Finite Group Databasebeta

← #18871

Subgroups — #18871

Order 256.

622 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 129 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#3246C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1642C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1631C212yesmaximal, complemented
8#1629C212yesmaximal, complemented
9C2×#331C211yesmaximal, characteristic, complemented
10#1480C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, 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#492C2212yescomplemented
28#486C2212yescomplemented
29#479—21no—
30#478—21no—
31#477C2211yescharacteristic, complemented
32#476C2211yescharacteristic, complemented
33#476C2212yescomplemented
34#475—21no—
35#475C2211yescharacteristic, complemented
36#474C2211yescharacteristic, complemented
37#465—22no—
38#461C2212yescomplemented
39#460—22no—
40#460—21no—
41#450—22no—
42#431C2211yescharacteristic, complemented
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic, complemented
45C2×C4⋊C8C2211yescharacteristic, complemented
46#416—21no—
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×SD16—24no—
68C2×SD16—22no—
69C22×C8—21no—
70C22×C8—21no—
71C4⋊Q8D411yescharacteristic
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74#127—22no—
75#127—22no—
76#127—22no—
77#127—21no—
78#127—21no—
79#127D411yescharacteristic
80#127D412yes—
81#127C2312yes—
82#125—21no—
83#124—22no—
84#124—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87#121—22no—
88#121—24no—
89#121—24no—
90#121—21no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121D412yes—
95#120—24no—
96#120—24no—
97#120—24no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8D411yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—22no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—22no—
106C4×D4—21no—
107C4×D4D411yescharacteristic
108C4×D4C2311yescharacteristic
109C4×D4C2312yes—
110#117—21no—
111#117—21no—
112#117D411yescharacteristic
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115#107—21no—
116#107—21no—
117#106—21no—
118#106—21no—
119#106—22no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2312yes—
126#103—21no—
127#103—21no—
128#103—22no—
129#102—21no—
130#102—21no—
131#102—24no—
132#102—22no—
133C22⋊C8—21no—
134C4×C8C2312yes—
135C24—24no—
136C24—22no—
137#41—22no—
138#41—48no—
139#41—44no—
140#41—44no—
141#41—42no—
142#41—42no—
143#41SD1614yescomplemented
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—416no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—82no—
170C2×D4—82no—
171C2×D4—82no—
172C2×D4—84no—
173C2×D4C2×D412yes—
174C2×D4C2×D411yescharacteristic
175C2×D4C2×D414yes—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184SD16—82no—
185SD16—84no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—42no—
191C2×C8—81no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D412yes—
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—22no—
209C22⋊C4—21no—
210C22⋊C4—22no—
211C22⋊C4—41no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42C2×D411yescharacteristic
217C42C2×D411yescharacteristic
218C42C2×D411yescharacteristic
219C42C2411yescharacteristic
220C23—21no—
221C23—22no—
222C23—24no—
223C23—48no—
224C23—44no—
225C23—82no—
226C23—82no—
227C23—84no—
228C23—84no—
229C23#12011yescharacteristic
230Q8—21no—
231Q8—42no—
232Q8—42no—
233Q8—41no—
234Q8—41no—
235Q8C2×SD1612yes—
236D4—24no—
237D4—21no—
238D4—22no—
239D4—22no—
240D4—44no—
241D4—48no—
242D4—44no—
243D4—42no—
244D4—48no—
245D4—44no—
246D4—42no—
247D4—42no—
248D4—42no—
249D4—42no—
250D4—42no—
251D4—44no—
252D4—82no—
253D4—84no—
254D4—84no—
255D4C2×SD1612yes—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4#12011yescharacteristic
273C2×C4#12011yescharacteristic
274C2×C4#12011yescharacteristic
275C2×C4C2×SD1612yes—
276C2×C4AGL(1,ℤ/8)12yes—
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C22×D411yescharacteristic
280C8—42no—
281C8—41no—
282C8—41no—
283C8—42no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—164no—
296C22—168no—
297C22#45012yes—
298C22C2×#12011yescharacteristic
299C4—22no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4#45012yes—
309C4C22×SD1611yescharacteristic
310C4C2×AGL(1,ℤ/8)11yescharacteristic
311C4#57611yescharacteristic
312C4#57711yescharacteristic
313C2—22no—
314C2—42no—
315C2—82no—
316C2—84no—
317C2C2×#45011yescharacteristic
318C2#301211yescharacteristic
319C2#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. #3246C2semidirect
3. #3208C2semidirect
4. #3076C2semidirect
5. #1642C2semidirect
6. #1639C2semidirect
7. #1631C2semidirect
8. #1629C2semidirect
9. C2×#331C2semidirect
10. #1480C2semidirect
11. #1477C2semidirect
21. #533C22semidirect
27. #492C22semidirect
28. #486C22semidirect
31. #477C22semidirect
32. #476C22semidirect
33. #476C22semidirect
35. #475C22semidirect
36. #474C22semidirect
38. #461C22semidirect
42. #431C22semidirect
44. C2×C4⋊C8C22semidirect
45. C2×C4⋊C8C22semidirect
47. #331C22semidirect
48. #331C22semidirect
49. Q8⋊C8C22semidirect
50. D4⋊C8C22semidirect
51. D4⋊C8C22semidirect
52. D4⋊C8C22semidirect
143. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#34982
#32462
#32082
#30762
#16422
#16392
#16312
#16292
C2×#3312
#14802
#14772