Finite Group Databasebeta

← #18576

Subgroups — #18576

Order 256.

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

normal subgroups only

This group has 490 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
2#3255C211yesmaximal, characteristic, complemented
3#3138C211yesmaximal, characteristic, complemented
4#3107C211yesmaximal, characteristic, complemented
5#1642C212yesmaximal, complemented
6#1641C212yesmaximal, complemented
7#1640C212yesmaximal, complemented
8#1640C212yesmaximal, complemented
9#1545C211yesmaximal, characteristic, complemented
10#1489C211yesmaximal, characteristic, complemented
11#1485C211yesmaximal, characteristic, complemented
12#559—21no—
13#559C2212yes—
14#550—21no—
15#550C2212yes—
16#546—22no—
17#546—22no—
18#540—22no—
19#535—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532—21no—
23#532—21no—
24C2×#127C2211yescharacteristic
25#518—21no—
26C4×#41C2211yescharacteristic
27#490—22no—
28#486C2212yes—
29#486C2212yes—
30#485—21no—
31#475—21no—
32#475C2211yescharacteristic, complemented
33#475C2212yescomplemented
34#474C2211yescharacteristic, complemented
35#470—22no—
36#462—21no—
37#462C2211yescharacteristic, complemented
38#462C2212yescomplemented
39#461C2211yescharacteristic, complemented
40#452—21no—
41C4×Q16—21no—
42C4×SD16—21no—
43#431C2211yescharacteristic
44#423C2211yescharacteristic
45#418—22no—
46C4×M16C2211yescharacteristic
47#331C2212yes—
48#331C2212yes—
49Q8⋊C8C2212yescomplemented
50Q8⋊C8C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—22no—
54C2×#41—22no—
55C2×#41D411yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4D412yes—
59C2×Q16—21no—
60C2×Q16—22no—
61C2×SD16—21no—
62C2×M16—21no—
63C2×M16—21no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68#127—22no—
69#127—21no—
70#127—21no—
71#127D412yes—
72#127C2312yes—
73#125—21no—
74#125—21no—
75#124—22no—
76#124—22no—
77#124—22no—
78#123—24no—
79#123—24no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8D412yes—
84#121—22no—
85#121—22no—
86#121—22no—
87#121—24no—
88#121—24no—
89#121—22no—
90#120—24no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8D411yescharacteristic
95C4×Q8C2311yescharacteristic
96C4×Q8C2312yes—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4D411yescharacteristic
106C4×D4C2311yescharacteristic
107#117—21no—
108#117—21no—
109#117—21no—
110#117D411yescharacteristic
111C2×C22⋊C4—24no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#106—21no—
116#106—22no—
117C4⋊C8—21no—
118C4⋊C8C2312yes—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—22no—
123#103—21no—
124#103—21no—
125#103—22no—
126#102—24no—
127#102—24no—
128#102—21no—
129C22⋊C8—21no—
130#97—21no—
131C4×C8—21no—
132C4×C8C2312yes—
133C24—22no—
134#41—41no—
135#41—41no—
136#41—82no—
137#41—82no—
138#41—84no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×D411yescharacteristic
143C2×Q8C2×D412yes—
144C2×D4—21no—
145C2×D4—24no—
146C2×D4—24no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—44no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D414yes—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167Q16—42no—
168Q16—82no—
169SD16—42no—
170M16—42no—
171M16—81no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4C2×D411yescharacteristic
190C4⋊C4C2×D412yes—
191C4⋊C4C2×D411yescharacteristic
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—24no—
196C22⋊C4—21no—
197C22⋊C4—22no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—22no—
205C42—41no—
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2411yescharacteristic
210C23—24no—
211C23—21no—
212C23—44no—
213C23—82no—
214C23—82no—
215C23#12011yescharacteristic
216Q8—21no—
217Q8—21no—
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225D4—21no—
226D4—21no—
227D4—44no—
228D4—44no—
229D4—42no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—44no—
235D4—82no—
236D4—84no—
237D4—84no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#12011yescharacteristic
257C2×C4#12011yescharacteristic
258C2×C4#12011yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C8—42no—
263C8—41no—
264C8—41no—
265C8—42no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—44no—
271C22—41no—
272C22—44no—
273C22—41no—
274C22—44no—
275C22—164no—
276C22C2×#12011yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—41no—
281C4—41no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—81no—
286C4—81no—
287C4#45512yes—
288C4#57511yescharacteristic
289C4#57511yescharacteristic
290C4#57611yescharacteristic
291C4#57711yescharacteristic
292C2—41no—
293C2—81no—
294C2—84no—
295C2#300511yescharacteristic
296C2#301211yescharacteristic
297C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3508C2
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. #3255C2semidirect
3. #3138C2semidirect
4. #3107C2semidirect
5. #1642C2semidirect
6. #1641C2semidirect
7. #1640C2semidirect
8. #1640C2semidirect
9. #1545C2semidirect
10. #1489C2semidirect
11. #1485C2semidirect
32. #475C22semidirect
33. #475C22semidirect
34. #474C22semidirect
37. #462C22semidirect
38. #462C22semidirect
39. #461C22semidirect
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
#32552
#31382
#31072
#16422
#16412
#16402
#15452
#14892
#14852