Finite Group Databasebeta

← C2×#2883 #60068

Subgroups — C2×#2883 #60068

Order 256.

2669 conjugacy classes of subgroups; only the 991 normal subgroups are shown.

normal subgroups only

This group has 2669 conjugacy classes of subgroups — too many to draw. Only its 991 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×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#518C212yesmaximal, complemented
3#2883C2116yesmaximal, direct product factor
4C2×#410C214yesmaximal, complemented
5C22×#99C218yesmaximal, complemented
6C2×H(5,𝔽2)—28no—
7C2×H(5,𝔽2)—28no—
8C2×H(5,𝔽2)C418yescomplemented
9C2×H(5,𝔽2)C2218yes—
10C22×#41C414yescomplemented
11C22×#41C2212yes—
12C23×D4C414yescomplemented
13C23×D4C2214yes—
14C23×D4C2211yescharacteristic
15#518—28no—
16#518C22116yescomplemented
17C2×C4×D4—28no—
18C2×#117C2214yescomplemented
19C22×C22⋊C4C2218yescomplemented
20#410C22132yescomplemented
21C2×#99C22116yescomplemented
22C2×#99C22164yescomplemented
23C25C2×C414yescomplemented
24C25C2312yes—
25H(5,𝔽2)—264no—
26H(5,𝔽2)—264no—
27C2×#41—216no—
28C2×#41—216no—
29C2×#41—28no—
30C2×#41—28no—
31C2×#41C2×C418yescomplemented
32C2×#41C2×C4132yescomplemented
33C2×#41C2318yes—
34C22×Q8C2×C412yescomplemented
35C22×D4—216no—
36C22×D4—216no—
37C22×D4—216no—
38C22×D4—216no—
39C22×D4C2×C414yes—
40C22×D4C2×C414yes—
41C22×D4C2×C412yescomplemented
42C22×D4C2×C418yescomplemented
43C22×D4C2×C4116yescomplemented
44C22×D4C2×C4132yescomplemented
45C22×D4D418yes—
46C22×D4D418yes—
47C22×D4C2314yes—
48C22×D4C23116yes—
49C22×D4C2314yes—
50C23×C4—28no—
51C23×C4C2×C414yescomplemented
52C23×C4C2×C414yescomplemented
53C23×C4C2311yescharacteristic
54C4×D4—264no—
55#117—28no—
56#117C23116yescomplemented
57C2×C4⋊C4—24no—
58C2×C22⋊C4—28no—
59C2×C22⋊C4—216no—
60C2×C22⋊C4C2318yes—
61C2×C22⋊C4C23132yescomplemented
62C2×C42—24no—
63#99C23164yescomplemented
64C24—28no—
65C24—28no—
66C24—84no—
67C24—88no—
68C24C22×C414yes—
69C24C22×C414yes—
70C24C22×C414yes—
71C24C22×C4116yescomplemented
72C24C2×D418yes—
73C24C2×D418yes—
74C24C2411yescharacteristic
75C24C2414yes—
76#41—2128no—
77#41—232no—
78#41—232no—
79C2×Q8—24no—
80C2×Q8—24no—
81C2×Q8C22×C418yescomplemented
82C2×D4—216no—
83C2×D4—216no—
84C2×D4—264no—
85C2×D4—28no—
86C2×D4—28no—
87C2×D4—264no—
88C2×D4—2128no—
89C2×D4—24no—
90C2×D4—24no—
91C2×D4C22⋊C4116yes—
92C2×D4C22⋊C4116yes—
93C2×D4C22⋊C4132yes—
94C2×D4C22×C418yes—
95C2×D4C22×C4116yes—
96C2×D4C22×C418yescomplemented
97C2×D4C22×C4132yescomplemented
98C2×D4C2×D4132yes—
99C2×D4C2418yes—
100C22×C4—28no—
101C22×C4—28no—
102C22×C4—28no—
103C22×C4—28no—
104C22×C4—28no—
105C22×C4—232no—
106C22×C4—816no—
107C22×C4C22×C414yes—
108C22×C4C22×C412yes—
109C22×C4C22×C412yes—
110C22×C4C22×C4116yescomplemented
111C22×C4C22×C4116yescomplemented
112C22×C4C2×D414yes—
113C22×C4C2×D414yes—
114C22×C4C2412yes—
115C4⋊C4—216no—
116C22⋊C4—28no—
117C22⋊C4—232no—
118C22⋊C4C24116yes—
119C42—216no—
120C23—24no—
121C23—24no—
122C23—232no—
123C23—416no—
124C23—48no—
125C23—42no—
126C23—48no—
127C23—816no—
128C23—84no—
129C23—88no—
130C23—84no—
131C23—816no—
132C23—816no—
133C23C2×C22⋊C418yes—
134C23C2×C22⋊C418yes—
135C23C2×C22⋊C4116yes—
136C23C23×C411yescharacteristic
137C23C23×C418yes—
138C23C23×C411yescharacteristic
139C23C23×C418yes—
140C23C22×D412yes—
141C23C22×D412yes—
142C23C22×D4116yes—
143C23C2511yescharacteristic
144Q8—216no—
145D4—216no—
146D4—264no—
147D4—216no—
148D4—216no—
149D4C2×C22⋊C4164yes—
150C2×C4—24no—
151C2×C4—232no—
152C2×C4—24no—
153C2×C4—232no—
154C2×C4—216no—
155C2×C4—416no—
156C2×C4—832no—
157C2×C4C2×C22⋊C414yes—
158C2×C4C2×C22⋊C418yes—
159C2×C4C2×C22⋊C414yes—
160C2×C4C23×C418yes—
161C2×C4C23×C414yes—
162C2×C4C22×D418yes—
163C22—24no—
164C22—22no—
165C22—216no—
166C22—28no—
167C22—416no—
168C22—48no—
169C22—41no—
170C22—48no—
171C22—42no—
172C22—41no—
173C22—416no—
174C22—48no—
175C22—48no—
176C22—48no—
177C22—832no—
178C22C22×C22⋊C411yescharacteristic
179C22C22×C22⋊C4116yes—
180C22C22×C22⋊C412yes—
181C22C22×C22⋊C411yescharacteristic
182C22C24×C411yescharacteristic
183C22C23×D412yes—
184C4—28no—
185C4—416no—
186C4C22×C22⋊C418yes—
187C2—22no—
188C2—21no—
189C2—21no—
190C2—22no—
191C2—216no—
192C2—48no—
193C2#288312yesdirect product factor
194C2C23×C22⋊C411yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC22C24×C4
Frattini, cosocleC23C25
prime radicalC22×H(5,𝔽2)C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×D4 ◁ G
LECSG ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#518C2semidirect
3. #2883C2direct (N × H)
4. C2×#410C2semidirect
5. C22×#99C2semidirect
8. C2×H(5,𝔽2)C4semidirect
10. C22×#41C4semidirect
12. C23×D4C4semidirect
16. #518C22semidirect
18. C2×#117C22semidirect
19. C22×C22⋊C4C22semidirect
20. #410C22semidirect
21. C2×#99C22semidirect
22. C2×#99C22semidirect
23. C25C2×C4semidirect
31. C2×#41C2×C4semidirect
32. C2×#41C2×C4semidirect
34. C22×Q8C2×C4semidirect
41. C22×D4C2×C4semidirect
42. C22×D4C2×C4semidirect
43. C22×D4C2×C4semidirect
44. C22×D4C2×C4semidirect
51. C23×C4C2×C4semidirect
52. C23×C4C2×C4semidirect
56. #117C23semidirect
61. C2×C22⋊C4C23semidirect
63. #99C23semidirect
71. C24C22×C4semidirect
81. C2×Q8C22×C4semidirect
96. C2×D4C22×C4semidirect
97. C2×D4C22×C4semidirect
110. C22×C4C22×C4semidirect
111. C22×C4C22×C4semidirect
193. C2#2883direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)2
C2×#5182
#28832
C2×#4102
C22×#992