Finite Group Databasebeta

← #18928

Subgroups — #18928

Order 256.

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

normal subgroups only

This group has 483 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
2#3229C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic, complemented
4C2×#461C211yesmaximal, characteristic, complemented
5#1654C214yesmaximal, complemented
6#1646C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×SD16—21no—
11#559—22no—
12#556—21no—
13#555C2212yes—
14#553—22no—
15#550—21no—
16#547C2212yes—
17#538—22no—
18#535—21no—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22#525—22no—
23#520—21no—
24C2×C4×D4C2211yescharacteristic
25#500—21no—
26#498C2212yescomplemented
27#487C2212yescomplemented
28#486—21no—
29#483—22no—
30#480—22no—
31#475—22no—
32#474C2214yescomplemented
33#461—22no—
34#461C2214yes—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41D4⋊C8C2214yes—
42D4⋊C8C2214yescomplemented
43C2×#41—22no—
44C2×#41—21no—
45C22×Q8—21no—
46C22×D4D411yescharacteristic
47C23×C4—21no—
48C2×SD16—24no—
49C2×SD16—44no—
50C22×C8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8D411yescharacteristic
55C4⋊Q8C2312yes—
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2311yescharacteristic
58#127—22no—
59#127D411yescharacteristic
60#126—24no—
61#125—21no—
62#124—22no—
63#124—24no—
64#124—21no—
65#123—24no—
66#123—24no—
67C22⋊Q8—21no—
68C22⋊Q8—22no—
69C22⋊Q8D412yes—
70#121—24no—
71#121—22no—
72#121—22no—
73#121—22no—
74#121—21no—
75#121D412yes—
76#120—24no—
77C4×Q8—22no—
78C4×Q8—21no—
79C4×Q8D412yes—
80C4×D4—24no—
81C4×D4—22no—
82C4×D4—24no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4D412yes—
87C4×D4C2312yes—
88C4×D4C2312yes—
89#117—21no—
90#117—22no—
91#117—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C22⋊C4—21no—
97C2×C42C2311yescharacteristic
98#107—24no—
99#106—24no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8C2312yes—
103C4⋊C8C2312yes—
104#103—24no—
105#103—24no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—21no—
112#41—42no—
113#41—42no—
114#41—84no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8C2×D412yes—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4C2×D412yes—
131C2×D4C2×D412yes—
132C22×C4—24no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—82no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142SD16—48no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—42no—
147C4⋊C4—24no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4C2×D412yes—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—24no—
171C42—22no—
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C42C2411yescharacteristic
176C23—22no—
177C23—22no—
178C23—42no—
179C23—81no—
180C23—81no—
181C23#12011yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187Q8—82no—
188D4—22no—
189D4—22no—
190D4—24no—
191D4—44no—
192D4—42no—
193D4—42no—
194D4—81no—
195D4—84no—
196D4—84no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#13512yes—
215C2×C4AGL(1,ℤ/8)12yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C8—42no—
220C8—42no—
221C8—42no—
222C22—21no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—84no—
231C22#44912yes—
232C22C2×#12011yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4#44912yes—
242C4C2×#13511yescharacteristic
243C4C2×AGL(1,ℤ/8)11yescharacteristic
244C4#57712yes—
245C2—22no—
246C2—82no—
247C2—82no—
248C2C2×#44911yescharacteristic
249C2#300911yescharacteristic
250C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3506C2
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. #3226C2semidirect
4. C2×#461C2semidirect
5. #1654C2semidirect
6. #1646C2semidirect
7. #1542C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
26. #498C22semidirect
27. #487C22semidirect
32. #474C22semidirect
39. #333C22semidirect
40. #332C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35062
#32292
#32262
C2×#4612
#16542
#16462
#15422
#14802
#14782