Finite Group Databasebeta

← C2×C3×#2060 #10642165

Subgroups — C2×C3×#2060 #10642165

Order 768.

1218 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#428C212yesmaximal
3C2×C3×#384C211yesmaximal, characteristic
4C3×#2060C218yesmaximal, direct product factor
5C2×C3×#343C211yesmaximal, characteristic
6C2×C3×#339C212yesmaximal
7C2×#2060C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×M16C2212yes—
9C3×#544—24no—
10C3×#544—22no—
11C3×#544C2214yes—
12C2×C3×C4⋊Q8—22no—
13C2×C3×#125—22no—
14C2×C3×C22⋊Q8—22no—
15C2×C3×#117C2212yes—
16C22×C3×C4⋊C4—22no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#428—24no—
21C3×#428C2218yes—
22C2×C3×#107—22no—
23C2×C3×#106—22no—
24C3×#384—22no—
25C3×#384C2214yes—
26C3×#343C2214yes—
27C3×#339C2218yes—
28C2×#544C2×C311yescharacteristic
29C2×#428C2×C312yes—
30C2×#384C2×C311yescharacteristic
31#2060C2×C318yesdirect product factor
32C2×#343C2×C311yescharacteristic
33C2×#339C2×C312yes—
34C22×C3×Q8—22no—
35C23×C3×C4—21no—
36C23×C3×C4—21no—
37C23×C3×C4D412yes—
38C23×C3×C4C2311yescharacteristic
39C2×C3×M16—22no—
40C2×C3×M16—24no—
41C2×C3×M16Q818yescomplemented
42C2×C3×M16C2314yes—
43C22×C3×C8—22no—
44C3×C4⋊Q8—28no—
45C3×C4⋊Q8—28no—
46C3×#125—28no—
47C3×#125—28no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—28no—
50C3×#117—22no—
51C3×#117—22no—
52C3×#117D414yes—
53C3×#117C2314yes—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4—48no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4D412yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C42—21no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42Q814yescomplemented
73C2×C3×C42C2312yes—
74C3×#107—28no—
75C3×#106—28no—
76C22×M16C22×C312yes—
77#544—24no—
78#544—22no—
79#544C22×C314yes—
80C2×C4⋊Q8—22no—
81C2×#125—22no—
82C2×C22⋊Q8—22no—
83C2×#117C22×C312yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4C22×C311yescharacteristic
86C22×C4⋊C4C22×C311yescharacteristic
87C22×C42C22×C311yescharacteristic
88#428—24no—
89#428C22×C318yes—
90C2×#107—22no—
91C2×#106—22no—
92#384—22no—
93#384C22×C314yes—
94#343C22×C314yes—
95#339C22×C318yes—
96C24×C3C2×D411yescharacteristic
97C2×C3×Q8—28no—
98C2×C3×Q8—84no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—44no—
109C22×C3×C4—44no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D414yes—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×Q814yes—
117C22×C3×C4C2×Q812yes—
118C22×C3×C4#4112yes—
119C22×C3×C4C2411yescharacteristic
120C3×M16—28no—
121C3×M16—24no—
122C3×M16C2×Q818yescomplemented
123C2×C3×C8—28no—
124C2×C3×C8—24no—
125C3×C4⋊C4—28no—
126C3×C4⋊C4—216no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—416no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4C2×D414yes—
135C3×C4⋊C4C2×D414yes—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C42—24no—
139C3×C42—24no—
140C3×C42—24no—
141C3×C42—22no—
142C3×C42C2×Q814yescomplemented
143C22×Q8—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C3×D412yes—
147C23×C4C23×C311yescharacteristic
148C2×M16—22no—
149C2×M16—24no—
150C2×M16C3×Q818yescomplemented
151C2×M16C23×C314yes—
152C22×C8—22no—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155#125—28no—
156#125—28no—
157C22⋊Q8—28no—
158C22⋊Q8—28no—
159#117—22no—
160#117—22no—
161#117C3×D414yes—
162#117C23×C314yes—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4C3×D412yes—
173C2×C4⋊C4C3×D412yes—
174C2×C4⋊C4C3×D412yes—
175C2×C4⋊C4C23×C312yes—
176C2×C4⋊C4C23×C312yes—
177C2×C22⋊C4—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42C3×Q814yescomplemented
182C2×C42C23×C312yes—
183#107—28no—
184#106—28no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3#12011yescharacteristic
188C23×C3#12011yescharacteristic
189C23×C3C22⋊Q812yes—
190C23×C3C4⋊Q812yes—
191C23×C3C22×D411yescharacteristic
192C3×Q8—88no—
193C2×C3×C4—24no—
194C2×C3×C4—216no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—42no—
206C2×C3×C4—82no—
207C2×C3×C4—82no—
208C2×C3×C4—82no—
209C2×C3×C4#12012yes—
210C2×C3×C4C22⋊Q814yes—
211C2×C3×C4C22⋊Q814yes—
212C2×C3×C4C22⋊Q812yes—
213C2×C3×C4C4⋊Q814yes—
214C2×C3×C4C4⋊Q814yes—
215C2×C3×C4C4⋊Q812yes—
216C2×C3×C4C22×D412yes—
217C2×C3×C4C22×Q812yes—
218C2×C3×C4C22×Q811yescharacteristic
219C2×C3×C4C2×#4111yescharacteristic
220C3×C8—28no—
221C24C2×C3×D411yescharacteristic
222C2×Q8—28no—
223C2×Q8—84no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C2×C3×D414yes—
238C22×C4C2×C3×D411yescharacteristic
239C22×C4C2×C3×D412yes—
240C22×C4C2×C3×D412yes—
241C22×C4C2×C3×Q814yes—
242C22×C4C2×C3×Q812yes—
243C22×C4C3×#4112yes—
244C22×C4C24×C311yescharacteristic
245M16—28no—
246M16—24no—
247M16C2×C3×Q818yescomplemented
248C2×C8—28no—
249C2×C8—24no—
250C4⋊C4—28no—
251C4⋊C4—216no—
252C4⋊C4—24no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—416no—
258C4⋊C4—42no—
259C4⋊C4C2×C3×D414yes—
260C4⋊C4C2×C3×D414yes—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—22no—
267C42C2×C3×Q814yescomplemented
268C22×C3—24no—
269C22×C3—24no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3#39512yes—
275C22×C3#39512yes—
276C22×C3#45312yes—
277C22×C3#45612yes—
278C22×C3C2×#12011yescharacteristic
279C22×C3C2×C22⋊Q811yescharacteristic
280C22×C3C2×C4⋊Q811yescharacteristic
281C3×C4—28no—
282C3×C4—44no—
283C3×C4—44no—
284C3×C4—44no—
285C3×C4—82no—
286C3×C4#39514yes—
287C3×C4C2×C22⋊Q812yes—
288C3×C4C2×C4⋊Q812yes—
289C23—22no—
290C23—22no—
291C23C3×#12011yescharacteristic
292C23C3×#12011yescharacteristic
293C23C3×C22⋊Q812yes—
294C23C3×C4⋊Q812yes—
295C23C22×C3×D411yescharacteristic
296Q8—88no—
297C2×C4—24no—
298C2×C4—216no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C3×#12012yes—
314C2×C4C3×C22⋊Q812yes—
315C2×C4C3×C22⋊Q814yes—
316C2×C4C3×C22⋊Q814yes—
317C2×C4C3×C4⋊Q812yes—
318C2×C4C3×C4⋊Q814yes—
319C2×C4C3×C4⋊Q814yes—
320C2×C4C22×C3×D412yes—
321C2×C4C22×C3×Q812yes—
322C2×C4C22×C3×Q811yescharacteristic
323C2×C4C2×C3×#4111yescharacteristic
324C8—28no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3#206014yesdirect product factor
329C2×C3C2×#39511yescharacteristic
330C2×C3C2×#45311yescharacteristic
331C2×C3C2×#45611yescharacteristic
332C22—24no—
333C22—24no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22C3×#39512yes—
339C22C3×#39512yes—
340C22C3×#45312yes—
341C22C3×#45612yes—
342C22C2×C3×#12011yescharacteristic
343C22C2×C3×C22⋊Q811yescharacteristic
344C22C2×C3×C4⋊Q811yescharacteristic
345C4—28no—
346C4—44no—
347C4—44no—
348C4—44no—
349C4—82no—
350C4C3×#39514yes—
351C4C2×C3×C22⋊Q812yes—
352C4C2×C3×C4⋊Q812yes—
353C3C2×#206011yescharacteristic, direct product factor, Sylow
354C2—21no—
355C2—22no—
356C2—21no—
357C2C3×#206014yesdirect product factor
358C2C2×C3×#39511yescharacteristic
359C2C2×C3×#45311yescharacteristic
360C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2060C2direct (N × H)
7. C2×#2060C3direct (N × H)
31. #2060C2×C3direct (N × H)
41. C2×C3×M16Q8semidirect
72. C2×C3×C42Q8semidirect
122. C3×M16C2×Q8semidirect
142. C3×C42C2×Q8semidirect
150. C2×M16C3×Q8semidirect
181. C2×C42C3×Q8semidirect
247. M16C2×C3×Q8semidirect
267. C42C2×C3×Q8semidirect
328. C2×C3#2060direct (N × H)
353. C3C2×#2060direct (N × H)
357. C2C3×#2060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#4282
C2×C3×#3842
C3×#20602
C2×C3×#3432
C2×C3×#3392
C2×#20603