Finite Group Databasebeta

← #18663

Subgroups — #18663

Order 256.

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

normal subgroups only

This group has 603 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#3496C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3167C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1635C212yesmaximal, complemented
8#1629C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1492C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, 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#498C2211yescharacteristic, complemented
27#492C2211yescharacteristic, complemented
28#487C2211yescharacteristic, complemented
29#486C2211yescharacteristic, complemented
30#478—22no—
31#476C2212yescomplemented
32#476C2212yescomplemented
33#475—21no—
34#474C2212yescomplemented
35#465—22no—
36#465—22no—
37#461C2212yescomplemented
38#460—21no—
39C4×SD16—22no—
40#428—21no—
41#423C2211yescharacteristic, complemented
42C2×C4⋊C8C2211yescharacteristic, complemented
43#418—21no—
44C2×C4×C8C2211yescharacteristic, complemented
45#332C2212yescomplemented
46#331C2212yescomplemented
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51H(5,𝔽2)—44no—
52H(5,𝔽2)—44no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—21no—
56C2×#41—21no—
57C22×D4—24no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4—22no—
61C22×D4D411yescharacteristic
62C23×C4—21no—
63C2×SD16—22no—
64C2×SD16—22no—
65C2×M16—21no—
66C22×C8—21no—
67C22×C8—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2311yescharacteristic
72#127D411yescharacteristic
73#127C2311yescharacteristic
74#125—21no—
75#124—21no—
76#124—21no—
77#123—24no—
78#123—24no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83#121—24no—
84#121—24no—
85#121—22no—
86#121—22no—
87#121—21no—
88#121—21no—
89#121—22no—
90#121—22no—
91#121D412yes—
92#121D412yes—
93#120—24no—
94#120—24no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4D412yes—
102C4×D4D412yes—
103C4×D4C2312yes—
104C4×D4C2312yes—
105#117—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4D411yescharacteristic
108C2×C22⋊C4—24no—
109C2×C22⋊C4—21no—
110C2×C42C2311yescharacteristic
111#107—22no—
112#106—22no—
113#106—22no—
114#106—22no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2312yes—
119#103—22no—
120#103—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125C4×C8—21no—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C24—21no—
129C24—24no—
130#41—44no—
131#41—48no—
132#41—44no—
133#41—42no—
134#41—42no—
135#41—42no—
136#41—42no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—416no—
155C2×D4—44no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4SD1614yescomplemented
159C2×D4C2×D412yes—
160C2×D4C2×D412yes—
161C2×D4C2×D412yes—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173SD16—44no—
174SD16—82no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—42no—
181C2×C8—81no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4C2×D412yes—
193C4⋊C4C2×D412yes—
194C4⋊C4C2×D412yes—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—41no—
204C42—41no—
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2411yescharacteristic
209C23—22no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—48no—
214C23—81no—
215C23—81no—
216C23—84no—
217C23—84no—
218C23#12011yescharacteristic
219Q8—42no—
220Q8—41no—
221Q8—41no—
222Q8—42no—
223Q8—41no—
224Q8—41no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—24no—
229D4—42no—
230D4—48no—
231D4—48no—
232D4—44no—
233D4—44no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—42no—
239D4—81no—
240D4—84no—
241D4C2×SD1614yes—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—48no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4C2×SD1612yes—
262C2×C4AGL(1,ℤ/8)12yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C8—22no—
267C8—41no—
268C8—42no—
269C8—42no—
270C22—21no—
271C22—21no—
272C22—24no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—84no—
282C22—168no—
283C22#45012yes—
284C22C2×#12011yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—81no—
292C4—81no—
293C4#45012yes—
294C4C22×SD1611yescharacteristic
295C4C2×AGL(1,ℤ/8)11yescharacteristic
296C4#57611yescharacteristic
297C4#57711yescharacteristic
298C2—22no—
299C2—44no—
300C2—82no—
301C2—82no—
302C2C2×#45011yescharacteristic
303C2#301211yescharacteristic
304C2#301811yescharacteristic
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. #3246C2semidirect
3. #3167C2semidirect
4. #3077C2semidirect
5. #1654C2semidirect
6. #1639C2semidirect
7. #1635C2semidirect
8. #1629C2semidirect
9. #1547C2semidirect
10. #1492C2semidirect
11. C2×D4⋊C8C2semidirect
21. #533C22semidirect
26. #498C22semidirect
27. #492C22semidirect
28. #487C22semidirect
29. #486C22semidirect
31. #476C22semidirect
32. #476C22semidirect
34. #474C22semidirect
37. #461C22semidirect
41. #423C22semidirect
42. C2×C4⋊C8C22semidirect
44. C2×C4×C8C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect
158. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#34962
#32462
#31672
#30772
#16542
#16392
#16352
#16292
#15472
#14922
C2×D4⋊C82