Finite Group Databasebeta

← #18568

Subgroups — #18568

Order 256.

533 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 115 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#3508C211yesmaximal, characteristic, complemented
2#3254C211yesmaximal, characteristic, complemented
3#3151C211yesmaximal, characteristic, complemented
4#3106C211yesmaximal, characteristic, complemented
5#1631C212yesmaximal, complemented
6#1630C212yesmaximal, complemented
7#1621C212yesmaximal, complemented
8#1620C212yesmaximal, complemented
9#1545C211yesmaximal, characteristic, complemented
10#1489C211yesmaximal, characteristic, complemented
11#1485C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—22no—
13#559—21no—
14#559C2212yes—
15#550—21no—
16#550C2212yes—
17#546—22no—
18#546—22no—
19#540—22no—
20#535—21no—
21#533C2211yescharacteristic
22#532—21no—
23#532—21no—
24#532C2211yescharacteristic
25C2×#127C2211yescharacteristic
26#518—21no—
27C4×#41C2211yescharacteristic
28#496—22no—
29#493C2212yes—
30#492C2212yes—
31#482—21no—
32#477C2211yescharacteristic, complemented
33#477C2212yescomplemented
34#476C2211yescharacteristic, complemented
35#474—21no—
36#468—22no—
37#461—21no—
38#460C2211yescharacteristic, complemented
39#460C2212yescomplemented
40#459C2211yescharacteristic, complemented
41#449—21no—
42C4×SD16—21no—
43C4×D8—21no—
44#431C2211yescharacteristic
45#423C2211yescharacteristic
46C4×M16C2211yescharacteristic
47#331C2212yes—
48#331C2212yes—
49Q8⋊C8C2212yescomplemented
50Q8⋊C8C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—21no—
56C2×#41D411yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4D412yes—
60AGL(1,ℤ/8)—48no—
61C2×SD16—21no—
62C2×SD16—24no—
63C2×SD16—22no—
64C2×D8—24no—
65C2×D8—21no—
66C2×M16—21no—
67C2×M16—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8D411yescharacteristic
70C4⋊Q8C2311yescharacteristic
71#127—22no—
72#127—21no—
73#127—21no—
74#127D412yes—
75#127C2311yescharacteristic
76#127C2312yes—
77#125—21no—
78#124—22no—
79#124—22no—
80#124—22no—
81#124—21no—
82#123—24no—
83#123—24no—
84C22⋊Q8—21no—
85C22⋊Q8D412yes—
86#121—22no—
87#121—22no—
88#121—22no—
89#121—24no—
90#121—24no—
91#121—22no—
92#121—21no—
93#121—21no—
94#120—24no—
95C4×Q8—21no—
96C4×Q8D411yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×Q8C2312yes—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4D411yescharacteristic
110C4×D4C2311yescharacteristic
111#117—21no—
112#117—21no—
113#117D411yescharacteristic
114C2×C22⋊C4—24no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117#107—21no—
118#107—22no—
119#106—21no—
120C4⋊C8—21no—
121C4⋊C8C2312yes—
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—22no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—22no—
130C22⋊C8—21no—
131#97—21no—
132C4×C8—21no—
133C4×C8C2312yes—
134C24—22no—
135#41—41no—
136#41—41no—
137#41—44no—
138#41—82no—
139#41—82no—
140#41—84no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C2×D411yescharacteristic
144C2×Q8C2×D412yes—
145C2×D4—21no—
146C2×D4—24no—
147C2×D4—24no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D414yes—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173SD16—48no—
174SD16—42no—
175SD16—82no—
176D8—48no—
177D8—42no—
178M16—24no—
179M16—81no—
180C2×C8—22no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D412yes—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—22no—
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42C2×D411yescharacteristic
213C42C2411yescharacteristic
214C23—24no—
215C23—21no—
216C23—44no—
217C23—41no—
218C23—82no—
219C23—82no—
220C23#12011yescharacteristic
221Q8—21no—
222Q8—21no—
223Q8—22no—
224Q8—22no—
225Q8—42no—
226Q8—42no—
227D4—21no—
228D4—21no—
229D4—44no—
230D4—44no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—41no—
236D4—41no—
237D4—42no—
238D4—41no—
239D4—42no—
240D4—41no—
241D4—44no—
242D4—82no—
243D4—84no—
244D4—84no—
245D4—82no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4#12011yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C8—24no—
271C8—41no—
272C8—41no—
273C8—42no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—44no—
279C22—41no—
280C22—44no—
281C22—41no—
282C22—44no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—164no—
287C22C2×#12011yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—42no—
294C4—41no—
295C4—42no—
296C4—81no—
297C4#45312yes—
298C4#57511yescharacteristic
299C4#57511yescharacteristic
300C4#57611yescharacteristic
301C4#57711yescharacteristic
302C2—41no—
303C2—81no—
304C2—84no—
305C2—161no—
306C2#300511yescharacteristic
307C2#301211yescharacteristic
308C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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
1. #3508C2semidirect
2. #3254C2semidirect
3. #3151C2semidirect
4. #3106C2semidirect
5. #1631C2semidirect
6. #1630C2semidirect
7. #1621C2semidirect
8. #1620C2semidirect
9. #1545C2semidirect
10. #1489C2semidirect
11. #1485C2semidirect
32. #477C22semidirect
33. #477C22semidirect
34. #476C22semidirect
38. #460C22semidirect
39. #460C22semidirect
40. #459C22semidirect
49. Q8⋊C8C22semidirect
50. Q8⋊C8C22semidirect
51. Q8⋊C8C22semidirect
52. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32542
#31512
#31062
#16312
#16302
#16212
#16202
#15452
#14892
#14852