Finite Group Databasebeta

← C3×#29651 #10648258

Subgroups — C3×#29651 #10648258

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal
3C3×#2504C211yesmaximal, characteristic
4C3×#2061C214yesmaximal, complemented
5C3×#2027C214yesmaximal, complemented
6C3×#1826C212yesmaximal, complemented
7C3×#1821C211yesmaximal, characteristic, complemented
8#29651C311yesmaximal, 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×#121C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C3×#482—24no—
19C3×#481—22no—
20C3×#480—22no—
21C3×#428C2214yes—
22C3×#418C2214yes—
23C3×#407C2212yes—
24C3×#397—22no—
25C3×#390C2212yes—
26C3×#386—22no—
27C3×#382C2212yes—
28C3×C4×C22⋊C4C2211yescharacteristic
29C3×#343C2214yes—
30C3×#339C2218yes—
31C2×#544C2×C311yescharacteristic, complemented
32#3094C2×C312yes—
33#2504C2×C311yescharacteristic
34#2061C2×C314yescomplemented
35#2027C2×C314yescomplemented
36#1826C2×C312yescomplemented
37#1821C2×C311yescharacteristic, complemented
38C22×C3×Q8—22no—
39C22×C3×D4—21no—
40C22×C3×D4C2311yescharacteristic
41C23×C3×C4D412yes—
42C23×C3×C4C2311yescharacteristic
43C2×C3×M16C2314yes—
44C3×C4⋊Q8—28no—
45C3×C4⋊Q8—28no—
46C3×#125—28no—
47C3×#125—28no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—28no—
50C3×#121—22no—
51C3×#121—22no—
52C3×#117—22no—
53C3×#117—22no—
54C3×#117D414yes—
55C3×#117C2314yes—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
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×C22⋊C4—24no—
69C2×C3×C42—22no—
70C2×C3×C42C2312yes—
71C3×#107—24no—
72C3×#106—24no—
73C3×#102—24no—
74C3×#102—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×#121C22×C311yescharacteristic
86C2×#117C22×C312yes—
87C22×C4⋊C4C22×C311yescharacteristic
88#482—24no—
89#481—22no—
90#480—22no—
91#428C22×C314yes—
92#418C22×C314yes—
93#407C22×C312yes—
94#397—22no—
95#390C22×C312yes—
96#386—22no—
97#382C22×C312yes—
98C4×C22⋊C4C22×C311yescharacteristic
99#343C22×C314yes—
100#339C22×C318yes—
101C24×C3—21no—
102C24×C3C2×D411yescharacteristic
103C2×C3×Q8—28no—
104C2×C3×Q8—84no—
105C2×C3×D4—22no—
106C2×C3×D4—24no—
107C2×C3×D4—44no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—82no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D414yes—
122C22×C3×C4C2411yescharacteristic
123C3×M16—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—22no—
132C3×C4⋊C4C2×D414yes—
133C3×C4⋊C4C2×D414yes—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—42no—
138C3×C22⋊C4—84no—
139C3×C42—24no—
140C3×C42—24no—
141C3×C42—24no—
142C22×Q8—22no—
143C22×D4—21no—
144C22×D4C23×C311yescharacteristic
145C23×C4C3×D412yes—
146C23×C4C23×C311yescharacteristic
147C2×M16C23×C314yes—
148C4⋊Q8—28no—
149C4⋊Q8—28no—
150#125—28no—
151#125—28no—
152C22⋊Q8—28no—
153C22⋊Q8—28no—
154#121—22no—
155#121—22no—
156#117—22no—
157#117—22no—
158#117C3×D414yes—
159#117C23×C314yes—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
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×C22⋊C4—24no—
173C2×C42—22no—
174C2×C42C23×C312yes—
175#107—24no—
176#106—24no—
177#102—24no—
178#102—24no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—21no—
182#95—24no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—41no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—41no—
189C23×C3—42no—
190C23×C3#12011yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3C22×D411yescharacteristic
193C3×Q8—88no—
194C3×D4—44no—
195C3×D4—82no—
196C3×D4—82no—
197C2×C3×C4—22no—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4#12012yes—
214C2×C3×C4C22×D412yes—
215C2×C3×C42−1+414yes—
216C3×C8—44no—
217C24—21no—
218C24C2×C3×D411yescharacteristic
219C2×Q8—28no—
220C2×Q8—84no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—44no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—82no—
234C22×C4C2×C3×D411yescharacteristic
235C22×C4C2×C3×D412yes—
236C22×C4C2×C3×D412yes—
237C22×C4C2×C3×D414yes—
238C22×C4C24×C311yescharacteristic
239M16—28no—
240C2×C8—24no—
241C4⋊C4—28no—
242C4⋊C4—216no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4C2×C3×D414yes—
249C4⋊C4C2×C3×D414yes—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—84no—
255C42—24no—
256C42—24no—
257C42—24no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—44no—
266C22×C3—84no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3—81no—
270C22×C3#45412yes—
271C22×C3C2×#12011yescharacteristic
272C22×C3#53612yes—
273C3×C4—22no—
274C3×C4—44no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—82no—
278C3×C4—81no—
279C3×C4#53614yes—
280C23—22no—
281C23—22no—
282C23—41no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—42no—
287C23C3×#12011yescharacteristic
288C23C3×#12011yescharacteristic
289C23C22×C3×D411yescharacteristic
290Q8—88no—
291D4—44no—
292D4—82no—
293D4—82no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4C3×#12012yes—
311C2×C4C22×C3×D412yes—
312C2×C4C3×2−1+414yes—
313C8—44no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—41no—
317C2×C3—161no—
318C2×C3#261411yescharacteristic
319C2×C3C2×#45411yescharacteristic
320C2×C3#301911yescharacteristic
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—84no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C3×#45412yes—
334C22C2×C3×#12011yescharacteristic
335C22C3×#53612yes—
336C4—22no—
337C4—44no—
338C4—44no—
339C4—44no—
340C4—82no—
341C4—81no—
342C4C3×#53614yes—
343C3#2965111yescharacteristic, direct product factor, Sylow
344C2—22no—
345C2—22no—
346C2—41no—
347C2—161no—
348C2C3×#261411yescharacteristic
349C2C2×C3×#45411yescharacteristic
350C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#544C2semidirect
4. C3×#2061C2semidirect
5. C3×#2027C2semidirect
6. C3×#1826C2semidirect
7. C3×#1821C2semidirect
8. #29651C3direct (N × H)
31. C2×#544C2×C3semidirect
34. #2061C2×C3semidirect
35. #2027C2×C3semidirect
36. #1826C2×C3semidirect
37. #1821C2×C3semidirect
343. C3#29651direct (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×#30942
C3×#25042
C3×#20612
C3×#20272
C3×#18262
C3×#18212
#296513