Finite Group Databasebeta

← C3×#29642 #10648249

Subgroups — C3×#29642 #10648249

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#484C212yesmaximal
3C3×#2855C211yesmaximal, characteristic
4C3×#2057C214yesmaximal, complemented
5C3×#2024C214yesmaximal, complemented
6C3×#1824C212yesmaximal
7C3×#1817C211yesmaximal, characteristic
8#29642C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8—21no—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8C2214yes—
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C3×#484—28no—
18C2×C3×#107C2214yes—
19C2×C3×#103C2214yes—
20C2×C3×C22⋊C8C2212yes—
21C3×#391C2212yes—
22C3×#389C2212yes—
23C3×#387—22no—
24C3×#385—21no—
25C3×#381—21no—
26C3×#380C2211yescharacteristic
27C3×#340C2218yes—
28C3×#336C2214yes—
29C22×C4⋊Q8C2×C311yescharacteristic
30C2×#484C2×C312yes—
31#2855C2×C311yescharacteristic
32#2057C2×C314yescomplemented
33#2024C2×C314yescomplemented
34#1824C2×C312yes—
35#1817C2×C311yescharacteristic
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—48no—
39C22×C3×Q8C2311yescharacteristic
40C23×C3×C4—22no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×C3×C8C2314yes—
45C3×C4⋊Q8—216no—
46C3×C4⋊Q8—216no—
47C3×C22⋊Q8—24no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D418yes—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C4⋊C4C2311yescharacteristic
57C2×C3×C4⋊C4C2314yes—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—24no—
61C2×C3×C42D412yes—
62C2×C3×C42C2312yes—
63C3×#107—28no—
64C3×#103—28no—
65C3×C22⋊C8—24no—
66C3×#95—21no—
67C3×#95—22no—
68C3×#95—22no—
69C3×#95—22no—
70C23×Q8—21no—
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C22×C314yes—
74C2×C22⋊Q8C22×C311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4C22×C312yes—
77C22×C42C22×C311yescharacteristic
78#484—28no—
79C2×#107C22×C314yes—
80C2×#103C22×C314yes—
81C2×C22⋊C8C22×C312yes—
82#391C22×C312yes—
83#389C22×C312yes—
84#387—22no—
85#385—21no—
86#381—21no—
87#380C22×C311yescharacteristic
88#340C22×C318yes—
89#336C22×C314yes—
90C24×C3C2×D411yescharacteristic
91C2×C3×Q8—216no—
92C2×C3×Q8—22no—
93C2×C3×Q8—24no—
94C2×C3×Q8—432no—
95C2×C3×Q8—48no—
96C2×C3×Q8—48no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—28no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—48no—
108C22×C3×C4Q1614yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D414yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2411yescharacteristic
114C2×C3×C8—24no—
115C2×C3×C8—28no—
116C3×C4⋊C4—216no—
117C3×C4⋊C4—28no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4C2×D418yes—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—42no—
126C3×C42—24no—
127C3×C42—24no—
128C22×Q8—28no—
129C22×Q8—24no—
130C22×Q8—48no—
131C22×Q8C23×C311yescharacteristic
132C23×C4—22no—
133C23×C4C3×D411yescharacteristic
134C23×C4C3×D411yescharacteristic
135C23×C4C23×C311yescharacteristic
136C22×C8C23×C314yes—
137C4⋊Q8—216no—
138C4⋊Q8—216no—
139C22⋊Q8—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C3×D418yes—
147C2×C4⋊C4C23×C311yescharacteristic
148C2×C4⋊C4C23×C312yes—
149C2×C4⋊C4C23×C314yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C42—24no—
153C2×C42C3×D412yes—
154C2×C42C23×C312yes—
155#107—28no—
156#103—28no—
157C22⋊C8—24no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3#12012yes—
165C23×C3C2×Q1612yes—
166C23×C3#13712yes—
167C23×C3C22×D411yescharacteristic
168C3×Q8—432no—
169C3×Q8—44no—
170C2×C3×C4—28no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—28no—
177C2×C3×C4—28no—
178C2×C3×C4—28no—
179C2×C3×C4—416no—
180C2×C3×C4—48no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4#12012yes—
189C2×C3×C4C2×Q1614yes—
190C2×C3×C4AGL(1,ℤ/8)14yes—
191C2×C3×C4C22×D412yes—
192C2×C3×C42−1+414yes—
193C3×C8—44no—
194C24C2×C3×D411yescharacteristic
195C2×Q8—216no—
196C2×Q8—22no—
197C2×Q8—24no—
198C2×Q8—432no—
199C2×Q8—48no—
200C2×Q8—48no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—48no—
212C22×C4C3×Q1614yes—
213C22×C4C2×C3×D411yescharacteristic
214C22×C4C2×C3×D414yes—
215C22×C4C2×C3×D412yes—
216C22×C4C2×C3×D412yes—
217C22×C4C24×C311yescharacteristic
218C2×C8—24no—
219C2×C8—28no—
220C4⋊C4—216no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4C2×C3×D418yes—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—24no—
231C42—24no—
232C22×C3—22no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—22no—
236C22×C3#45114yes—
237C22×C3C2×#12011yescharacteristic
238C22×C3#53612yes—
239C22×C3C22×Q1611yescharacteristic
240C22×C3C2×AGL(1,ℤ/8)12yes—
241C22×C3C2×#13711yescharacteristic
242C3×C4—22no—
243C3×C4—24no—
244C3×C4—24no—
245C3×C4—48no—
246C3×C4—81no—
247C3×C4—82no—
248C3×C4—81no—
249C3×C4#53614yes—
250C23—22no—
251C23—22no—
252C23C3×#12012yes—
253C23C2×C3×Q1612yes—
254C23C3×#13712yes—
255C23C22×C3×D411yescharacteristic
256Q8—432no—
257Q8—44no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—416no—
268C2×C4—48no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4C3×#12012yes—
277C2×C4C2×C3×Q1614yes—
278C2×C4C3×AGL(1,ℤ/8)14yes—
279C2×C4C22×C3×D412yes—
280C2×C4C3×2−1+414yes—
281C8—44no—
282C2×C3—24no—
283C2×C3#261411yescharacteristic
284C2×C3C2×#45111yescharacteristic
285C2×C3#300311yescharacteristic
286C2×C3#323012yes—
287C2×C3#323912yes—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22C3×#45114yes—
293C22C2×C3×#12011yescharacteristic
294C22C3×#53612yes—
295C22C22×C3×Q1611yescharacteristic
296C22C2×C3×AGL(1,ℤ/8)12yes—
297C22C2×C3×#13711yescharacteristic
298C4—24no—
299C4—24no—
300C4—22no—
301C4—48no—
302C4—82no—
303C4—81no—
304C4—81no—
305C4C3×#53614yes—
306C3#2964211yescharacteristic, direct product factor, Sylow
307C2—24no—
308C2C3×#261411yescharacteristic
309C2C2×C3×#45111yescharacteristic
310C2C3×#300311yescharacteristic
311C2C3×#323012yes—
312C2C3×#323912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2057C2semidirect
5. C3×#2024C2semidirect
8. #29642C3direct (N × H)
32. #2057C2×C3semidirect
33. #2024C2×C3semidirect
306. C3#29642direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4842
C3×#28552
C3×#20572
C3×#20242
C3×#18242
C3×#18172
#296423