Finite Group Databasebeta

← #18892

Subgroups — #18892

Order 256.

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

normal subgroups only

This group has 651 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, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4C2×#459C211yesmaximal, characteristic, complemented
5#1625C214yesmaximal, complemented
6#1619C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C22×D8—21no—
12#547—21no—
13#547C2212yescomplemented
14#546—22no—
15#546—22no—
16D42—21no—
17D42C2212yescomplemented
18#537—21no—
19#534—22no—
20#532C2211yescharacteristic
21#532C2211yescharacteristic
22C2×#127C2211yescharacteristic, complemented
23C2×#121—22no—
24C2×C4×D4C2211yescharacteristic
25#493—21no—
26#493C2212yescomplemented
27#486—21no—
28#486C2212yescomplemented
29#459—22no—
30#459—22no—
31#459C2214yescomplemented
32#459C2214yescomplemented
33#447—22no—
34#447—22no—
35#431C2211yescharacteristic, complemented
36#431C2211yescharacteristic, complemented
37C2×C4⋊C8C2211yescharacteristic, complemented
38C2×#102—21no—
39#332C2212yescomplemented
40#331C2212yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43H(5,𝔽2)—44no—
44H(5,𝔽2)—44no—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—21no—
48C2×#41—21no—
49C22×D4—24no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4D411yescharacteristic
57C23×C4—21no—
58C2×D8—24no—
59C2×D8—24no—
60C2×D8—24no—
61C2×D8—44no—
62C22×C8—21no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2311yescharacteristic
65#127—21no—
66#127—21no—
67#127—21no—
68#127D411yescharacteristic
69#127C2312yes—
70#127C2311yescharacteristic
71#124—21no—
72#124—21no—
73#123—24no—
74#123—24no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—24no—
78#121—24no—
79#121—22no—
80#121—22no—
81#121—21no—
82#121—21no—
83#121—22no—
84#121—22no—
85#121D412yes—
86#121D412yes—
87#120—24no—
88#120—24no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4D412yes—
94C4×D4D412yes—
95C4×D4C2312yes—
96C4×D4C2312yes—
97C2×C4⋊C4—22no—
98C2×C4⋊C4D411yescharacteristic
99C2×C22⋊C4—24no—
100C2×C22⋊C4—21no—
101C2×C42C2311yescharacteristic
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8C2312yes—
105C4⋊C8C2312yes—
106#102—24no—
107#102—24no—
108#102—24no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C4×C8—21no—
112C4×C8C2312yes—
113C24—21no—
114C24—24no—
115C24—41no—
116#41—44no—
117#41—48no—
118#41—44no—
119#41—42no—
120#41—42no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—24no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—44no—
140C2×D4—48no—
141C2×D4—416no—
142C2×D4—44no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—82no—
149C2×D4—84no—
150C2×D4—82no—
151C2×D4D814yescomplemented
152C2×D4C2×D412yes—
153C2×D4C2×D412yes—
154C2×D4C2×D412yes—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—82no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165D8—48no—
166D8—84no—
167D8—84no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—42no—
172C4⋊C4—24no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C4⋊C4C2×D412yes—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2411yescharacteristic
191C23—22no—
192C23—24no—
193C23—22no—
194C23—22no—
195C23—48no—
196C23—42no—
197C23—81no—
198C23—81no—
199C23—82no—
200C23—82no—
201C23—84no—
202C23—84no—
203C23—82no—
204C23#12011yescharacteristic
205Q8—41no—
206Q8—41no—
207Q8—42no—
208D4—22no—
209D4—24no—
210D4—24no—
211D4—42no—
212D4—44no—
213D4—48no—
214D4—48no—
215D4—44no—
216D4—44no—
217D4—41no—
218D4—42no—
219D4—41no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—42no—
224D4—81no—
225D4—82no—
226D4—82no—
227D4—84no—
228D4—82no—
229D4C2×D814yes—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—48no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4#12011yescharacteristic
245C2×C4C2×D812yes—
246C2×C4AGL(1,ℤ/8)12yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C8—42no—
251C8—42no—
252C8—42no—
253C22—21no—
254C22—21no—
255C22—24no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—84no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—164no—
269C22—168no—
270C22#44712yes—
271C22C2×#12011yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—42no—
278C4#44712yes—
279C4C22×D811yescharacteristic
280C4C2×AGL(1,ℤ/8)11yescharacteristic
281C4#57612yes—
282C2—22no—
283C2—44no—
284C2—82no—
285C2—82no—
286C2—162no—
287C2C2×#44711yescharacteristic
288C2#300811yescharacteristic
289C2#302111yescharacteristic
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. #3496C2semidirect
2. #3205C2semidirect
3. #3205C2semidirect
4. C2×#459C2semidirect
5. #1625C2semidirect
6. #1619C2semidirect
7. #1540C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
13. #547C22semidirect
17. D42C22semidirect
22. C2×#127C22semidirect
26. #493C22semidirect
28. #486C22semidirect
31. #459C22semidirect
32. #459C22semidirect
35. #431C22semidirect
36. #431C22semidirect
37. C2×C4⋊C8C22semidirect
39. #332C22semidirect
40. #331C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect
151. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
#34962
#32052
C2×#4592
#16252
#16192
#15402
#14802
#14782