Finite Group Databasebeta

← C3×#20347 #10638954

Subgroups — C3×#20347 #10638954

Order 768.

2270 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic
2C22×C3×#117C212yesmaximal
3C3×#1736C218yesmaximal, complemented
4C3×#1735C214yesmaximal, complemented
5#20347C311yesmaximal, characteristic, direct product factor, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—28no—
8C22×C3×M16C418yescomplemented
9C22×C3×M16C2214yes—
10C23×C3×C8C2212yes—
11C2×C3×#117—28no—
12C2×C3×#117—22no—
13C2×C3×#117—22no—
14C2×C3×#117C4116yescomplemented
15C2×C3×#117C2214yes—
16C2×C3×#117C2214yes—
17C22×C3×C4⋊C4C2214yes—
18C22×C3×C22⋊C4—22no—
19C22×C3×C42—22no—
20C3×#340C22116yescomplemented
21C23×M16C2×C311yescharacteristic
22C22×#117C2×C312yes—
23#1736C2×C318yescomplemented
24#1735C2×C314yescomplemented
25C25×C3D411yescharacteristic
26C23×C3×C4—24no—
27C23×C3×C4C2×C414yes—
28C23×C3×C4C2×C418yes—
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D412yes—
31C23×C3×C4D418yes—
32C23×C3×C4Q814yes—
33C23×C3×C4C2312yes—
34C23×C3×C4C2311yescharacteristic
35C2×C3×M16—232no—
36C2×C3×M16—28no—
37C2×C3×M16—28no—
38C2×C3×M16C2×C4116yescomplemented
39C22×C3×C8—28no—
40C22×C3×C8C2×C418yescomplemented
41C22×C3×C8C2314yes—
42C3×#117—216no—
43C3×#117—216no—
44C3×#117—28no—
45C3×#117—28no—
46C3×#117C2×C4116yes—
47C3×#117C2×C4116yescomplemented
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4C2×C4116yescomplemented
51C2×C3×C4⋊C4C2318yes—
52C2×C3×C22⋊C4—216no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C42—216no—
56C2×C3×C42—24no—
57C2×C3×C42—24no—
58C24×C4C22×C311yescharacteristic
59C22×M16—28no—
60C22×M16C3×C418yescomplemented
61C22×M16C22×C314yes—
62C23×C8C22×C312yes—
63C2×#117—28no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117C3×C4116yescomplemented
67C2×#117C22×C314yes—
68C2×#117C22×C314yes—
69C22×C4⋊C4C22×C314yes—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72#340C22×C3116yescomplemented
73C24×C3—44no—
74C24×C3C22⋊C418yes—
75C24×C3C4⋊C414yes—
76C24×C3C2×D411yescharacteristic
77C24×C3C2×D412yes—
78C22×C3×C4—216no—
79C22×C3×C4—28no—
80C22×C3×C4—416no—
81C22×C3×C4C42116yes—
82C22×C3×C4C22⋊C418yes—
83C22×C3×C4C22⋊C418yes—
84C22×C3×C4C22⋊C418yes—
85C22×C3×C4C22⋊C4116yes—
86C22×C3×C4C4⋊C414yes—
87C22×C3×C4C4⋊C418yes—
88C22×C3×C4C4⋊C4132yes—
89C22×C3×C4C22×C412yes—
90C22×C3×C4C22×C414yes—
91C22×C3×C4C22×C414yes—
92C22×C3×C4C22×C414yes—
93C22×C3×C4C22×C414yes—
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D414yes—
96C22×C3×C4C2×D414yes—
97C22×C3×C4C2×D414yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×Q814yes—
100C22×C3×C4C2×Q812yes—
101C22×C3×C4C2411yescharacteristic
102C3×M16—232no—
103C2×C3×C8—216no—
104C2×C3×C8—28no—
105C2×C3×C8C22×C418yes—
106C3×C4⋊C4—216no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4C22×C4116yes—
109C3×C22⋊C4—216no—
110C3×C22⋊C4—216no—
111C3×C42—216no—
112C3×C42—216no—
113C25C3×D411yescharacteristic
114C23×C4—24no—
115C23×C4C2×C3×C414yes—
116C23×C4C2×C3×C418yes—
117C23×C4C3×D411yescharacteristic
118C23×C4C3×D412yes—
119C23×C4C3×D418yes—
120C23×C4C3×Q814yes—
121C23×C4C23×C312yes—
122C23×C4C23×C311yescharacteristic
123C2×M16—232no—
124C2×M16—28no—
125C2×M16—28no—
126C2×M16C2×C3×C4116yescomplemented
127C22×C8—28no—
128C22×C8C2×C3×C418yescomplemented
129C22×C8C23×C314yes—
130#117—216no—
131#117—216no—
132#117—28no—
133#117—28no—
134#117C2×C3×C4116yes—
135#117C2×C3×C4116yescomplemented
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4C2×C3×C4116yescomplemented
139C2×C4⋊C4C23×C318yes—
140C2×C22⋊C4—216no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C42—216no—
144C2×C42—24no—
145C2×C42—24no—
146C23×C3—28no—
147C23×C3—24no—
148C23×C3—416no—
149C23×C3—48no—
150C23×C3#95116yes—
151C23×C3C2×C22⋊C414yes—
152C23×C3C2×C22⋊C414yes—
153C23×C3C2×C22⋊C414yes—
154C23×C3C2×C4⋊C412yes—
155C23×C3C2×C4⋊C414yes—
156C23×C3C22×D411yescharacteristic
157C2×C3×C4—216no—
158C2×C3×C4—432no—
159C2×C3×C4—416no—
160C2×C3×C4#95132yes—
161C2×C3×C4#95116yes—
162C2×C3×C4C2×C4214yes—
163C2×C3×C4C2×C4218yes—
164C2×C3×C4C2×C22⋊C414yes—
165C2×C3×C4C2×C22⋊C418yes—
166C2×C3×C4C2×C22⋊C414yes—
167C2×C3×C4C2×C22⋊C414yes—
168C2×C3×C4C2×C22⋊C414yes—
169C2×C3×C4C2×C4⋊C418yes—
170C2×C3×C4C2×C4⋊C418yes—
171C2×C3×C4C2×C4⋊C418yes—
172C2×C3×C4C2×C4⋊C414yes—
173C2×C3×C4C2×C4⋊C412yes—
174C2×C3×C4C23×C411yescharacteristic
175C2×C3×C4C23×C412yes—
176C2×C3×C4C22×D412yes—
177C2×C3×C4C22×Q811yescharacteristic
178C3×C8—28no—
179C24—44no—
180C24C3×C22⋊C418yes—
181C24C3×C4⋊C414yes—
182C24C2×C3×D411yescharacteristic
183C24C2×C3×D412yes—
184C22×C4—216no—
185C22×C4—28no—
186C22×C4—416no—
187C22×C4C3×C42116yes—
188C22×C4C3×C22⋊C418yes—
189C22×C4C3×C22⋊C418yes—
190C22×C4C3×C22⋊C418yes—
191C22×C4C3×C22⋊C4116yes—
192C22×C4C3×C4⋊C414yes—
193C22×C4C3×C4⋊C418yes—
194C22×C4C3×C4⋊C4132yes—
195C22×C4C22×C3×C412yes—
196C22×C4C22×C3×C414yes—
197C22×C4C22×C3×C414yes—
198C22×C4C22×C3×C414yes—
199C22×C4C22×C3×C414yes—
200C22×C4C2×C3×D411yescharacteristic
201C22×C4C2×C3×D412yes—
202C22×C4C2×C3×D414yes—
203C22×C4C2×C3×D414yes—
204C22×C4C2×C3×D414yes—
205C22×C4C2×C3×Q814yes—
206C22×C4C2×C3×Q812yes—
207C22×C4C24×C311yescharacteristic
208M16—232no—
209C2×C8—216no—
210C2×C8—28no—
211C2×C8C22×C3×C418yes—
212C4⋊C4—216no—
213C4⋊C4—28no—
214C4⋊C4C22×C3×C4116yes—
215C22⋊C4—216no—
216C22⋊C4—216no—
217C42—216no—
218C42—216no—
219C22×C3—28no—
220C22×C3—28no—
221C22×C3—28no—
222C22×C3—28no—
223C22×C3—24no—
224C22×C3—416no—
225C22×C3C2×#9514yes—
226C22×C3C2×#9518yes—
227C22×C3C22×C22⋊C412yes—
228C22×C3C22×C4⋊C411yescharacteristic
229C22×C3#57514yes—
230C3×C4—416no—
231C3×C4C2×#9514yes—
232C3×C4C2×#9518yes—
233C3×C4C22×C4211yescharacteristic
234C3×C4C22×C22⋊C411yescharacteristic
235C3×C4C22×C4⋊C412yes—
236C23—28no—
237C23—24no—
238C23—416no—
239C23—48no—
240C23C3×#95116yes—
241C23C2×C3×C22⋊C414yes—
242C23C2×C3×C22⋊C414yes—
243C23C2×C3×C22⋊C414yes—
244C23C2×C3×C4⋊C412yes—
245C23C2×C3×C4⋊C414yes—
246C23C22×C3×D411yescharacteristic
247C2×C4—216no—
248C2×C4—432no—
249C2×C4—416no—
250C2×C4C3×#95132yes—
251C2×C4C3×#95116yes—
252C2×C4C2×C3×C4214yes—
253C2×C4C2×C3×C4218yes—
254C2×C4C2×C3×C22⋊C414yes—
255C2×C4C2×C3×C22⋊C414yes—
256C2×C4C2×C3×C22⋊C414yes—
257C2×C4C2×C3×C22⋊C414yes—
258C2×C4C2×C3×C22⋊C418yes—
259C2×C4C2×C3×C4⋊C412yes—
260C2×C4C2×C3×C4⋊C414yes—
261C2×C4C2×C3×C4⋊C418yes—
262C2×C4C2×C3×C4⋊C418yes—
263C2×C4C2×C3×C4⋊C418yes—
264C2×C4C23×C3×C411yescharacteristic
265C2×C4C23×C3×C412yes—
266C2×C4C22×C3×D412yes—
267C2×C4C22×C3×Q811yescharacteristic
268C8—28no—
269C2×C3—28no—
270C2×C3—24no—
271C2×C3C22×#9511yescharacteristic
272C2×C3#289614yes—
273C2×C3#291112yes—
274C22—28no—
275C22—28no—
276C22—28no—
277C22—28no—
278C22—24no—
279C22—416no—
280C22C2×C3×#9514yes—
281C22C2×C3×#9518yes—
282C22C22×C3×C22⋊C412yes—
283C22C22×C3×C4⋊C411yescharacteristic
284C22C3×#57514yes—
285C4—416no—
286C4C2×C3×#9514yes—
287C4C2×C3×#9518yes—
288C4C22×C3×C4211yescharacteristic
289C4C22×C3×C22⋊C411yescharacteristic
290C4C22×C3×C4⋊C412yes—
291C3#2034711yescharacteristic, direct product factor, Sylow
292C2—28no—
293C2—24no—
294C2C22×C3×#9511yescharacteristic
295C2C3×#289614yes—
296C2C3×#291112yes—
subgroupG/H
centerC22×C3×C4C2×D4
commutatorC4C22×C3×C42
Frattini, cosocleC22×C4C24×C3
socleC23×C3C22×D4
prime radicalC25×C3D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1736C2semidirect
4. C3×#1735C2semidirect
5. #20347C3direct (N × H)
8. C22×C3×M16C4semidirect
14. C2×C3×#117C4semidirect
20. C3×#340C22semidirect
23. #1736C2×C3semidirect
24. #1735C2×C3semidirect
38. C2×C3×M16C2×C4semidirect
40. C22×C3×C8C2×C4semidirect
47. C3×#117C2×C4semidirect
50. C2×C3×C4⋊C4C2×C4semidirect
60. C22×M16C3×C4semidirect
66. C2×#117C3×C4semidirect
72. #340C22×C3semidirect
126. C2×M16C2×C3×C4semidirect
128. C22×C8C2×C3×C4semidirect
135. #117C2×C3×C4semidirect
138. C2×C4⋊C4C2×C3×C4semidirect
291. C3#20347direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C22×C3×#1172
C3×#17362
C3×#17352
#203473