Finite Group Databasebeta

← C3×#29652 #10648259

Subgroups — C3×#29652 #10648259

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C3×#3095C212yesmaximal
3C3×#2503C211yesmaximal, characteristic
4C3×#2061C214yesmaximal, complemented
5C3×#2028C214yesmaximal, complemented
6C3×#1826C212yesmaximal
7C3×#1821C211yesmaximal, characteristic
8#29652C311yesmaximal, 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×#400—22no—
25C3×#391C2212yes—
26C3×#382C2212yes—
27C3×#381—22no—
28C3×C4×C22⋊C4C2211yescharacteristic
29C3×#343C2214yes—
30C3×#339C2218yes—
31C2×#544C2×C311yescharacteristic
32#3095C2×C312yes—
33#2503C2×C311yescharacteristic
34#2061C2×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⋊C4D412yes—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C42—22no—
69C2×C3×C42C2312yes—
70C3×#107—24no—
71C3×#106—24no—
72C3×#103—24no—
73C3×#103—24no—
74C3×C22⋊C8—22no—
75C3×C22⋊C8—22no—
76C3×#95—21no—
77C3×#95—24no—
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#400—22no—
95#391C22×C312yes—
96#382C22×C312yes—
97#381—22no—
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—44no—
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⋊C4C2×D414yes—
131C3×C4⋊C4C2×D414yes—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—24no—
138C3×C42—24no—
139C3×C42—24no—
140C22×Q8—22no—
141C22×Q8C23×C311yescharacteristic
142C23×C4C3×D412yes—
143C23×C4C23×C311yescharacteristic
144C2×M16C23×C314yes—
145C4⋊Q8—28no—
146C4⋊Q8—28no—
147#125—28no—
148#125—28no—
149C22⋊Q8—28no—
150C22⋊Q8—28no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153#117—22no—
154#117—22no—
155#117C3×D414yes—
156#117C23×C314yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C3×D412yes—
163C2×C4⋊C4C3×D412yes—
164C2×C4⋊C4C3×D412yes—
165C2×C4⋊C4C23×C311yescharacteristic
166C2×C4⋊C4C23×C312yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—22no—
171C2×C42C23×C312yes—
172#107—24no—
173#106—24no—
174#103—24no—
175#103—24no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178#95—21no—
179#95—24no—
180#95—24no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—41no—
184C23×C3#12011yescharacteristic
185C23×C3#12011yescharacteristic
186C23×C3C22×D411yescharacteristic
187C3×Q8—44no—
188C3×Q8—88no—
189C2×C3×C4—22no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—44no—
196C2×C3×C4—44no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—82no—
207C2×C3×C4#12012yes—
208C2×C3×C4C22×D412yes—
209C2×C3×C42−1+414yes—
210C3×C8—44no—
211C24C2×C3×D411yescharacteristic
212C2×Q8—28no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—84no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—44no—
225C22×C4C2×C3×D411yescharacteristic
226C22×C4C2×C3×D412yes—
227C22×C4C2×C3×D412yes—
228C22×C4C2×C3×D414yes—
229C22×C4C24×C311yescharacteristic
230M16—28no—
231C2×C8—24no—
232C4⋊C4—28no—
233C4⋊C4—216no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4C2×C3×D414yes—
241C4⋊C4C2×C3×D414yes—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—42no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C42—24no—
248C42—24no—
249C42—24no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—44no—
258C22×C3#45512yes—
259C22×C3C2×#12011yescharacteristic
260C22×C3#53612yes—
261C3×C4—22no—
262C3×C4—44no—
263C3×C4—44no—
264C3×C4—44no—
265C3×C4—82no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4#53614yes—
269C23—22no—
270C23—22no—
271C23—41no—
272C23C3×#12011yescharacteristic
273C23C3×#12011yescharacteristic
274C23C22×C3×D411yescharacteristic
275Q8—44no—
276Q8—88no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4C3×#12012yes—
296C2×C4C22×C3×D412yes—
297C2×C4C3×2−1+414yes—
298C8—44no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3—41no—
302C2×C3#261411yescharacteristic
303C2×C3C2×#45511yescharacteristic
304C2×C3#302011yescharacteristic
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22C3×#45512yes—
314C22C2×C3×#12011yescharacteristic
315C22C3×#53612yes—
316C4—22no—
317C4—44no—
318C4—44no—
319C4—44no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4C3×#53614yes—
324C3#2965211yescharacteristic, direct product factor, Sylow
325C2—22no—
326C2—22no—
327C2—41no—
328C2C3×#261411yescharacteristic
329C2C2×C3×#45511yescharacteristic
330C2C3×#302011yescharacteristic
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×#2061C2semidirect
5. C3×#2028C2semidirect
8. #29652C3direct (N × H)
34. #2061C2×C3semidirect
35. #2028C2×C3semidirect
324. C3#29652direct (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×#25032
C3×#20612
C3×#20282
C3×#18262
C3×#18212
#296523