Finite Group Databasebeta

← C22×C3×#568 #10681681

Subgroups — C22×C3×#568 #10681681

Order 768.

11118 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#41C211yesmaximal, characteristic
2C2×C3×#568C2148yesmaximal, direct product factor
3C22×C3×#131C218yesmaximal, complemented
4C23×C3×M16C216yesmaximal, complemented
5C22×#568C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×#41C414yes—
7C22×C3×#41C4124yes—
8C22×C3×#41C2214yes—
9C22×C3×#41C22124yes—
10C23×C3×Q8C411yescharacteristic
11C23×C3×D4C413yes—
12C24×C3×C4C2213yes—
13C3×#568C221256yesdirect product factor
14C2×C3×#131C221192yescomplemented
15C22×C3×M16C2214yes—
16C22×C3×M16C22112yescomplemented
17C22×C3×M16C221144yescomplemented
18C23×C3×C8C22112yescomplemented
19C23×#41C2×C311yescharacteristic
20C2×#568C2×C3148yesdirect product factor
21C22×#131C2×C318yescomplemented
22C23×M16C2×C316yescomplemented
23C25×C3C2×C413yes—
24C2×C3×#41C2×C4148yes—
25C2×C3×#41C2×C4196yes—
26C2×C3×#41C2×C41192yes—
27C2×C3×#41C23148yes—
28C2×C3×#41C23164yes—
29C22×C3×Q8C2×C414yes—
30C22×C3×Q8C2×C4124yes—
31C22×C3×D4C2×C4112yes—
32C22×C3×D4C2×C4172yes—
33C23×C3×C4C2×C413yes—
34C23×C3×C4C2×C416yes—
35C23×C3×C4C2×C4136yes—
36C23×C3×C4C23136yes—
37C23×C3×C4C2316yes—
38C23×C3×C4C2311yescharacteristic
39C3×#131C231512yescomplemented
40C2×C3×M16C23148yes—
41C2×C3×M16C231144yescomplemented
42C2×C3×M16C231384yescomplemented
43C22×C3×C8C2318yes—
44C22×C3×C8C231144yescomplemented
45C22×#41C3×C414yes—
46C22×#41C3×C4124yes—
47C22×#41C22×C314yes—
48C22×#41C22×C3124yes—
49C23×Q8C3×C411yescharacteristic
50C23×D4C3×C413yes—
51C24×C4C22×C313yes—
52#568C22×C31256yesdirect product factor
53C2×#131C22×C31192yescomplemented
54C22×M16C22×C314yes—
55C22×M16C22×C3112yescomplemented
56C22×M16C22×C31144yescomplemented
57C23×C8C22×C3112yescomplemented
58C24×C3—412no—
59C24×C3C22×C411yescharacteristic
60C24×C3C22×C416yes—
61C24×C3C22×C4136yes—
62C3×#41C22×C41192yes—
63C3×#41C22×C41256yes—
64C3×#41C24164yes—
65C2×C3×Q8C22×C4148yes—
66C2×C3×Q8C22×C4164yes—
67C2×C3×D4C22×C41144yes—
68C2×C3×D4C22×C41192yes—
69C22×C3×C4C22×C411yescharacteristic
70C22×C3×C4C22×C416yes—
71C22×C3×C4C22×C4136yes—
72C22×C3×C4C22×C416yes—
73C22×C3×C4C22×C4172yes—
74C22×C3×C4C22×C41144yes—
75C22×C3×C4C22×C4136yes—
76C22×C3×C4C24136yes—
77C22×C3×C4C2416yes—
78C22×C3×C4C24148yes—
79C22×C3×C4C2411yescharacteristic
80C3×M16C24164yes—
81C3×M16C241192yescomplemented
82C2×C3×C8C24148yes—
83C2×C3×C8C241192yescomplemented
84C25C2×C3×C413yes—
85C2×#41C2×C3×C4148yes—
86C2×#41C2×C3×C4196yes—
87C2×#41C2×C3×C41192yes—
88C2×#41C23×C3148yes—
89C2×#41C23×C3164yes—
90C22×Q8C2×C3×C414yes—
91C22×Q8C2×C3×C4124yes—
92C22×D4C2×C3×C4112yes—
93C22×D4C2×C3×C4172yes—
94C23×C4C2×C3×C413yes—
95C23×C4C2×C3×C416yes—
96C23×C4C2×C3×C4136yes—
97C23×C4C23×C316yes—
98C23×C4C23×C311yescharacteristic
99C23×C4C23×C3136yes—
100#131C23×C31512yescomplemented
101C2×M16C23×C3148yes—
102C2×M16C23×C31144yescomplemented
103C2×M16C23×C31384yescomplemented
104C22×C8C23×C318yes—
105C22×C8C23×C31144yescomplemented
106C23×C3—24no—
107C23×C3—224no—
108C23×C3—472no—
109C23×C3C23×C416yes—
110C23×C3C23×C4136yes—
111C23×C3C23×C4148yes—
112C23×C3C23×C411yescharacteristic
113C3×Q8C23×C4164yes—
114C3×D4C23×C41192yes—
115C2×C3×C4C23×C4112yes—
116C2×C3×C4C23×C416yes—
117C2×C3×C4C23×C413yes—
118C2×C3×C4C23×C4148yes—
119C2×C3×C4C23×C4136yes—
120C2×C3×C4C23×C4172yes—
121C2×C3×C4C23×C4196yes—
122C2×C3×C4C2513yes—
123C2×C3×C4C25124yes—
124C2×C3×C4C2514yes—
125C3×C8C25132yes—
126C24—412no—
127C24C22×C3×C411yescharacteristic
128C24C22×C3×C416yes—
129C24C22×C3×C4136yes—
130#41C22×C3×C41192yes—
131#41C22×C3×C41256yes—
132#41C24×C3164yes—
133C2×Q8C22×C3×C4148yes—
134C2×Q8C22×C3×C4164yes—
135C2×D4C22×C3×C41144yes—
136C2×D4C22×C3×C41192yes—
137C22×C4C22×C3×C416yes—
138C22×C4C22×C3×C411yescharacteristic
139C22×C4C22×C3×C4136yes—
140C22×C4C22×C3×C416yes—
141C22×C4C22×C3×C4136yes—
142C22×C4C22×C3×C4172yes—
143C22×C4C22×C3×C41144yes—
144C22×C4C24×C311yescharacteristic
145C22×C4C24×C3136yes—
146C22×C4C24×C316yes—
147C22×C4C24×C3148yes—
148M16C24×C3164yes—
149M16C24×C31192yescomplemented
150C2×C8C24×C3148yes—
151C2×C8C24×C31192yescomplemented
152C22×C3—212no—
153C22×C3—272no—
154C22×C3—448no—
155C22×C3#56814yesdirect product factor
156C22×C3C24×C414yes—
157C22×C3C24×C4124yes—
158C22×C3C24×C413yes—
159C3×C4C24×C414yes—
160C3×C4C24×C413yes—
161C3×C4C24×C4124yes—
162C3×C4C2611yescharacteristic
163C23—24no—
164C23—224no—
165C23—472no—
166C23C23×C3×C416yes—
167C23C23×C3×C4136yes—
168C23C23×C3×C4148yes—
169C23C23×C3×C411yescharacteristic
170Q8C23×C3×C4164yes—
171D4C23×C3×C41192yes—
172C2×C4C23×C3×C4112yes—
173C2×C4C23×C3×C413yes—
174C2×C4C23×C3×C416yes—
175C2×C4C23×C3×C4148yes—
176C2×C4C23×C3×C4136yes—
177C2×C4C23×C3×C4172yes—
178C2×C4C23×C3×C4196yes—
179C2×C4C25×C313yes—
180C2×C4C25×C3124yes—
181C2×C4C25×C314yes—
182C8C25×C3132yes—
183C2×C3—24no—
184C2×C3—224no—
185C2×C3C2×#56816yesdirect product factor
186C2×C3C25×C411yescharacteristic
187C22—212no—
188C22—272no—
189C22—448no—
190C22C3×#56814yesdirect product factor
191C22C24×C3×C414yes—
192C22C24×C3×C4124yes—
193C22C24×C3×C413yes—
194C4C24×C3×C413yes—
195C4C24×C3×C414yes—
196C4C24×C3×C4124yes—
197C4C26×C311yescharacteristic
198C3C22×#56811yescharacteristic, direct product factor, Sylow
199C2—24no—
200C2—224no—
201C2C2×C3×#56816yesdirect product factor
202C2C25×C3×C411yescharacteristic
subgroupG/H
centerC22×C3×C4C24
commutatorC2C25×C3×C4
Frattini, cosocleC4C26×C3
socleC23×C3C23×C4
prime radicalC23×C3×#41C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#568C2direct (N × H)
3. C22×C3×#131C2semidirect
4. C23×C3×M16C2semidirect
5. C22×#568C3direct (N × H)
13. C3×#568C22direct (N × H)
14. C2×C3×#131C22semidirect
16. C22×C3×M16C22semidirect
17. C22×C3×M16C22semidirect
18. C23×C3×C8C22semidirect
20. C2×#568C2×C3direct (N × H)
21. C22×#131C2×C3semidirect
22. C23×M16C2×C3semidirect
39. C3×#131C23semidirect
41. C2×C3×M16C23semidirect
42. C2×C3×M16C23semidirect
44. C22×C3×C8C23semidirect
52. #568C22×C3direct (N × H)
53. C2×#131C22×C3semidirect
55. C22×M16C22×C3semidirect
56. C22×M16C22×C3semidirect
57. C23×C8C22×C3semidirect
81. C3×M16C24semidirect
83. C2×C3×C8C24semidirect
100. #131C23×C3semidirect
102. C2×M16C23×C3semidirect
103. C2×M16C23×C3semidirect
105. C22×C8C23×C3semidirect
149. M16C24×C3semidirect
151. C2×C8C24×C3semidirect
155. C22×C3#568direct (N × H)
185. C2×C3C2×#568direct (N × H)
190. C22C3×#568direct (N × H)
198. C3C22×#568direct (N × H)
201. C2C2×C3×#568direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#412
C2×C3×#5682
C22×C3×#1312
C23×C3×M162
C22×#5683