Finite Group Databasebeta

← C2×C3×#2061 #10642166

Subgroups — C2×C3×#2061 #10642166

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 358 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×#382C211yesmaximal, characteristic
4C3×#2061C218yesmaximal, direct product factor
5C2×C3×#343C211yesmaximal, characteristic
6C2×C3×#339C212yesmaximal
7C2×#2061C311yesmaximal, 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⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#428—24no—
20C3×#428C2218yes—
21C2×C3×#107—22no—
22C2×C3×#106—22no—
23C3×#382—22no—
24C3×#382C2214yes—
25C2×C3×#95—22no—
26C3×#343C2214yes—
27C3×#339C2218yes—
28C2×#544C2×C311yescharacteristic
29C2×#428C2×C312yes—
30C2×#382C2×C311yescharacteristic
31#2061C2×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⋊C4D412yes—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—21no—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C3×C42Q814yescomplemented
71C2×C3×C42C2312yes—
72C3×#107—28no—
73C3×#106—28no—
74C3×#95—28no—
75C22×M16C22×C312yes—
76#544—24no—
77#544—22no—
78#544C22×C314yes—
79C2×C4⋊Q8—22no—
80C2×#125—22no—
81C2×C22⋊Q8—22no—
82C2×#117C22×C312yes—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C4⋊C4C22×C311yescharacteristic
85C22×C42C22×C311yescharacteristic
86#428—24no—
87#428C22×C318yes—
88C2×#107—22no—
89C2×#106—22no—
90#382—22no—
91#382C22×C314yes—
92C2×#95—22no—
93#343C22×C314yes—
94#339C22×C318yes—
95C24×C3C2×D411yescharacteristic
96C2×C3×Q8—28no—
97C2×C3×Q8—84no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—44no—
108C22×C3×C4—44no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D414yes—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×Q814yes—
116C22×C3×C4C2×Q812yes—
117C22×C3×C4#4112yes—
118C22×C3×C4C2411yescharacteristic
119C3×M16—28no—
120C3×M16—24no—
121C3×M16C2×Q818yescomplemented
122C2×C3×C8—28no—
123C2×C3×C8—24no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—216no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4C2×D414yes—
133C3×C4⋊C4C2×D414yes—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C42—24no—
137C3×C42—24no—
138C3×C42—24no—
139C3×C42—22no—
140C3×C42C2×Q814yescomplemented
141C22×Q8—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C3×D412yes—
145C23×C4C23×C311yescharacteristic
146C2×M16—22no—
147C2×M16—24no—
148C2×M16C3×Q818yescomplemented
149C2×M16C23×C314yes—
150C22×C8—22no—
151C4⋊Q8—28no—
152C4⋊Q8—28no—
153#125—28no—
154#125—28no—
155C22⋊Q8—28no—
156C22⋊Q8—28no—
157#117—22no—
158#117—22no—
159#117C3×D414yes—
160#117C23×C314yes—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C3×D412yes—
169C2×C4⋊C4C3×D412yes—
170C2×C4⋊C4C3×D412yes—
171C2×C4⋊C4C23×C312yes—
172C2×C4⋊C4C23×C312yes—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42C3×Q814yescomplemented
178C2×C42C23×C312yes—
179#107—28no—
180#106—28no—
181#95—28no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3#12011yescharacteristic
185C23×C3#12011yescharacteristic
186C23×C3C22⋊Q812yes—
187C23×C3C4⋊Q812yes—
188C23×C3C22×D411yescharacteristic
189C3×Q8—88no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—48no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4—82no—
205C2×C3×C4—82no—
206C2×C3×C4#12012yes—
207C2×C3×C4C22⋊Q814yes—
208C2×C3×C4C22⋊Q814yes—
209C2×C3×C4C22⋊Q812yes—
210C2×C3×C4C4⋊Q814yes—
211C2×C3×C4C4⋊Q814yes—
212C2×C3×C4C4⋊Q812yes—
213C2×C3×C4C22×D412yes—
214C2×C3×C4C22×Q812yes—
215C2×C3×C4C22×Q811yescharacteristic
216C2×C3×C4C2×#4111yescharacteristic
217C3×C8—28no—
218C24C2×C3×D411yescharacteristic
219C2×Q8—28no—
220C2×Q8—84no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C2×C3×D414yes—
235C22×C4C2×C3×D411yescharacteristic
236C22×C4C2×C3×D412yes—
237C22×C4C2×C3×D412yes—
238C22×C4C2×C3×Q814yes—
239C22×C4C2×C3×Q812yes—
240C22×C4C3×#4112yes—
241C22×C4C24×C311yescharacteristic
242M16—28no—
243M16—24no—
244M16C2×C3×Q818yescomplemented
245C2×C8—28no—
246C2×C8—24no—
247C4⋊C4—28no—
248C4⋊C4—216no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4C2×C3×D414yes—
256C4⋊C4C2×C3×D414yes—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C42—24no—
260C42—24no—
261C42—24no—
262C42—22no—
263C42C2×C3×Q814yescomplemented
264C22×C3—24no—
265C22×C3—24no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39512yes—
271C22×C3#39512yes—
272C22×C3#45412yes—
273C22×C3#45512yes—
274C22×C3C2×#12011yescharacteristic
275C22×C3C2×C22⋊Q811yescharacteristic
276C22×C3C2×C4⋊Q811yescharacteristic
277C3×C4—44no—
278C3×C4—44no—
279C3×C4—44no—
280C3×C4—44no—
281C3×C4—82no—
282C3×C4#39514yes—
283C3×C4C2×C22⋊Q812yes—
284C3×C4C2×C4⋊Q812yes—
285C23—22no—
286C23—22no—
287C23C3×#12011yescharacteristic
288C23C3×#12011yescharacteristic
289C23C3×C22⋊Q812yes—
290C23C3×C4⋊Q812yes—
291C23C22×C3×D411yescharacteristic
292Q8—88no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—48no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C3×#12012yes—
310C2×C4C3×C22⋊Q812yes—
311C2×C4C3×C22⋊Q814yes—
312C2×C4C3×C22⋊Q814yes—
313C2×C4C3×C4⋊Q812yes—
314C2×C4C3×C4⋊Q814yes—
315C2×C4C3×C4⋊Q814yes—
316C2×C4C22×C3×D412yes—
317C2×C4C22×C3×Q812yes—
318C2×C4C22×C3×Q811yescharacteristic
319C2×C4C2×C3×#4111yescharacteristic
320C8—28no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—22no—
324C2×C3#206114yesdirect product factor
325C2×C3C2×#39511yescharacteristic
326C2×C3C2×#45511yescharacteristic
327C2×C3C2×#45411yescharacteristic
328C22—24no—
329C22—24no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22C3×#39512yes—
335C22C3×#39512yes—
336C22C3×#45412yes—
337C22C3×#45512yes—
338C22C2×C3×#12011yescharacteristic
339C22C2×C3×C22⋊Q811yescharacteristic
340C22C2×C3×C4⋊Q811yescharacteristic
341C4—44no—
342C4—44no—
343C4—44no—
344C4—44no—
345C4—82no—
346C4C3×#39514yes—
347C4C2×C3×C22⋊Q812yes—
348C4C2×C3×C4⋊Q812yes—
349C3C2×#206111yescharacteristic, direct product factor, Sylow
350C2—21no—
351C2—22no—
352C2—21no—
353C2C3×#206114yesdirect product factor
354C2C2×C3×#39511yescharacteristic
355C2C2×C3×#45511yescharacteristic
356C2C2×C3×#45411yescharacteristic
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×#2061C2direct (N × H)
7. C2×#2061C3direct (N × H)
31. #2061C2×C3direct (N × H)
41. C2×C3×M16Q8semidirect
70. C2×C3×C42Q8semidirect
121. C3×M16C2×Q8semidirect
140. C3×C42C2×Q8semidirect
148. C2×M16C3×Q8semidirect
177. C2×C42C3×Q8semidirect
244. M16C2×C3×Q8semidirect
263. C42C2×C3×Q8semidirect
324. C2×C3#2061direct (N × H)
349. C3C2×#2061direct (N × H)
353. C2C3×#2061direct (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×#3822
C3×#20612
C2×C3×#3432
C2×C3×#3392
C2×#20613