Finite Group Databasebeta

← #23914

Subgroups — #23914

Order 256.

853 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 853 conjugacy classes of subgroups — too many to draw. Only its 143 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#3485C211yesmaximal, characteristic
2C2×#536C211yesmaximal, characteristic, complemented
3C2×#455C212yesmaximal, complemented
4#3004C212yesmaximal, complemented
5#2005C214yesmaximal, complemented
6#1897C212yesmaximal, complemented
7#1885C212yesmaximal, complemented
8#1785C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic
10C22×#41C2211yescharacteristic
11C23×Q8C2211yescharacteristic, complemented
12C2×#137C2214yes—
13#547—24no—
14#546—24no—
15#536—28no—
16#536C2212yes—
17#534—22no—
18C2×C4⋊Q8C2212yescomplemented
19C2×#124C2212yes—
20#525C2212yes—
21C2×#123—22no—
22C2×#123—22no—
23C2×C22⋊Q8—22no—
24#518C2211yescharacteristic
25C2×#117C2211yescharacteristic, complemented
26#455—28no—
27#451—24no—
28#448—24no—
29C2×#104C2214yes—
30#419C2214yescomplemented
31C2×#101C2212yescomplemented
32C2×#100C2212yes—
33#407C2212yescomplemented
34#339C2214yescomplemented
35H(5,𝔽2)—24no—
36H(5,𝔽2)—42no—
37C2×#41—22no—
38C2×#41—24no—
39C2×#41—24no—
40C2×#41—42no—
41C2×#41C2312yes—
42C22×Q8—24no—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—42no—
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×D4—22no—
49C22×D4—22no—
50C22×D4D412yes—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4D412yes—
54C23×C4C2311yescharacteristic
55#137—28no—
56#137—44no—
57C2×Q16—24no—
58C2×SD16—24no—
59C2×M16C2314yes—
60C4⋊Q8—24no—
61C4⋊Q8—22no—
62C4⋊Q8—24no—
63#124—24no—
64#124—22no—
65#124D418yescomplemented
66#123—216no—
67#123—28no—
68#123—24no—
69#123—22no—
70#123—24no—
71C22⋊Q8—216no—
72C22⋊Q8—24no—
73C22⋊Q8D414yes—
74#121—28no—
75#121D414yes—
76#120—24no—
77#120—22no—
78C4×D4—24no—
79C4×D4—24no—
80#117—24no—
81#117C2312yes—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C42C2312yes—
90#104—28no—
91#103—28no—
92#101—42no—
93#100—24no—
94C22⋊C8—24no—
95C24—22no—
96C24C2×D412yes—
97C24C2×D411yescharacteristic
98#41—24no—
99#41—24no—
100#41—416no—
101#41—48no—
102C2×Q8—24no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—24no—
108C2×Q8—24no—
109C2×Q8—416no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8C2×D412yes—
113C2×Q8C2×D414yes—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—42no—
122C2×D4—48no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4C2×D412yes—
126C2×D4C2×D414yes—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D414yes—
143C22×C4C2411yescharacteristic
144Q16—48no—
145SD16—48no—
146M16—24no—
147M16—24no—
148C2×C8—24no—
149C4⋊C4—24no—
150C4⋊C4—28no—
151C4⋊C4—216no—
152C4⋊C4—24no—
153C4⋊C4C2×D414yes—
154C22⋊C4—24no—
155C22⋊C4—24no—
156C22⋊C4—28no—
157C22⋊C4—216no—
158C22⋊C4—24no—
159C22⋊C4—22no—
160C22⋊C4—24no—
161C42—22no—
162C42—22no—
163C42C2×D414yes—
164C23—21no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—22no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23—84no—
173C23#12012yes—
174C23#12014yes—
175C23C22×D411yescharacteristic
176Q8—28no—
177Q8—24no—
178Q8—48no—
179Q8—44no—
180Q8—44no—
181D4—28no—
182D4—24no—
183D4—48no—
184D4—48no—
185D4—42no—
186D4—42no—
187D4—84no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—28no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—84no—
206C2×C4#12012yes—
207C2×C4C22×D412yes—
208C2×C4C22×D412yes—
209C2×C4H(5,𝔽2)12yes—
210C8—44no—
211C22—22no—
212C22—22no—
213C22—24no—
214C22—24no—
215C22—44no—
216C22—42no—
217C22—42no—
218C22—44no—
219C22—42no—
220C22—44no—
221C22—82no—
222C22—164no—
223C22C2×#12011yescharacteristic
224C22C2×#12011yescharacteristic
225C22#53411yescharacteristic
226C4—22no—
227C4—44no—
228C4—42no—
229C4—42no—
230C4—44no—
231C4—44no—
232C4D4214yes—
233C2—21no—
234C2—21no—
235C2—42no—
236C2—44no—
237C2—82no—
238C2#240311yescharacteristic
239C2#302212yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3485C2
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. C2×#536C2semidirect
3. C2×#455C2semidirect
4. #3004C2semidirect
5. #2005C2semidirect
6. #1897C2semidirect
7. #1885C2semidirect
8. #1785C2semidirect
11. C23×Q8C22semidirect
18. C2×C4⋊Q8C22semidirect
25. C2×#117C22semidirect
30. #419C22semidirect
31. C2×#101C22semidirect
33. #407C22semidirect
34. #339C22semidirect
65. #124D4semidirect

Maximal subgroups

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

subgroupindex
#34852
C2×#5362
C2×#4552
#30042
#20052
#18972
#18852
#17852