Finite Group Databasebeta

← #18664

Subgroups — #18664

Order 256.

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

normal subgroups only

This group has 490 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#3501C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3167C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic
5#1651C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1622C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—21no—
13#554—21no—
14#554—22no—
15#554—22no—
16#554C2212yes—
17D4×Q8—21no—
18D4×Q8C2212yes—
19#544—21no—
20#543—22no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23#532C2211yescharacteristic
24C2×C22⋊Q8—22no—
25#518—21no—
26C2×C4×D4C2211yescharacteristic
27#498C2211yescharacteristic, complemented
28#492C2211yescharacteristic, complemented
29#487C2211yescharacteristic, complemented
30#486C2211yescharacteristic, complemented
31#476—21no—
32#475C2212yescomplemented
33#465—22no—
34#465—22no—
35#463—22no—
36#461—21no—
37#461C2212yes—
38#460C2212yescomplemented
39#459C2212yes—
40C4×SD16—22no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#417—21no—
44C2×C4×C8C2211yescharacteristic
45#333C2212yescomplemented
46#332C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8—22no—
53C22×D4D411yescharacteristic
54C23×C4—21no—
55AGL(1,ℤ/8)—44no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×SD16—22no—
59C2×D8—22no—
60C2×M16—21no—
61C22×C8—21no—
62C22×C8—21no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8D411yescharacteristic
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69C4⋊Q8C2311yescharacteristic
70#127C2311yescharacteristic
71#125—22no—
72#125—22no—
73#125—21no—
74#125—21no—
75#124—21no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—22no—
85C22⋊Q8—24no—
86C22⋊Q8D412yes—
87C22⋊Q8D412yes—
88#121—21no—
89#121—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D412yes—
93C4×Q8C2312yes—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4D412yes—
103C4×D4C2312yes—
104#117—21no—
105#117—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4D411yescharacteristic
109C2×C22⋊C4—24no—
110C2×C22⋊C4—21no—
111C2×C42C2311yescharacteristic
112#106—22no—
113#106—22no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2312yes—
117C4⋊C8C2312yes—
118#103—22no—
119#103—22no—
120#103—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—
130C2×Q8—24no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8C2×D412yes—
137C2×Q8C2×D412yes—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—41no—
144C2×D4SD1614yescomplemented
145C2×D4C2×D412yes—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157SD16—44no—
158SD16—44no—
159SD16—82no—
160D8—44no—
161M16—42no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—81no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
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—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C42—22no—
196C42—22no—
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2411yescharacteristic
201C23—22no—
202C23—24no—
203C23—41no—
204C23—81no—
205C23—81no—
206C23#12011yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214Q8—81no—
215Q8—84no—
216D4—22no—
217D4—24no—
218D4—42no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—82no—
224D4C2×SD1614yes—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—84no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4C2×SD1612yes—
245C2×C4#13712yes—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C8—22no—
250C8—41no—
251C8—42no—
252C8—42no—
253C22—21no—
254C22—21no—
255C22—24no—
256C22—42no—
257C22—42no—
258C22—84no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22#44812yes—
263C22C2×#12011yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—42no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—42no—
272C4—81no—
273C4#44812yes—
274C4C22×SD1611yescharacteristic
275C4C2×#13711yescharacteristic
276C4#57611yescharacteristic
277C4#57711yescharacteristic
278C2—22no—
279C2—44no—
280C2—161no—
281C2C2×#44811yescharacteristic
282C2#301211yescharacteristic
283C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3036C2
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. #3501C2semidirect
2. #3246C2semidirect
3. #3167C2semidirect
5. #1651C2semidirect
6. #1643C2semidirect
7. #1627C2semidirect
8. #1622C2semidirect
9. #1550C2semidirect
10. #1493C2semidirect
11. C2×D4⋊C8C2semidirect
27. #498C22semidirect
28. #492C22semidirect
29. #487C22semidirect
30. #486C22semidirect
32. #475C22semidirect
38. #460C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect
144. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#35012
#32462
#31672
#30362
#16512
#16432
#16272
#16222
#15502
#14932
C2×D4⋊C82