Finite Group Databasebeta

← #33520

Subgroups — #33520

Order 256.

596 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 129 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×#135C211yesmaximal, characteristic, complemented
2C2×#509C212yesmaximal, complemented
3C2×#466C211yesmaximal, characteristic, complemented
4#2186C214yesmaximal, complemented
5#2185C212yesmaximal, complemented
6#2184C212yesmaximal, complemented
7#2139C212yesmaximal, complemented
8#2112C211yesmaximal, characteristic, complemented
9C22×#41—21no—
10C2×#135—28no—
11C2×#135—21no—
12C2×#135—21no—
13C2×#135C2212yes—
14C2×#135C2212yescomplemented
15C22×Q16C2211yescharacteristic, complemented
16C22×SD16—21no—
17C22×D8C2211yescharacteristic, complemented
18C22×D8C2211yescharacteristic, complemented
19C23×C8C2211yescharacteristic
20C2×#121—21no—
21#509—28no—
22C2×SD32C2214yescomplemented
23C2×D16C2214yescomplemented
24C2×M32C2212yescomplemented
25#466—21no—
26#466—21no—
27#466C2212yes—
28#466C2212yescomplemented
29C2×#107C2211yescharacteristic, complemented
30C2×#102—21no—
31#358C2214yescomplemented
32#357C2214yescomplemented
33C22⋊C16C2212yescomplemented
34C22⋊C16C2212yescomplemented
35C2×#41—28no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—44no—
39C2×#41—41no—
40C2×#41—41no—
41C22×Q8—21no—
42C22×D4—21no—
43C22×D4—21no—
44C22×D4—41no—
45C22×D4—41no—
46C23×C4—41no—
47C23×C4D411yescharacteristic
48#135—28no—
49#135—28no—
50#135—24no—
51#135—24no—
52#135—24no—
53#135D418yescomplemented
54C2×Q16—24no—
55C2×Q16—21no—
56C2×Q16C2312yes—
57C2×SD16—28no—
58C2×SD16—22no—
59C2×SD16—22no—
60C2×D8—24no—
61C2×D8—21no—
62C2×D8—24no—
63C2×D8—21no—
64C2×D8C2312yes—
65C2×D8C2312yes—
66C22×C8—24no—
67C22×C8D411yescharacteristic
68C22×C8D411yescharacteristic
69C22×C8D411yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#121—22no—
74#121—22no—
75#121—41no—
76#121—41no—
77C2×C4⋊C4—21no—
78C2×C22⋊C4—41no—
79SD32—28no—
80D16—28no—
81M32—24no—
82C2×C16C2314yes—
83#107—21no—
84#107C2312yes—
85#102—22no—
86#102—22no—
87C24—41no—
88C24—41no—
89C24D811yescharacteristic
90#41—28no—
91#41—28no—
92#41—48no—
93#41—44no—
94#41—48no—
95#41—44no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×Q8—41no—
100C2×D4—24no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—42no—
104C2×D4—48no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—84no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—48no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—41no—
121C22×C4D811yescharacteristic
122C22×C4D811yescharacteristic
123C22×C4D811yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127Q16—24no—
128Q16—22no—
129Q16C2×D414yescomplemented
130SD16—28no—
131SD16—28no—
132D8—24no—
133D8—22no—
134D8—24no—
135D8—42no—
136D8C2×D414yescomplemented
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D411yescharacteristic
144C2×C8C2×D411yescharacteristic
145C2×C8C2×D414yes—
146C2×C8C2411yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—41no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C16—24no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—44no—
156C23—42no—
157C23—42no—
158C23—82no—
159C23—82no—
160C23—82no—
161C23—82no—
162C23—82no—
163C23C2×D811yescharacteristic
164C23C2×D811yescharacteristic
165C23C2×D811yescharacteristic
166Q8—24no—
167Q8—44no—
168Q8—42no—
169D4—24no—
170D4—44no—
171D4—48no—
172D4—48no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—84no—
177D4—84no—
178D4—82no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—48no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4#12012yes—
193C2×C4C2×D811yescharacteristic
194C2×C4C2×D811yescharacteristic
195C2×C4C2×D811yescharacteristic
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4C22×D411yescharacteristic
198C8—21no—
199C8—21no—
200C8#12012yes—
201C8C22×D412yes—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—44no—
211C22—88no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—164no—
218C22#44712yes—
219C22#50912yescomplemented
220C22C22×D811yescharacteristic
221C4—21no—
222C4—21no—
223C4—44no—
224C4—82no—
225C4#44712yes—
226C4C2×#12011yescharacteristic
227C4C2×AGL(1,ℤ/8)11yescharacteristic
228C2—22no—
229C2—22no—
230C2—41no—
231C2—84no—
232C2—162no—
233C2C2×#44711yescharacteristic
234C2C2×#50911yescharacteristic
235C2#341411yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#135C2semidirect
2. C2×#509C2semidirect
3. C2×#466C2semidirect
4. #2186C2semidirect
5. #2185C2semidirect
6. #2184C2semidirect
7. #2139C2semidirect
8. #2112C2semidirect
14. C2×#135C22semidirect
15. C22×Q16C22semidirect
17. C22×D8C22semidirect
18. C22×D8C22semidirect
22. C2×SD32C22semidirect
23. C2×D16C22semidirect
24. C2×M32C22semidirect
28. #466C22semidirect
29. C2×#107C22semidirect
31. #358C22semidirect
32. #357C22semidirect
33. C22⋊C16C22semidirect
34. C22⋊C16C22semidirect
53. #135D4semidirect
129. Q16C2×D4semidirect
136. D8C2×D4semidirect
219. C22#509semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#5092
C2×#4662
#21862
#21852
#21842
#21392
#21122