Finite Group Databasebeta

← C2×C3×#1807 #10639263

Subgroups — C2×C3×#1807 #10639263

Order 768.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3×C8C211yesmaximal, characteristic
2C22×C3×#106C212yesmaximal, complemented
3C2×C3×#380C212yesmaximal
4C3×#1807C218yesmaximal, direct product factor
5C2×C3×#340C212yesmaximal, complemented
6C2×#1807C311yesmaximal, characteristic, direct product factor, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×C3×C8—28no—
9C23×C3×C8C418yescomplemented
10C23×C3×C8C2211yescharacteristic
11C23×C3×C8C2214yes—
12C23×C3×C8C2211yescharacteristic
13C22×C3×C4⋊C4C2214yes—
14C22×C3×C22⋊C4—22no—
15C2×C3×#106—28no—
16C2×C3×#106C2218yescomplemented
17C3×#380—24no—
18C3×#380C2218yes—
19C2×C3×#95—22no—
20C3×#340C2218yescomplemented
21C24×C8C2×C311yescharacteristic
22C22×#106C2×C312yescomplemented
23C2×#380C2×C312yes—
24#1807C2×C318yesdirect product factor
25C2×#340C2×C312yescomplemented
26C25×C3D411yescharacteristic
27C23×C3×C4—24no—
28C23×C3×C4—24no—
29C23×C3×C4—24no—
30C23×C3×C4C2×C414yes—
31C23×C3×C4D412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4Q814yes—
34C23×C3×C4C2312yes—
35C23×C3×C4C2311yescharacteristic
36C22×C3×C8—24no—
37C22×C3×C8—232no—
38C22×C3×C8—216no—
39C22×C3×C8C2×C414yes—
40C22×C3×C8C2×C414yes—
41C22×C3×C8C2×C4116yescomplemented
42C22×C3×C8D418yes—
43C22×C3×C8C2312yes—
44C22×C3×C8C2312yes—
45C2×C3×C4⋊C4—28no—
46C2×C3×C4⋊C4—28no—
47C2×C3×C4⋊C4—24no—
48C2×C3×C4⋊C4C2318yes—
49C2×C3×C22⋊C4—28no—
50C2×C3×C22⋊C4—48no—
51C3×#106—216no—
52C3×#95—28no—
53C24×C4C22×C311yescharacteristic
54C23×C8—28no—
55C23×C8C3×C418yescomplemented
56C23×C8C22×C311yescharacteristic
57C23×C8C22×C314yes—
58C23×C8C22×C311yescharacteristic
59C22×C4⋊C4C22×C314yes—
60C22×C22⋊C4—22no—
61C2×#106—28no—
62C2×#106C22×C318yescomplemented
63#380—24no—
64#380C22×C318yes—
65C2×#95—22no—
66#340C22×C318yescomplemented
67C24×C3—24no—
68C24×C3—24no—
69C24×C3C4⋊C414yes—
70C24×C3SD1618yes—
71C24×C3C2×D412yes—
72C24×C3C2×D411yescharacteristic
73C22×C3×C4—24no—
74C22×C3×C4—28no—
75C22×C3×C4—28no—
76C22×C3×C4—28no—
77C22×C3×C4—216no—
78C22×C3×C4—24no—
79C22×C3×C4—24no—
80C22×C3×C4—28no—
81C22×C3×C4—48no—
82C22×C3×C4—48no—
83C22×C3×C4—48no—
84C22×C3×C4C4⋊C414yes—
85C22×C3×C4C4⋊C418yes—
86C22×C3×C4C22×C414yes—
87C22×C3×C4C22×C412yes—
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D414yes—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×Q814yes—
92C22×C3×C4C2×Q812yes—
93C22×C3×C4#4114yes—
94C22×C3×C4C2411yescharacteristic
95C2×C3×C8—28no—
96C2×C3×C8—24no—
97C2×C3×C8—232no—
98C2×C3×C8C22⋊C4116yescomplemented
99C2×C3×C8C22×C414yes—
100C2×C3×C8C22×C414yes—
101C2×C3×C8C2×D418yes—
102C3×C4⋊C4—28no—
103C3×C4⋊C4—28no—
104C3×C4⋊C4—48no—
105C3×C22⋊C4—416no—
106C25C3×D411yescharacteristic
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4C2×C3×C414yes—
111C23×C4C3×D411yescharacteristic
112C23×C4C3×D412yes—
113C23×C4C3×Q814yes—
114C23×C4C23×C312yes—
115C23×C4C23×C311yescharacteristic
116C22×C8—24no—
117C22×C8—232no—
118C22×C8—216no—
119C22×C8C2×C3×C414yes—
120C22×C8C2×C3×C414yes—
121C22×C8C2×C3×C4116yescomplemented
122C22×C8C3×D418yes—
123C22×C8C23×C312yes—
124C22×C8C23×C312yes—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4C23×C318yes—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—48no—
131#106—216no—
132#95—28no—
133C23×C3—24no—
134C23×C3—24no—
135C23×C3—28no—
136C23×C3—28no—
137C23×C3—232no—
138C23×C3#106116yescomplemented
139C23×C3C2×C4⋊C414yes—
140C23×C3C2×C4⋊C412yes—
141C23×C3#12114yes—
142C23×C3C2×SD1618yes—
143C23×C3C2×SD1614yes—
144C23×C3#13514yes—
145C23×C3C22×D411yescharacteristic
146C2×C3×C4—24no—
147C2×C3×C4—28no—
148C2×C3×C4—24no—
149C2×C3×C4—24no—
150C2×C3×C4—28no—
151C2×C3×C4—216no—
152C2×C3×C4—48no—
153C2×C3×C4—48no—
154C2×C3×C4—48no—
155C2×C3×C4—816no—
156C2×C3×C4C2×C22⋊C414yes—
157C2×C3×C4C2×C4⋊C414yes—
158C2×C3×C4C2×C4⋊C412yes—
159C2×C3×C4#11714yes—
160C2×C3×C4#12114yes—
161C2×C3×C4C22⋊Q818yes—
162C2×C3×C4C23×C411yescharacteristic
163C2×C3×C4C22×D412yes—
164C2×C3×C4C22×Q811yescharacteristic
165C2×C3×C4C2×#4112yes—
166C3×C8—24no—
167C3×C8C2×C22⋊C418yescomplemented
168C24—24no—
169C24—24no—
170C24C3×C4⋊C414yes—
171C24C3×SD1618yes—
172C24C2×C3×D412yes—
173C24C2×C3×D411yescharacteristic
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—216no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4—48no—
185C22×C4C3×C4⋊C414yes—
186C22×C4C3×C4⋊C418yes—
187C22×C4C22×C3×C414yes—
188C22×C4C22×C3×C412yes—
189C22×C4C2×C3×D412yes—
190C22×C4C2×C3×D414yes—
191C22×C4C2×C3×D411yescharacteristic
192C22×C4C2×C3×Q814yes—
193C22×C4C2×C3×Q812yes—
194C22×C4C3×#4114yes—
195C22×C4C24×C311yescharacteristic
196C2×C8—28no—
197C2×C8—24no—
198C2×C8—232no—
199C2×C8C3×C22⋊C4116yescomplemented
200C2×C8C22×C3×C414yes—
201C2×C8C22×C3×C414yes—
202C2×C8C2×C3×D418yes—
203C4⋊C4—28no—
204C4⋊C4—28no—
205C4⋊C4—48no—
206C22⋊C4—416no—
207C22×C3—232no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—216no—
211C22×C3#38014yes—
212C22×C3C2×#10614yes—
213C22×C3C2×#10618yescomplemented
214C22×C3#42714yes—
215C22×C3#465116yes—
216C22×C3C22×C4⋊C411yescharacteristic
217C22×C3C2×#12112yes—
218C22×C3C22×SD1612yes—
219C22×C3C2×#13512yes—
220C3×C4—24no—
221C3×C4—88no—
222C3×C4#38014yes—
223C3×C4C22×C22⋊C411yescharacteristic
224C3×C4C2×#11711yescharacteristic
225C3×C4C2×C22⋊Q812yes—
226C23—24no—
227C23—28no—
228C23—28no—
229C23—232no—
230C23—24no—
231C23C3×#106116yescomplemented
232C23C2×C3×C4⋊C414yes—
233C23C2×C3×C4⋊C412yes—
234C23C3×#12114yes—
235C23C2×C3×SD1618yes—
236C23C2×C3×SD1614yes—
237C23C3×#13514yes—
238C23C22×C3×D411yescharacteristic
239C2×C4—28no—
240C2×C4—24no—
241C2×C4—216no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—28no—
245C2×C4—48no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—816no—
249C2×C4C2×C3×C22⋊C414yes—
250C2×C4C2×C3×C4⋊C414yes—
251C2×C4C2×C3×C4⋊C412yes—
252C2×C4C3×#11714yes—
253C2×C4C3×#12114yes—
254C2×C4C3×C22⋊Q818yes—
255C2×C4C23×C3×C411yescharacteristic
256C2×C4C22×C3×D412yes—
257C2×C4C22×C3×Q811yescharacteristic
258C2×C4C2×C3×#4112yes—
259C8—24no—
260C8C2×C3×C22⋊C418yescomplemented
261C2×C3—28no—
262C2×C3#180718yesdirect product factor
263C2×C3C2×#38011yescharacteristic
264C2×C3C22×#10611yescharacteristic
265C2×C3C2×#42711yescharacteristic
266C2×C3C2×#46514yes—
267C22—24no—
268C22—232no—
269C22—24no—
270C22—216no—
271C22C3×#38014yes—
272C22C2×C3×#10614yes—
273C22C2×C3×#10618yescomplemented
274C22C3×#42714yes—
275C22C3×#465116yes—
276C22C22×C3×C4⋊C411yescharacteristic
277C22C2×C3×#12112yes—
278C22C22×C3×SD1612yes—
279C22C2×C3×#13512yes—
280C4—24no—
281C4—88no—
282C4C3×#38014yes—
283C4C22×C3×C22⋊C411yescharacteristic
284C4C2×C3×#11711yescharacteristic
285C4C2×C3×C22⋊Q812yes—
286C3C2×#180711yescharacteristic, direct product factor, Sylow
287C2—28no—
288C2C3×#180718yesdirect product factor
289C2C2×C3×#38011yescharacteristic
290C2C22×C3×#10611yescharacteristic
291C2C2×C3×#42711yescharacteristic
292C2C2×C3×#46514yes—
subgroupG/H
center, socleC24×C3C2×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#106C2semidirect
4. C3×#1807C2direct (N × H)
5. C2×C3×#340C2semidirect
6. C2×#1807C3direct (N × H)
9. C23×C3×C8C4semidirect
16. C2×C3×#106C22semidirect
20. C3×#340C22semidirect
22. C22×#106C2×C3semidirect
24. #1807C2×C3direct (N × H)
25. C2×#340C2×C3semidirect
41. C22×C3×C8C2×C4semidirect
55. C23×C8C3×C4semidirect
62. C2×#106C22×C3semidirect
66. #340C22×C3semidirect
98. C2×C3×C8C22⋊C4semidirect
121. C22×C8C2×C3×C4semidirect
138. C23×C3#106semidirect
167. C3×C8C2×C22⋊C4semidirect
199. C2×C8C3×C22⋊C4semidirect
213. C22×C3C2×#106semidirect
231. C23C3×#106semidirect
260. C8C2×C3×C22⋊C4semidirect
262. C2×C3#1807direct (N × H)
273. C22C2×C3×#106semidirect
286. C3C2×#1807direct (N × H)
288. C2C3×#1807direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C82
C22×C3×#1062
C2×C3×#3802
C3×#18072
C2×C3×#3402
C2×#18073