Finite Group Databasebeta

← C2×#3003 #60354

Subgroups — C2×#3003 #60354

Order 256.

3557 conjugacy classes of subgroups; only the 583 normal subgroups are shown.

normal subgroups only

This group has 3557 conjugacy classes of subgroups — too many to draw. Only its 583 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
2C22×AGL(1,ℤ/8)C212yesmaximal, complemented
3C2×#525C211yesmaximal, characteristic, complemented
4#3003C2116yesmaximal, direct product factor
5C2×#450C214yesmaximal, complemented
6C2×#447C214yesmaximal, complemented
7C2×#418C212yesmaximal, complemented
8C2×#407C211yesmaximal, characteristic, complemented
9C26—21no—
10C22×#41C2211yescharacteristic, complemented
11C23×D4—216no—
12C23×D4—22no—
13C23×D4—21no—
14C23×D4—24no—
15C23×D4C2214yes—
16C23×D4C2212yes—
17C23×D4C2218yescomplemented
18C24×C4C2211yescharacteristic
19C2×AGL(1,ℤ/8)—216no—
20C2×AGL(1,ℤ/8)C22116yescomplemented
21C22×SD16C2214yescomplemented
22C22×D8C2214yescomplemented
23C22×M16C2212yescomplemented
24#525—24no—
25#525C2218yescomplemented
26C2×#123—21no—
27C2×C22⋊Q8C2212yescomplemented
28C2×#121C2212yescomplemented
29C2×#120—21no—
30C2×C4×D4—22no—
31C2×#117C2211yescharacteristic, complemented
32#450C22132yescomplemented
33#447C22132yescomplemented
34#418C22116yescomplemented
35C2×#102C2218yescomplemented
36#407C2218yescomplemented
37C2×C22⋊C8C2214yescomplemented
38C25—216no—
39C25—24no—
40C25—22no—
41C25—28no—
42C25—416no—
43C25D411yescharacteristic
44C2×#41—22no—
45C2×#41—28no—
46C2×#41C2314yescomplemented
47C22×Q8C2311yescharacteristic, complemented
48C22×D4—216no—
49C22×D4—232no—
50C22×D4—264no—
51C22×D4—232no—
52C22×D4—264no—
53C22×D4—22no—
54C22×D4—21no—
55C22×D4—28no—
56C22×D4—24no—
57C22×D4—24no—
58C22×D4—22no—
59C22×D4—28no—
60C22×D4—216no—
61C22×D4—28no—
62C22×D4D4116yescomplemented
63C22×D4C23116yes—
64C22×D4C2314yes—
65C22×D4C2318yes—
66C22×D4C2311yescharacteristic, complemented
67C23×C4—28no—
68C23×C4—21no—
69C23×C4D412yes—
70C23×C4D411yescharacteristic
71C23×C4D414yes—
72C23×C4C2312yes—
73C23×C4C2314yes—
74C23×C4C2311yescharacteristic
75AGL(1,ℤ/8)—264no—
76C2×SD16—216no—
77C2×SD16C23116yescomplemented
78C2×D8—216no—
79C2×D8C23116yescomplemented
80C2×M16—28no—
81C2×M16C2318yescomplemented
82C22×C8C2314yescomplemented
83#123—28no—
84C22⋊Q8—24no—
85C22⋊Q8C2318yescomplemented
86#121—24no—
87#121C2318yescomplemented
88#120—28no—
89C4×D4—216no—
90#117—22no—
91#117C2314yescomplemented
92C2×C4⋊C4—22no—
93C2×C4⋊C4C2312yescomplemented
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C42—21no—
98#102C23132yescomplemented
99C22⋊C8C23116yescomplemented
100C24—24no—
101C24—28no—
102C24—232no—
103C24—264no—
104C24—216no—
105C24—216no—
106C24—28no—
107C24—24no—
108C24—21no—
109C24—4128no—
110C24—416no—
111C24—416no—
112C24—464no—
113C24—416no—
114C24C2×D414yes—
115C24C2×D411yescharacteristic
116C24C2×D412yes—
117#41—216no—
118#41—48no—
119C2×Q8—21no—
120C2×Q8—24no—
121C2×Q8C2412yescomplemented
122C2×D4—232no—
123C2×D4—264no—
124C2×D4—264no—
125C2×D4—2128no—
126C2×D4—264no—
127C2×D4—28no—
128C2×D4—24no—
129C2×D4—216no—
130C2×D4—232no—
131C2×D4—216no—
132C2×D4—28no—
133C2×D4—24no—
134C2×D4—28no—
135C2×D4—216no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—44no—
139C2×D4—48no—
140C2×D4C2×D4116yes—
141C2×D4C2×D4132yescomplemented
142C2×D4C2418yes—
143C2×D4C2412yescomplemented
144C22×C4—24no—
145C22×C4—28no—
146C22×C4—216no—
147C22×C4—28no—
148C22×C4—216no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—44no—
153C22×C4C2×D418yes—
154C22×C4C2×D414yes—
155C22×C4C2×D414yes—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D418yes—
160C22×C4C2414yes—
161C22×C4C2412yes—
162C22×C4C2411yescharacteristic
163SD16—232no—
164D8—232no—
165M16—216no—
166C2×C8—28no—
167C2×C8C2418yescomplemented
168C4⋊C4—28no—
169C4⋊C4—22no—
170C4⋊C4C2414yescomplemented
171C22⋊C4—24no—
172C22⋊C4—28no—
173C22⋊C4—28no—
174C42—24no—
175C23—232no—
176C23—28no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—216no—
181C23—264no—
182C23—216no—
183C23—232no—
184C23—28no—
185C23—22no—
186C23—4256no—
187C23—464no—
188C23—4128no—
189C23—432no—
190C23—432no—
191C23—416no—
192C23—432no—
193C23—42no—
194C23—44no—
195C23#12014yes—
196C23AGL(1,ℤ/8)18yescomplemented
197C23C22×D412yes—
198C23C22×D414yes—
199C23C22×D411yescharacteristic
200Q8—24no—
201Q8—42no—
202D4—264no—
203D4—216no—
204D4—232no—
205D4—28no—
206D4—24no—
207D4—48no—
208D4—416no—
209D4—42no—
210D4C22×D4116yes—
211C2×C4—28no—
212C2×C4—24no—
213C2×C4—216no—
214C2×C4—28no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4#12014yes—
222C2×C4#12018yes—
223C2×C4C22×D414yes—
224C2×C4C22×D412yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D414yes—
227C2×C4C22×D418yes—
228C2×C4C2511yescharacteristic
229C8—28no—
230C22—232no—
231C22—24no—
232C22—28no—
233C22—216no—
234C22—24no—
235C22—24no—
236C22—216no—
237C22—464no—
238C22—4128no—
239C22—432no—
240C22—416no—
241C22—42no—
242C22—44no—
243C22—84no—
244C22C2×#12014yes—
245C22C2×#12012yes—
246C22C2×AGL(1,ℤ/8)14yes—
247C22C2×AGL(1,ℤ/8)18yescomplemented
248C22C23×D411yescharacteristic
249C4—24no—
250C4—42no—
251C4—44no—
252C4C2×#12014yes—
253C4C2×#12012yes—
254C4C23×D412yes—
255C2—24no—
256C2—28no—
257C2—416no—
258C2—82no—
259C2#300314yesdirect product factor
260C2C22×#12011yescharacteristic
261C2C22×AGL(1,ℤ/8)12yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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. C22×AGL(1,ℤ/8)C2semidirect
3. C2×#525C2semidirect
4. #3003C2direct (N × H)
5. C2×#450C2semidirect
6. C2×#447C2semidirect
7. C2×#418C2semidirect
8. C2×#407C2semidirect
10. C22×#41C22semidirect
17. C23×D4C22semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
21. C22×SD16C22semidirect
22. C22×D8C22semidirect
23. C22×M16C22semidirect
25. #525C22semidirect
27. C2×C22⋊Q8C22semidirect
28. C2×#121C22semidirect
31. C2×#117C22semidirect
32. #450C22semidirect
33. #447C22semidirect
34. #418C22semidirect
35. C2×#102C22semidirect
36. #407C22semidirect
37. C2×C22⋊C8C22semidirect
46. C2×#41C23semidirect
47. C22×Q8C23semidirect
62. C22×D4D4semidirect
66. C22×D4C23semidirect
77. C2×SD16C23semidirect
79. C2×D8C23semidirect
81. C2×M16C23semidirect
82. C22×C8C23semidirect
85. C22⋊Q8C23semidirect
87. #121C23semidirect
91. #117C23semidirect
93. C2×C4⋊C4C23semidirect
98. #102C23semidirect
99. C22⋊C8C23semidirect
121. C2×Q8C24semidirect
141. C2×D4C2×D4semidirect
143. C2×D4C24semidirect
167. C2×C8C24semidirect
170. C4⋊C4C24semidirect
196. C23AGL(1,ℤ/8)semidirect
247. C22C2×AGL(1,ℤ/8)semidirect
259. C2#3003direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D42
C22×AGL(1,ℤ/8)2
C2×#5252
#30032
C2×#4502
C2×#4472
C2×#4182
C2×#4072