Finite Group Databasebeta

← #18846

Subgroups — #18846

Order 256.

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

normal subgroups only

This group has 591 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#3496C211yesmaximal, characteristic
2#3254C211yesmaximal, characteristic, complemented
3#3228C211yesmaximal, characteristic, complemented
4#3110C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1647C212yesmaximal, complemented
7#1639C212yesmaximal, complemented
8#1639C212yesmaximal, complemented
9#1557C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic, complemented
11#1486C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#547—21no—
14#547C2212yes—
15#546—22no—
16#546—22no—
17D42—21no—
18D42C2212yes—
19#537—21no—
20#534—22no—
21#533C2211yescharacteristic, complemented
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24C2×#121—22no—
25C2×C4×D4C2211yescharacteristic
26#500C2211yescharacteristic, complemented
27#498C2211yescharacteristic, complemented
28#486C2211yescharacteristic, complemented
29#486C2211yescharacteristic, complemented
30#484—21no—
31#482—22no—
32#478—22no—
33#477—21no—
34#474C2212yescomplemented
35#474C2212yescomplemented
36#468—22no—
37#461C2212yescomplemented
38#461C2212yescomplemented
39#460—21no—
40#450—21no—
41#440—22no—
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44#423C2211yescharacteristic, complemented
45#332C2212yescomplemented
46#331C2212yescomplemented
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51H(5,𝔽2)—48no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41—21no—
56C22×D4—24no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C2×SD16—22no—
63C2×SD16—22no—
64C2×M16—21no—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69#127D411yescharacteristic
70#127C2311yescharacteristic
71#125—21no—
72#124—21no—
73#124—21no—
74#123—24no—
75#123—24no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80#121—24no—
81#121—24no—
82#121—22no—
83#121—22no—
84#121—21no—
85#121—21no—
86#121—22no—
87#121—22no—
88#121D412yes—
89#121D412yes—
90#120—24no—
91#120—24no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D412yes—
99C4×D4D412yes—
100C4×D4C2312yes—
101C4×D4C2312yes—
102#117—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4D411yescharacteristic
105C2×C22⋊C4—24no—
106C2×C22⋊C4—21no—
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#107—22no—
111#106—22no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C2312yes—
115C4⋊C8C2312yes—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122C22⋊C8—21no—
123C22⋊C8—21no—
124#97—21no—
125C4×C8C2311yescharacteristic
126C4×C8C2311yescharacteristic
127C24—21no—
128C24—24no—
129#41—44no—
130#41—44no—
131#41—48no—
132#41—44no—
133#41—44no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—48no—
150C2×D4—44no—
151C2×D4—416no—
152C2×D4—44no—
153C2×D4—82no—
154C2×D4—82no—
155C2×D4C2×D412yes—
156C2×D4C2×D412yes—
157C2×D4C2×D412yes—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169SD16—44no—
170SD16—82no—
171M16—81no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—22no—
176C4⋊C4—24no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—41no—
198C42—41no—
199C42C2×D411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42C2411yescharacteristic
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—48no—
208C23—81no—
209C23—81no—
210C23—84no—
211C23—84no—
212C23#12011yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—41no—
217Q8—41no—
218D4—24no—
219D4—22no—
220D4—22no—
221D4—22no—
222D4—42no—
223D4—42no—
224D4—44no—
225D4—44no—
226D4—48no—
227D4—48no—
228D4—42no—
229D4—41no—
230D4—41no—
231D4—81no—
232D4—84no—
233D4AGL(1,ℤ/8)14yes—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4#12011yescharacteristic
252C2×C4#12011yescharacteristic
253C2×C4#12011yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C8—41no—
258C8—41no—
259C8—42no—
260C8—42no—
261C22—21no—
262C22—24no—
263C22—21no—
264C22—21no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—84no—
274C22—168no—
275C22C2×#12011yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—81no—
284C4—81no—
285C4C2×AGL(1,ℤ/8)12yes—
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—41no—
289C2—44no—
290C2—82no—
291C2—82no—
292C2#300311yescharacteristic
293C2#301211yescharacteristic
294C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3496C2
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. #3254C2semidirect
3. #3228C2semidirect
4. #3110C2semidirect
5. #1654C2semidirect
6. #1647C2semidirect
7. #1639C2semidirect
8. #1639C2semidirect
9. #1557C2semidirect
10. #1496C2semidirect
11. #1486C2semidirect
21. #533C22semidirect
26. #500C22semidirect
27. #498C22semidirect
28. #486C22semidirect
29. #486C22semidirect
34. #474C22semidirect
35. #474C22semidirect
37. #461C22semidirect
38. #461C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. #423C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#32542
#32282
#31102
#16542
#16472
#16392
#15572
#14962
#14862