Finite Group Databasebeta

← #18631

Subgroups — #18631

Order 256.

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

normal subgroups only

This group has 631 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#3206C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1642C214yesmaximal, complemented
6#1639C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#550—21no—
12#550C2212yes—
13D42—22no—
14D42—22no—
15D42—21no—
16D42C2212yes—
17#535—22no—
18#535—21no—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic, complemented
22C2×#127—21no—
23C2×#127C2211yescharacteristic
24C4×#41C2211yescharacteristic
25#492—21no—
26#487—21no—
27#486—22no—
28#486C2214yescomplemented
29#477—21no—
30#476—21no—
31#475C2212yescomplemented
32#474C2212yescomplemented
33#461C2214yescomplemented
34#448—22no—
35#431C2211yescharacteristic, complemented
36#428—21no—
37#423C2211yescharacteristic, complemented
38#417—21no—
39C2×#102—22no—
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—24no—
61C2×M16—21no—
62C22×C8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D411yescharacteristic
67C4⋊Q8C2312yes—
68#127—22no—
69#127—22no—
70#127—22no—
71#127—21no—
72#127—21no—
73#127D411yescharacteristic
74#127D412yes—
75#127C2312yes—
76#124—22no—
77#124—21no—
78#121—22no—
79#121—24no—
80#121—24no—
81#121—21no—
82#121—22no—
83#121—22no—
84#121—22no—
85#121D412yes—
86#120—24no—
87#120—24no—
88#120—24no—
89C4×Q8D411yescharacteristic
90C4×Q8C2311yescharacteristic
91C4×D4—22no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—22no—
95C4×D4—21no—
96C4×D4D411yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2312yes—
99#117—21no—
100#117D411yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107C4⋊C8—21no—
108C4⋊C8C2312yes—
109C4⋊C8C2312yes—
110#103—22no—
111#103—24no—
112#102—28no—
113#102—22no—
114C22⋊C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2312yes—
118C24—24no—
119C24—22no—
120#41—22no—
121#41—48no—
122#41—44no—
123#41—44no—
124#41—42no—
125#41—42no—
126#41SD1614yescomplemented
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—82no—
131C2×Q8C2×D411yescharacteristic
132C2×D4—21no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—48no—
146C2×D4—44no—
147C2×D4—416no—
148C2×D4—44no—
149C2×D4—44no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—82no—
153C2×D4—84no—
154C2×D4—84no—
155C2×D4C2×D412yes—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D414yes—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167SD16—84no—
168M16—42no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D411yescharacteristic
182C4⋊C4C2×D411yescharacteristic
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—21no—
189C22⋊C4—22no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2411yescharacteristic
197C23—21no—
198C23—22no—
199C23—24no—
200C23—48no—
201C23—44no—
202C23—82no—
203C23—82no—
204C23—84no—
205C23—84no—
206C23#12011yescharacteristic
207Q8—21no—
208Q8—42no—
209Q8—44no—
210Q8—82no—
211Q8C2×SD1612yes—
212D4—24no—
213D4—21no—
214D4—24no—
215D4—44no—
216D4—48no—
217D4—48no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—44no—
222D4—42no—
223D4—44no—
224D4—44no—
225D4—42no—
226D4—82no—
227D4—84no—
228D4—84no—
229D4C2×SD1612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4C2×SD1612yes—
248C2×C4AGL(1,ℤ/8)12yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C8—42no—
253C8—42no—
254C8—42no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—44no—
264C22—44no—
265C22—42no—
266C22—164no—
267C22—168no—
268C22#45012yes—
269C22C2×#12011yescharacteristic
270C4—22no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4—82no—
277C4#45012yes—
278C4#45512yes—
279C4C22×SD1611yescharacteristic
280C4C2×AGL(1,ℤ/8)11yescharacteristic
281C4#57612yes—
282C2—22no—
283C2—42no—
284C2—82no—
285C2—84no—
286C2C2×#45011yescharacteristic
287C2#300811yescharacteristic
288C2C2×#45511yescharacteristic
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. #3206C2semidirect
3. C2×#486C2semidirect
4. #3079C2semidirect
5. #1642C2semidirect
6. #1639C2semidirect
7. #1545C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
21. C2×C4⋊Q8C22semidirect
28. #486C22semidirect
31. #475C22semidirect
32. #474C22semidirect
33. #461C22semidirect
35. #431C22semidirect
37. #423C22semidirect
40. C2×C4×C8C22semidirect
41. #331C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect
126. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#34982
#32062
C2×#4862
#30792
#16422
#16392
#15452
#14922
#14762