Finite Group Databasebeta

← C3×#29650 #10648257

Subgroups — C3×#29650 #10648257

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal
3C3×#2501C211yesmaximal, characteristic
4C3×#2060C214yesmaximal, complemented
5C3×#2027C214yesmaximal, complemented
6C3×#1826C212yesmaximal, complemented
7C3×#1821C211yesmaximal, characteristic, complemented
8#29650C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#544—24no—
10C3×#544—22no—
11C3×#544C2214yes—
12C2×C3×C4⋊Q8—22no—
13C2×C3×#125—22no—
14C2×C3×#123—22no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×#121C2211yescharacteristic
17C2×C3×#117C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#482—24no—
20C3×#481—22no—
21C3×#480—22no—
22C3×#428C2214yes—
23C3×#418C2214yes—
24C3×#407C2212yes—
25C3×#399—22no—
26C3×#390C2212yes—
27C3×#384C2212yes—
28C3×C4×C22⋊C4C2211yescharacteristic
29C3×#343C2214yes—
30C3×#339C2218yes—
31C2×#544C2×C311yescharacteristic, complemented
32#3094C2×C312yes—
33#2501C2×C311yescharacteristic
34#2060C2×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×#123—48no—
49C3×C22⋊Q8—28no—
50C3×C22⋊Q8—28no—
51C3×#121—22no—
52C3×#121—22no—
53C3×#117—22no—
54C3×#117—22no—
55C3×#117D414yes—
56C3×#117C2314yes—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42—22no—
72C2×C3×C42C2312yes—
73C3×#107—24no—
74C3×#106—24no—
75C3×#102—24no—
76C3×#102—24no—
77C3×C22⋊C8—22no—
78C3×C22⋊C8—22no—
79C3×#95—21no—
80#544—24no—
81#544—22no—
82#544C22×C314yes—
83C2×C4⋊Q8—22no—
84C2×#125—22no—
85C2×#123—22no—
86C2×C22⋊Q8—22no—
87C2×#121C22×C311yescharacteristic
88C2×#117C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90#482—24no—
91#481—22no—
92#480—22no—
93#428C22×C314yes—
94#418C22×C314yes—
95#407C22×C312yes—
96#399—22no—
97#390C22×C312yes—
98#384C22×C312yes—
99C4×C22⋊C4C22×C311yescharacteristic
100#343C22×C314yes—
101#339C22×C318yes—
102C24×C3—21no—
103C24×C3C2×D411yescharacteristic
104C2×C3×Q8—28no—
105C2×C3×Q8—84no—
106C2×C3×D4—22no—
107C2×C3×D4—24no—
108C2×C3×D4—44no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C22×C3×C4—24no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—44no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D414yes—
123C22×C3×C4C2411yescharacteristic
124C3×M16—28no—
125C2×C3×C8—24no—
126C3×C4⋊C4—28no—
127C3×C4⋊C4—216no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—48no—
134C3×C4⋊C4C2×D414yes—
135C3×C4⋊C4C2×D414yes—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—48no—
140C3×C22⋊C4—42no—
141C3×C42—24no—
142C3×C42—24no—
143C3×C42—24no—
144C22×Q8—22no—
145C22×D4—21no—
146C22×D4C23×C311yescharacteristic
147C23×C4C3×D412yes—
148C23×C4C23×C311yescharacteristic
149C2×M16C23×C314yes—
150C4⋊Q8—28no—
151C4⋊Q8—28no—
152#125—28no—
153#125—28no—
154#123—48no—
155C22⋊Q8—28no—
156C22⋊Q8—28no—
157#121—22no—
158#121—22no—
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—24no—
168C2×C4⋊C4C3×D412yes—
169C2×C4⋊C4C3×D412yes—
170C2×C4⋊C4C3×D412yes—
171C2×C4⋊C4C23×C311yescharacteristic
172C2×C4⋊C4C23×C312yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C42—22no—
178C2×C42C23×C312yes—
179#107—24no—
180#106—24no—
181#102—24no—
182#102—24no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185#95—21no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—41no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—42no—
193C23×C3#12011yescharacteristic
194C23×C3#12011yescharacteristic
195C23×C3C22×D411yescharacteristic
196C3×Q8—88no—
197C3×D4—44no—
198C3×D4—82no—
199C3×D4—82no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—42no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—82no—
216C2×C3×C4#12012yes—
217C2×C3×C4C22×D412yes—
218C2×C3×C42−1+414yes—
219C3×C8—44no—
220C24—21no—
221C24C2×C3×D411yescharacteristic
222C2×Q8—28no—
223C2×Q8—84no—
224C2×D4—22no—
225C2×D4—24no—
226C2×D4—44no—
227C2×D4—41no—
228C2×D4—41no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—44no—
237C22×C4C2×C3×D411yescharacteristic
238C22×C4C2×C3×D412yes—
239C22×C4C2×C3×D412yes—
240C22×C4C2×C3×D414yes—
241C22×C4C24×C311yescharacteristic
242M16—28no—
243C2×C8—24no—
244C4⋊C4—28no—
245C4⋊C4—216no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—48no—
252C4⋊C4C2×C3×D414yes—
253C4⋊C4C2×C3×D414yes—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—48no—
258C22⋊C4—42no—
259C42—24no—
260C42—24no—
261C42—24no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—44no—
270C22×C3—84no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3#45312yes—
275C22×C3C2×#12011yescharacteristic
276C22×C3#53612yes—
277C3×C4—22no—
278C3×C4—44no—
279C3×C4—44no—
280C3×C4—44no—
281C3×C4—44no—
282C3×C4—81no—
283C3×C4#53614yes—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—41no—
290C23—42no—
291C23C3×#12011yescharacteristic
292C23C3×#12011yescharacteristic
293C23C22×C3×D411yescharacteristic
294Q8—88no—
295D4—44no—
296D4—82no—
297D4—82no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4C3×#12012yes—
315C2×C4C22×C3×D412yes—
316C2×C4C3×2−1+414yes—
317C8—44no—
318C2×C3—22no—
319C2×C3—22no—
320C2×C3—41no—
321C2×C3—161no—
322C2×C3#261411yescharacteristic
323C2×C3C2×#45311yescharacteristic
324C2×C3#302011yescharacteristic
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—44no—
333C22—84no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C3×#45312yes—
338C22C2×C3×#12011yescharacteristic
339C22C3×#53612yes—
340C4—22no—
341C4—44no—
342C4—44no—
343C4—44no—
344C4—44no—
345C4—81no—
346C4C3×#53614yes—
347C3#2965011yescharacteristic, direct product factor, Sylow
348C2—22no—
349C2—22no—
350C2—41no—
351C2—161no—
352C2C3×#261411yescharacteristic
353C2C2×C3×#45311yescharacteristic
354C2C3×#302011yescharacteristic
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×#2060C2semidirect
5. C3×#2027C2semidirect
6. C3×#1826C2semidirect
7. C3×#1821C2semidirect
8. #29650C3direct (N × H)
31. C2×#544C2×C3semidirect
34. #2060C2×C3semidirect
35. #2027C2×C3semidirect
36. #1826C2×C3semidirect
37. #1821C2×C3semidirect
347. C3#29650direct (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×#25012
C3×#20602
C3×#20272
C3×#18262
C3×#18212
#296503