Finite Group Databasebeta

← #18621

Subgroups — #18621

Order 256.

657 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 131 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#3498C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3C2×#492C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1631C214yesmaximal, complemented
6#1629C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C22×SD16—22no—
12#550—21no—
13#550C2212yes—
14D42—22no—
15D42—22no—
16D42—21no—
17D42C2212yes—
18#535—22no—
19#535—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic, complemented
23C2×#127—21no—
24C2×#127C2211yescharacteristic
25C4×#41C2211yescharacteristic
26#498—21no—
27#492—22no—
28#492C2214yescomplemented
29#486—21no—
30#481—22no—
31#477C2212yescomplemented
32#476C2212yescomplemented
33#476C2214yescomplemented
34#475—21no—
35#474—21no—
36#431C2211yescharacteristic, complemented
37#428—21no—
38#423C2211yescharacteristic, complemented
39#417—21no—
40C2×C4×C8C2211yescharacteristic, complemented
41#331C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43D4⋊C8C2212yescomplemented
44D4⋊C8C2214yescomplemented
45H(5,𝔽2)—44no—
46H(5,𝔽2)—44no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—21no—
50C2×#41D411yescharacteristic
51C22×Q8—21no—
52C22×D4—22no—
53C22×D4—24no—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4D412yes—
60C2×SD16—28no—
61C2×SD16—48no—
62C2×M16—21no—
63C22×C8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8C2312yes—
69#127—22no—
70#127—22no—
71#127—22no—
72#127—21no—
73#127—21no—
74#127D411yescharacteristic
75#127D412yes—
76#127C2312yes—
77#124—22no—
78#124—21no—
79#121—22no—
80#121—24no—
81#121—24no—
82#121—21no—
83#121—22no—
84#121—22no—
85#121—22no—
86#121D412yes—
87#120—24no—
88#120—24no—
89#120—24no—
90C4×Q8D411yescharacteristic
91C4×Q8C2311yescharacteristic
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4D411yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2312yes—
100#117—21no—
101#117D411yescharacteristic
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106#107—22no—
107#106—22no—
108#106—24no—
109C4⋊C8—21no—
110C4⋊C8C2312yes—
111C4⋊C8C2312yes—
112#103—22no—
113#102—22no—
114#102—24no—
115C22⋊C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8C2312yes—
119C24—24no—
120C24—22no—
121#41—22no—
122#41—48no—
123#41—44no—
124#41—44no—
125#41—42no—
126#41—42no—
127#41SD1614yescomplemented
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—41no—
132C2×Q8C2×D411yescharacteristic
133C2×D4—21no—
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—48no—
147C2×D4—44no—
148C2×D4—416no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—82no—
154C2×D4—84no—
155C2×D4—84no—
156C2×D4C2×D412yes—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D414yes—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168SD16—416no—
169M16—42no—
170C2×C8—24no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4C2×D411yescharacteristic
182C4⋊C4C2×D412yes—
183C4⋊C4C2×D411yescharacteristic
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2×D411yescharacteristic
197C42C2411yescharacteristic
198C23—21no—
199C23—22no—
200C23—24no—
201C23—48no—
202C23—44no—
203C23—82no—
204C23—82no—
205C23—84no—
206C23—84no—
207C23#12011yescharacteristic
208Q8—21no—
209Q8—44no—
210Q8—42no—
211Q8—82no—
212Q8C2×SD1612yescomplemented
213D4—24no—
214D4—21no—
215D4—24no—
216D4—44no—
217D4—48no—
218D4—48no—
219D4—44no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—42no—
224D4—44no—
225D4—44no—
226D4—42no—
227D4—82no—
228D4—84no—
229D4—84no—
230D4C2×SD1612yescomplemented
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—48no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4#12011yescharacteristic
248C2×C4C2×SD1612yes—
249C2×C4AGL(1,ℤ/8)12yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C8—24no—
254C8—42no—
255C8—42no—
256C22—21no—
257C22—21no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—44no—
266C22—42no—
267C22—164no—
268C22—168no—
269C22#45012yes—
270C22C2×#12011yescharacteristic
271C4—22no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4#45012yes—
279C4#45312yes—
280C4C22×SD1611yescharacteristic
281C4C2×AGL(1,ℤ/8)11yescharacteristic
282C4#57712yes—
283C2—22no—
284C2—42no—
285C2—82no—
286C2—84no—
287C2C2×#45011yescharacteristic
288C2#300911yescharacteristic
289C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3498C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3227C2semidirect
3. C2×#492C2semidirect
4. #3079C2semidirect
5. #1631C2semidirect
6. #1629C2semidirect
7. #1545C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
22. C2×C4⋊Q8C22semidirect
28. #492C22semidirect
31. #477C22semidirect
32. #476C22semidirect
33. #476C22semidirect
36. #431C22semidirect
38. #423C22semidirect
40. C2×C4×C8C22semidirect
41. #331C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect
127. #41SD16semidirect
212. Q8C2×SD16semidirect
230. D4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#34982
#32272
C2×#4922
#30792
#16312
#16292
#15452
#14922
#14762