Finite Group Databasebeta

← #18815

Subgroups — #18815

Order 256.

614 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 117 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#3235C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3116C211yesmaximal, characteristic, complemented
5#1641C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1631C212yesmaximal, complemented
8#1629C212yesmaximal, complemented
9#1555C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#550—21no—
14#550C2212yes—
15D42—22no—
16D42—22no—
17D42—21no—
18D42C2212yes—
19#535—22no—
20#535—21no—
21#533C2211yescharacteristic, complemented
22#532—21no—
23#532C2211yescharacteristic
24C2×#127—21no—
25C2×#127C2211yescharacteristic
26C4×#41C2211yescharacteristic
27#501—21no—
28#492C2212yescomplemented
29#490—21no—
30#486C2212yescomplemented
31#485—21no—
32#482—22no—
33#477C2211yescharacteristic, complemented
34#476C2211yescharacteristic, complemented
35#476C2212yescomplemented
36#474C2212yescomplemented
37#462C2211yescharacteristic, complemented
38#461C2211yescharacteristic, complemented
39#460—22no—
40#452—21no—
41#450—22no—
42#441—21no—
43#440—21no—
44#432C2211yescharacteristic, complemented
45#432C2211yescharacteristic, complemented
46#431C2211yescharacteristic, complemented
47#331C2212yescomplemented
48#331C2212yescomplemented
49Q8⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51D4⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53H(5,𝔽2)—44no—
54H(5,𝔽2)—44no—
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—21no—
58C2×#41D411yescharacteristic
59C22×D4—22no—
60C22×D4—24no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4D412yes—
67C2×Q16—21no—
68C2×SD16—21no—
69C2×SD16—24no—
70C4⋊Q8D411yescharacteristic
71C4⋊Q8C2311yescharacteristic
72C4⋊Q8C2311yescharacteristic
73#127—22no—
74#127—22no—
75#127—22no—
76#127—21no—
77#127—21no—
78#127D411yescharacteristic
79#127D412yes—
80#127C2312yes—
81#125—21no—
82#124—22no—
83#124—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86#121—22no—
87#121—24no—
88#121—24no—
89#121—21no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121D412yes—
94#120—24no—
95#120—24no—
96#120—24no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8D411yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—22no—
105C4×D4—21no—
106C4×D4D411yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2312yes—
109#117—21no—
110#117—21no—
111#117D411yescharacteristic
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#107—22no—
116#106—21no—
117#106—22no—
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2312yes—
122#103—21no—
123#103—21no—
124#103—21no—
125#102—21no—
126#102—24no—
127#102—22no—
128#102—22no—
129C22⋊C8—21no—
130C22⋊C8—21no—
131C22⋊C8—21no—
132#97—21no—
133#97—21no—
134C4×C8C2312yes—
135C24—24no—
136C24—22no—
137#41—22no—
138#41—22no—
139#41—48no—
140#41—42no—
141#41—42no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8C2×D411yescharacteristic
148C2×D4—21no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—48no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—416no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—82no—
168C2×D4—84no—
169C2×D4—82no—
170C2×D4—84no—
171C2×D4C2×D412yes—
172C2×D4C2×D411yescharacteristic
173C2×D4C2×D414yes—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182Q16—42no—
183SD16—42no—
184SD16—84no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D412yes—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—22no—
208C22⋊C4—41no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42C2×D411yescharacteristic
214C42C2×D411yescharacteristic
215C42C2×D411yescharacteristic
216C42C2411yescharacteristic
217C23—21no—
218C23—22no—
219C23—24no—
220C23—44no—
221C23—48no—
222C23—82no—
223C23—82no—
224C23—84no—
225C23—84no—
226C23#12011yescharacteristic
227Q8—21no—
228Q8—42no—
229Q8—42no—
230Q8—41no—
231Q8—41no—
232Q8AGL(1,ℤ/8)12yes—
233D4—24no—
234D4—21no—
235D4—24no—
236D4—44no—
237D4—44no—
238D4—42no—
239D4—48no—
240D4—44no—
241D4—42no—
242D4—44no—
243D4—48no—
244D4—42no—
245D4—42no—
246D4—44no—
247D4—82no—
248D4—84no—
249D4—84no—
250D4AGL(1,ℤ/8)12yes—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C8—42no—
274C8—41no—
275C8—41no—
276C8—42no—
277C22—21no—
278C22—22no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—44no—
284C22—42no—
285C22—42no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—164no—
290C22—168no—
291C22C2×#12011yescharacteristic
292C4—21no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4C2×AGL(1,ℤ/8)11yescharacteristic
303C4C2×AGL(1,ℤ/8)11yescharacteristic
304C4#57712yes—
305C2—41no—
306C2—42no—
307C2—82no—
308C2—84no—
309C2#300311yescharacteristic
310C2#300911yescharacteristic
311C2#301911yescharacteristic
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. #3235C2semidirect
3. #3208C2semidirect
4. #3116C2semidirect
5. #1641C2semidirect
6. #1639C2semidirect
7. #1631C2semidirect
8. #1629C2semidirect
9. #1555C2semidirect
10. #1496C2semidirect
11. #1488C2semidirect
21. #533C22semidirect
28. #492C22semidirect
30. #486C22semidirect
33. #477C22semidirect
34. #476C22semidirect
35. #476C22semidirect
36. #474C22semidirect
37. #462C22semidirect
38. #461C22semidirect
44. #432C22semidirect
45. #432C22semidirect
46. #431C22semidirect
47. #331C22semidirect
48. #331C22semidirect
49. Q8⋊C8C22semidirect
50. D4⋊C8C22semidirect
51. D4⋊C8C22semidirect
52. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34982
#32352
#32082
#31162
#16412
#16392
#16312
#16292
#15552
#14962
#14882