Finite Group Databasebeta

← C3×#14658 #10633265

Subgroups — C3×#14658 #10633265

Order 768.

1898 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C2×C3×#380C212yesmaximal, complemented
3C3×#1441C214yesmaximal, complemented
4C3×#1440C212yesmaximal, complemented
5C3×#1439C214yesmaximal, complemented
6C3×#1438C212yesmaximal
7#14658C311yesmaximal, characteristic, direct product factor, Sylow
8C23×C3×D4C2211yescharacteristic
9C24×C3×C4C2211yescharacteristic
10C2×C3×#123—28no—
11C2×C3×#123C418yescomplemented
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2214yes—
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2214yes—
17C22×C3×C22⋊C4C2214yes—
18C3×#380—28no—
19C2×C3×#95C2214yes—
20C2×C3×#95C2214yes—
21C2×C3×#95C2218yes—
22C22×#123C2×C311yescharacteristic
23C2×#380C2×C312yescomplemented
24#1441C2×C314yescomplemented
25#1440C2×C312yescomplemented
26#1439C2×C314yescomplemented
27#1438C2×C312yes—
28C25×C3C2311yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—24no—
31C22×C3×D4—44no—
32C22×C3×D4C2×C414yes—
33C23×C3×C4—44no—
34C23×C3×C4C2×C414yes—
35C23×C3×C4D418yes—
36C23×C3×C4C2318yes—
37C23×C3×C4C2311yescharacteristic
38C23×C3×C4C2312yes—
39C23×C3×C4C2312yes—
40C3×#123—232no—
41C2×C3×C4⋊C4—24no—
42C2×C3×C4⋊C4—24no—
43C2×C3×C4⋊C4—28no—
44C2×C3×C4⋊C4—48no—
45C2×C3×C4⋊C4C2×C418yes—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4—24no—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4—28no—
50C2×C3×C22⋊C4—48no—
51C2×C3×C22⋊C4—48no—
52C2×C3×C22⋊C4—44no—
53C2×C3×C22⋊C4C2×C414yes—
54C2×C3×C22⋊C4C2×C418yes—
55C3×#95—28no—
56C3×#95—48no—
57C3×#95—44no—
58C23×D4C22×C311yescharacteristic
59C24×C4C22×C311yescharacteristic
60C2×#123—28no—
61C2×#123C3×C418yescomplemented
62C22×C4⋊C4C22×C312yes—
63C22×C4⋊C4C22×C314yes—
64C22×C22⋊C4C22×C312yes—
65C22×C22⋊C4C22×C311yescharacteristic
66C22×C22⋊C4C22×C314yes—
67C22×C22⋊C4C22×C314yes—
68#380—28no—
69C2×#95C22×C314yes—
70C2×#95C22×C314yes—
71C2×#95C22×C318yes—
72C24×C3—24no—
73C24×C3—24no—
74C24×C3—24no—
75C24×C3—22no—
76C24×C3—24no—
77C24×C3—24no—
78C24×C3—24no—
79C24×C3C22×C412yes—
80C24×C3C22×C412yes—
81C24×C3C2×D414yes—
82C24×C3C2411yescharacteristic
83C2×C3×D4—28no—
84C2×C3×D4—48no—
85C2×C3×D4—416no—
86C22×C3×C4—28no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—24no—
93C22×C3×C4—28no—
94C22×C3×C4—28no—
95C22×C3×C4—28no—
96C22×C3×C4—28no—
97C22×C3×C4—28no—
98C22×C3×C4—28no—
99C22×C3×C4—28no—
100C22×C3×C4—48no—
101C22×C3×C4—416no—
102C22×C3×C4C22⋊C4116yescomplemented
103C22×C3×C4C22×C414yes—
104C22×C3×C4C22×C412yes—
105C22×C3×C4C22×C414yes—
106C22×C3×C4C2×D414yes—
107C22×C3×C4C2×D414yes—
108C3×C4⋊C4—216no—
109C3×C4⋊C4—416no—
110C3×C22⋊C4—28no—
111C3×C22⋊C4—216no—
112C3×C22⋊C4—416no—
113C3×C22⋊C4—416no—
114C25C23×C311yescharacteristic
115C25C23×C311yescharacteristic
116C22×D4—24no—
117C22×D4—44no—
118C22×D4C2×C3×C414yes—
119C23×C4—44no—
120C23×C4C2×C3×C414yes—
121C23×C4C3×D418yes—
122C23×C4C23×C318yes—
123C23×C4C23×C312yes—
124C23×C4C23×C311yescharacteristic
125C23×C4C23×C312yes—
126#123—232no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4C2×C3×C418yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—48no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4C2×C3×C414yes—
140C2×C22⋊C4C2×C3×C418yes—
141#95—28no—
142#95—48no—
143#95—44no—
144C23×C3—24no—
145C23×C3—28no—
146C23×C3—28no—
147C23×C3—24no—
148C23×C3—24no—
149C23×C3—28no—
150C23×C3—22no—
151C23×C3—24no—
152C23×C3—28no—
153C23×C3—44no—
154C23×C3—48no—
155C23×C3—44no—
156C23×C3—44no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—44no—
160C23×C3—44no—
161C23×C3—48no—
162C23×C3—44no—
163C23×C3C2×C22⋊C414yes—
164C23×C3C23×C411yescharacteristic
165C23×C3C22×D412yes—
166C23×C3H(5,𝔽2)14yes—
167C23×C3H(5,𝔽2)14yes—
168C23×C32−1+414yes—
169C3×D4—416no—
170C2×C3×C4—28no—
171C2×C3×C4—28no—
172C2×C3×C4—28no—
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—28no—
177C2×C3×C4—416no—
178C2×C3×C4—44no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—44no—
182C2×C3×C4—48no—
183C2×C3×C4—48no—
184C2×C3×C4—48no—
185C2×C3×C4—48no—
186C2×C3×C4—48no—
187C2×C3×C4—48no—
188C2×C3×C4—48no—
189C2×C3×C4C2×C22⋊C418yes—
190C24—24no—
191C24—24no—
192C24—24no—
193C24—22no—
194C24—24no—
195C24—24no—
196C24—24no—
197C24C22×C3×C412yes—
198C24C22×C3×C412yes—
199C24C2×C3×D414yes—
200C24C24×C311yescharacteristic
201C2×D4—28no—
202C2×D4—48no—
203C2×D4—416no—
204C22×C4—28no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—48no—
219C22×C4—416no—
220C22×C4C3×C22⋊C4116yescomplemented
221C22×C4C22×C3×C414yes—
222C22×C4C22×C3×C412yes—
223C22×C4C22×C3×C414yes—
224C22×C4C2×C3×D414yes—
225C22×C4C2×C3×D414yes—
226C4⋊C4—216no—
227C4⋊C4—416no—
228C22⋊C4—28no—
229C22⋊C4—216no—
230C22⋊C4—416no—
231C22⋊C4—416no—
232C22×C3—24no—
233C22×C3—28no—
234C22×C3—48no—
235C22×C3—48no—
236C22×C3—48no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—48no—
240C22×C3—84no—
241C22×C3—84no—
242C22×C3C22×C22⋊C411yescharacteristic
243C22×C3#51812yes—
244C22×C3#52014yes—
245C22×C3#53414yes—
246C22×C3#53512yes—
247C22×C3#53612yes—
248C22×C3#53714yes—
249C22×C3#56014yes—
250C22×C3#56218yes—
251C22×C3#56314yes—
252C3×C4—28no—
253C3×C4—42no—
254C3×C4—44no—
255C3×C4—88no—
256C23—24no—
257C23—28no—
258C23—28no—
259C23—24no—
260C23—24no—
261C23—28no—
262C23—22no—
263C23—24no—
264C23—28no—
265C23—44no—
266C23—48no—
267C23—44no—
268C23—44no—
269C23—48no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—42no—
274C23—44no—
275C23C2×C3×C22⋊C414yes—
276C23C23×C3×C411yescharacteristic
277C23C22×C3×D412yes—
278C23C3×H(5,𝔽2)14yes—
279C23C3×H(5,𝔽2)14yes—
280C23C3×2−1+414yes—
281D4—416no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—28no—
289C2×C4—416no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4C2×C3×C22⋊C418yes—
302C2×C3—44no—
303C2×C3—82no—
304C2×C3#230811yescharacteristic
305C2×C3#231112yes—
306C2×C3#237614yes—
307C2×C3#281314yes—
308C2×C3#281414yes—
309C22—24no—
310C22—28no—
311C22—48no—
312C22—48no—
313C22—48no—
314C22—48no—
315C22—42no—
316C22—44no—
317C22—84no—
318C22—84no—
319C22C22×C3×C22⋊C411yescharacteristic
320C22C3×#51812yes—
321C22C3×#52014yes—
322C22C3×#53414yes—
323C22C3×#53512yes—
324C22C3×#53612yes—
325C22C3×#53714yes—
326C22C3×#56014yes—
327C22C3×#56218yes—
328C22C3×#56314yes—
329C4—28no—
330C4—42no—
331C4—44no—
332C4—88no—
333C3#1465811yescharacteristic, direct product factor, Sylow
334C2—44no—
335C2—82no—
336C2C3×#230811yescharacteristic
337C2C3×#231112yes—
338C2C3×#237614yes—
339C2C3×#281314yes—
340C2C3×#281414yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#380C2semidirect
3. C3×#1441C2semidirect
4. C3×#1440C2semidirect
5. C3×#1439C2semidirect
7. #14658C3direct (N × H)
11. C2×C3×#123C4semidirect
23. C2×#380C2×C3semidirect
24. #1441C2×C3semidirect
25. #1440C2×C3semidirect
26. #1439C2×C3semidirect
61. C2×#123C3×C4semidirect
102. C22×C3×C4C22⋊C4semidirect
220. C22×C4C3×C22⋊C4semidirect
333. C3#14658direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C2×C3×#3802
C3×#14412
C3×#14402
C3×#14392
C3×#14382
#146583