Finite Group Databasebeta

← #18818

Subgroups — #18818

Order 256.

502 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 113 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#3216C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4#3115C211yesmaximal, characteristic, complemented
5#1641C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1621C212yesmaximal, complemented
8#1620C212yesmaximal, complemented
9#1555C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, 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#532—21no—
22#532—21no—
23#532C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#518—21no—
26C4×#41C2211yescharacteristic
27#497—21no—
28#493C2212yes—
29#489—21no—
30#486C2212yes—
31#482—21no—
32#475C2211yescharacteristic, complemented
33#474C2211yescharacteristic, complemented
34#462C2212yescomplemented
35#460C2211yescharacteristic, complemented
36#460C2212yescomplemented
37#459—22no—
38#459C2211yescharacteristic, complemented
39#452—22no—
40#449—21no—
41#447—22no—
42#442—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
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—
60C2×Q16—22no—
61C2×SD16—21no—
62C2×SD16—22no—
63C2×D8—21no—
64C2×D8—24no—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8D411yescharacteristic
67C4⋊Q8C2311yescharacteristic
68#127—22no—
69#127—21no—
70#127—21no—
71#127D412yes—
72#127C2311yescharacteristic
73#127C2312yes—
74#125—21no—
75#124—22no—
76#124—22no—
77#124—22no—
78#124—21no—
79#123—24no—
80#123—24no—
81C22⋊Q8—21no—
82C22⋊Q8D412yes—
83#121—22no—
84#121—22no—
85#121—22no—
86#121—24no—
87#121—24no—
88#121—22no—
89#121—21no—
90#121—21no—
91#120—24no—
92C4×Q8—21no—
93C4×Q8D411yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×Q8C2312yes—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4D411yescharacteristic
107C4×D4C2311yescharacteristic
108#117—21no—
109#117—21no—
110#117D411yescharacteristic
111C2×C22⋊C4—24no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#106—21no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—22no—
122#102—21no—
123#102—24no—
124#102—21no—
125#102—21no—
126#102—22no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133C24—22no—
134#41—41no—
135#41—41no—
136#41—82no—
137#41—82no—
138#41—82no—
139#41—84no—
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—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—82no—
159C2×D4—82no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D414yes—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171Q16—82no—
172SD16—42no—
173SD16—82no—
174D8—42no—
175D8—84no—
176C2×C8—22no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D411yescharacteristic
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—
201C22⋊C4—41no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—22no—
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2411yescharacteristic
210C23—24no—
211C23—21no—
212C23—44no—
213C23—41no—
214C23—82no—
215C23—82no—
216C23#12011yescharacteristic
217Q8—21no—
218Q8—21no—
219Q8—22no—
220Q8—22no—
221Q8—42no—
222Q8—42no—
223D4—21no—
224D4—21no—
225D4—44no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—42no—
232D4—41no—
233D4—41no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—82no—
238D4—84no—
239D4—84no—
240D4—82no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4#12011yescharacteristic
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C8—42no—
264C8—41no—
265C8—41no—
266C8—42no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—44no—
272C22—41no—
273C22—44no—
274C22—41no—
275C22—44no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—164no—
280C22C2×#12011yescharacteristic
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—81no—
291C4#57511yescharacteristic
292C4#57511yescharacteristic
293C4#57612yes—
294C2—41no—
295C2—81no—
296C2—84no—
297C2—161no—
298C2#300511yescharacteristic
299C2#300811yescharacteristic
300C2#301911yescharacteristic
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. #3216C2semidirect
3. #3205C2semidirect
4. #3115C2semidirect
5. #1641C2semidirect
6. #1640C2semidirect
7. #1621C2semidirect
8. #1620C2semidirect
9. #1555C2semidirect
10. #1497C2semidirect
11. #1488C2semidirect
32. #475C22semidirect
33. #474C22semidirect
34. #462C22semidirect
35. #460C22semidirect
36. #460C22semidirect
38. #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
#32162
#32052
#31152
#16412
#16402
#16212
#16202
#15552
#14972
#14882