Finite Group Databasebeta

← #29621

Subgroups — #29621

Order 256.

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

normal subgroups only

This group has 641 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
2#3007C212yesmaximal, complemented
3#2499C211yesmaximal, characteristic, complemented
4#2040C212yesmaximal, complemented
5#2039C212yesmaximal, complemented
6#2007C212yesmaximal, complemented
7#2007C212yesmaximal, complemented
8#1822C211yesmaximal, characteristic, complemented
9#1789C212yesmaximal, complemented
10C22×#41C2212yes—
11C2×#137C2214yescomplemented
12#537C2212yes—
13#537C2212yes—
14#536—22no—
15#535—22no—
16#534—22no—
17#532—22no—
18#525—24no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#452—24no—
22#451—22no—
23#450—22no—
24C2×#104C2214yescomplemented
25C2×#104C2214yescomplemented
26#417C2214yescomplemented
27#407C2212yes—
28#397—22no—
29#391C2211yescharacteristic, complemented
30#391C2211yescharacteristic, complemented
31#387—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×Q8C2311yescharacteristic
45C22×D4—22no—
46C22×D4D412yes—
47C23×C4—22no—
48C23×C4C2311yescharacteristic
49#137—44no—
50#137—44no—
51C2×Q16—24no—
52C2×SD16—24no—
53C2×M16C2314yes—
54C4⋊Q8—22no—
55#127—22no—
56#124—24no—
57#123—28no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62#121—24no—
63#121—24no—
64#121—24no—
65#121—24no—
66#120—28no—
67C4×D4—24no—
68C4×D4—24no—
69#117—22no—
70#117—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4D412yes—
79C2×C42D412yes—
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#104—44no—
83#104—44no—
84#103—24no—
85#102—24no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—22no—
92C24—22no—
93C24C2×D411yescharacteristic
94#41—24no—
95#41—24no—
96#41—24no—
97#41—24no—
98#41—44no—
99#41—44no—
100#41—44no—
101#41—44no—
102#41—48no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—44no—
108C2×Q8C2×D414yes—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—22no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4C2×D414yes—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—48no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2411yescharacteristic
134Q16—48no—
135SD16—48no—
136M16—24no—
137M16—24no—
138C2×C8—24no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—44no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C22⋊C4—28no—
148C22⋊C4—28no—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C23—21no—
157C23—21no—
158C23—24no—
159C23—41no—
160C23—41no—
161C23—44no—
162C23—82no—
163C23—82no—
164C23#12012yes—
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—24no—
168Q8—44no—
169Q8—44no—
170D4—24no—
171D4—24no—
172D4—44no—
173D4—48no—
174D4—44no—
175D4—44no—
176D4—84no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—41no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C22×D412yes—
205C2×C4H(5,𝔽2)12yes—
206C2×C4H(5,𝔽2)12yes—
207C8—44no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—44no—
214C22—44no—
215C22—44no—
216C22—81no—
217C22—81no—
218C22—88no—
219C22C2×#12011yescharacteristic
220C22#53511yescharacteristic
221C22#53511yescharacteristic
222C4—21no—
223C4—21no—
224C4—42no—
225C4—44no—
226C4—42no—
227C4—42no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4#53414yes—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—84no—
236C2#261311yescharacteristic
237C2#301811yescharacteristic
238C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3449C2
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
2. #3007C2semidirect
3. #2499C2semidirect
4. #2040C2semidirect
5. #2039C2semidirect
6. #2007C2semidirect
7. #2007C2semidirect
8. #1822C2semidirect
9. #1789C2semidirect
11. C2×#137C22semidirect
24. C2×#104C22semidirect
25. C2×#104C22semidirect
26. #417C22semidirect
29. #391C22semidirect
30. #391C22semidirect
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
#30072
#24992
#20402
#20392
#20072
#18222
#17892