Finite Group Databasebeta

← C3×#14668 #10633275

Subgroups — C3×#14668 #10633275

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C3×#1446C212yesmaximal
3C3×#1443C214yesmaximal
4C3×#1442C214yesmaximal
5C3×#1441C212yesmaximal
6C3×#1434C212yesmaximal
7#14668C311yesmaximal, characteristic, direct product factor, Sylow
8C23×C3×Q8C2211yescharacteristic
9C22×C3×C4⋊C4C2212yes—
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C3×#391C418yescomplemented
13C2×C3×#95C2214yes—
14C2×C3×#95C2212yes—
15C2×C3×#95C2214yes—
16C2×C3×#95C2212yes—
17C2×C3×#95C2214yes—
18C2×C3×#95C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2214yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2214yes—
23C2×#391C2×C311yescharacteristic
24#1446C2×C312yes—
25#1443C2×C314yes—
26#1442C2×C314yes—
27#1441C2×C312yes—
28#1434C2×C312yes—
29C22×C3×Q8—24no—
30C22×C3×Q8—42no—
31C22×C3×Q8—41no—
32C22×C3×Q8—41no—
33C22×C3×Q8C2×C414yes—
34C23×C3×C4C2314yes—
35C23×C3×C4C2314yes—
36C23×C3×C4C2312yes—
37C23×C3×C4C2312yes—
38C23×C3×C4C2312yes—
39C23×C3×C4C2311yescharacteristic
40C2×C3×C4⋊C4—41no—
41C2×C3×C4⋊C4—41no—
42C2×C3×C4⋊C4—44no—
43C2×C3×C4⋊C4—42no—
44C2×C3×C4⋊C4C2×C414yes—
45C2×C3×C42—42no—
46C2×C3×C42C2×C414yes—
47C3×#95—28no—
48C3×#95—42no—
49C3×#95—42no—
50C3×#95—44no—
51C3×#95—42no—
52C3×#95—44no—
53C3×#95—44no—
54C3×#95—42no—
55C3×#95—42no—
56C3×#95C2×C4116yes—
57C23×Q8C22×C311yescharacteristic
58C22×C4⋊C4C22×C312yes—
59C22×C4⋊C4C22×C311yescharacteristic
60C22×C42C22×C311yescharacteristic
61#391C3×C418yescomplemented
62C2×#95C22×C314yes—
63C2×#95C22×C312yes—
64C2×#95C22×C314yes—
65C2×#95C22×C312yes—
66C2×#95C22×C314yes—
67C2×#95C22×C312yes—
68C2×#95C22×C312yes—
69C2×#95C22×C314yes—
70C2×#95C22×C312yes—
71C2×#95C22×C314yes—
72C24×C3C2411yescharacteristic
73C2×C3×Q8—28no—
74C2×C3×Q8—44no—
75C2×C3×Q8—44no—
76C2×C3×Q8—48no—
77C2×C3×Q8—48no—
78C22×C3×C4—22no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—24no—
82C22×C3×C4—22no—
83C22×C3×C4—24no—
84C22×C3×C4—22no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—24no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—22no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4C22×C414yes—
108C22×C3×C4C22×C414yes—
109C22×C3×C4C22×C414yes—
110C22×C3×C4C22×C412yes—
111C22×C3×C4#4114yes—
112C3×C4⋊C4—44no—
113C3×C4⋊C4—48no—
114C3×C42—44no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8—41no—
118C22×Q8—41no—
119C22×Q8C2×C3×C414yes—
120C23×C4C23×C314yes—
121C23×C4C23×C314yes—
122C23×C4C23×C312yes—
123C23×C4C23×C312yes—
124C23×C4C23×C312yes—
125C23×C4C23×C311yescharacteristic
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×C3×C414yes—
131C2×C42—42no—
132C2×C42C2×C3×C414yes—
133#95—28no—
134#95—42no—
135#95—42no—
136#95—44no—
137#95—42no—
138#95—44no—
139#95—44no—
140#95—42no—
141#95—42no—
142#95C2×C3×C4116yes—
143C23×C3C23×C411yescharacteristic
144C23×C3C2×#4112yes—
145C23×C3H(5,𝔽2)11yescharacteristic
146C23×C3H(5,𝔽2)12yes—
147C23×C3H(5,𝔽2)11yescharacteristic
148C23×C3H(5,𝔽2)12yes—
149C23×C32−1+414yes—
150C23×C32−1+412yes—
151C3×Q8—416no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—24no—
163C2×C3×C4—28no—
164C2×C3×C4—44no—
165C2×C3×C4—44no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—44no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4—44no—
175C2×C3×C4—44no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×C3×C4—44no—
179C2×C3×C4—44no—
180C2×C3×C4—44no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—44no—
185C2×C3×C4#11714yes—
186C24C24×C311yescharacteristic
187C2×Q8—28no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—48no—
191C2×Q8—48no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4C22×C3×C414yes—
222C22×C4C22×C3×C414yes—
223C22×C4C22×C3×C414yes—
224C22×C4C22×C3×C412yes—
225C22×C4C3×#4114yes—
226C4⋊C4—44no—
227C4⋊C4—48no—
228C42—44no—
229C22×C3C2×#11711yescharacteristic
230C22×C3#51811yescharacteristic
231C22×C3#51811yescharacteristic
232C22×C3#51912yes—
233C22×C3#52012yes—
234C22×C3#53812yes—
235C22×C3#53812yes—
236C22×C3#54114yes—
237C22×C3#54214yes—
238C22×C3H(3,𝔽4)11yescharacteristic
239C22×C3#56212yes—
240C22×C3#56214yes—
241C22×C3#56312yes—
242C22×C3#56314yes—
243C22×C3#56312yes—
244C22×C3#56311yescharacteristic
245C3×C4—24no—
246C3×C4—44no—
247C3×C4—44no—
248C3×C4—44no—
249C3×C4—84no—
250C3×C4—84no—
251C23C23×C3×C411yescharacteristic
252C23C2×C3×#4112yes—
253C23C3×H(5,𝔽2)12yes—
254C23C3×H(5,𝔽2)11yescharacteristic
255C23C3×H(5,𝔽2)12yes—
256C23C3×H(5,𝔽2)11yescharacteristic
257C23C3×2−1+414yes—
258C23C3×2−1+412yes—
259Q8—416no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—28no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4C3×#11714yes—
294C2×C3#231211yescharacteristic
295C2×C3#232012yes—
296C2×C3#237812yes—
297C2×C3#237911yescharacteristic
298C2×C3#238111yescharacteristic
299C2×C3#283114yes—
300C2×C3#283212yes—
301C2×C3#283312yes—
302C22C2×C3×#11711yescharacteristic
303C22C3×#51811yescharacteristic
304C22C3×#51811yescharacteristic
305C22C3×#51912yes—
306C22C3×#52012yes—
307C22C3×#53812yes—
308C22C3×#53812yes—
309C22C3×#54114yes—
310C22C3×#54214yes—
311C22C3×H(3,𝔽4)11yescharacteristic
312C22C3×#56214yes—
313C22C3×#56212yes—
314C22C3×#56314yes—
315C22C3×#56311yescharacteristic
316C22C3×#56312yes—
317C22C3×#56312yes—
318C4—24no—
319C4—44no—
320C4—44no—
321C4—44no—
322C4—84no—
323C4—84no—
324C3#1466811yescharacteristic, direct product factor, Sylow
325C2C3×#231211yescharacteristic
326C2C3×#232012yes—
327C2C3×#237812yes—
328C2C3×#237911yescharacteristic
329C2C3×#238111yescharacteristic
330C2C3×#283114yes—
331C2C3×#283212yes—
332C2C3×#283312yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
7. #14668C3direct (N × H)
12. C3×#391C4semidirect
61. #391C3×C4semidirect
324. C3#14668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C3×#14462
C3×#14432
C3×#14422
C3×#14412
C3×#14342
#146683