Finite Group Databasebeta

← #12793

Subgroups — #12793

Order 256.

404 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×#120C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)C212yesmaximal
3#1800C211yesmaximal, characteristic, complemented
4#1799C211yesmaximal, characteristic, complemented
5#1390C212yesmaximal, complemented
6C24×C4C411yescharacteristic, complemented
7C2×#120C2211yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C2211yescharacteristic
10#423—21no—
11C2×#100C2212yes—
12C2×#99C2212yes—
13#386—21no—
14#386C2211yescharacteristic
15#385—21no—
16#385C411yescharacteristic, complemented
17#380—21no—
18#379C411yescharacteristic, complemented
19C4×C22⋊C4—21no—
20C4×C22⋊C4C411yescharacteristic, complemented
21AΣL(1,𝔽16)—24no—
22#343—21no—
23#342—21no—
24C25C2×C411yescharacteristic
25C22×D4—21no—
26C22×D4C2311yescharacteristic
27C23×C4—21no—
28C23×C4—21no—
29C23×C4—44no—
30C23×C4—41no—
31C23×C4—41no—
32C23×C4C2×C411yescharacteristic
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C2×M16—22no—
36#120—21no—
37#120—21no—
38#120D412yes—
39#120Q812yes—
40C4×D4—41no—
41C4×D4—41no—
42C4×D4—44no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—41no—
45C2×C4⋊C4D411yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4C2×C411yescharacteristic
52C2×C22⋊C4C2×C411yescharacteristic
53C2×C22⋊C4C2×C411yescharacteristic
54C2×C22⋊C4D411yescharacteristic
55C2×C42—21no—
56C2×C42D411yescharacteristic
57C4⋊C8—41no—
58C4⋊C8—41no—
59#100—22no—
60#100—22no—
61#99—22no—
62#99—22no—
63#95—21no—
64#95—21no—
65#95—21no—
66#95—41no—
67C24—21no—
68C24—21no—
69C24—42no—
70C24—41no—
71C24—42no—
72C24—41no—
73C24C4⋊C412yescomplemented
74C24C22×C411yescharacteristic
75C24C2×D411yescharacteristic
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4C2×D411yescharacteristic
83C2×D4C2×Q811yescharacteristic
84C2×D4#4111yescharacteristic
85C2×D4#4111yescharacteristic
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—44no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4C22⋊C411yescharacteristic
110C22×C4C22⋊C411yescharacteristic
111C22×C4C22⋊C411yescharacteristic
112C22×C4C22⋊C411yescharacteristic
113C22×C4C2×D411yescharacteristic
114M16—42no—
115M16—42no—
116C2×C8—42no—
117C4⋊C4—41no—
118C4⋊C4—42no—
119C4⋊C4—82no—
120C22⋊C4—21no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—41no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4C4⋊C412yescomplemented
132C42—21no—
133C42—21no—
134C42—42no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—41no—
143C23—44no—
144C23—42no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—44no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—48no—
156C23—42no—
157C23—41no—
158C23—41no—
159C23C2×C22⋊C411yescharacteristic
160C23C2×C4⋊C411yescharacteristic
161C23#11711yescharacteristic
162C23#12111yescharacteristic
163C23C22⋊Q811yescharacteristic
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—44no—
169D4—44no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#9911yescharacteristic
201C2×C4#9911yescharacteristic
202C2×C4#12111yescharacteristic
203C2×C4C22⋊Q811yescharacteristic
204C8—82no—
205C22—21no—
206C22—21no—
207C22—24no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—41no—
231C22—81no—
232C22#38011yescharacteristic
233C22C2×#9911yescharacteristic
234C22#41011yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—162no—
245C4AΣL(1,𝔽16)12yescomplemented
246C2—21no—
247C2—21no—
248C2—41no—
249C2—44no—
250C2—41no—
251C2—41no—
252C2—41no—
253C2#179911yescharacteristic
254C2C2×AΣL(1,𝔽16)11yescharacteristic
255C2#212011yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1800C2semidirect
4. #1799C2semidirect
5. #1390C2semidirect
6. C24×C4C4semidirect
16. #385C4semidirect
18. #379C4semidirect
20. C4×C22⋊C4C4semidirect
73. C24C4⋊C4semidirect
131. C22⋊C4C4⋊C4semidirect
245. C4AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C4×#1202
C2×AΣL(1,𝔽16)2
#18002
#17992
#13902