Finite Group Databasebeta

← #18630

Subgroups — #18630

Order 256.

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

normal subgroups only

This group has 641 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#3226C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1641C214yesmaximal, complemented
6#1639C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C2×#137—21no—
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#500—21no—
27#486—22no—
28#486—21no—
29#486C2214yescomplemented
30#480—22no—
31#474C2214yescomplemented
32#462—21no—
33#462C2212yescomplemented
34#461—21no—
35#461C2212yescomplemented
36#431C2211yescharacteristic, complemented
37#423C2211yescharacteristic, complemented
38#419—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—
60#137—44no—
61C2×Q16—22no—
62C2×SD16—22no—
63C2×M16—21no—
64C22×C8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2312yes—
70#127—22no—
71#127—22no—
72#127—22no—
73#127—21no—
74#127—21no—
75#127D411yescharacteristic
76#127D412yes—
77#127C2312yes—
78#124—22no—
79#124—21no—
80#121—22no—
81#121—24no—
82#121—24no—
83#121—21no—
84#121—22no—
85#121—22no—
86#121—22no—
87#121D412yes—
88#120—24no—
89#120—24no—
90#120—24no—
91C4×Q8D411yescharacteristic
92C4×Q8C2311yescharacteristic
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4D411yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2312yes—
101#117—21no—
102#117D411yescharacteristic
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—24no—
108C4⋊C8—21no—
109C4⋊C8C2312yes—
110C4⋊C8C2312yes—
111#103—22no—
112#103—22no—
113#102—28no—
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#41D814yes—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8C2×D411yescharacteristic
134C2×D4—21no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—48no—
148C2×D4—44no—
149C2×D4—416no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—82no—
155C2×D4—84no—
156C2×D4—84no—
157C2×D4C2×D412yes—
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D414yes—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169Q16—44no—
170SD16—44no—
171M16—42no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—42no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4C2×D411yescharacteristic
184C4⋊C4C2×D411yescharacteristic
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C22⋊C4—21no—
192C22⋊C4—22no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2411yescharacteristic
200C23—21no—
201C23—22no—
202C23—24no—
203C23—48no—
204C23—44no—
205C23—82no—
206C23—82no—
207C23—84no—
208C23—84no—
209C23#12011yescharacteristic
210Q8—21no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214Q8—82no—
215Q8C2×D812yes—
216D4—24no—
217D4—21no—
218D4—24no—
219D4—44no—
220D4—48no—
221D4—48no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—44no—
226D4—42no—
227D4—44no—
228D4—44no—
229D4—42no—
230D4—82no—
231D4—84no—
232D4—84no—
233D4C2×D812yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—48no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4C2×D812yes—
252C2×C4AGL(1,ℤ/8)12yes—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C8—42no—
257C8—42no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—44no—
269C22—42no—
270C22—164no—
271C22—168no—
272C22#44712yes—
273C22C2×#12011yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—82no—
281C4#44712yes—
282C4#45512yes—
283C4C22×D811yescharacteristic
284C4C2×AGL(1,ℤ/8)11yescharacteristic
285C4#57712yes—
286C2—22no—
287C2—42no—
288C2—82no—
289C2—84no—
290C2C2×#44711yescharacteristic
291C2#300911yescharacteristic
292C2C2×#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. #3226C2semidirect
3. C2×#486C2semidirect
4. #3039C2semidirect
5. #1641C2semidirect
6. #1639C2semidirect
7. #1545C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
22. C2×C4⋊Q8C22semidirect
29. #486C22semidirect
31. #474C22semidirect
33. #462C22semidirect
35. #461C22semidirect
36. #431C22semidirect
37. #423C22semidirect
40. C2×C4×C8C22semidirect
41. #331C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34982
#32262
C2×#4862
#30392
#16412
#16392
#15452
#14922
#14762