Finite Group Databasebeta

← #18203

Subgroups — #18203

Order 256.

631 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 129 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×#481C212yesmaximal, complemented
3#2487C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1613C214yesmaximal, complemented
6#1519C212yesmaximal, complemented
7#1459C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9#547—22no—
10#546—24no—
11D42—22no—
12#537C2212yes—
13#534—21no—
14#534C2212yes—
15#532—21no—
16C2×#121—21no—
17C2×#121C2212yes—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic, complemented
20#481—24no—
21#481C2218yescomplemented
22C2×#106—22no—
23C2×#102—22no—
24C2×C22⋊C8C2212yescomplemented
25#400C2212yescomplemented
26#399C2212yescomplemented
27#389—21no—
28#386—21no—
29#385—21no—
30#380—21no—
31#380C2211yescharacteristic, complemented
32#327C2218yescomplemented
33#324C2212yescomplemented
34#323C2212yescomplemented
35H(5,𝔽2)—44no—
36H(5,𝔽2)—44no—
37C2×#41—24no—
38C2×#41—22no—
39C22×D4—24no—
40C22×D4—22no—
41C22×D4—22no—
42C22×D4—21no—
43C22×D4D411yescharacteristic
44C22×D4D412yes—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8—22no—
49C4⋊Q8—21no—
50#127—21no—
51#124—22no—
52#123—24no—
53#123—24no—
54C22⋊Q8—24no—
55#121—22no—
56#121—22no—
57#121—24no—
58#121—22no—
59#121—22no—
60#121—22no—
61#121—22no—
62#121D414yes—
63#121C2314yes—
64#120—24no—
65#120—24no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D412yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4D411yescharacteristic
84C2×C22⋊C4C2311yescharacteristic
85C2×C42D411yescharacteristic
86#106—28no—
87#102—28no—
88C22⋊C8—22no—
89C22⋊C8C2314yes—
90#95—22no—
91#95—22no—
92#95C2312yes—
93C24—22no—
94C24—22no—
95C24C2×D411yescharacteristic
96C24C2×D411yescharacteristic
97#41—44no—
98#41—416no—
99#41—44no—
100C2×Q8—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—44no—
111C2×D4—416no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—82no—
117C2×D4SD1614yes—
118C2×D4C2×D414yes—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2411yescharacteristic
137C2×C8—24no—
138C2×C8—44no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—42no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—82no—
148C4⋊C4C2×D414yes—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—42no—
156C22⋊C4—82no—
157C42—21no—
158C42—21no—
159C23—21no—
160C23—21no—
161C23—24no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—44no—
167C23—44no—
168C23—82no—
169C23—82no—
170C23—82no—
171C23—82no—
172C23#12012yes—
173C23C2×SD1612yes—
174C23C2×SD1612yes—
175C23AGL(1,ℤ/8)12yes—
176C23C22×D411yescharacteristic
177Q8—44no—
178D4—28no—
179D4—44no—
180D4—44no—
181D4—48no—
182D4—44no—
183D4—44no—
184D4—48no—
185D4—44no—
186D4—82no—
187D4—82no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—48no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—84no—
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4C2×SD1612yes—
214C2×C4C22×D412yes—
215C8—44no—
216C22—21no—
217C22—22no—
218C22—21no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—44no—
227C22—44no—
228C22—44no—
229C22—81no—
230C22—164no—
231C22—164no—
232C22#45014yes—
233C22C2×#12011yescharacteristic
234C22C22×SD1611yescharacteristic
235C22C2×AGL(1,ℤ/8)11yescharacteristic
236C4—22no—
237C4—22no—
238C4—21no—
239C4—44no—
240C4—41no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4#45812yes—
245C2—21no—
246C2—21no—
247C2—24no—
248C2—42no—
249C2—84no—
250C2C2×#45011yescharacteristic
251C2#302111yescharacteristic
252C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#481C2semidirect
3. #2487C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1519C2semidirect
7. #1459C2semidirect
19. C22×C4⋊C4C22semidirect
21. #481C22semidirect
24. C2×C22⋊C8C22semidirect
25. #400C22semidirect
26. #399C22semidirect
31. #380C22semidirect
32. #327C22semidirect
33. #324C22semidirect
34. #323C22semidirect

Maximal subgroups

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

subgroupindex
#34962
C2×#4812
#24872
#16162
#16132
#15192
#14592