Finite Group Databasebeta

← #18678

Subgroups — #18678

Order 256.

500 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 121 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#3506C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1650C212yesmaximal, complemented
6#1642C212yesmaximal, complemented
7#1628C212yesmaximal, complemented
8#1621C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—21no—
13#559—22no—
14#556C2212yes—
15#555—21no—
16#553—22no—
17#550C2212yes—
18#547—21no—
19#538—22no—
20#535—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24#525—22no—
25#520—21no—
26C2×C4×D4C2211yescharacteristic
27#500C2211yescharacteristic, complemented
28#493C2211yescharacteristic, complemented
29#486C2211yescharacteristic, complemented
30#486C2211yescharacteristic, complemented
31#476—21no—
32#475C2212yescomplemented
33#466—22no—
34#465—22no—
35#463—22no—
36#461—21no—
37#461C2212yes—
38#460C2212yescomplemented
39#459C2212yes—
40C4×D8—22no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#417—21no—
44C2×C4×C8C2211yescharacteristic
45#332C2212yescomplemented
46#331C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—22no—
52C2×#41—21no—
53C2×#41—21no—
54C22×D4D411yescharacteristic
55C23×C4—21no—
56AGL(1,ℤ/8)—44no—
57C2×SD16—22no—
58C2×SD16—22no—
59C2×D8—22no—
60C2×D8—22no—
61C2×M16—21no—
62C22×C8—21no—
63C22×C8—21no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68#127—22no—
69#127C2311yescharacteristic
70#127C2311yescharacteristic
71#126—24no—
72#125—21no—
73#124—24no—
74#124—22no—
75#124—21no—
76#124—21no—
77#123—24no—
78#123—24no—
79C22⋊Q8—22no—
80C22⋊Q8—21no—
81C22⋊Q8D412yes—
82#121—22no—
83#121—22no—
84#121—24no—
85#121—21no—
86#121—22no—
87#121—21no—
88#121—21no—
89#121D412yes—
90#120—24no—
91C4×Q8—22no—
92C4×Q8—21no—
93C4×Q8C2312yes—
94C4×D4—24no—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4D412yes—
103C4×D4D412yes—
104C4×D4C2312yes—
105#117—22no—
106#117—21no—
107#117—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4D411yescharacteristic
110C2×C22⋊C4—21no—
111C2×C42C2311yescharacteristic
112#107—22no—
113#106—22no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2312yes—
117C4⋊C8C2312yes—
118#103—22no—
119#103—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124C4×C8—21no—
125C4×C8C2311yescharacteristic
126C4×C8C2311yescharacteristic
127C24—21no—
128#41—42no—
129#41—42no—
130#41—42no—
131#41—42no—
132#41—82no—
133#41—82no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8C2×D412yes—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—41no—
150C2×D4C2×D412yes—
151C2×D4C2×D412yes—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163SD16—44no—
164SD16—82no—
165D8—44no—
166D8—44no—
167M16—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—81no—
174C4⋊C4—24no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C42—24no—
198C42—22no—
199C42C2×D411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42C2411yescharacteristic
203C23—22no—
204C23—22no—
205C23—42no—
206C23—41no—
207C23—81no—
208C23—81no—
209C23#12011yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—81no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—42no—
221D4—44no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—42no—
231D4—84no—
232D4—84no—
233D4—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4#12011yescharacteristic
252C2×C4#13512yes—
253C2×C4AGL(1,ℤ/8)12yes—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C8—22no—
258C8—41no—
259C8—42no—
260C8—42no—
261C22—21no—
262C22—21no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22#44912yes—
274C22C2×#12011yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—81no—
283C4#44912yes—
284C4C2×#13511yescharacteristic
285C4C2×AGL(1,ℤ/8)11yescharacteristic
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—22no—
289C2—82no—
290C2—82no—
291C2—161no—
292C2C2×#44911yescharacteristic
293C2#301211yescharacteristic
294C2#301811yescharacteristic
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. #3506C2semidirect
2. #3246C2semidirect
3. #3169C2semidirect
4. #3036C2semidirect
5. #1650C2semidirect
6. #1642C2semidirect
7. #1628C2semidirect
8. #1621C2semidirect
9. #1547C2semidirect
10. #1493C2semidirect
11. C2×D4⋊C8C2semidirect
27. #500C22semidirect
28. #493C22semidirect
29. #486C22semidirect
30. #486C22semidirect
32. #475C22semidirect
38. #460C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35062
#32462
#31692
#30362
#16502
#16422
#16282
#16212
#15472
#14932
C2×D4⋊C82