Finite Group Databasebeta

← C22×C3×#576 #10681691

Subgroups — C22×C3×#576 #10681691

Order 768.

11462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C212yesmaximal, complemented
2C2×C3×#576C2148yesmaximal, direct product factor
3C22×C3×AGL(1,ℤ/8)C216yesmaximal, complemented
4C22×C3×#135C213yesmaximal, complemented
5C23×C3×D8C213yesmaximal, complemented
6C22×C3×#131C211yesmaximal, characteristic, complemented
7C22×#576C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×H(5,𝔽2)—224no—
9C2×C3×H(5,𝔽2)C22148yescomplemented
10C22×C3×#41—22no—
11C22×C3×#41C2211yescharacteristic
12C22×C3×#41C2216yescomplemented
13C23×C3×D4—26no—
14C23×C3×D4C2216yescomplemented
15C3×#576C221256yesdirect product factor
16C2×C3×AGL(1,ℤ/8)C221144yescomplemented
17C2×C3×#135C22172yescomplemented
18C22×C3×Q16C2211yescharacteristic, complemented
19C22×C3×SD16C2216yescomplemented
20C22×C3×D8C2213yescomplemented
21C22×C3×D8C2216yescomplemented
22C22×C3×D8C22172yescomplemented
23C2×C3×#131C22124yescomplemented
24C22×C3×M16C2213yescomplemented
25C23×C3×C8C2213yescomplemented
26C22×H(5,𝔽2)C2×C312yescomplemented
27C2×#576C2×C3148yesdirect product factor
28C22×AGL(1,ℤ/8)C2×C316yescomplemented
29C22×#135C2×C313yescomplemented
30C23×D8C2×C313yescomplemented
31C22×#131C2×C311yescharacteristic, complemented
32C25×C3—26no—
33C3×H(5,𝔽2)—2192no—
34C3×H(5,𝔽2)C231128yescomplemented
35C2×C3×#41—248no—
36C2×C3×#41—236no—
37C2×C3×#41D4112yes—
38C2×C3×#41C23112yes—
39C2×C3×#41C23172yescomplemented
40C22×C3×Q8D411yescharacteristic
41C22×C3×Q8C2312yes—
42C22×C3×D4—26no—
43C22×C3×D4—2144no—
44C22×C3×D4—26no—
45C22×C3×D4—236no—
46C22×C3×D4D413yes—
47C22×C3×D4C2316yes—
48C22×C3×D4C23172yescomplemented
49C23×C3×C4—26no—
50C23×C3×C4C2313yes—
51C3×AGL(1,ℤ/8)C231384yescomplemented
52C3×#135C231192yescomplemented
53C2×C3×Q16C23112yescomplemented
54C2×C3×SD16C23172yescomplemented
55C2×C3×D8C23136yescomplemented
56C2×C3×D8C23172yescomplemented
57C2×C3×D8C231192yescomplemented
58C3×#131C23164yescomplemented
59C2×C3×M16C23136yescomplemented
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2313yes—
62C22×C3×C8C23136yescomplemented
63C2×H(5,𝔽2)—224no—
64C2×H(5,𝔽2)C22×C3148yescomplemented
65C22×#41—22no—
66C22×#41C22×C311yescharacteristic
67C22×#41C22×C316yescomplemented
68C23×D4—26no—
69C23×D4C22×C316yescomplemented
70#576C22×C31256yesdirect product factor
71C2×AGL(1,ℤ/8)C22×C31144yescomplemented
72C2×#135C22×C3172yescomplemented
73C22×Q16C22×C311yescharacteristic, complemented
74C22×SD16C22×C316yescomplemented
75C22×D8C22×C313yescomplemented
76C22×D8C22×C316yescomplemented
77C22×D8C22×C3172yescomplemented
78C2×#131C22×C3124yescomplemented
79C22×M16C22×C313yescomplemented
80C23×C8C22×C313yescomplemented
81C24×C3—26no—
82C24×C3—272no—
83C24×C3—824no—
84C24×C3C2×D413yes—
85C3×#41—2128no—
86C3×#41—2144no—
87C3×#41C2×D4148yes—
88C3×#41C24116yes—
89C3×#41C24196yescomplemented
90C2×C3×Q8—26no—
91C2×C3×Q8C2×D4112yes—
92C2×C3×Q8C24112yes—
93C2×C3×D4—272no—
94C2×C3×D4—2384no—
95C2×C3×D4—218no—
96C2×C3×D4—272no—
97C2×C3×D4—2144no—
98C2×C3×D4C2×D4136yes—
99C2×C3×D4C24136yes—
100C2×C3×D4C24196yescomplemented
101C22×C3×C4—22no—
102C22×C3×C4—272no—
103C22×C3×C4C2×D4118yes—
104C22×C3×C4C2×D413yes—
105C22×C3×C4C24118yes—
106C22×C3×C4C2411yescharacteristic
107C3×Q16C24116yescomplemented
108C3×SD16C24196yescomplemented
109C3×D8C24148yescomplemented
110C3×D8C24196yescomplemented
111C3×M16C24148yescomplemented
112C2×C3×C8C2416yes—
113C2×C3×C8C24118yes—
114C2×C3×C8C24148yescomplemented
115C25—26no—
116H(5,𝔽2)—2192no—
117H(5,𝔽2)C23×C31128yescomplemented
118C2×#41—248no—
119C2×#41—236no—
120C2×#41C3×D4112yes—
121C2×#41C23×C3112yes—
122C2×#41C23×C3172yescomplemented
123C22×Q8C3×D411yescharacteristic
124C22×Q8C23×C312yes—
125C22×D4—26no—
126C22×D4—2144no—
127C22×D4—26no—
128C22×D4—236no—
129C22×D4C3×D413yes—
130C22×D4C23×C316yes—
131C22×D4C23×C3172yescomplemented
132C23×C4—26no—
133C23×C4C23×C313yes—
134AGL(1,ℤ/8)C23×C31384yescomplemented
135#135C23×C31192yescomplemented
136C2×Q16C23×C3112yescomplemented
137C2×SD16C23×C3172yescomplemented
138C2×D8C23×C3136yescomplemented
139C2×D8C23×C3172yescomplemented
140C2×D8C23×C31192yescomplemented
141#131C23×C3164yescomplemented
142C2×M16C23×C3136yescomplemented
143C22×C8C23×C311yescharacteristic
144C22×C8C23×C313yes—
145C22×C8C23×C3136yescomplemented
146C23×C3—212no—
147C23×C3—236no—
148C23×C3—296no—
149C23×C3—424no—
150C23×C3—8144no—
151C23×C3C22×D4118yes—
152C23×C3C22×D411yescharacteristic
153C3×Q8—212no—
154C3×Q8C22×D4116yes—
155C3×Q8C2518yes—
156C3×D4—296no—
157C3×D4—236no—
158C3×D4—296no—
159C3×D4C22×D4148yes—
160C3×D4C25124yes—
161C2×C3×C4—212no—
162C2×C3×C4—296no—
163C2×C3×C4C22×D4118yes—
164C2×C3×C4C22×D413yes—
165C2×C3×C4C22×D4136yes—
166C2×C3×C4C2513yes—
167C2×C3×C4C25112yes—
168C3×C8C2514yes—
169C3×C8C25112yes—
170C24—26no—
171C24—272no—
172C24—824no—
173C24C2×C3×D413yes—
174#41—2128no—
175#41—2144no—
176#41C2×C3×D4148yes—
177#41C24×C3116yes—
178#41C24×C3196yescomplemented
179C2×Q8—26no—
180C2×Q8C2×C3×D4112yes—
181C2×Q8C24×C3112yes—
182C2×D4—272no—
183C2×D4—2384no—
184C2×D4—218no—
185C2×D4—272no—
186C2×D4—2144no—
187C2×D4C2×C3×D4136yes—
188C2×D4C24×C3136yes—
189C2×D4C24×C3196yescomplemented
190C22×C4—22no—
191C22×C4—272no—
192C22×C4C2×C3×D413yes—
193C22×C4C2×C3×D4118yes—
194C22×C4C24×C311yescharacteristic
195C22×C4C24×C3118yes—
196Q16C24×C3116yescomplemented
197SD16C24×C3196yescomplemented
198D8C24×C3148yescomplemented
199D8C24×C3196yescomplemented
200M16C24×C3148yescomplemented
201C2×C8C24×C316yes—
202C2×C8C24×C3118yes—
203C2×C8C24×C3148yescomplemented
204C22×C3—236no—
205C22×C3—224no—
206C22×C3—472no—
207C22×C3—896no—
208C22×C3#57614yesdirect product factor
209C22×C3C23×D4112yes—
210C22×C3C23×D413yes—
211C3×C4—28no—
212C3×C4C23×D4112yes—
213C3×C4C23×D413yes—
214C3×C4C2611yescharacteristic
215C23—212no—
216C23—236no—
217C23—296no—
218C23—424no—
219C23—8144no—
220C23C22×C3×D4118yes—
221C23C22×C3×D411yescharacteristic
222Q8—212no—
223Q8C22×C3×D4116yes—
224Q8C25×C318yes—
225D4—296no—
226D4—236no—
227D4—296no—
228D4C22×C3×D4148yes—
229D4C25×C3124yes—
230C2×C4—212no—
231C2×C4—296no—
232C2×C4C22×C3×D4118yes—
233C2×C4C22×C3×D413yes—
234C2×C4C22×C3×D4136yes—
235C2×C4C25×C313yes—
236C2×C4C25×C3112yes—
237C8C25×C314yes—
238C8C25×C3112yes—
239C2×C3—212no—
240C2×C3—424no—
241C2×C3C2×#57616yesdirect product factor
242C2×C3C24×D411yescharacteristic
243C22—236no—
244C22—224no—
245C22—472no—
246C22—896no—
247C22C3×#57614yesdirect product factor
248C22C23×C3×D4112yes—
249C22C23×C3×D413yes—
250C4—28no—
251C4C23×C3×D4112yes—
252C4C23×C3×D413yes—
253C4C26×C311yescharacteristic
254C3C22×#57611yescharacteristic, direct product factor, Sylow
255C2—212no—
256C2—424no—
257C2C2×C3×#57616yesdirect product factor
258C2C24×C3×D411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC4C26×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×H(5,𝔽2)C2semidirect
2. C2×C3×#576C2direct (N × H)
3. C22×C3×AGL(1,ℤ/8)C2semidirect
4. C22×C3×#135C2semidirect
5. C23×C3×D8C2semidirect
6. C22×C3×#131C2semidirect
7. C22×#576C3direct (N × H)
9. C2×C3×H(5,𝔽2)C22semidirect
12. C22×C3×#41C22semidirect
14. C23×C3×D4C22semidirect
15. C3×#576C22direct (N × H)
16. C2×C3×AGL(1,ℤ/8)C22semidirect
17. C2×C3×#135C22semidirect
18. C22×C3×Q16C22semidirect
19. C22×C3×SD16C22semidirect
20. C22×C3×D8C22semidirect
21. C22×C3×D8C22semidirect
22. C22×C3×D8C22semidirect
23. C2×C3×#131C22semidirect
24. C22×C3×M16C22semidirect
25. C23×C3×C8C22semidirect
26. C22×H(5,𝔽2)C2×C3semidirect
27. C2×#576C2×C3direct (N × H)
28. C22×AGL(1,ℤ/8)C2×C3semidirect
29. C22×#135C2×C3semidirect
30. C23×D8C2×C3semidirect
31. C22×#131C2×C3semidirect
34. C3×H(5,𝔽2)C23semidirect
39. C2×C3×#41C23semidirect
48. C22×C3×D4C23semidirect
51. C3×AGL(1,ℤ/8)C23semidirect
52. C3×#135C23semidirect
53. C2×C3×Q16C23semidirect
54. C2×C3×SD16C23semidirect
55. C2×C3×D8C23semidirect
56. C2×C3×D8C23semidirect
57. C2×C3×D8C23semidirect
58. C3×#131C23semidirect
59. C2×C3×M16C23semidirect
62. C22×C3×C8C23semidirect
64. C2×H(5,𝔽2)C22×C3semidirect
67. C22×#41C22×C3semidirect
69. C23×D4C22×C3semidirect
70. #576C22×C3direct (N × H)
71. C2×AGL(1,ℤ/8)C22×C3semidirect
72. C2×#135C22×C3semidirect
73. C22×Q16C22×C3semidirect
74. C22×SD16C22×C3semidirect
75. C22×D8C22×C3semidirect
76. C22×D8C22×C3semidirect
77. C22×D8C22×C3semidirect
78. C2×#131C22×C3semidirect
79. C22×M16C22×C3semidirect
80. C23×C8C22×C3semidirect
89. C3×#41C24semidirect
100. C2×C3×D4C24semidirect
107. C3×Q16C24semidirect
108. C3×SD16C24semidirect
109. C3×D8C24semidirect
110. C3×D8C24semidirect
111. C3×M16C24semidirect
114. C2×C3×C8C24semidirect
117. H(5,𝔽2)C23×C3semidirect
122. C2×#41C23×C3semidirect
131. C22×D4C23×C3semidirect
134. AGL(1,ℤ/8)C23×C3semidirect
135. #135C23×C3semidirect
136. C2×Q16C23×C3semidirect
137. C2×SD16C23×C3semidirect
138. C2×D8C23×C3semidirect
139. C2×D8C23×C3semidirect
140. C2×D8C23×C3semidirect
141. #131C23×C3semidirect
142. C2×M16C23×C3semidirect
145. C22×C8C23×C3semidirect
178. #41C24×C3semidirect
189. C2×D4C24×C3semidirect
196. Q16C24×C3semidirect
197. SD16C24×C3semidirect
198. D8C24×C3semidirect
199. D8C24×C3semidirect
200. M16C24×C3semidirect
203. C2×C8C24×C3semidirect
208. C22×C3#576direct (N × H)
241. C2×C3C2×#576direct (N × H)
247. C22C3×#576direct (N × H)
254. C3C22×#576direct (N × H)
257. C2C2×C3×#576direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×C3×#5762
C22×C3×AGL(1,ℤ/8)2
C22×C3×#1352
C23×C3×D82
C22×C3×#1312
C22×#5763