Finite Group Databasebeta

← #18350

Subgroups — #18350

Order 256.

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

normal subgroups only

This group has 638 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#3496C211yesmaximal, characteristic, complemented
2#3192C212yesmaximal, complemented
3#2635C211yesmaximal, characteristic, complemented
4#1616C212yesmaximal, complemented
5#1613C212yesmaximal, complemented
6#1598C212yesmaximal, complemented
7#1595C212yesmaximal, complemented
8#1510C212yesmaximal, complemented
9#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#547—22no—
12#546—24no—
13D42—22no—
14#537C2212yes—
15#534—21no—
16#534C2212yes—
17#532—21no—
18C2×#121—21no—
19C2×#121C2212yes—
20C2×#121C2211yescharacteristic, complemented
21C2×C4×D4C2211yescharacteristic
22#481—22no—
23#481C2214yescomplemented
24#469—24no—
25#447—22no—
26#447C2214yescomplemented
27#407C2212yescomplemented
28#400C2211yescharacteristic, complemented
29#399C2211yescharacteristic, complemented
30#394—21no—
31#394C2211yescharacteristic, complemented
32#392—21no—
33#392C2211yescharacteristic, complemented
34#388—21no—
35#386—21no—
36#381C2211yescharacteristic, complemented
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yescomplemented
41H(5,𝔽2)—44no—
42H(5,𝔽2)—44no—
43C2×#41—24no—
44C2×#41—22no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C22×D4D412yescomplemented
53C22×D4C2311yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4C2311yescharacteristic
56C2×D8—24no—
57C2×M16—22no—
58C4⋊Q8—21no—
59#127—21no—
60#124—22no—
61#123—24no—
62#123—24no—
63C22⋊Q8—24no—
64#121—22no—
65#121—22no—
66#121—24no—
67#121—22no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121—22no—
72#121—22no—
73#121D414yescomplemented
74#121C2314yes—
75#120—24no—
76#120—24no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D412yescomplemented
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4C2311yescharacteristic
96C2×C42D411yescharacteristic
97#106—24no—
98#102—24no—
99#102—24no—
100C22⋊C8—22no—
101C22⋊C8C2314yes—
102#95—21no—
103#95—21no—
104#95—21no—
105#95C2311yescharacteristic
106#95C2311yescharacteristic
107C24—22no—
108C24—22no—
109C24—21no—
110C24C2×D411yescharacteristic
111C24C2×D411yescharacteristic
112#41—44no—
113#41—416no—
114#41—44no—
115C2×Q8—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—416no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—81no—
135C2×D4—82no—
136C2×D4—82no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4—81no—
140C2×D4—81no—
141C2×D4C2×D414yes—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159D8—84no—
160M16—82no—
161C2×C8—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—81no—
168C4⋊C4C2×D414yes—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—22no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C42—21no—
183C42—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—24no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—44no—
194C23—44no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—82no—
203C23—82no—
204C23—82no—
205C23—82no—
206C23#12012yes—
207C23AGL(1,ℤ/8)12yes—
208C23C22×D411yescharacteristic
209Q8—44no—
210D4—28no—
211D4—44no—
212D4—44no—
213D4—44no—
214D4—44no—
215D4—48no—
216D4—48no—
217D4—44no—
218D4—44no—
219D4—82no—
220D4—82no—
221D4—84no—
222D4—84no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4AGL(1,ℤ/8)12yes—
251C2×C4C22×D412yes—
252C8—44no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—44no—
263C22—44no—
264C22—44no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—164no—
276C22—164no—
277C22C2×#12011yescharacteristic
278C22C2×AGL(1,ℤ/8)11yescharacteristic
279C22C2×AGL(1,ℤ/8)11yescharacteristic
280C4—22no—
281C4—22no—
282C4—21no—
283C4—21no—
284C4—44no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2—84no—
295C2—161no—
296C2#300311yescharacteristic
297C2#302111yescharacteristic
298C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
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
1. #3496C2semidirect
2. #3192C2semidirect
3. #2635C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1598C2semidirect
7. #1595C2semidirect
8. #1510C2semidirect
9. #1465C2semidirect
20. C2×#121C22semidirect
23. #481C22semidirect
26. #447C22semidirect
27. #407C22semidirect
28. #400C22semidirect
29. #399C22semidirect
31. #394C22semidirect
33. #392C22semidirect
36. #381C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
40. #323C22semidirect
52. C22×D4D4semidirect
73. #121D4semidirect
83. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#34962
#31922
#26352
#16162
#16132
#15982
#15952
#15102
#14652