Finite Group Databasebeta

← #18666

Subgroups — #18666

Order 256.

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

normal subgroups only

This group has 468 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#3507C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic, complemented
5#1648C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1626C212yesmaximal, complemented
8#1620C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13#559C2212yes—
14#557C2212yes—
15#556—21no—
16#556—22no—
17#554—21no—
18#552—22no—
19#539—22no—
20#536—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24#529—22no—
25#520—21no—
26C2×C4×Q8C2211yescharacteristic
27#500C2211yescharacteristic
28#493C2211yescharacteristic
29#486C2211yescharacteristic
30#486C2211yescharacteristic
31#476—21no—
32#475C2212yes—
33#466—22no—
34#465—22no—
35#464—22no—
36#462C2212yes—
37#461—21no—
38#460C2212yes—
39#460C2212yes—
40C4×D8—22no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#417—21no—
44C2×C4×C8C2211yescharacteristic
45#332C2212yes—
46#331C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×#41—21no—
52C2×#41—21no—
53C22×Q8D411yescharacteristic
54#137—44no—
55C2×Q16—22no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×D8—22no—
59C2×M16—21no—
60C22×C8—21no—
61C22×C8—21no—
62C4⋊Q8—22no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#127C2311yescharacteristic
68#127C2311yescharacteristic
69#126—24no—
70#126—24no—
71#125—22no—
72#125—24no—
73#125—21no—
74#124—22no—
75#124—22no—
76#124—21no—
77#124—21no—
78#123—24no—
79#123—24no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8—21no—
84C22⋊Q8D412yes—
85#121—21no—
86#121—21no—
87#121—21no—
88#121D412yes—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8D412yes—
95C4×Q8C2312yes—
96C4×Q8C2312yes—
97C4×D4—24no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4D412yes—
104#117—22no—
105#117—24no—
106#117—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C42—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
127#41—42no—
128#41—42no—
129#41—42no—
130#41—42no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—21no—
137C2×Q8C2×D412yes—
138C2×Q8C2×D412yes—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—41no—
146C2×D4C2×D412yes—
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155Q16—44no—
156SD16—44no—
157SD16—82no—
158D8—44no—
159M16—42no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—81no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C42—24no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42C2×D411yescharacteristic
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C42C2411yescharacteristic
199C23—22no—
200C23—41no—
201C23#12011yescharacteristic
202Q8—22no—
203Q8—22no—
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—41no—
210Q8—41no—
211Q8—81no—
212D4—42no—
213D4—42no—
214D4—44no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—41no—
235C2×C4—81no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4#12011yescharacteristic
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#13512yes—
242C2×C4AGL(1,ℤ/8)12yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C8—22no—
247C8—41no—
248C8—42no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#44912yes—
259C22C2×#12011yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4#44912yes—
270C4C2×#13511yescharacteristic
271C4C2×AGL(1,ℤ/8)11yescharacteristic
272C4#57711yescharacteristic
273C4#57811yescharacteristic
274C2—22no—
275C2—82no—
276C2—161no—
277C2C2×#44911yescharacteristic
278C2#301311yescharacteristic
279C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3169C2
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. #3507C2semidirect
2. #3246C2semidirect
4. #3037C2semidirect
5. #1648C2semidirect
6. #1640C2semidirect
7. #1626C2semidirect
8. #1620C2semidirect
9. #1547C2semidirect
10. #1493C2semidirect
11. C2×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32462
#31692
#30372
#16482
#16402
#16262
#16202
#15472
#14932
C2×Q8⋊C82