Finite Group Databasebeta

← C3×#25814 #10644421

Subgroups — C3×#25814 #10644421

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#499C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic
4C3×#2058C212yesmaximal, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#1991C212yesmaximal, complemented
7C3×#1854C214yesmaximal
8C3×C4×C22⋊C8C211yesmaximal, characteristic
9#25814C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#533—28no—
11C2×C3×C4⋊Q8C2212yes—
12C2×C3×#125C2211yescharacteristic
13C2×C3×#125C2212yes—
14C2×C3×C22⋊Q8—22no—
15C2×C3×C4×Q8—22no—
16C2×C3×#117C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#499—28no—
19C2×C3×#107C2214yes—
20C2×C3×#106C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C2×C3×C4×C8C2212yes—
23C3×#389C2212yes—
24C3×#387—22no—
25C3×#385—22no—
26C3×#382C2211yescharacteristic
27C3×#380C2212yes—
28C3×#340C2214yes—
29C3×#340C2214yes—
30C3×#336C2211yescharacteristic
31C3×#336C2211yescharacteristic
32C2×#533C2×C311yescharacteristic
33C2×#499C2×C312yescomplemented
34#2851C2×C311yescharacteristic
35#2058C2×C312yescomplemented
36#2057C2×C312yescomplemented
37#1991C2×C312yescomplemented
38#1854C2×C314yes—
39C4×C22⋊C8C2×C311yescharacteristic
40C22×C3×Q8—22no—
41C23×C3×C4D412yes—
42C23×C3×C4C2311yescharacteristic
43C22×C3×C8C2312yes—
44C22×C3×C8C2312yes—
45C3×C4⋊Q8—28no—
46C3×#125—24no—
47C3×#125—24no—
48C3×#125D418yes—
49C3×C22⋊Q8—216no—
50C3×C4×Q8—216no—
51C3×#117—28no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2314yes—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42—22no—
62C2×C3×C42—24no—
63C2×C3×C42D412yes—
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C3×#107—28no—
67C3×#106—28no—
68C3×C22⋊C8Q818yescomplemented
69C3×C4×C8—24no—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73#533—28no—
74C2×C4⋊Q8C22×C312yes—
75C2×#125C22×C311yescharacteristic
76C2×#125C22×C312yes—
77C2×C22⋊Q8—22no—
78C2×C4×Q8—22no—
79C2×#117C22×C312yes—
80C22×C42C22×C311yescharacteristic
81#499—28no—
82C2×#107C22×C314yes—
83C2×#106C22×C314yes—
84C2×C22⋊C8C22×C312yes—
85C2×C4×C8C22×C312yes—
86#389C22×C312yes—
87#387—22no—
88#385—22no—
89#382C22×C311yescharacteristic
90#380C22×C312yes—
91#340C22×C314yes—
92#340C22×C314yes—
93#336C22×C311yescharacteristic
94#336C22×C311yescharacteristic
95C24×C3C2×D411yescharacteristic
96C2×C3×Q8—28no—
97C2×C3×Q8—44no—
98C2×C3×Q8—44no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×Q814yes—
112C22×C3×C4C2411yescharacteristic
113C2×C3×C8—22no—
114C2×C3×C8—22no—
115C2×C3×C8—22no—
116C2×C3×C8—22no—
117C2×C3×C8C2×Q814yes—
118C2×C3×C8C2×Q814yes—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—216no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—44no—
126C3×C4⋊C4—44no—
127C3×C4⋊C4C2×D418yes—
128C3×C22⋊C4—28no—
129C3×C22⋊C4—44no—
130C3×C42—28no—
131C3×C42—24no—
132C3×C42—22no—
133C3×C42C2×D414yes—
134C22×Q8—22no—
135C23×C4C3×D412yes—
136C23×C4C23×C311yescharacteristic
137C22×C8C23×C312yes—
138C22×C8C23×C312yes—
139C4⋊Q8—28no—
140#125—24no—
141#125—24no—
142#125C3×D418yes—
143C22⋊Q8—216no—
144C4×Q8—216no—
145#117—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C23×C312yes—
151C2×C4⋊C4C23×C312yes—
152C2×C4⋊C4C23×C314yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42C3×D412yes—
158C2×C42C23×C311yescharacteristic
159C2×C42C23×C311yescharacteristic
160#107—28no—
161#106—28no—
162C22⋊C8C3×Q818yescomplemented
163C4×C8—24no—
164#95—22no—
165#95—22no—
166#95—22no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3C4⋊Q812yes—
172C23×C3C22×D411yescharacteristic
173C3×Q8—416no—
174C2×C3×C4—21no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—28no—
183C2×C3×C4—48no—
184C2×C3×C4—44no—
185C2×C3×C4—44no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4#12014yes—
193C2×C3×C4C4⋊Q812yes—
194C2×C3×C4#13514yes—
195C2×C3×C4C22×D412yes—
196C2×C3×C4C22×Q812yes—
197C2×C3×C42−1+411yescharacteristic
198C2×C3×C42−1+411yescharacteristic
199C3×C8—24no—
200C3×C8—24no—
201C24C2×C3×D411yescharacteristic
202C2×Q8—28no—
203C2×Q8—44no—
204C2×Q8—44no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4C2×C3×D412yes—
215C22×C4C2×C3×D411yescharacteristic
216C22×C4C2×C3×D412yes—
217C22×C4C2×C3×Q814yes—
218C22×C4C24×C311yescharacteristic
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8C2×C3×Q814yes—
224C2×C8C2×C3×Q814yes—
225C4⋊C4—28no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4C2×C3×D418yes—
234C22⋊C4—28no—
235C22⋊C4—44no—
236C42—28no—
237C42—24no—
238C42—22no—
239C42C2×C3×D414yes—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3C2×#12011yescharacteristic
246C22×C3C2×C4⋊Q811yescharacteristic
247C22×C3#53611yescharacteristic
248C22×C3C2×#13512yes—
249C22×C3#57512yes—
250C3×C4—22no—
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—48no—
254C3×C4—82no—
255C3×C4—82no—
256C3×C4D4×Q812yes—
257C3×C4D4×Q812yes—
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23C3×C4⋊Q812yes—
263C23C22×C3×D411yescharacteristic
264Q8—416no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—28no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4C3×#12014yes—
284C2×C4C3×C4⋊Q812yes—
285C2×C4C3×#13514yes—
286C2×C4C22×C3×D412yes—
287C2×C4C22×C3×Q812yes—
288C2×C4C3×2−1+411yescharacteristic
289C2×C4C3×2−1+411yescharacteristic
290C8—24no—
291C8—24no—
292C2×C3—42no—
293C2×C3#243411yescharacteristic
294C2×C3#300212yes—
295C2×C3#316011yescharacteristic
296C2×C3#316611yescharacteristic
297C2×C3#323312yes—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22C2×C3×#12011yescharacteristic
304C22C2×C3×C4⋊Q811yescharacteristic
305C22C3×#53611yescharacteristic
306C22C2×C3×#13512yes—
307C22C3×#57512yes—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—48no—
312C4—82no—
313C4—82no—
314C4C3×D4×Q812yes—
315C4C3×D4×Q812yes—
316C3#2581411yescharacteristic, direct product factor, Sylow
317C2—42no—
318C2C3×#243411yescharacteristic
319C2C3×#300212yes—
320C2C3×#316011yescharacteristic
321C2C3×#316611yescharacteristic
322C2C3×#323312yes—
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
2. C2×C3×#499C2semidirect
4. C3×#2058C2semidirect
5. C3×#2057C2semidirect
6. C3×#1991C2semidirect
9. #25814C3direct (N × H)
33. C2×#499C2×C3semidirect
35. #2058C2×C3semidirect
36. #2057C2×C3semidirect
37. #1991C2×C3semidirect
68. C3×C22⋊C8Q8semidirect
162. C22⋊C8C3×Q8semidirect
316. C3#25814direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4992
C3×#28512
C3×#20582
C3×#20572
C3×#19912
C3×#18542
C3×C4×C22⋊C82
#258143