Finite Group Databasebeta

← C3×#20354 #10638961

Subgroups — C3×#20354 #10638961

Order 768.

2410 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C212yesmaximal
2C2×C3×#518C211yesmaximal, characteristic
3C3×#1740C214yesmaximal, complemented
4C2×C3×#342C218yesmaximal, complemented
5#20354C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×D4C2211yescharacteristic
7C3×#518—28no—
8C3×#518—28no—
9C3×#518C4116yescomplemented
10C2×C3×C4×D4—28no—
11C2×C3×C4×D4C418yescomplemented
12C2×C3×#117C2214yes—
13C2×C3×#117C2212yes—
14C22×C3×C22⋊C4C2214yes—
15C22×C3×C22⋊C4C2218yes—
16C3×#342C22116yes—
17C2×#518C2×C312yes—
18C2×#518C2×C311yescharacteristic
19#1740C2×C314yescomplemented
20C2×#342C2×C318yescomplemented
21C25×C3C2312yes—
22C22×C3×D4C2×C414yes—
23C22×C3×D4C2×C418yes—
24C22×C3×D4D418yes—
25C22×C3×D4D414yes—
26C22×C3×D4Q814yes—
27C23×C3×C4—28no—
28C23×C3×C4C2×C418yescomplemented
29C23×C3×C4C2311yescharacteristic
30C3×C4×D4—264no—
31C3×C4×D4—232no—
32C3×#117—28no—
33C3×#117—28no—
34C3×#117C2×C4116yescomplemented
35C2×C3×C4⋊C4—24no—
36C2×C3×C4⋊C4C2×C414yescomplemented
37C2×C3×C22⋊C4—28no—
38C2×C3×C22⋊C4—216no—
39C2×C3×C22⋊C4—216no—
40C2×C3×C22⋊C4C2×C418yescomplemented
41C2×C3×C22⋊C4C2×C4132yescomplemented
42C2×C3×C22⋊C4C2314yes—
43C2×C3×C22⋊C4C2318yes—
44C2×C3×C42—24no—
45C2×C3×C42C2×C414yescomplemented
46C23×D4C22×C311yescharacteristic
47#518—28no—
48#518—28no—
49#518C3×C4116yescomplemented
50C2×C4×D4—28no—
51C2×C4×D4C3×C418yescomplemented
52C2×#117C22×C314yes—
53C2×#117C22×C312yes—
54C22×C22⋊C4C22×C314yes—
55C22×C22⋊C4C22×C318yes—
56#342C22×C3116yes—
57C24×C3—84no—
58C24×C3C22×C414yes—
59C24×C3C22×C418yes—
60C24×C3C2×D418yes—
61C24×C3C2×D414yes—
62C24×C3C2×Q814yes—
63C24×C3C2411yescharacteristic
64C2×C3×D4C42116yes—
65C2×C3×D4C22⋊C4132yes—
66C2×C3×D4C22⋊C4116yes—
67C2×C3×D4C4⋊C4116yes—
68C2×C3×D4C4⋊C4132yes—
69C22×C3×C4—28no—
70C22×C3×C4—216no—
71C22×C3×C4—232no—
72C22×C3×C4—816no—
73C22×C3×C4—88no—
74C22×C3×C4C22×C418yes—
75C22×C3×C4C22×C412yes—
76C22×C3×C4C22×C414yes—
77C22×C3×C4C2×D414yes—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×Q812yes—
80C3×C4⋊C4—216no—
81C3×C4⋊C4—28no—
82C3×C22⋊C4—232no—
83C3×C22⋊C4—216no—
84C3×C22⋊C4—28no—
85C3×C22⋊C4—28no—
86C3×C22⋊C4C22×C4116yes—
87C3×C42—216no—
88C3×C42—28no—
89C25C23×C312yes—
90C22×D4C2×C3×C414yes—
91C22×D4C2×C3×C418yes—
92C22×D4C3×D418yes—
93C22×D4C3×D414yes—
94C22×D4C3×Q814yes—
95C23×C4—28no—
96C23×C4C2×C3×C418yescomplemented
97C23×C4C23×C311yescharacteristic
98C4×D4—264no—
99C4×D4—232no—
100#117—28no—
101#117—28no—
102#117C2×C3×C4116yescomplemented
103C2×C4⋊C4—24no—
104C2×C4⋊C4C2×C3×C414yescomplemented
105C2×C22⋊C4—28no—
106C2×C22⋊C4—216no—
107C2×C22⋊C4—216no—
108C2×C22⋊C4C2×C3×C418yescomplemented
109C2×C22⋊C4C2×C3×C4132yescomplemented
110C2×C22⋊C4C23×C314yes—
111C2×C22⋊C4C23×C318yes—
112C2×C42—24no—
113C2×C42C2×C3×C414yescomplemented
114C23×C3—48no—
115C23×C3—416no—
116C23×C3—42no—
117C23×C3—88no—
118C23×C3—84no—
119C23×C3—84no—
120C23×C3C2×C4218yes—
121C23×C3C2×C22⋊C418yes—
122C23×C3C2×C22⋊C4116yes—
123C23×C3C2×C4⋊C418yes—
124C23×C3C2×C4⋊C4116yes—
125C23×C3C23×C411yescharacteristic
126C23×C3C23×C412yes—
127C23×C3C22×D412yes—
128C23×C3C22×D411yescharacteristic
129C23×C3C22×Q811yescharacteristic
130C3×D4#95164yes—
131C2×C3×C4—216no—
132C2×C3×C4—28no—
133C2×C3×C4—416no—
134C2×C3×C4—48no—
135C2×C3×C4—816no—
136C2×C3×C4—832no—
137C2×C3×C4C2×C4214yes—
138C2×C3×C4C2×C22⋊C418yes—
139C2×C3×C4C2×C22⋊C414yes—
140C2×C3×C4C2×C4⋊C414yes—
141C2×C3×C4C2×C4⋊C418yes—
142C24—84no—
143C24C22×C3×C414yes—
144C24C22×C3×C418yes—
145C24C2×C3×D418yes—
146C24C2×C3×D414yes—
147C24C2×C3×Q814yes—
148C24C24×C311yescharacteristic
149C2×D4C3×C42116yes—
150C2×D4C3×C22⋊C4116yes—
151C2×D4C3×C22⋊C4132yes—
152C2×D4C3×C4⋊C4116yes—
153C2×D4C3×C4⋊C4132yes—
154C22×C4—28no—
155C22×C4—216no—
156C22×C4—232no—
157C22×C4—816no—
158C22×C4—88no—
159C22×C4C22×C3×C418yes—
160C22×C4C22×C3×C412yes—
161C22×C4C22×C3×C414yes—
162C22×C4C2×C3×D414yes—
163C22×C4C2×C3×D412yes—
164C22×C4C2×C3×Q812yes—
165C4⋊C4—216no—
166C4⋊C4—28no—
167C22⋊C4—216no—
168C22⋊C4—232no—
169C22⋊C4—28no—
170C22⋊C4—28no—
171C22⋊C4C22×C3×C4116yes—
172C42—216no—
173C42—28no—
174C22×C3—22no—
175C22×C3—24no—
176C22×C3—216no—
177C22×C3—416no—
178C22×C3—42no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—48no—
182C22×C3—48no—
183C22×C3—416no—
184C22×C3C2×#95116yes—
185C22×C3C22×C4211yescharacteristic
186C22×C3C22×C22⋊C412yes—
187C22×C3C22×C22⋊C411yescharacteristic
188C22×C3C22×C4⋊C411yescharacteristic
189C22×C3C22×C4⋊C412yes—
190C3×C4—416no—
191C3×C4—48no—
192C3×C4C2×#9518yes—
193C23—48no—
194C23—416no—
195C23—42no—
196C23—88no—
197C23—84no—
198C23—84no—
199C23C2×C3×C4218yes—
200C23C2×C3×C22⋊C418yes—
201C23C2×C3×C22⋊C4116yes—
202C23C2×C3×C4⋊C418yes—
203C23C2×C3×C4⋊C4116yes—
204C23C23×C3×C411yescharacteristic
205C23C23×C3×C412yes—
206C23C22×C3×D412yes—
207C23C22×C3×D411yescharacteristic
208C23C22×C3×Q811yescharacteristic
209D4C3×#95164yes—
210C2×C4—28no—
211C2×C4—216no—
212C2×C4—416no—
213C2×C4—48no—
214C2×C4—816no—
215C2×C4—832no—
216C2×C4C2×C3×C4214yes—
217C2×C4C2×C3×C22⋊C418yes—
218C2×C4C2×C3×C22⋊C414yes—
219C2×C4C2×C3×C4⋊C414yes—
220C2×C4C2×C3×C4⋊C418yes—
221C2×C3—22no—
222C2×C3—21no—
223C2×C3—22no—
224C2×C3—21no—
225C2×C3—216no—
226C2×C3C22×#9511yescharacteristic
227C2×C3#288312yes—
228C22—22no—
229C22—24no—
230C22—216no—
231C22—416no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—48no—
236C22—48no—
237C22—416no—
238C22C2×C3×#95116yes—
239C22C22×C3×C4211yescharacteristic
240C22C22×C3×C22⋊C412yes—
241C22C22×C3×C22⋊C411yescharacteristic
242C22C22×C3×C4⋊C411yescharacteristic
243C22C22×C3×C4⋊C412yes—
244C4—416no—
245C4—48no—
246C4C2×C3×#9518yes—
247C3#2035411yescharacteristic, direct product factor, Sylow
248C2—22no—
249C2—21no—
250C2—21no—
251C2—22no—
252C2—216no—
253C2C22×C3×#9511yescharacteristic
254C2C3×#288312yes—
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC22C22×C3×C42
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1740C2semidirect
4. C2×C3×#342C2semidirect
5. #20354C3direct (N × H)
9. C3×#518C4semidirect
11. C2×C3×C4×D4C4semidirect
19. #1740C2×C3semidirect
20. C2×#342C2×C3semidirect
28. C23×C3×C4C2×C4semidirect
34. C3×#117C2×C4semidirect
36. C2×C3×C4⋊C4C2×C4semidirect
40. C2×C3×C22⋊C4C2×C4semidirect
41. C2×C3×C22⋊C4C2×C4semidirect
45. C2×C3×C42C2×C4semidirect
49. #518C3×C4semidirect
51. C2×C4×D4C3×C4semidirect
96. C23×C4C2×C3×C4semidirect
102. #117C2×C3×C4semidirect
104. C2×C4⋊C4C2×C3×C4semidirect
108. C2×C22⋊C4C2×C3×C4semidirect
109. C2×C22⋊C4C2×C3×C4semidirect
113. C2×C42C2×C3×C4semidirect
247. C3#20354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C3×#17402
C2×C3×#3422
#203543