Finite Group Databasebeta

← C3×#29653 #10648260

Subgroups — C3×#29653 #10648260

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C3×#3095C212yesmaximal
3C3×#2502C211yesmaximal, characteristic
4C3×#2060C214yesmaximal, complemented
5C3×#2028C214yesmaximal, complemented
6C3×#1826C212yesmaximal
7C3×#1821C211yesmaximal, characteristic
8#29653C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#544—24no—
10C3×#544—22no—
11C3×#544C2214yes—
12C2×C3×C4⋊Q8—22no—
13C2×C3×#125—22no—
14C2×C3×C22⋊Q8—22no—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C3×#485—24no—
19C3×#484—22no—
20C3×#483—22no—
21C3×#428C2214yes—
22C3×#419C2214yes—
23C3×#407C2212yes—
24C3×#398—22no—
25C3×#391C2212yes—
26C3×#385—22no—
27C3×#384C2212yes—
28C3×C4×C22⋊C4C2211yescharacteristic
29C3×#343C2214yes—
30C3×#339C2218yes—
31C2×#544C2×C311yescharacteristic
32#3095C2×C312yes—
33#2502C2×C311yescharacteristic
34#2060C2×C314yescomplemented
35#2028C2×C314yescomplemented
36#1826C2×C312yes—
37#1821C2×C311yescharacteristic
38C22×C3×Q8—22no—
39C22×C3×Q8C2311yescharacteristic
40C23×C3×C4D412yes—
41C23×C3×C4C2311yescharacteristic
42C2×C3×M16C2314yes—
43C3×C4⋊Q8—28no—
44C3×C4⋊Q8—28no—
45C3×#125—28no—
46C3×#125—28no—
47C3×C22⋊Q8—28no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—22no—
50C3×C22⋊Q8—22no—
51C3×#117—22no—
52C3×#117—22no—
53C3×#117D414yes—
54C3×#117C2314yes—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—22no—
70C2×C3×C42C2312yes—
71C3×#107—24no—
72C3×#106—24no—
73C3×#103—24no—
74C3×#103—24no—
75C3×C22⋊C8—22no—
76C3×C22⋊C8—22no—
77C3×#95—21no—
78C3×#95—24no—
79#544—24no—
80#544—22no—
81#544C22×C314yes—
82C2×C4⋊Q8—22no—
83C2×#125—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8C22×C311yescharacteristic
86C2×#117C22×C312yes—
87C22×C4⋊C4C22×C311yescharacteristic
88#485—24no—
89#484—22no—
90#483—22no—
91#428C22×C314yes—
92#419C22×C314yes—
93#407C22×C312yes—
94#398—22no—
95#391C22×C312yes—
96#385—22no—
97#384C22×C312yes—
98C4×C22⋊C4C22×C311yescharacteristic
99#343C22×C314yes—
100#339C22×C318yes—
101C24×C3C2×D411yescharacteristic
102C2×C3×Q8—28no—
103C2×C3×Q8—22no—
104C2×C3×Q8—24no—
105C2×C3×Q8—84no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—82no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D414yes—
119C22×C3×C4C2411yescharacteristic
120C3×M16—28no—
121C2×C3×C8—24no—
122C3×C4⋊C4—28no—
123C3×C4⋊C4—216no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—84no—
131C3×C4⋊C4C2×D414yes—
132C3×C4⋊C4C2×D414yes—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—24no—
139C3×C42—24no—
140C3×C42—24no—
141C22×Q8—22no—
142C22×Q8C23×C311yescharacteristic
143C23×C4C3×D412yes—
144C23×C4C23×C311yescharacteristic
145C2×M16C23×C314yes—
146C4⋊Q8—28no—
147C4⋊Q8—28no—
148#125—28no—
149#125—28no—
150C22⋊Q8—28no—
151C22⋊Q8—28no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154#117—22no—
155#117—22no—
156#117C3×D414yes—
157#117C23×C314yes—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C3×D412yes—
165C2×C4⋊C4C3×D412yes—
166C2×C4⋊C4C3×D412yes—
167C2×C4⋊C4C23×C311yescharacteristic
168C2×C4⋊C4C23×C312yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—22no—
173C2×C42C23×C312yes—
174#107—24no—
175#106—24no—
176#103—24no—
177#103—24no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180#95—21no—
181#95—24no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—41no—
185C23×C3#12011yescharacteristic
186C23×C3#12011yescharacteristic
187C23×C3C22×D411yescharacteristic
188C3×Q8—44no—
189C3×Q8—88no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4#12012yes—
209C2×C3×C4C22×D412yes—
210C2×C3×C42−1+414yes—
211C3×C8—44no—
212C24C2×C3×D411yescharacteristic
213C2×Q8—28no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—84no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—82no—
226C22×C4C2×C3×D411yescharacteristic
227C22×C4C2×C3×D412yes—
228C22×C4C2×C3×D412yes—
229C22×C4C2×C3×D414yes—
230C22×C4C24×C311yescharacteristic
231M16—28no—
232C2×C8—24no—
233C4⋊C4—28no—
234C4⋊C4—216no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4C2×C3×D414yes—
243C4⋊C4C2×C3×D414yes—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C42—24no—
250C42—24no—
251C42—24no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—44no—
260C22×C3#45612yes—
261C22×C3C2×#12011yescharacteristic
262C22×C3#53612yes—
263C3×C4—22no—
264C3×C4—44no—
265C3×C4—44no—
266C3×C4—44no—
267C3×C4—44no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4#53614yes—
271C23—22no—
272C23—22no—
273C23—41no—
274C23C3×#12011yescharacteristic
275C23C3×#12011yescharacteristic
276C23C22×C3×D411yescharacteristic
277Q8—44no—
278Q8—88no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4C3×#12012yes—
298C2×C4C22×C3×D412yes—
299C2×C4C3×2−1+414yes—
300C8—44no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—41no—
304C2×C3#261411yescharacteristic
305C2×C3C2×#45611yescharacteristic
306C2×C3#301911yescharacteristic
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—44no—
315C22C3×#45612yes—
316C22C2×C3×#12011yescharacteristic
317C22C3×#53612yes—
318C4—22no—
319C4—44no—
320C4—44no—
321C4—44no—
322C4—44no—
323C4—81no—
324C4—81no—
325C4C3×#53614yes—
326C3#2965311yescharacteristic, direct product factor, Sylow
327C2—22no—
328C2—22no—
329C2—41no—
330C2C3×#261411yescharacteristic
331C2C2×C3×#45611yescharacteristic
332C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2060C2semidirect
5. C3×#2028C2semidirect
8. #29653C3direct (N × H)
34. #2060C2×C3semidirect
35. #2028C2×C3semidirect
326. C3#29653direct (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
C3×#30952
C3×#25022
C3×#20602
C3×#20282
C3×#18262
C3×#18212
#296533