Finite Group Databasebeta

← C22×#447 #60347

Subgroups — C22×#447 #60347

Order 256.

3735 conjugacy classes of subgroups; only the 647 normal subgroups are shown.

normal subgroups only

This group has 3735 conjugacy classes of subgroups — too many to draw. Only its 647 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
1C24×D4C211yesmaximal, characteristic, complemented
2C23×D8C212yesmaximal, complemented
3C22×#121C211yesmaximal, characteristic, complemented
4C2×#447C2124yesmaximal, direct product factor
5C22×#102C212yesmaximal, complemented
6C22×C22⋊C8C211yesmaximal, characteristic, complemented
7C26—21no—
8C23×D4—216no—
9C23×D4—21no—
10C23×D4—26no—
11C23×D4—21no—
12C23×D4C2212yes—
13C23×D4C22112yescomplemented
14C23×D4C2211yescharacteristic, complemented
15C24×C4C2211yescharacteristic
16C22×D8—216no—
17C22×D8C22124yescomplemented
18C23×C8C2212yescomplemented
19C2×#121—26no—
20C2×#121C22112yescomplemented
21C22×C4⋊C4C2211yescharacteristic, complemented
22C22×C22⋊C4—21no—
23#447C22164yesdirect product factor
24C2×#102C22124yescomplemented
25C2×C22⋊C8C22112yescomplemented
26C25—216no—
27C25—212no—
28C25—22no—
29C25—21no—
30C25—48no—
31C25—48no—
32C25D411yescharacteristic
33C22×D4—216no—
34C22×D4—296no—
35C22×D4—296no—
36C22×D4—212no—
37C22×D4—26no—
38C22×D4—28no—
39C22×D4—212no—
40C22×D4—224no—
41C22×D4—28no—
42C22×D4—23no—
43C22×D4—48no—
44C22×D4D4116yescomplemented
45C22×D4C23112yes—
46C22×D4C23116yescomplemented
47C22×D4C2316yescomplemented
48C23×C4—28no—
49C23×C4—21no—
50C23×C4D416yes—
51C23×C4D411yescharacteristic
52C23×C4C2316yes—
53C23×C4C2311yescharacteristic
54C2×D8—296no—
55C2×D8C23132yescomplemented
56C22×C8—28no—
57C22×C8C23112yescomplemented
58#121—224no—
59#121C23116yescomplemented
60C2×C4⋊C4—23no—
61C2×C4⋊C4C2316yescomplemented
62C2×C22⋊C4—212no—
63#102C23132yescomplemented
64C22⋊C8C23116yescomplemented
65C24—24no—
66C24—24no—
67C24—296no—
68C24—216no—
69C24—212no—
70C24—216no—
71C24—26no—
72C24—448no—
73C24—4128no—
74C24—48no—
75C24—448no—
76C24—48no—
77C24—44no—
78C24—44no—
79C24—44no—
80C24D818yescomplemented
81C24C2×D416yes—
82C24C2×D411yescharacteristic
83C2×D4—296no—
84C2×D4—2192no—
85C2×D4—264no—
86C2×D4—216no—
87C2×D4—248no—
88C2×D4—212no—
89C2×D4—224no—
90C2×D4—216no—
91C2×D4—224no—
92C2×D4—26no—
93C2×D4—448no—
94C2×D4—412no—
95C2×D4C2×D4148yescomplemented
96C2×D4C2418yes—
97C2×D4C2414yescomplemented
98C22×C4—24no—
99C22×C4—224no—
100C22×C4—224no—
101C22×C4—26no—
102C22×C4—44no—
103C22×C4C2×D416yes—
104C22×C4C2×D4112yes—
105C22×C4C2×D413yes—
106C22×C4C2×D418yes—
107C22×C4C2414yes—
108C22×C4C2413yes—
109D8—264no—
110C2×C8—224no—
111C2×C8C2418yescomplemented
112C4⋊C4—26no—
113C4⋊C4C2414yescomplemented
114C22⋊C4—216no—
115C23—212no—
116C23—212no—
117C23—232no—
118C23—264no—
119C23—248no—
120C23—28no—
121C23—24no—
122C23—4384no—
123C23—432no—
124C23—424no—
125C23—432no—
126C23—424no—
127C23—464no—
128C23—412no—
129C23—412no—
130C23—412no—
131C23—816no—
132C23#12014yes—
133C23C2×D814yes—
134C23C2×D8124yescomplemented
135C23AGL(1,ℤ/8)14yes—
136C23C22×D414yes—
137C23C22×D413yes—
138D4—264no—
139D4—248no—
140D4—28no—
141D4—28no—
142D4—432no—
143D4—412no—
144D4C22×D4116yescomplemented
145C2×C4—212no—
146C2×C4—224no—
147C2×C4—28no—
148C2×C4—24no—
149C2×C4—412no—
150C2×C4#120112yes—
151C2×C4C22×D414yes—
152C2×C4C22×D413yes—
153C2×C4C22×D4112yes—
154C2×C4C2511yescharacteristic
155C8—28no—
156C22—24no—
157C22—248no—
158C22—24no—
159C22—216no—
160C22—4128no—
161C22—496no—
162C22—48no—
163C22—48no—
164C22—44no—
165C22—44no—
166C22—44no—
167C22—824no—
168C22#447116yesdirect product factor
169C22C2×#12016yes—
170C22C22×D816yes—
171C22C22×D818yescomplemented
172C22C2×AGL(1,ℤ/8)16yes—
173C22C23×D411yescharacteristic
174C4—24no—
175C4—44no—
176C4C2×#12016yes—
177C4C23×D412yes—
178C2—28no—
179C2—416no—
180C2—84no—
181C2C2×#447112yesdirect product factor
182C2C22×#12011yescharacteristic
183C2C23×D811yescharacteristic
184C2C22×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC24C2×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×D4C2semidirect
2. C23×D8C2semidirect
3. C22×#121C2semidirect
4. C2×#447C2direct (N × H)
5. C22×#102C2semidirect
6. C22×C22⋊C8C2semidirect
13. C23×D4C22semidirect
14. C23×D4C22semidirect
17. C22×D8C22semidirect
18. C23×C8C22semidirect
20. C2×#121C22semidirect
21. C22×C4⋊C4C22semidirect
23. #447C22direct (N × H)
24. C2×#102C22semidirect
25. C2×C22⋊C8C22semidirect
44. C22×D4D4semidirect
46. C22×D4C23semidirect
47. C22×D4C23semidirect
55. C2×D8C23semidirect
57. C22×C8C23semidirect
59. #121C23semidirect
61. C2×C4⋊C4C23semidirect
63. #102C23semidirect
64. C22⋊C8C23semidirect
80. C24D8semidirect
95. C2×D4C2×D4semidirect
97. C2×D4C24semidirect
111. C2×C8C24semidirect
113. C4⋊C4C24semidirect
134. C23C2×D8semidirect
144. D4C22×D4semidirect
168. C22#447direct (N × H)
171. C22C22×D8semidirect
181. C2C2×#447direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D42
C23×D82
C22×#1212
C2×#4472
C22×#1022
C22×C22⋊C82