Finite Group Databasebeta

← #12689

Subgroups — #12689

Order 256.

580 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#525C211yesmaximal, characteristic
2C2×#352C212yesmaximal, complemented
3C2×AΣL(1,𝔽16)C212yesmaximal, complemented
4#1785C211yesmaximal, characteristic, complemented
5#1782C211yesmaximal, characteristic, complemented
6C22×#41C2211yescharacteristic
7C24×C4C411yescharacteristic, complemented
8#525—24no—
9#525—24no—
10C2×#123C411yescharacteristic, complemented
11C2×#123C2211yescharacteristic
12C2×C22⋊Q8—21no—
13C2×#121—21no—
14C2×#120C411yescharacteristic, complemented
15C2×#120C2211yescharacteristic
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×#117C411yescharacteristic, complemented
19C2×#101—21no—
20C2×#100C2212yes—
21C2×#99—21no—
22C2×#99C2212yes—
23#407—21no—
24#381—21no—
25#352—24no—
26AΣL(1,𝔽16)—24no—
27C25C2×C411yescharacteristic
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—42no—
31C2×#41—42no—
32C22×Q8D411yescharacteristic
33C22×D4—21no—
34C22×D4—21no—
35C22×D4C2×C411yescharacteristic
36C22×D4D411yescharacteristic
37C22×D4C2311yescharacteristic
38C23×C4—22no—
39C23×C4—21no—
40C23×C4—22no—
41C23×C4—21no—
42C23×C4—42no—
43C23×C4—42no—
44C23×C4C2×C411yescharacteristic
45C23×C4C2×C411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C2×M16—22no—
49#123—22no—
50#123—22no—
51#123—22no—
52#123D414yes—
53C22⋊Q8—28no—
54#121—28no—
55#120—22no—
56#120—22no—
57#120—22no—
58#120D414yes—
59C4×D4—28no—
60C4×D4—28no—
61#117—22no—
62#117—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4C2×C411yescharacteristic
72C2×C22⋊C4C2×C411yescharacteristic
73C2×C42—21no—
74#101—42no—
75#100—22no—
76#100—22no—
77#99—22no—
78#99—22no—
79#99—42no—
80C22⋊C8—42no—
81#95—42no—
82C24—21no—
83C24—21no—
84C24—41no—
85C24—41no—
86C24—44no—
87C24C22⋊C411yescharacteristic
88C24C22⋊C412yes—
89C24C22×C411yescharacteristic
90C24C2×D411yescharacteristic
91#41—48no—
92#41—48no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—41no—
96C2×D4—24no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4C2×D412yes—
110C2×D4C2×D412yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—48no—
128C22×C4—48no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4C22⋊C411yescharacteristic
137C22×C4C22⋊C411yescharacteristic
138C22×C4C22⋊C412yes—
139C22×C4C22⋊C411yescharacteristic
140C22×C4C2×D411yescharacteristic
141M16—42no—
142M16—42no—
143C2×C8—42no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4C22⋊C412yes—
156C22⋊C4C22⋊C412yes—
157C42—24no—
158C23—24no—
159C23—22no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—44no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—44no—
173C23—42no—
174C23—42no—
175C23—44no—
176C23—48no—
177C23—44no—
178C23—41no—
179C23—41no—
180C23C2×C22⋊C411yescharacteristic
181C23C2×C22⋊C411yescharacteristic
182C23C2×C22⋊C411yescharacteristic
183C23#12012yes—
184Q8—44no—
185D4—42no—
186D4—42no—
187D4—44no—
188D4—48no—
189D4—48no—
190D4—42no—
191D4—42no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—48no—
209C2×C4—48no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4#9912yes—
219C2×C4#9912yes—
220C2×C4#12012yes—
221C8—82no—
222C22—22no—
223C22—24no—
224C22—21no—
225C22—21no—
226C22—22no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22—48no—
237C22—42no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—81no—
244C22—81no—
245C22#37911yescharacteristic
246C22C2×#9911yescharacteristic
247C22C2×#9911yescharacteristic
248C4—22no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—41no—
254C4—42no—
255C4—162no—
256C2—21no—
257C2—21no—
258C2—42no—
259C2—44no—
260C2—41no—
261C2—41no—
262C2—81no—
263C2#178111yescharacteristic
264C2#212012yes—
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#525C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#352C2semidirect
3. C2×AΣL(1,𝔽16)C2semidirect
4. #1785C2semidirect
5. #1782C2semidirect
7. C24×C4C4semidirect
10. C2×#123C4semidirect
14. C2×#120C4semidirect
18. C2×#117C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#3522
C2×AΣL(1,𝔽16)2
#17852
#17822