Finite Group Databasebeta

← #18339

Subgroups — #18339

Order 256.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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#3449C211yesmaximal, characteristic
2#3234C212yesmaximal, complemented
3#2611C211yesmaximal, characteristic, complemented
4#1616C212yesmaximal, complemented
5#1613C212yesmaximal, complemented
6#1610C212yesmaximal, complemented
7#1607C212yesmaximal, complemented
8#1505C212yesmaximal, complemented
9#1464C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#537—21no—
12#537C2212yes—
13#536—22no—
14#535—22no—
15#534—21no—
16#534C2212yes—
17#532—21no—
18#532C2212yes—
19C2×#125C2211yescharacteristic, complemented
20#525—24no—
21C2×#117C2211yescharacteristic
22#501—22no—
23#490—22no—
24#481C2214yescomplemented
25#480C2214yescomplemented
26#478—24no—
27#432C2212yescomplemented
28#400C2211yescharacteristic, complemented
29#399C2211yescharacteristic
30#398C2211yescharacteristic, complemented
31#397C2211yescharacteristic
32#389—21no—
33#388—21no—
34#387—21no—
35#385—21no—
36#376C2211yescharacteristic, complemented
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yes—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—44no—
46C2×#41—44no—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—22no—
50C23×C4—22no—
51C23×C4D411yescharacteristic
52C4⋊Q8—21no—
53C4⋊Q8D412yes—
54#127—21no—
55#127D412yes—
56#125—22no—
57#125—22no—
58#124—22no—
59#124—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64#121—24no—
65#121—24no—
66#121—22no—
67#121—22no—
68#121—22no—
69#121C2314yes—
70#120—24no—
71#120—24no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4D414yes—
76#117—22no—
77#117—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4D411yescharacteristic
89C2×C22⋊C4C2311yescharacteristic
90C2×C42—21no—
91C2×C42D411yescharacteristic
92C2×C42C2311yescharacteristic
93#107—24no—
94#106—24no—
95C4⋊C8—22no—
96#102—24no—
97#102—24no—
98C22⋊C8C2314yes—
99#97—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95C2311yescharacteristic
106#95C2311yescharacteristic
107C24—22no—
108C24C2×D411yescharacteristic
109#41—48no—
110#41—48no—
111#41—44no—
112#41—44no—
113#41—44no—
114#41—44no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—44no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4C2×D414yes—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—48no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147C2×C8—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4C2×D414yes—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—24no—
165C22⋊C4—22no—
166C22⋊C4—24no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—21no—
170C42—21no—
171C42—42no—
172C42C2×D412yes—
173C42C2×D412yes—
174C23—21no—
175C23—21no—
176C23—24no—
177C23—41no—
178C23—41no—
179C23—44no—
180C23—82no—
181C23—82no—
182C23AGL(1,ℤ/8)12yes—
183C23C22×D411yescharacteristic
184Q8—44no—
185Q8—44no—
186D4—44no—
187D4—44no—
188D4—44no—
189D4—48no—
190D4—44no—
191D4—44no—
192D4—82no—
193D4—82no—
194D4—82no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4#12011yescharacteristic
224C2×C4#12011yescharacteristic
225C2×C4#12012yes—
226C2×C4AGL(1,ℤ/8)12yes—
227C2×C4C22×D412yes—
228C8—44no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—44no—
236C22—44no—
237C22—44no—
238C22—81no—
239C22—88no—
240C22C2×#12011yescharacteristic
241C22C2×AGL(1,ℤ/8)11yescharacteristic
242C22C2×AGL(1,ℤ/8)11yescharacteristic
243C4—21no—
244C4—22no—
245C4—21no—
246C4—22no—
247C4—22no—
248C4—42no—
249C4—44no—
250C4—41no—
251C4—41no—
252C4—81no—
253C4—81no—
254C4—81no—
255C2—21no—
256C2—21no—
257C2—42no—
258C2—84no—
259C2#300311yescharacteristic
260C2#301811yescharacteristic
261C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3234C2semidirect
3. #2611C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1610C2semidirect
7. #1607C2semidirect
8. #1505C2semidirect
9. #1464C2semidirect
19. C2×#125C22semidirect
24. #481C22semidirect
25. #480C22semidirect
27. #432C22semidirect
28. #400C22semidirect
30. #398C22semidirect
36. #376C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect

Maximal subgroups

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

subgroupindex
#34492
#32342
#26112
#16162
#16132
#16102
#16072
#15052
#14642