Finite Group Databasebeta

← C2×#14521791 #16859831

Subgroups — C2×#14521791 #16859831

Order 3968.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#14521791C2116yesmaximal, direct product factor
2C2×#11758524C212yesmaximal, complemented
3C2×#11758452C212yesmaximal, complemented
4C2×#11758462C214yesmaximal, complemented
5C2×#11758459C214yesmaximal, complemented
6C2×#11758514C212yesmaximal, complemented
7C2×C31×#124C211yesmaximal, characteristic, complemented
8C24×D31C2212yescomplemented
9C22×Q8×C31C2211yescharacteristic, complemented
10C22×D4×C31C2211yescharacteristic
11C22×C22⋊D31C2214yes—
12C22×D124C2212yes—
13C22×D124C2212yes—
14C22×Dic62C2212yescomplemented
15C31×#124C2218yescomplemented
16C2×C22⋊C4×C31C2214yescomplemented
17C2×C42×C31C2211yescharacteristic, complemented
18C2×C22⋊Dic31C2212yescomplemented
19#11758524C22116yescomplemented
20#11758514C22116yescomplemented
21C2×(C2×D31)⋊C4C2214yescomplemented
22C2×(C2×D31)⋊C4C2214yescomplemented
23C2×(C2×D31)⋊C4C2214yescomplemented
24C2×C4×Dic31C2212yescomplemented
25#11758462C22132yescomplemented
26#11758459C22132yescomplemented
27#11758452C22116yescomplemented
28C24×C31C2312yes—
29C23×D31—44no—
30C23×D31—48no—
31C23×D31C2314yes—
32C23×D31C2318yescomplemented
33C2×Q8×C31—42no—
34C2×Q8×C31C2314yescomplemented
35C2×D4×C31—42no—
36C2×D4×C31C2314yes—
37C22×C4×C31C2312yes—
38C22×C4×C31C2311yescharacteristic
39C22×C4×C31C2312yes—
40C2×C22⋊D31—48no—
41C2×C22⋊D31C23116yes—
42C22×Dic31C2314yes—
43C2×D124—44no—
44C2×D124—44no—
45C2×D124C2318yes—
46C2×D124C2318yes—
47C2×Dic62—44no—
48C2×Dic62C2318yescomplemented
49C22⋊C4×C31C23116yescomplemented
50C42×C31C2314yescomplemented
51C22⋊Dic31C2318yescomplemented
52(C2×D31)⋊C4C23116yescomplemented
53(C2×D31)⋊C4C23116yescomplemented
54(C2×D31)⋊C4C23116yescomplemented
55C4×Dic31C2318yescomplemented
56C23×C31—28no—
57C23×C31—24no—
58C23×C31C2414yes—
59C23×C31C2411yescharacteristic
60C22×D31—28no—
61C22×D31—28no—
62C22×D31—28no—
63C22×D31—48no—
64C22×D31—416no—
65C22×D31—432no—
66C22×D31C2418yes—
67Q8×C31—44no—
68D4×C31—44no—
69C2×C4×C31—24no—
70C2×C4×C31—24no—
71C2×C4×C31—22no—
72C2×C4×C31C2414yes—
73C2×C4×C31C2412yes—
74C2×C4×C31C2414yes—
75C22⋊D31—416no—
76C2×Dic31—28no—
77C2×Dic31C2418yes—
78D124—48no—
79D124—48no—
80Dic62—48no—
81C2×H(3,𝔽4)—311nomaximal, Sylow
82C22×C31—28no—
83C22×C31—24no—
84C22×C31—48no—
85C22×C31H(5,𝔽2)12yes—
86C22×C31H(5,𝔽2)14yes—
87C22×C31C2511yescharacteristic
88C2×D31—28no—
89C2×D31—28no—
90C2×D31—28no—
91C2×D31—432no—
92C2×D31—416no—
93C4×C31—24no—
94C4×C31—24no—
95C4×C31—22no—
96Dic31—28no—
97H(3,𝔽4)—3116no—
98C2×#124—312no—
99C2×#124—314no—
100C2×#124—312no—
101C2×#124D3111yescharacteristic, complemented
102C2×#120—312no—
103C2×#120—314no—
104C2×C31—44no—
105C2×C31H(3,𝔽4)14yescomplemented
106C2×C31C2×H(5,𝔽2)11yescharacteristic
107C2×C31C2×H(5,𝔽2)12yes—
108D31—48no—
109C25—312no—
110C22×Q8—312no—
111C22×Q8C2×D3111yescharacteristic, complemented
112C22×D4—314no—
113C22×D4—312no—
114C22×D4—312no—
115C22×D4C2×D3111yescharacteristic
116#124—3116no—
117#124—3132no—
118#124—3116no—
119#124C2×D3118yescomplemented
120#120—3116no—
121#120—3132no—
122C2×C22⋊C4—312no—
123C2×C22⋊C4—314no—
124C2×C22⋊C4—314no—
125C2×C22⋊C4—314no—
126C2×C22⋊C4C2×D3114yescomplemented
127C2×C42—312no—
128C2×C42C2×D3111yescharacteristic, complemented
129C31C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
130C24—314no—
131C24—318no—
132C24—1244no—
133C24—1248no—
134C24C22×D3112yes—
135C2×Q8—42no—
136C2×Q8—318no—
137C2×Q8—1244no—
138C2×Q8C22×D3114yescomplemented
139C2×D4—42no—
140C2×D4—3116no—
141C2×D4—318no—
142C2×D4—318no—
143C2×D4—1244no—
144C2×D4—1244no—
145C2×D4—1248no—
146C2×D4C22×D3114yes—
147C22×C4—314no—
148C22×C4C22×D3112yes—
149C22×C4C22×D3111yescharacteristic
150C22×C4C22×D3112yes—
151C22⋊C4—318no—
152C22⋊C4—3116no—
153C22⋊C4—3116no—
154C22⋊C4—3116no—
155C22⋊C4C22×D31116yescomplemented
156C42—318no—
157C42C22×D3114yescomplemented
158C23—24no—
159C23—28no—
160C23—318no—
161C23—628no—
162C23—628no—
163C23—628no—
164C23—1248no—
165C23—12416no—
166C23—12432no—
167C23C23×D3114yes—
168C23C23×D3111yescharacteristic
169Q8—44no—
170Q8—1248no—
171D4—44no—
172D4—1248no—
173D4—1248no—
174D4—12416no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—318no—
179C2×C4—628no—
180C2×C4C23×D3114yes—
181C2×C4C23×D3112yes—
182C2×C4C23×D3114yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—628no—
187C22—628no—
188C22—628no—
189C22—12432no—
190C22—12416no—
191C22#1175856112yes—
192C22#1175856614yes—
193C22C24×D3111yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—628no—
198C2—44no—
199C2—1248no—
200C2#1452179114yesdirect product factor
201C2C2×#1175856111yescharacteristic
202C2C2×#1175856612yes—
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC2×C31×#124C2
socleC23×C31C24
hypercenterC2×#124D31
hypocenterC31C2×H(3,𝔽4)
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14521791C2direct (N × H)
2. C2×#11758524C2semidirect
3. C2×#11758452C2semidirect
4. C2×#11758462C2semidirect
5. C2×#11758459C2semidirect
6. C2×#11758514C2semidirect
7. C2×C31×#124C2semidirect
8. C24×D31C22semidirect
9. C22×Q8×C31C22semidirect
14. C22×Dic62C22semidirect
15. C31×#124C22semidirect
16. C2×C22⋊C4×C31C22semidirect
17. C2×C42×C31C22semidirect
18. C2×C22⋊Dic31C22semidirect
19. #11758524C22semidirect
20. #11758514C22semidirect
21. C2×(C2×D31)⋊C4C22semidirect
22. C2×(C2×D31)⋊C4C22semidirect
23. C2×(C2×D31)⋊C4C22semidirect
24. C2×C4×Dic31C22semidirect
25. #11758462C22semidirect
26. #11758459C22semidirect
27. #11758452C22semidirect
32. C23×D31C23semidirect
34. C2×Q8×C31C23semidirect
48. C2×Dic62C23semidirect
49. C22⋊C4×C31C23semidirect
50. C42×C31C23semidirect
51. C22⋊Dic31C23semidirect
52. (C2×D31)⋊C4C23semidirect
53. (C2×D31)⋊C4C23semidirect
54. (C2×D31)⋊C4C23semidirect
55. C4×Dic31C23semidirect
101. C2×#124D31semidirect
105. C2×C31H(3,𝔽4)semidirect
111. C22×Q8C2×D31semidirect
119. #124C2×D31semidirect
126. C2×C22⋊C4C2×D31semidirect
128. C2×C42C2×D31semidirect
129. C31C2×H(3,𝔽4)semidirect
138. C2×Q8C22×D31semidirect
155. C22⋊C4C22×D31semidirect
157. C42C22×D31semidirect
200. C2#14521791direct (N × H)

Maximal subgroups

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

subgroupindex
#145217912
C2×#117585242
C2×#117584522
C2×#117584622
C2×#117584592
C2×#117585142
C2×C31×#1242
C2×H(3,𝔽4)31