Finite Group Databasebeta

← #25772

Subgroups — #25772

Order 256.

809 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 145 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#3496C211yesmaximal, characteristic
2C2×#458C214yesmaximal, complemented
3#2501C212yesmaximal, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2022C214yesmaximal, complemented
6#1964C212yesmaximal, complemented
7C4×#99C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)C2212yes—
9#547—22no—
10#546—24no—
11D42—22no—
12#537—21no—
13#534—21no—
14#534C2212yes—
15#532—21no—
16C2×#123C2212yes—
17C2×#123C2218yescomplemented
18C2×#121—21no—
19C2×#121C2212yes—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic, complemented
22#458—216no—
23C2×#99C2212yes—
24C2×#99C2218yescomplemented
25#399—24no—
26#398—21no—
27#397—21no—
28#390—22no—
29#389—21no—
30#384—22no—
31#381C2212yescomplemented
32#380—21no—
33#380C2211yescharacteristic, complemented
34C4×C22⋊C4C2212yescomplemented
35#342C2212yescomplemented
36H(5,𝔽2)—28no—
37H(5,𝔽2)—44no—
38C2×#41—22no—
39C2×#41—24no—
40C22×D4—22no—
41C22×D4—24no—
42C22×D4—21no—
43C22×D4D412yes—
44C22×D4C2314yes—
45C22×D4C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C4⋊Q8—21no—
50#127—21no—
51#124—22no—
52#123—24no—
53#123—24no—
54#123—28no—
55#123—216no—
56#123—48no—
57C22⋊Q8—24no—
58#121—22no—
59#121—22no—
60#121—24no—
61#121—22no—
62#121—22no—
63#121—22no—
64#121—22no—
65#121—22no—
66#121—22no—
67#120—24no—
68#120—24no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D412yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—28no—
86C2×C22⋊C4D411yescharacteristic
87C2×C22⋊C4D414yes—
88C2×C22⋊C4C2312yes—
89C2×C22⋊C4C2311yescharacteristic
90C2×C22⋊C4C2314yes—
91C2×C42—22no—
92C2×C42D411yescharacteristic
93#99—216no—
94#99D418yescomplemented
95#95—22no—
96#95—22no—
97#95—21no—
98#95—22no—
99C24—22no—
100C24C2×D414yes—
101C24C2×D411yescharacteristic
102C24C2411yescharacteristic
103#41—416no—
104#41—44no—
105#41—44no—
106C2×Q8—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—28no—
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—28no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—44no—
120C2×D4—416no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—82no—
124C2×D4C2×D414yes—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4C2×D414yes—
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—416no—
151C4⋊C4—42no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—44no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—416no—
165C22⋊C4—44no—
166C22⋊C4C2×D418yescomplemented
167C42—21no—
168C42—21no—
169C42—44no—
170C23—21no—
171C23—21no—
172C23—24no—
173C23—28no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—44no—
179C23—44no—
180C23—44no—
181C23—82no—
182C23—82no—
183C23#12712yes—
184C23C22×D411yescharacteristic
185C23C22×D412yes—
186C23C22×D412yes—
187C23H(5,𝔽2)12yes—
188Q8—44no—
189D4—28no—
190D4—44no—
191D4—48no—
192D4—44no—
193D4—44no—
194D4—48no—
195D4—44no—
196D4—82no—
197D4—82no—
198D4—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—28no—
207C2×C4—28no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—48no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—48no—
224C2×C4—81no—
225C2×C4#12011yescharacteristic
226C2×C4#12012yes—
227C2×C4#12011yescharacteristic
228C2×C4#12712yes—
229C22—21no—
230C22—22no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—44no—
240C22—44no—
241C22—44no—
242C22—81no—
243C22—88no—
244C22—164no—
245C22C2×#12011yescharacteristic
246C22C2×#12711yescharacteristic
247C22#53511yescharacteristic
248C22D4214yes—
249C4—22no—
250C4—21no—
251C4—22no—
252C4—48no—
253C4—44no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4#45812yescomplemented
259C2—24no—
260C2—21no—
261C2—21no—
262C2—42no—
263C2—84no—
264C2#243311yescharacteristic
265C2C2×#45811yescharacteristic
266C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#458C2semidirect
3. #2501C2semidirect
4. #2486C2semidirect
5. #2022C2semidirect
6. #1964C2semidirect
7. C4×#99C2semidirect
17. C2×#123C22semidirect
21. C22×C4⋊C4C22semidirect
24. C2×#99C22semidirect
31. #381C22semidirect
33. #380C22semidirect
34. C4×C22⋊C4C22semidirect
35. #342C22semidirect
94. #99D4semidirect
166. C22⋊C4C2×D4semidirect
258. C4#458semidirect

Maximal subgroups

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

subgroupindex
#34962
C2×#4582
#25012
#24862
#20222
#19642
C4×#992