Finite Group Databasebeta

← #29620

Subgroups — #29620

Order 256.

662 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 119 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#3449C211yesmaximal, characteristic, complemented
2#3006C212yesmaximal, complemented
3#2500C211yesmaximal, characteristic, complemented
4#2040C212yesmaximal, complemented
5#2039C212yesmaximal, complemented
6#2006C212yesmaximal, complemented
7#2006C212yesmaximal, complemented
8#1822C211yesmaximal, characteristic, complemented
9#1789C212yesmaximal, complemented
10C22×#41C2212yes—
11C2×AGL(1,ℤ/8)C2214yescomplemented
12#537C2212yes—
13#537C2212yes—
14#536—22no—
15#535—22no—
16#534—22no—
17#532—22no—
18#525—24no—
19C2×#121C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#449—24no—
22#448—22no—
23#447—22no—
24C2×#104C2214yescomplemented
25C2×#104C2214yescomplemented
26#417C2214yescomplemented
27#407C2212yes—
28#394—22no—
29#390C2211yescharacteristic, complemented
30#390C2211yescharacteristic, complemented
31#388—22no—
32#384C2211yescharacteristic, complemented
33#382C2211yescharacteristic, complemented
34C4×C22⋊C4C2211yescharacteristic
35#343C2212yescomplemented
36#343C2212yescomplemented
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—44no—
40C2×#41—44no—
41C2×#41D414yes—
42C2×#41C2314yes—
43C22×Q8D412yes—
44C22×D4—22no—
45C22×D4—21no—
46C22×D4D412yes—
47C22×D4C2311yescharacteristic
48C23×C4—22no—
49C23×C4C2311yescharacteristic
50AGL(1,ℤ/8)—44no—
51AGL(1,ℤ/8)—44no—
52C2×SD16—24no—
53C2×D8—24no—
54C2×M16C2314yes—
55C4⋊Q8—22no—
56#127—22no—
57#124—24no—
58#123—28no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61#121—24no—
62#121—24no—
63#121—24no—
64#121—24no—
65#121—22no—
66#121—22no—
67#120—28no—
68C4×D4—24no—
69C4×D4—24no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4D412yes—
81C2×C42D412yes—
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#104—44no—
85#104—44no—
86#103—24no—
87#102—24no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90#95—21no—
91#95—22no—
92C24—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95#41—24no—
96#41—24no—
97#41—24no—
98#41—24no—
99#41—44no—
100#41—44no—
101#41—44no—
102#41—44no—
103#41—48no—
104C2×Q8—22no—
105C2×Q8—44no—
106C2×Q8C2×D414yes—
107C2×D4—24no—
108C2×D4—24no—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—42no—
119C2×D4—82no—
120C2×D4C2×D414yes—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—48no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136SD16—48no—
137D8—48no—
138M16—24no—
139M16—24no—
140C2×C8—24no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—82no—
147C22⋊C4—28no—
148C22⋊C4—28no—
149C22⋊C4—24no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—41no—
153C22⋊C4—41no—
154C22⋊C4—82no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—22no—
159C23—21no—
160C23—21no—
161C23—24no—
162C23—41no—
163C23—41no—
164C23—44no—
165C23—42no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—82no—
171C23—82no—
172C23#12012yes—
173C23C22×D411yescharacteristic
174Q8—24no—
175Q8—24no—
176Q8—44no—
177D4—24no—
178D4—24no—
179D4—44no—
180D4—48no—
181D4—44no—
182D4—44no—
183D4—44no—
184D4—84no—
185D4—84no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—48no—
196C2×C4—41no—
197C2×C4—44no—
198C2×C4—41no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4#12011yescharacteristic
209C2×C4#12011yescharacteristic
210C2×C4C22×D412yes—
211C2×C4H(5,𝔽2)12yes—
212C2×C4H(5,𝔽2)12yes—
213C8—44no—
214C22—21no—
215C22—21no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—44no—
220C22—44no—
221C22—44no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—82no—
226C22—88no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22C2×#12011yescharacteristic
231C22#53511yescharacteristic
232C22#53511yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4#53414yes—
242C2—21no—
243C2—21no—
244C2—42no—
245C2—84no—
246C2—161no—
247C2#261311yescharacteristic
248C2#301811yescharacteristic
249C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3449C2semidirect
2. #3006C2semidirect
3. #2500C2semidirect
4. #2040C2semidirect
5. #2039C2semidirect
6. #2006C2semidirect
7. #2006C2semidirect
8. #1822C2semidirect
9. #1789C2semidirect
11. C2×AGL(1,ℤ/8)C22semidirect
24. C2×#104C22semidirect
25. C2×#104C22semidirect
26. #417C22semidirect
29. #390C22semidirect
30. #390C22semidirect
32. #384C22semidirect
33. #382C22semidirect
35. #343C22semidirect
36. #343C22semidirect

Maximal subgroups

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

subgroupindex
#34492
#30062
#25002
#20402
#20392
#20062
#18222
#17892