Finite Group Databasebeta

← #18634

Subgroups — #18634

Order 256.

512 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 123 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
2#3229C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1642C214yesmaximal, complemented
6#1640C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×#137—21no—
11#559—21no—
12#559C2212yes—
13#550—21no—
14#550C2212yes—
15#546—22no—
16#546—22no—
17#540—22no—
18#535—21no—
19#533C2211yescharacteristic
20#532—21no—
21#532—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#127C2211yescharacteristic
24#518—21no—
25C4×#41C2211yescharacteristic
26#498—21no—
27#487—21no—
28#486—22no—
29#486C2214yes—
30#483—22no—
31#475C2214yescomplemented
32#462—21no—
33#462C2212yescomplemented
34#461—21no—
35#461C2212yescomplemented
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#419—21no—
39C2×#102—22no—
40C2×C4×C8C2211yescharacteristic
41#331C2214yes—
42Q8⋊C8C2212yescomplemented
43Q8⋊C8C2214yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41—22no—
46C2×#41—22no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D412yes—
52#137—44no—
53C2×Q16—22no—
54C2×SD16—22no—
55C2×M16—21no—
56C22×C8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2312yes—
63#127—22no—
64#127—21no—
65#127—21no—
66#127D412yes—
67#127C2312yes—
68#125—21no—
69#124—22no—
70#124—22no—
71#124—22no—
72#123—24no—
73#123—24no—
74C22⋊Q8—21no—
75C22⋊Q8D412yes—
76#121—22no—
77#121—22no—
78#121—22no—
79#121—24no—
80#121—24no—
81#121—22no—
82#120—24no—
83C4×Q8—21no—
84C4×Q8D411yescharacteristic
85C4×Q8C2311yescharacteristic
86C4×Q8C2312yes—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4D411yescharacteristic
96C4×D4C2311yescharacteristic
97#117—21no—
98#117—21no—
99#117D411yescharacteristic
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C22⋊C4—24no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#106—24no—
106C4⋊C8—21no—
107C4⋊C8C2312yes—
108C4⋊C8C2312yes—
109#103—24no—
110#103—22no—
111#103—22no—
112#102—28no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117C24—22no—
118#41—22no—
119#41—22no—
120#41—84no—
121#41—84no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—41no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8C2×D412yes—
128C2×Q8C2×D411yescharacteristic
129C2×D4—21no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—44no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—82no—
141C2×D4C2×D411yescharacteristic
142C2×D4C2×D414yes—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152Q16—44no—
153SD16—44no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4C2×D411yescharacteristic
171C4⋊C4C2×D411yescharacteristic
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—21no—
178C22⋊C4—22no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2411yescharacteristic
189C23—24no—
190C23—21no—
191C23—44no—
192C23—82no—
193C23—82no—
194C23#12011yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—24no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—82no—
202D4—21no—
203D4—21no—
204D4—44no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—44no—
209D4—44no—
210D4—44no—
211D4—82no—
212D4—84no—
213D4—84no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—82no—
228C2×C4#12011yescharacteristic
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#13512yes—
232C2×C4AGL(1,ℤ/8)12yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C8—42no—
237C8—42no—
238C8—42no—
239C22—21no—
240C22—21no—
241C22—41no—
242C22—44no—
243C22—41no—
244C22—44no—
245C22—41no—
246C22—44no—
247C22—164no—
248C22#44912yes—
249C22C2×#12011yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—41no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—82no—
258C4#44912yes—
259C4#45512yes—
260C4C2×#13511yescharacteristic
261C4C2×AGL(1,ℤ/8)11yescharacteristic
262C4#57712yes—
263C2—22no—
264C2—81no—
265C2—84no—
266C2C2×#44911yescharacteristic
267C2#300911yescharacteristic
268C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3508C2
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
2. #3229C2semidirect
3. C2×#486C2semidirect
4. #3039C2semidirect
5. #1642C2semidirect
6. #1640C2semidirect
7. #1545C2semidirect
8. #1493C2semidirect
9. #1476C2semidirect
31. #475C22semidirect
33. #462C22semidirect
35. #461C22semidirect
42. Q8⋊C8C22semidirect
43. Q8⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32292
C2×#4862
#30392
#16422
#16402
#15452
#14932
#14762