dic3—32no—
16C2×C3×C4⋊Dic3—62no—
17C22×C3×C3⋊C8—32no—
18C22×C3×C3⋊C8—61no—
19#63574—34no—
20#63573—38no—
21C23×C33D411yescharacteristic
22C2×C33×C4C2×C412yes—
23C2×C33×C4D411yescharacteristic
24C2×C33×C4Q812yes—
25C2×C33×C4C2311yescharacteristic
26C2×C3×C3⋊Dic3—24no—
27C2×C3×C3⋊Dic3—44no—
28C3×C3⋊(C3⋊C8)C2×C414yescomplemented
29C2×#817—91no—
30C2×#808—92no—
31C22×C32×C4—21no—
32C22×C32×C4—22no—
33C22×C32×C4—41no—
34C22×C32×C4C2×D312yes—
35C22×C32×C4C2×D311yescharacteristic
36C22×C3⋊Dic3—22no—
37C22×C3×Dic3—64no—
38C22×C3×Dic3—62no—
39C22×C3×Dic3—62no—
40C4⋊(C3⋊Dic3)—22no—
41C4⋊(C3⋊Dic3)C2×D314yes—
42C2×C3⋊(C3⋊C8)Dic314yescomplemented
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×C4⋊Dic3—34no—
45C3×C4⋊Dic3—34no—
46C3×C4⋊Dic3—64no—
47C3×C4⋊Dic3—64no—
48C3×C4⋊Dic3—62no—
49C3×C4⋊Dic3—62no—
50C2×C3×C3⋊C8—38no—
51C2×C3×C3⋊C8—34no—
52C2×C3×C3⋊C8—64no—
53C2×C3×C3⋊C8—62no—
54C22×C33C4⋊C412yes—
55C22×C33SD1614yes—
56C22×C33C2×D411yescharacteristic
57C33×C4C4⋊C412yes—
58C33×C4C22×C411yescharacteristic
59C33×C4C2×Q811yescharacteristic
60C3×C3⋊Dic3—44no—
61C22×C3×C8—91no—
62C2×C3×C4⋊C4—92no—
63C2×C4⋊Dic3—32no—
64C2×C4⋊Dic3—32no—
65C2×C4⋊Dic3—62no—
66C22×C3⋊C8—32no—
67C22×C3⋊C8—61no—
68#817—94no—
69#808—98no—
70C23×C32—21no—
71C23×C32—22no—
72C23×C32—41no—
73C23×C32D1211yescharacteristic
74C23×C32#6712yes—
75C2×C32×C4—24no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4—22no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—22no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C32×C4—41no—
87C2×C32×C4Dic614yes—
88C2×C32×C4Dic612yes—
89C2×C32×C4C4×D314yes—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4C2×Dic312yes—
92C2×C32×C4#6712yes—
93C2×C32×C4C22×D312yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—24no—
96C2×C3⋊Dic3—44no—
97C2×C3×Dic3—68no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—128no—
101C2×C3×Dic3—124no—
102C2×C3×Dic3—124no—
103C3⋊(C3⋊C8)C2×Dic314yescomplemented
104C3×C3⋊C8—38no—
105C3×C3⋊C8—64no—
106C2×#106—271noSylow
107C2×C33#10614yescomplemented
108C2×C33C2×C4⋊C411yescharacteristic
109C2×C33C2×SD1612yes—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—41no—
113C22×C3×C4—182no—
114C22×C3×C4D3212yes—
115C22×C3×C4D3211yescharacteristic
116C22×Dic3—64no—
117C22×Dic3—62no—
118C22×Dic3—62no—
119C2×C3×C8—94no—
120C2×C3×C8—92no—
121C3×C4⋊C4—94no—
122C3×C4⋊C4—182no—
123C4⋊Dic3—34no—
124C4⋊Dic3—34no—
125C4⋊Dic3—64no—
126C4⋊Dic3—64no—
127C4⋊Dic3—62no—
128C4⋊Dic3—62no—
129C2×C3⋊C8—38no—
130C2×C3⋊C8—34no—
131C2×C3⋊C8—64no—
132C2×C3⋊C8—62no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—22no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—42no—
141C22×C32—44no—
142C22×C32—41no—
143C22×C32#20414yes—
144C22×C32Dic3⋊C414yes—
145C22×C32C4⋊Dic312yes—
146C22×C32#21414yes—
147C22×C32#21514yes—
148C22×C32C2×D1211yescharacteristic
149C22×C32C2×#6712yes—
150C32×C4—24no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—21no—
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4Dic3⋊C414yes—
160C32×C4C4⋊Dic312yes—
161C32×C4C2×Dic612yes—
162C32×C4C2×Dic611yescharacteristic
163C32×C4C2×C4×D312yes—
164C32×C4C22×Dic311yescharacteristic
165C3⋊Dic3—44no—
166C3×Dic3—128no—
167C3×Dic3—124no—
168C3×Dic3—124no—
169C22×C8—91no—
170C2×C4⋊C4—92no—
171#106—274no—
172C33C2×#10611yescharacteristic, complemented, Sylow
173C23×C3—21no—
174C23×C3—22no—
175C23×C3—41no—
176C23×C3#62511yescharacteristic
177C23×C3#62612yes—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—21no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—184no—
191C2×C3×C4—364no—
192C2×C3×C4D3×Dic314yes—
193C2×C3×C4#62412yes—
194C2×C3×C4#62511yescharacteristic
195C2×C3×C4#62612yes—
196C2×C3×C4#62714yes—
197C2×C3×C4#62712yes—
198C2×C3×C4C2×D3212yes—
199C2×C3×C4C2×D3211yescharacteristic
200C2×Dic3—68no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—128no—
204C2×Dic3—124no—
205C2×Dic3—124no—
206C3×C8—94no—
207C3⋊C8—38no—
208C3⋊C8—64no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—28no—
212C2×C32—24no—
213C2×C32—22no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—44no—
217C2×C32—41no—
218C2×C32—42no—
219C2×C32#80818yescomplemented
220C2×C32#81714yescomplemented
221C2×C32C2×#20412yes—
222C2×C32C2×Dic3⋊C412yes—
223C2×C32C2×C4⋊Dic311yescharacteristic
224C2×C32C2×#21412yes—
225C2×C32C2×#21512yes—
226C22×C4—182no—
227C22×C4#111311yescharacteristic
228C2×C8—94no—
229C2×C8—92no—
230C4⋊C4—94no—
231C4⋊C4—182no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—22no—
239C22×C3—41no—
240C22×C3—44no—
241C22×C3—42no—
242C22×C3#371514yes—
243C22×C3#371614yes—
244C22×C3#371714yes—
245C22×C3Dic3⋊Dic314yes—
246C22×C3#372412yes—
247C22×C3C2×#62511yescharacteristic
248C22×C3C2×#62612yes—
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—24no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—42no—
258C3×C4—364no—
259C3×C4Dic3⋊Dic314yes—
260C3×C4#372412yes—
261C3×C4C2×D3×Dic312yes—
262C3×C4C2×#62411yescharacteristic
263C3×C4C2×#62712yes—
264C3×C4C2×#62711yescharacteristic
265Dic3—128no—
266Dic3—124no—
267Dic3—124no—
268C32—22no—
269C32—21no—
270C32—41no—
271C32C2×#80812yescomplemented
272C32C2×#81711yescharacteristic, complemented
273C23#629511yescharacteristic
274C2×C4—184no—
275C2×C4—364no—
276C2×C4#629412yes—
277C2×C4#629511yescharacteristic
278C2×C4#629612yes—
279C2×C4C2×#111311yescharacteristic
280C8—94no—
281C2×C3—28no—
282C2×C3—24no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—44no—
289C2×C3—42no—
290C2×C3—41no—
291C2×C3#6357318yescomplemented
292C2×C3#6357414yescomplemented
293C2×C3C2×#371512yes—
294C2×C3C2×#371612yes—
295C2×C3C2×#371712yes—
296C2×C3C2×Dic3⋊Dic312yes—
297C2×C3C2×#372411yescharacteristic
298C22#8881114yes—
299C22#8881512yes—
300C22C2×#629511yescharacteristic
301C4—364no—
302C4#8881512yes—
303C4C2×#629411yescharacteristic
304C4C2×#629611yescharacteristic
305C3—22no—
306C3—21no—
307C3—41no—
308C3C2×#6357312yescomplemented
309C3C2×#6357411yescharacteristic, complemented
310C2#1172207914yesdirect product factor
311C2C2×#8881112yes—
312C2C2×#8881511yescharacteristic
subgroupG/H
centerC23#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socle, prime radicalC23×C33D4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33C2×#106
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722079C2direct (N × H)
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
28. C3×C3⋊(C3⋊C8)C2×C4semidirect
42. C2×C3⋊(C3⋊C8)Dic3semidirect
103. C3⋊(C3⋊C8)C2×Dic3semidirect
107. C2×C33#106semidirect
172. C33C2×#106semidirect
219. C2×C32#808semidirect
220. C2×C32#817semidirect
271. C32C2×#808semidirect
272. C32C2×#817semidirect
291. C2×C3#63573semidirect
292. C2×C3#63574semidirect
308. C3C2×#63573semidirect
309. C3C2×#63574semidirect
310. C2#11722079direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C22×C3×C3⋊(C3⋊C8)2
#117220792
C2×#635743
C2×#635733