Finite Group Databasebeta

← C3×#14651 #10633258

Subgroups — C3×#14651 #10633258

Order 768.

1810 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#392C211yesmaximal, characteristic
2C3×#1443C212yesmaximal, complemented
3C3×#1440C213yesmaximal, complemented
4C3×#1438C216yesmaximal, complemented
5C3×#1437C213yesmaximal, complemented
6#14651C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C2213yes—
8C22×C3×C22⋊C4C2216yes—
9C22×C3×C22⋊C4C2213yes—
10C22×C3×C22⋊C4C2213yes—
11C22×C3×C22⋊C4C2213yes—
12C22×C3×C42C2211yescharacteristic
13C3×#392C418yescomplemented
14C2×C3×#95C2216yes—
15C2×C3×#95C2213yes—
16C2×C3×#95C2216yes—
17C2×C3×#95C2211yescharacteristic
18C2×#392C2×C311yescharacteristic
19#1443C2×C312yescomplemented
20#1440C2×C313yescomplemented
21#1438C2×C316yescomplemented
22#1437C2×C313yescomplemented
23C25×C3C2313yes—
24C22×C3×D4—46no—
25C22×C3×D4—46no—
26C22×C3×D4—43no—
27C22×C3×D4—43no—
28C22×C3×D4C2×C4112yes—
29C23×C3×C4C2316yes—
30C23×C3×C4C2313yes—
31C23×C3×C4C2311yescharacteristic
32C23×C3×C4C2312yes—
33C2×C3×C22⋊C4—212no—
34C2×C3×C22⋊C4—43no—
35C2×C3×C22⋊C4—43no—
36C2×C3×C22⋊C4—46no—
37C2×C3×C22⋊C4—46no—
38C2×C3×C22⋊C4—46no—
39C2×C3×C22⋊C4—46no—
40C2×C3×C22⋊C4—43no—
41C2×C3×C22⋊C4—43no—
42C2×C3×C22⋊C4C2×C4112yes—
43C2×C3×C42—43no—
44C3×#95—46no—
45C3×#95—43no—
46C3×#95—46no—
47C3×#95C2×C414yes—
48C23×D4C22×C313yes—
49C22×C22⋊C4C22×C316yes—
50C22×C22⋊C4C22×C313yes—
51C22×C22⋊C4C22×C313yes—
52C22×C22⋊C4C22×C313yes—
53C22×C42C22×C311yescharacteristic
54#392C3×C418yescomplemented
55C2×#95C22×C316yes—
56C2×#95C22×C313yes—
57C2×#95C22×C316yes—
58C2×#95C22×C311yescharacteristic
59C24×C3—26no—
60C24×C3—23no—
61C24×C3—23no—
62C24×C3—26no—
63C24×C3—26no—
64C24×C3—26no—
65C24×C3—26no—
66C24×C3—26no—
67C24×C3C22×C416yes—
68C24×C3C2411yescharacteristic
69C2×C3×D4—412no—
70C2×C3×D4—424no—
71C2×C3×D4—424no—
72C2×C3×D4—412no—
73C2×C3×D4—412no—
74C22×C3×C4—23no—
75C22×C3×C4—26no—
76C22×C3×C4—26no—
77C22×C3×C4—26no—
78C22×C3×C4—23no—
79C22×C3×C4—26no—
80C22×C3×C4—23no—
81C22×C3×C4—23no—
82C22×C3×C4—26no—
83C22×C3×C4—26no—
84C22×C3×C4—26no—
85C22×C3×C4—26no—
86C22×C3×C4—26no—
87C22×C3×C4—26no—
88C22×C3×C4—26no—
89C22×C3×C4C22×C416yes—
90C22×C3×C4C22×C412yes—
91C22×C3×C4#4114yes—
92C3×C22⋊C4—412no—
93C3×C22⋊C4—424no—
94C3×C22⋊C4—412no—
95C3×C22⋊C4—412no—
96C3×C42—44no—
97C25C23×C313yes—
98C22×D4—46no—
99C22×D4—46no—
100C22×D4—43no—
101C22×D4—43no—
102C22×D4C2×C3×C4112yes—
103C23×C4C23×C316yes—
104C23×C4C23×C311yescharacteristic
105C23×C4C23×C313yes—
106C23×C4C23×C312yes—
107C2×C22⋊C4—212no—
108C2×C22⋊C4—43no—
109C2×C22⋊C4—43no—
110C2×C22⋊C4—46no—
111C2×C22⋊C4—46no—
112C2×C22⋊C4—46no—
113C2×C22⋊C4—46no—
114C2×C22⋊C4—43no—
115C2×C22⋊C4—43no—
116C2×C22⋊C4C2×C3×C4112yes—
117C2×C42—43no—
118#95—46no—
119#95—43no—
120#95—46no—
121#95C2×C3×C414yes—
122C23×C3—26no—
123C23×C3—26no—
124C23×C3—26no—
125C23×C3—212no—
126C23×C3—212no—
127C23×C3—46no—
128C23×C3—46no—
129C23×C3—46no—
130C23×C3—46no—
131C23×C3—46no—
132C23×C3—46no—
133C23×C3—43no—
134C23×C3—46no—
135C23×C3—46no—
136C23×C3—46no—
137C23×C3—46no—
138C23×C3—43no—
139C23×C3—43no—
140C23×C3—46no—
141C23×C3—43no—
142C23×C3—46no—
143C23×C3C23×C411yescharacteristic
144C23×C3C2×#4112yes—
145C23×C3H(5,𝔽2)13yes—
146C23×C3H(5,𝔽2)16yes—
147C23×C3H(5,𝔽2)13yes—
148C3×D4—448no—
149C2×C3×C4—26no—
150C2×C3×C4—26no—
151C2×C3×C4—212no—
152C2×C3×C4—212no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—46no—
156C2×C3×C4—43no—
157C2×C3×C4—46no—
158C2×C3×C4—43no—
159C2×C3×C4—43no—
160C2×C3×C4—46no—
161C2×C3×C4—46no—
162C2×C3×C4—46no—
163C2×C3×C4—46no—
164C2×C3×C4—46no—
165C2×C3×C4—46no—
166C2×C3×C4—46no—
167C24—26no—
168C24—23no—
169C24—23no—
170C24—26no—
171C24—26no—
172C24—26no—
173C24—26no—
174C24—26no—
175C24C22×C3×C416yes—
176C24C24×C311yescharacteristic
177C2×D4—412no—
178C2×D4—424no—
179C2×D4—424no—
180C2×D4—412no—
181C2×D4—412no—
182C22×C4—23no—
183C22×C4—26no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—23no—
187C22×C4—26no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—26no—
191C22×C4—26no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—26no—
195C22×C4—26no—
196C22×C4—26no—
197C22×C4C22×C3×C416yes—
198C22×C4C22×C3×C412yes—
199C22×C4C3×#4114yes—
200C22⋊C4—412no—
201C22⋊C4—424no—
202C22⋊C4—412no—
203C22⋊C4—412no—
204C42—44no—
205C22×C3—46no—
206C22×C3—412no—
207C22×C3—412no—
208C22×C3—46no—
209C22×C3—46no—
210C22×C3—86no—
211C22×C3—86no—
212C22×C3—86no—
213C22×C3—86no—
214C22×C3C4×#4111yescharacteristic
215C22×C3#51813yes—
216C22×C3#51813yes—
217C22×C3#53816yes—
218C22×C3#54016yes—
219C22×C3#56011yescharacteristic
220C22×C3#56016yes—
221C22×C3#56013yes—
222C22×C3#56013yes—
223C22×C3H(3,𝔽4)13yes—
224C3×C4—44no—
225C3×C4—46no—
226C3×C4—42no—
227C3×C4—86no—
228C23—26no—
229C23—26no—
230C23—26no—
231C23—212no—
232C23—212no—
233C23—43no—
234C23—46no—
235C23—46no—
236C23—46no—
237C23—46no—
238C23—43no—
239C23—43no—
240C23—46no—
241C23—43no—
242C23—46no—
243C23—46no—
244C23—46no—
245C23—46no—
246C23—46no—
247C23—46no—
248C23—46no—
249C23C23×C3×C411yescharacteristic
250C23C2×C3×#4112yes—
251C23C3×H(5,𝔽2)13yes—
252C23C3×H(5,𝔽2)16yes—
253C23C3×H(5,𝔽2)13yes—
254D4—448no—
255C2×C4—26no—
256C2×C4—26no—
257C2×C4—212no—
258C2×C4—212no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—46no—
262C2×C4—43no—
263C2×C4—43no—
264C2×C4—46no—
265C2×C4—43no—
266C2×C4—46no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—46no—
271C2×C4—46no—
272C2×C4—46no—
273C2×C3—86no—
274C2×C3#233313yes—
275C2×C3#237513yes—
276C2×C3#238011yescharacteristic
277C2×C3#282512yes—
278C2×C3#282816yes—
279C22—412no—
280C22—46no—
281C22—46no—
282C22—46no—
283C22—412no—
284C22—86no—
285C22—86no—
286C22—86no—
287C22—86no—
288C22C3×C4×#4111yescharacteristic
289C22C3×#51813yes—
290C22C3×#51813yes—
291C22C3×#53816yes—
292C22C3×#54016yes—
293C22C3×#56016yes—
294C22C3×#56013yes—
295C22C3×#56011yescharacteristic
296C22C3×#56013yes—
297C22C3×H(3,𝔽4)13yes—
298C4—46no—
299C4—42no—
300C4—44no—
301C4—86no—
302C3#1465111yescharacteristic, direct product factor, Sylow
303C2—86no—
304C2C3×#233313yes—
305C2C3×#237513yes—
306C2C3×#238011yescharacteristic
307C2C3×#282512yes—
308C2C3×#282816yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#1443C2semidirect
3. C3×#1440C2semidirect
4. C3×#1438C2semidirect
5. C3×#1437C2semidirect
6. #14651C3direct (N × H)
13. C3×#392C4semidirect
19. #1443C2×C3semidirect
20. #1440C2×C3semidirect
21. #1438C2×C3semidirect
22. #1437C2×C3semidirect
54. #392C3×C4semidirect
302. C3#14651direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3922
C3×#14432
C3×#14402
C3×#14382
C3×#14372
#146513