Finite Group Databasebeta

← C3×#14657 #10633264

Subgroups — C3×#14657 #10633264

Order 768.

3274 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 3274 conjugacy classes of subgroups — too many to draw. Only its 462 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
1C22×C3×#120C211yesmaximal, characteristic
2C2×C3×#379C212yesmaximal, complemented
3C3×#1440C216yesmaximal, complemented
4C3×#1438C216yesmaximal, complemented
5#14657C311yesmaximal, characteristic, direct product factor, Sylow
6C26×C3C2211yescharacteristic
7C23×C3×D4C2213yes—
8C2×C3×#120—28no—
9C2×C3×#120C418yescomplemented
10C22×C3×C22⋊C4C2213yes—
11C22×C3×C22⋊C4C22112yes—
12C22×C3×C22⋊C4C2214yescomplemented
13C3×#379—28no—
14C2×C3×#95C22112yescomplemented
15C22×#120C2×C311yescharacteristic
16C2×#379C2×C312yescomplemented
17#1440C2×C316yescomplemented
18#1438C2×C316yescomplemented
19C25×C3—412no—
20C25×C3C2×C414yes—
21C25×C3D418yes—
22C25×C3C2311yescharacteristic
23C25×C3C2313yes—
24C22×C3×D4—212no—
25C22×C3×D4—412no—
26C22×C3×D4C2×C4112yes—
27C23×C3×C4C2318yes—
28C23×C3×C4C2313yes—
29C3×#120—232no—
30C2×C3×C22⋊C4—212no—
31C2×C3×C22⋊C4—224no—
32C2×C3×C22⋊C4—424no—
33C2×C3×C22⋊C4—412no—
34C2×C3×C22⋊C4C2×C4112yes—
35C3×#95—412no—
36C26C22×C311yescharacteristic
37C23×D4C22×C313yes—
38C2×#120—28no—
39C2×#120C3×C418yescomplemented
40C22×C22⋊C4C22×C313yes—
41C22×C22⋊C4C22×C3112yes—
42C22×C22⋊C4C22×C314yescomplemented
43#379—28no—
44C2×#95C22×C3112yescomplemented
45C24×C3—224no—
46C24×C3—212no—
47C24×C3—22no—
48C24×C3—212no—
49C24×C3—424no—
50C24×C3—448no—
51C24×C3—448no—
52C24×C3—416no—
53C24×C3C22⋊C4116yescomplemented
54C24×C3C22×C412yes—
55C24×C3C22×C416yes—
56C24×C3C2×D4112yes—
57C24×C3C2411yescharacteristic
58C2×C3×D4—224no—
59C2×C3×D4—424no—
60C2×C3×D4—448no—
61C22×C3×C4—26no—
62C22×C3×C4—212no—
63C22×C3×C4—28no—
64C22×C3×C4—224no—
65C22×C3×C4—224no—
66C22×C3×C4C22×C416yes—
67C3×C22⋊C4—224no—
68C3×C22⋊C4—448no—
69C3×C22⋊C4—416no—
70C25—412no—
71C25C2×C3×C414yes—
72C25C3×D418yes—
73C25C23×C311yescharacteristic
74C25C23×C313yes—
75C22×D4—212no—
76C22×D4—412no—
77C22×D4C2×C3×C4112yes—
78C23×C4C23×C318yes—
79C23×C4C23×C313yes—
80#120—232no—
81C2×C22⋊C4—212no—
82C2×C22⋊C4—224no—
83C2×C22⋊C4—424no—
84C2×C22⋊C4—412no—
85C2×C22⋊C4C2×C3×C4112yes—
86#95—412no—
87C23×C3—212no—
88C23×C3—224no—
89C23×C3—224no—
90C23×C3—224no—
91C23×C3—224no—
92C23×C3—22no—
93C23×C3—212no—
94C23×C3—448no—
95C23×C3—464no—
96C23×C3—4128no—
97C23×C3—436no—
98C23×C3—412no—
99C23×C3—412no—
100C23×C3—412no—
101C23×C3—44no—
102C23×C3C2×C22⋊C4112yes—
103C23×C3C23×C411yescharacteristic
104C23×C3C22×D412yes—
105C23×C3H(5,𝔽2)112yes—
106C3×D4—448no—
107C2×C3×C4—26no—
108C2×C3×C4—212no—
109C2×C3×C4—412no—
110C2×C3×C4—424no—
111C2×C3×C4—424no—
112C2×C3×C4—48no—
113C24—224no—
114C24—212no—
115C24—22no—
116C24—212no—
117C24—424no—
118C24—448no—
119C24—448no—
120C24—416no—
121C24C3×C22⋊C4116yescomplemented
122C24C22×C3×C412yes—
123C24C22×C3×C416yes—
124C24C2×C3×D4112yes—
125C24C24×C311yescharacteristic
126C2×D4—224no—
127C2×D4—424no—
128C2×D4—448no—
129C22×C4—26no—
130C22×C4—212no—
131C22×C4—28no—
132C22×C4—224no—
133C22×C4—224no—
134C22×C4C22×C3×C416yes—
135C22⋊C4—224no—
136C22⋊C4—448no—
137C22⋊C4—416no—
138C22×C3—212no—
139C22×C3—224no—
140C22×C3—464no—
141C22×C3—424no—
142C22×C3—448no—
143C22×C3—46no—
144C22×C3—48no—
145C22×C3—88no—
146C22×C3C22×C22⋊C411yescharacteristic
147C22×C3#51816yes—
148C22×C3#534112yes—
149C22×C3#560112yes—
150C22×C3H(3,𝔽4)14yes—
151C3×C4—46no—
152C3×C4—88no—
153C23—212no—
154C23—224no—
155C23—224no—
156C23—224no—
157C23—224no—
158C23—22no—
159C23—212no—
160C23—448no—
161C23—464no—
162C23—4128no—
163C23—436no—
164C23—412no—
165C23—412no—
166C23—44no—
167C23—412no—
168C23C2×C3×C22⋊C4112yes—
169C23C23×C3×C411yescharacteristic
170C23C22×C3×D412yes—
171C23C3×H(5,𝔽2)112yes—
172D4—448no—
173C2×C4—26no—
174C2×C4—212no—
175C2×C4—412no—
176C2×C4—424no—
177C2×C4—48no—
178C2×C4—424no—
179C2×C3—412no—
180C2×C3—82no—
181C2×C3#230813yes—
182C2×C3#237514yes—
183C2×C3#281218yes—
184C22—212no—
185C22—224no—
186C22—464no—
187C22—424no—
188C22—448no—
189C22—48no—
190C22—46no—
191C22—88no—
192C22C22×C3×C22⋊C411yescharacteristic
193C22C3×#51816yes—
194C22C3×#534112yes—
195C22C3×#560112yes—
196C22C3×H(3,𝔽4)14yes—
197C4—46no—
198C4—88no—
199C3#1465711yescharacteristic, direct product factor, Sylow
200C2—412no—
201C2—82no—
202C2C3×#230813yes—
203C2C3×#237514yes—
204C2C3×#281218yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC22×C3×#120C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#379C2semidirect
3. C3×#1440C2semidirect
4. C3×#1438C2semidirect
5. #14657C3direct (N × H)
9. C2×C3×#120C4semidirect
12. C22×C3×C22⋊C4C22semidirect
14. C2×C3×#95C22semidirect
16. C2×#379C2×C3semidirect
17. #1440C2×C3semidirect
18. #1438C2×C3semidirect
39. C2×#120C3×C4semidirect
42. C22×C22⋊C4C22×C3semidirect
44. C2×#95C22×C3semidirect
53. C24×C3C22⋊C4semidirect
121. C24C3×C22⋊C4semidirect
199. C3#14657direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1202
C2×C3×#3792
C3×#14402
C3×#14382
#146573