Finite Group Databasebeta

← #18676

Subgroups — #18676

Order 256.

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

normal subgroups only

This group has 456 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#3502C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic
3#3167C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic
5#1653C212yesmaximal, complemented
6#1645C212yesmaximal, complemented
7#1626C212yesmaximal, complemented
8#1623C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13Q82—21no—
14Q82C2212yes—
15#557—21no—
16#557—22no—
17#557—22no—
18#557C2212yes—
19#544—22no—
20#544—21no—
21#533—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#532C2211yescharacteristic
25#519—21no—
26C2×C4×Q8C2211yescharacteristic
27#498C2211yescharacteristic
28#492C2211yescharacteristic
29#487C2211yescharacteristic
30#486C2211yescharacteristic
31#476—21no—
32#475C2212yes—
33#465—22no—
34#465—22no—
35#464—22no—
36#462C2212yes—
37#461—21no—
38#460C2212yes—
39#460C2212yes—
40C4×SD16—22no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#417—21no—
44C2×C4×C8C2211yescharacteristic
45#333C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8D411yescharacteristic
53#137—44no—
54C2×Q16—22no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×M16—21no—
59C22×C8—21no—
60C22×C8—21no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—24no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2311yescharacteristic
72#127C2311yescharacteristic
73#125—24no—
74#125—24no—
75#125—22no—
76#125—24no—
77#125—22no—
78#125—21no—
79#125—21no—
80#124—21no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8D412yes—
85C22⋊Q8D412yes—
86#121—21no—
87#121—21no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—24no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8D412yes—
100C4×Q8D412yes—
101C4×Q8C2312yes—
102C4×Q8C2312yes—
103C4×D4—21no—
104C4×D4—21no—
105#117—24no—
106#117—21no—
107#117—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C42—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
127#41—42no—
128#41—42no—
129C2×Q8—24no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—22no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8SD1614yescomplemented
136C2×Q8C2×D412yes—
137C2×Q8C2×D412yes—
138C2×Q8C2×D412yes—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149Q16—44no—
150SD16—44no—
151SD16—44no—
152SD16—82no—
153M16—42no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—81no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C42—24no—
188C42—24no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2411yescharacteristic
197C23—41no—
198C23#12011yescharacteristic
199Q8—22no—
200Q8—24no—
201Q8—22no—
202Q8—22no—
203Q8—42no—
204Q8—41no—
205Q8—41no—
206Q8—41no—
207Q8—41no—
208Q8—81no—
209Q8—84no—
210Q8C2×SD1614yes—
211D4—42no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—82no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C2×SD1612yes—
234C2×C4#13712yes—
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C8—22no—
239C8—41no—
240C8—42no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22#44812yes—
248C22C2×#12011yescharacteristic
249C4—24no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4#44812yes—
260C4C22×SD1611yescharacteristic
261C4C2×#13711yescharacteristic
262C4#57711yescharacteristic
263C4#57811yescharacteristic
264C2—22no—
265C2—161no—
266C2C2×#44811yescharacteristic
267C2#301311yescharacteristic
268C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
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. #3502C2semidirect
5. #1653C2semidirect
6. #1645C2semidirect
7. #1626C2semidirect
8. #1623C2semidirect
9. #1550C2semidirect
10. #1493C2semidirect
11. C2×Q8⋊C8C2semidirect
135. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32462
#31672
#30372
#16532
#16452
#16262
#16232
#15502
#14932
C2×Q8⋊C82