Finite Group Databasebeta

← C3×#21795 #10640402

Subgroups — C3×#21795 #10640402

Order 768.

1538 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic
2C3×#2312C212yesmaximal, complemented
3C3×#1964C218yesmaximal, complemented
4C3×#1781C214yesmaximal, complemented
5#21795C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×D4C2212yes—
7C3×#534—28no—
8C2×C3×#123C414yescomplemented
9C2×C3×#121C2214yes—
10C2×C3×#120C414yescomplemented
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×C3×#99C22116yescomplemented
13C3×#388—28no—
14C3×#380C2214yescomplemented
15C3×#379C2214yes—
16C3×C4×C22⋊C4C2214yescomplemented
17C2×#534C2×C311yescharacteristic
18#2312C2×C312yescomplemented
19#1964C2×C318yescomplemented
20#1781C2×C314yescomplemented
21C25×C3C2×C412yes—
22C25×C3C2311yescharacteristic
23C22×C3×D4—28no—
24C22×C3×D4—48no—
25C22×C3×D4C2×C414yes—
26C22×C3×D4C2×C414yes—
27C22×C3×D4C2314yes—
28C23×C3×C4C2×C412yes—
29C23×C3×C4C2312yes—
30C3×#123—216no—
31C3×#121—28no—
32C3×#121—28no—
33C3×#120—216no—
34C2×C3×C4⋊C4—24no—
35C2×C3×C4⋊C4C2×C414yes—
36C2×C3×C22⋊C4—24no—
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4C2×C418yes—
39C2×C3×C22⋊C4C2×C414yes—
40C2×C3×C22⋊C4D418yes—
41C2×C3×C22⋊C4C2318yes—
42C2×C3×C42—24no—
43C3×#99—416no—
44C3×#95—24no—
45C3×#95—24no—
46C23×D4C22×C312yes—
47#534—28no—
48C2×#123C3×C414yescomplemented
49C2×#121C22×C314yes—
50C2×#120C3×C414yescomplemented
51C22×C22⋊C4C22×C311yescharacteristic
52C2×#99C22×C3116yescomplemented
53#388—28no—
54#380C22×C314yescomplemented
55#379C22×C314yes—
56C4×C22⋊C4C22×C314yescomplemented
57C24×C3—24no—
58C24×C3—24no—
59C24×C3—44no—
60C24×C3—82no—
61C24×C3—84no—
62C24×C3C22×C414yes—
63C24×C3C22×C412yes—
64C24×C3C2×D414yes—
65C24×C3C2411yescharacteristic
66C2×C3×D4—28no—
67C2×C3×D4—28no—
68C2×C3×D4—216no—
69C2×C3×D4—28no—
70C2×C3×D4—48no—
71C2×C3×D4—432no—
72C2×C3×D4—84no—
73C22×C3×C4—24no—
74C22×C3×C4—28no—
75C22×C3×C4—44no—
76C22×C3×C4—82no—
77C22×C3×C4C22×C414yes—
78C22×C3×C4C22×C414yes—
79C22×C3×C4C2×D414yes—
80C22×C3×C4C2×D414yes—
81C3×C4⋊C4—28no—
82C3×C4⋊C4—84no—
83C3×C22⋊C4—216no—
84C3×C22⋊C4—216no—
85C3×C22⋊C4—84no—
86C3×C22⋊C4—84no—
87C3×C22⋊C4C22⋊C4116yes—
88C3×C42—84no—
89C25C2×C3×C412yes—
90C25C23×C311yescharacteristic
91C22×D4—28no—
92C22×D4—48no—
93C22×D4C2×C3×C414yes—
94C22×D4C2×C3×C414yes—
95C22×D4C23×C314yes—
96C23×C4C2×C3×C412yes—
97C23×C4C23×C312yes—
98#123—216no—
99#121—28no—
100#121—28no—
101#120—216no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4C2×C3×C414yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4C2×C3×C418yes—
107C2×C22⋊C4C2×C3×C414yes—
108C2×C22⋊C4C3×D418yes—
109C2×C22⋊C4C23×C318yes—
110C2×C42—24no—
111#99—416no—
112#95—24no—
113#95—24no—
114C23×C3—24no—
115C23×C3—28no—
116C23×C3—24no—
117C23×C3—28no—
118C23×C3—42no—
119C23×C3—44no—
120C23×C3—44no—
121C23×C3—44no—
122C23×C3—48no—
123C23×C3—48no—
124C23×C3—88no—
125C23×C3—88no—
126C23×C3—816no—
127C23×C3C2×C22⋊C414yes—
128C23×C3C23×C411yescharacteristic
129C23×C3C22×D412yes—
130C23×C3H(5,𝔽2)14yes—
131C3×D4—416no—
132C3×D4—416no—
133C3×D4—88no—
134C2×C3×C4—24no—
135C2×C3×C4—28no—
136C2×C3×C4—44no—
137C2×C3×C4—44no—
138C2×C3×C4—44no—
139C2×C3×C4—48no—
140C2×C3×C4—48no—
141C2×C3×C4—48no—
142C2×C3×C4—48no—
143C2×C3×C4C2×C22⋊C418yes—
144C24—24no—
145C24—24no—
146C24—44no—
147C24—82no—
148C24—84no—
149C24C22×C3×C414yes—
150C24C22×C3×C412yes—
151C24C2×C3×D414yes—
152C24C24×C311yescharacteristic
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—216no—
156C2×D4—28no—
157C2×D4—48no—
158C2×D4—432no—
159C2×D4—84no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—44no—
163C22×C4—82no—
164C22×C4C22×C3×C414yes—
165C22×C4C22×C3×C414yes—
166C22×C4C2×C3×D414yes—
167C22×C4C2×C3×D414yes—
168C4⋊C4—28no—
169C4⋊C4—84no—
170C22⋊C4—216no—
171C22⋊C4—216no—
172C22⋊C4—84no—
173C22⋊C4—84no—
174C22⋊C4C3×C22⋊C4116yes—
175C42—84no—
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—24no—
180C22×C3—28no—
181C22×C3—44no—
182C22×C3—48no—
183C22×C3—44no—
184C22×C3—48no—
185C22×C3—88no—
186C22×C3—88no—
187C22×C3—816no—
188C22×C3C22×C22⋊C411yescharacteristic
189C22×C3#51812yes—
190C22×C3#53514yes—
191C3×C4—28no—
192C3×C4—44no—
193C3×C4—88no—
194C23—24no—
195C23—28no—
196C23—24no—
197C23—28no—
198C23—42no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—48no—
203C23—48no—
204C23—88no—
205C23—88no—
206C23—816no—
207C23C2×C3×C22⋊C414yes—
208C23C23×C3×C411yescharacteristic
209C23C22×C3×D412yes—
210C23C3×H(5,𝔽2)14yes—
211D4—416no—
212D4—416no—
213D4—88no—
214C2×C4—24no—
215C2×C4—28no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4C2×C3×C22⋊C418yes—
224C2×C3—24no—
225C2×C3—22no—
226C2×C3—44no—
227C2×C3—84no—
228C2×C3#230911yescharacteristic
229C2×C3#288312yes—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—28no—
234C22—24no—
235C22—44no—
236C22—48no—
237C22—48no—
238C22—44no—
239C22—88no—
240C22—88no—
241C22—816no—
242C22C22×C3×C22⋊C411yescharacteristic
243C22C3×#51812yes—
244C22C3×#53514yes—
245C4—28no—
246C4—44no—
247C4—88no—
248C3#2179511yescharacteristic, direct product factor, Sylow
249C2—24no—
250C2—22no—
251C2—44no—
252C2—84no—
253C2C3×#230911yescharacteristic
254C2C3×#288312yes—
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#534C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2312C2semidirect
3. C3×#1964C2semidirect
4. C3×#1781C2semidirect
5. #21795C3direct (N × H)
8. C2×C3×#123C4semidirect
10. C2×C3×#120C4semidirect
12. C2×C3×#99C22semidirect
14. C3×#380C22semidirect
16. C3×C4×C22⋊C4C22semidirect
18. #2312C2×C3semidirect
19. #1964C2×C3semidirect
20. #1781C2×C3semidirect
48. C2×#123C3×C4semidirect
50. C2×#120C3×C4semidirect
52. C2×#99C22×C3semidirect
54. #380C22×C3semidirect
56. C4×C22⋊C4C22×C3semidirect
248. C3#21795direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C3×#23122
C3×#19642
C3×#17812
#217953