Finite Group Databasebeta

← C3×#20335 #10638942

Subgroups — C3×#20335 #10638942

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C212yesmaximal
2C22×C3×#117C211yesmaximal, characteristic
3C3×#1729C218yesmaximal, complemented
4C3×#1728C214yesmaximal, complemented
5#20335C311yesmaximal, characteristic, direct product factor, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—28no—
8C22×C3×M16—28no—
9C22×C3×M16C4116yescomplemented
10C22×C3×M16C2214yes—
11C22×C3×M16C2214yes—
12C23×C3×C8C2214yes—
13C2×C3×#117—28no—
14C2×C3×#117C414yes—
15C2×C3×#117C2214yes—
16C22×C3×C4⋊C4C412yes—
17C22×C3×C22⋊C4C412yes—
18C22×C3×C42C2212yes—
19C3×#336C22116yescomplemented
20C23×M16C2×C312yes—
21C22×#117C2×C311yescharacteristic
22#1729C2×C318yescomplemented
23#1728C2×C314yescomplemented
24C25×C3C2×C411yescharacteristic
25C23×C3×C4C2×C414yes—
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4C2×C412yes—
28C23×C3×C4C2×C418yes—
29C23×C3×C4D414yes—
30C23×C3×C4D418yes—
31C23×C3×C4Q814yes—
32C23×C3×C4C2312yes—
33C23×C3×C4C2311yescharacteristic
34C2×C3×M16—264no—
35C2×C3×M16—28no—
36C2×C3×M16—28no—
37C2×C3×M16—28no—
38C2×C3×M16—28no—
39C2×C3×M16C2×C4116yes—
40C2×C3×M16C2×C4116yescomplemented
41C22×C3×C8—28no—
42C22×C3×C8—28no—
43C22×C3×C8C2×C4116yescomplemented
44C22×C3×C8C2318yes—
45C3×#117—216no—
46C3×#117—28no—
47C3×#117—28no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4C2×C418yes—
50C2×C3×C22⋊C4—28no—
51C2×C3×C22⋊C4C2×C418yes—
52C2×C3×C42—28no—
53C2×C3×C42C2×C414yes—
54C2×C3×C42C2314yes—
55C24×C4C22×C311yescharacteristic
56C22×M16—28no—
57C22×M16—28no—
58C22×M16C3×C4116yescomplemented
59C22×M16C22×C314yes—
60C22×M16C22×C314yes—
61C23×C8C22×C314yes—
62C2×#117—28no—
63C2×#117C3×C414yes—
64C2×#117C22×C314yes—
65C22×C4⋊C4C3×C412yes—
66C22×C22⋊C4C3×C412yes—
67C22×C42C22×C312yes—
68#336C22×C3116yescomplemented
69C24×C3—44no—
70C24×C3C4214yes—
71C24×C3C22⋊C414yes—
72C24×C3C4⋊C414yes—
73C24×C3C22×C412yes—
74C24×C3C22×C411yescharacteristic
75C22×C3×C4—28no—
76C22×C3×C4—48no—
77C22×C3×C4C4214yes—
78C22×C3×C4C4218yes—
79C22×C3×C4C22⋊C414yes—
80C22×C3×C4C22⋊C418yes—
81C22×C3×C4C22⋊C4132yes—
82C22×C3×C4C4⋊C414yes—
83C22×C3×C4C4⋊C418yes—
84C22×C3×C4C4⋊C4132yes—
85C22×C3×C4C22×C418yes—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C22×C412yes—
88C22×C3×C4C22×C414yes—
89C22×C3×C4C22×C414yes—
90C22×C3×C4C22×C414yes—
91C22×C3×C4C2×D414yes—
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D414yes—
94C22×C3×C4C2×D414yes—
95C22×C3×C4C2×D414yes—
96C22×C3×C4C2×Q814yes—
97C22×C3×C4C2×Q812yes—
98C22×C3×C4C2411yescharacteristic
99C3×M16—232no—
100C3×M16—232no—
101C2×C3×C8—232no—
102C2×C3×C8—28no—
103C2×C3×C8—28no—
104C2×C3×C8C22×C4116yes—
105C3×C4⋊C4—216no—
106C3×C22⋊C4—216no—
107C3×C42—28no—
108C3×C42—28no—
109C25C2×C3×C411yescharacteristic
110C23×C4C2×C3×C414yes—
111C23×C4C2×C3×C411yescharacteristic
112C23×C4C2×C3×C412yes—
113C23×C4C2×C3×C418yes—
114C23×C4C3×D414yes—
115C23×C4C3×D418yes—
116C23×C4C3×Q814yes—
117C23×C4C23×C312yes—
118C23×C4C23×C311yescharacteristic
119C2×M16—264no—
120C2×M16—28no—
121C2×M16—28no—
122C2×M16—28no—
123C2×M16—28no—
124C2×M16C2×C3×C4116yes—
125C2×M16C2×C3×C4116yescomplemented
126C22×C8—28no—
127C22×C8—28no—
128C22×C8C2×C3×C4116yescomplemented
129C22×C8C23×C318yes—
130#117—216no—
131#117—28no—
132#117—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4C2×C3×C418yes—
135C2×C22⋊C4—28no—
136C2×C22⋊C4C2×C3×C418yes—
137C2×C42—28no—
138C2×C42C2×C3×C414yes—
139C2×C42C23×C314yes—
140C23×C3—24no—
141C23×C3—28no—
142C23×C3—48no—
143C23×C3—48no—
144C23×C3—48no—
145C23×C3#95116yes—
146C23×C3C2×C4214yes—
147C23×C3C2×C4212yes—
148C23×C3C2×C22⋊C414yes—
149C23×C3C2×C22⋊C412yes—
150C23×C3C2×C4⋊C414yes—
151C23×C3C2×C4⋊C412yes—
152C23×C3C23×C411yescharacteristic
153C2×C3×C4—28no—
154C2×C3×C4—216no—
155C2×C3×C4—416no—
156C2×C3×C4#95116yes—
157C2×C3×C4#95132yes—
158C2×C3×C4C2×C4212yes—
159C2×C3×C4C2×C4214yes—
160C2×C3×C4C2×C22⋊C418yes—
161C2×C3×C4C2×C22⋊C418yes—
162C2×C3×C4C2×C22⋊C418yes—
163C2×C3×C4C2×C22⋊C412yes—
164C2×C3×C4C2×C22⋊C414yes—
165C2×C3×C4C2×C4⋊C418yes—
166C2×C3×C4C2×C4⋊C418yes—
167C2×C3×C4C2×C4⋊C418yes—
168C2×C3×C4C2×C4⋊C412yes—
169C2×C3×C4C2×C4⋊C414yes—
170C2×C3×C4C23×C412yes—
171C2×C3×C4C22×D411yescharacteristic
172C2×C3×C4C22×D412yes—
173C2×C3×C4C22×Q811yescharacteristic
174C3×C8—216no—
175C24—44no—
176C24C3×C4214yes—
177C24C3×C22⋊C414yes—
178C24C3×C4⋊C414yes—
179C24C22×C3×C412yes—
180C24C22×C3×C411yescharacteristic
181C22×C4—28no—
182C22×C4—48no—
183C22×C4C3×C4214yes—
184C22×C4C3×C4218yes—
185C22×C4C3×C22⋊C414yes—
186C22×C4C3×C22⋊C418yes—
187C22×C4C3×C22⋊C4132yes—
188C22×C4C3×C4⋊C414yes—
189C22×C4C3×C4⋊C418yes—
190C22×C4C3×C4⋊C4132yes—
191C22×C4C22×C3×C418yes—
192C22×C4C22×C3×C411yescharacteristic
193C22×C4C22×C3×C412yes—
194C22×C4C22×C3×C414yes—
195C22×C4C22×C3×C414yes—
196C22×C4C22×C3×C414yes—
197C22×C4C2×C3×D414yes—
198C22×C4C2×C3×D412yes—
199C22×C4C2×C3×D414yes—
200C22×C4C2×C3×D414yes—
201C22×C4C2×C3×D414yes—
202C22×C4C2×C3×Q814yes—
203C22×C4C2×C3×Q812yes—
204C22×C4C24×C311yescharacteristic
205M16—232no—
206M16—232no—
207C2×C8—232no—
208C2×C8—28no—
209C2×C8—28no—
210C2×C8C22×C3×C4116yes—
211C4⋊C4—216no—
212C22⋊C4—216no—
213C42—28no—
214C42—28no—
215C22×C3—24no—
216C22×C3—28no—
217C22×C3—24no—
218C22×C3—28no—
219C22×C3—24no—
220C22×C3—28no—
221C22×C3—416no—
222C22×C3C2×#9518yes—
223C22×C3C2×#9514yes—
224C22×C3C22×C4211yescharacteristic
225C22×C3C22×C22⋊C411yescharacteristic
226C22×C3C22×C4⋊C411yescharacteristic
227C22×C3#56814yes—
228C3×C4—48no—
229C3×C4C2×#9518yes—
230C3×C4C2×#9514yes—
231C3×C4C22×C22⋊C412yes—
232C3×C4C22×C4⋊C412yes—
233C23—24no—
234C23—28no—
235C23—48no—
236C23—48no—
237C23—48no—
238C23C3×#95116yes—
239C23C2×C3×C4214yes—
240C23C2×C3×C4212yes—
241C23C2×C3×C22⋊C414yes—
242C23C2×C3×C22⋊C412yes—
243C23C2×C3×C4⋊C414yes—
244C23C2×C3×C4⋊C412yes—
245C23C23×C3×C411yescharacteristic
246C2×C4—216no—
247C2×C4—28no—
248C2×C4—416no—
249C2×C4C3×#95116yes—
250C2×C4C3×#95132yes—
251C2×C4C2×C3×C4212yes—
252C2×C4C2×C3×C4214yes—
253C2×C4C2×C3×C22⋊C412yes—
254C2×C4C2×C3×C22⋊C414yes—
255C2×C4C2×C3×C22⋊C418yes—
256C2×C4C2×C3×C22⋊C418yes—
257C2×C4C2×C3×C22⋊C418yes—
258C2×C4C2×C3×C4⋊C412yes—
259C2×C4C2×C3×C4⋊C414yes—
260C2×C4C2×C3×C4⋊C418yes—
261C2×C4C2×C3×C4⋊C418yes—
262C2×C4C2×C3×C4⋊C418yes—
263C2×C4C23×C3×C412yes—
264C2×C4C22×C3×D411yescharacteristic
265C2×C4C22×C3×D412yes—
266C2×C4C22×C3×Q811yescharacteristic
267C8—216no—
268C2×C3—24no—
269C2×C3—28no—
270C2×C3C22×#9511yescharacteristic
271C2×C3#287312yes—
272C2×C3#287912yes—
273C2×C3#290512yes—
274C22—24no—
275C22—28no—
276C22—24no—
277C22—28no—
278C22—24no—
279C22—28no—
280C22—416no—
281C22C2×C3×#9518yes—
282C22C2×C3×#9514yes—
283C22C22×C3×C4211yescharacteristic
284C22C22×C3×C22⋊C411yescharacteristic
285C22C22×C3×C4⋊C411yescharacteristic
286C22C3×#56814yes—
287C4—48no—
288C4C2×C3×#9514yes—
289C4C2×C3×#9518yes—
290C4C22×C3×C22⋊C412yes—
291C4C22×C3×C4⋊C412yes—
292C3#2033511yescharacteristic, direct product factor, Sylow
293C2—24no—
294C2—28no—
295C2C22×C3×#9511yescharacteristic
296C2C3×#287312yes—
297C2C3×#287912yes—
298C2C3×#290512yes—
subgroupG/H
centerC22×C3×C4C24
commutatorC22C22×C3×C42
Frattini, cosocleC22×C4C24×C3
socleC23×C3C23×C4
prime radicalC25×C3C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1729C2semidirect
4. C3×#1728C2semidirect
5. #20335C3direct (N × H)
9. C22×C3×M16C4semidirect
19. C3×#336C22semidirect
22. #1729C2×C3semidirect
23. #1728C2×C3semidirect
40. C2×C3×M16C2×C4semidirect
43. C22×C3×C8C2×C4semidirect
58. C22×M16C3×C4semidirect
68. #336C22×C3semidirect
125. C2×M16C2×C3×C4semidirect
128. C22×C8C2×C3×C4semidirect
292. C3#20335direct (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×#17292
C3×#17282
#203353