Finite Group Databasebeta

← C3×#29641 #10648248

Subgroups — C3×#29641 #10648248

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#483C212yesmaximal
3C3×#2855C211yesmaximal, characteristic
4C3×#2058C214yesmaximal, complemented
5C3×#2024C214yesmaximal, complemented
6C3×#1823C212yesmaximal
7C3×#1817C211yesmaximal, characteristic
8#29641C311yesmaximal, 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×#483—28no—
18C2×C3×#106C2214yes—
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×#483C2×C312yes—
31#2855C2×C311yescharacteristic
32#2058C2×C314yescomplemented
33#2024C2×C314yescomplemented
34#1823C2×C312yes—
35#1817C2×C311yescharacteristic
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—44no—
39C22×C3×Q8—44no—
40C22×C3×Q8C2311yescharacteristic
41C23×C3×C4—22no—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×C3×C8C2314yes—
46C3×C4⋊Q8—216no—
47C3×C4⋊Q8—216no—
48C3×C22⋊Q8—24no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D418yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2311yescharacteristic
58C2×C3×C4⋊C4C2314yes—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—24no—
62C2×C3×C42D412yes—
63C2×C3×C42C2312yes—
64C3×#106—28no—
65C3×#103—28no—
66C3×C22⋊C8—24no—
67C3×#95—21no—
68C3×#95—22no—
69C3×#95—22no—
70C3×#95—22no—
71C23×Q8—21no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C22×C314yes—
75C2×C22⋊Q8C22×C311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4C22×C312yes—
78C22×C42C22×C311yescharacteristic
79#483—28no—
80C2×#106C22×C314yes—
81C2×#103C22×C314yes—
82C2×C22⋊C8C22×C312yes—
83#391C22×C312yes—
84#389C22×C312yes—
85#387—22no—
86#385—21no—
87#381—21no—
88#380C22×C311yescharacteristic
89#340C22×C318yes—
90#336C22×C314yes—
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—216no—
93C2×C3×Q8—22no—
94C2×C3×Q8—24no—
95C2×C3×Q8—48no—
96C2×C3×Q8—432no—
97C2×C3×Q8—48no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—28no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—44no—
109C22×C3×C4—44no—
110C22×C3×C4SD1614yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D414yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2411yescharacteristic
116C2×C3×C8—24no—
117C2×C3×C8—28no—
118C3×C4⋊C4—216no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—44no—
125C3×C4⋊C4C2×D418yes—
126C3×C22⋊C4—42no—
127C3×C22⋊C4—42no—
128C3×C42—24no—
129C3×C42—24no—
130C22×Q8—28no—
131C22×Q8—24no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8C23×C311yescharacteristic
135C23×C4—22no—
136C23×C4C3×D411yescharacteristic
137C23×C4C3×D411yescharacteristic
138C23×C4C23×C311yescharacteristic
139C22×C8C23×C314yes—
140C4⋊Q8—216no—
141C4⋊Q8—216no—
142C22⋊Q8—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C3×D418yes—
150C2×C4⋊C4C23×C311yescharacteristic
151C2×C4⋊C4C23×C312yes—
152C2×C4⋊C4C23×C314yes—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C42—24no—
156C2×C42C3×D412yes—
157C2×C42C23×C312yes—
158#106—28no—
159#103—28no—
160C22⋊C8—24no—
161#95—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3#12012yes—
168C23×C3C2×SD1612yes—
169C23×C3AGL(1,ℤ/8)12yes—
170C23×C3C22×D411yescharacteristic
171C3×Q8—432no—
172C3×Q8—44no—
173C2×C3×C4—28no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—28no—
180C2×C3×C4—28no—
181C2×C3×C4—28no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—416no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4#12012yes—
193C2×C3×C4C2×SD1614yes—
194C2×C3×C4#13714yes—
195C2×C3×C4C22×D412yes—
196C2×C3×C42−1+414yes—
197C3×C8—44no—
198C24C2×C3×D411yescharacteristic
199C2×Q8—216no—
200C2×Q8—22no—
201C2×Q8—24no—
202C2×Q8—48no—
203C2×Q8—432no—
204C2×Q8—48no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4C3×SD1614yes—
218C22×C4C2×C3×D411yescharacteristic
219C22×C4C2×C3×D414yes—
220C22×C4C2×C3×D412yes—
221C22×C4C2×C3×D412yes—
222C22×C4C24×C311yescharacteristic
223C2×C8—24no—
224C2×C8—28no—
225C4⋊C4—216no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4C2×C3×D418yes—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—24no—
236C42—24no—
237C22×C3—22no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3#45014yes—
242C22×C3C2×#12011yescharacteristic
243C22×C3#53612yes—
244C22×C3C22×SD1611yescharacteristic
245C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C3C2×#13712yes—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—48no—
251C3×C4—81no—
252C3×C4—82no—
253C3×C4—81no—
254C3×C4#53614yes—
255C23—22no—
256C23—22no—
257C23C3×#12012yes—
258C23C2×C3×SD1612yes—
259C23C3×AGL(1,ℤ/8)12yes—
260C23C22×C3×D411yescharacteristic
261Q8—432no—
262Q8—44no—
263C2×C4—28no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—416no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4C3×#12012yes—
283C2×C4C2×C3×SD1614yes—
284C2×C4C3×#13714yes—
285C2×C4C22×C3×D412yes—
286C2×C4C3×2−1+414yes—
287C8—44no—
288C2×C3—24no—
289C2×C3#261411yescharacteristic
290C2×C3C2×#45011yescharacteristic
291C2×C3#300411yescharacteristic
292C2×C3#322912yes—
293C2×C3#323812yes—
294C22—24no—
295C22—24no—
296C22—22no—
297C22—22no—
298C22C3×#45014yes—
299C22C2×C3×#12011yescharacteristic
300C22C3×#53612yes—
301C22C22×C3×SD1611yescharacteristic
302C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
303C22C2×C3×#13712yes—
304C4—24no—
305C4—24no—
306C4—22no—
307C4—48no—
308C4—82no—
309C4—81no—
310C4—81no—
311C4C3×#53614yes—
312C3#2964111yescharacteristic, direct product factor, Sylow
313C2—24no—
314C2C3×#261411yescharacteristic
315C2C2×C3×#45011yescharacteristic
316C2C3×#300411yescharacteristic
317C2C3×#322912yes—
318C2C3×#323812yes—
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×#2058C2semidirect
5. C3×#2024C2semidirect
8. #29641C3direct (N × H)
32. #2058C2×C3semidirect
33. #2024C2×C3semidirect
312. C3#29641direct (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×#4832
C3×#28552
C3×#20582
C3×#20242
C3×#18232
C3×#18172
#296413