Finite Group Databasebeta

← #25593

Subgroups — #25593

Order 256.

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

normal subgroups only

This group has 617 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, complemented
2#3140C211yesmaximal, characteristic, complemented
3#3040C211yesmaximal, characteristic, complemented
4#2657C211yesmaximal, characteristic, complemented
5#2004C212yesmaximal, complemented
6#2002C212yesmaximal, complemented
7#1995C211yesmaximal, characteristic, complemented
8#1965C211yesmaximal, characteristic, complemented
9#1887C212yesmaximal, complemented
10#1885C212yesmaximal, complemented
11#1759C211yesmaximal, characteristic, complemented
12C2×#137C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2211yescharacteristic
14#559—22no—
15#550—22no—
16#546—24no—
17#540—22no—
18#535C2212yes—
19#533C2211yescharacteristic
20#532—22no—
21C2×#127C2212yes—
22C2×#127C2211yescharacteristic
23C2×#124C2212yes—
24C2×#124C2211yescharacteristic
25#518C2212yes—
26C4×#41C2211yescharacteristic
27#489—22no—
28#488—22no—
29#464—22no—
30#463—22no—
31#423C2211yescharacteristic
32C2×#104C2212yes—
33#419C2212yes—
34#419C2211yescharacteristic
35#418C2212yes—
36#418C2211yescharacteristic
37C2×#100C2212yes—
38C2×#100C2212yes—
39C4×M16C2211yescharacteristic
40#392—24no—
41#376C2211yescharacteristic
42#339C2212yes—
43#339C2212yes—
44#339C2212yes—
45C2×#41—24no—
46C2×#41C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—24no—
49C22×D4D414yes—
50C22×D4C2311yescharacteristic
51C22×D4C2312yes—
52#137—42no—
53AGL(1,ℤ/8)—42no—
54C2×Q16—21no—
55C2×SD16—21no—
56C2×SD16—21no—
57C2×D8—21no—
58C2×M16C2311yescharacteristic
59C2×M16C2311yescharacteristic
60C2×M16C2312yes—
61C4⋊Q8D412yescomplemented
62#127—22no—
63#127—21no—
64#127—21no—
65#127—22no—
66#127—24no—
67#127—42no—
68#125D412yescomplemented
69#124—24no—
70#124—22no—
71#124—22no—
72#124—21no—
73#124—21no—
74#124—24no—
75#124—42no—
76#123—28no—
77C22⋊Q8—22no—
78#121—24no—
79#121—28no—
80#121—24no—
81#120—24no—
82C4×Q8—22no—
83C4×Q8D412yescomplemented
84C4×D4—24no—
85C4×D4—22no—
86C4×D4—24no—
87C4×D4—24no—
88C4×D4D412yescomplemented
89#117—21no—
90#117C2311yescharacteristic
91#117C2312yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—22no—
95C2×C42—21no—
96C2×C42C2312yes—
97C2×C42C2311yescharacteristic
98C4⋊C8—22no—
99#104—24no—
100#103—22no—
101#103—24no—
102#102—22no—
103#102—24no—
104#100—42no—
105#100—42no—
106#97—21no—
107C4×C8—21no—
108#95—22no—
109C24—21no—
110C24C2×D414yes—
111#41—22no—
112#41—22no—
113#41—88no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8C2×D411yescharacteristic
120C2×D4—21no—
121C2×D4—28no—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—48no—
135C2×D4—84no—
136C2×D4—88no—
137C2×D4C2×D411yescharacteristic
138C22×C4—22no—
139C22×C4—28no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4C2×D412yes—
144C22×C4C2411yescharacteristic
145Q16—42no—
146SD16—42no—
147SD16—42no—
148D8—42no—
149M16—22no—
150M16—22no—
151M16—22no—
152M16—22no—
153C2×C8—22no—
154C2×C8—21no—
155C2×C8—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—24no—
159C4⋊C4C2×D412yes—
160C4⋊C4C2×D414yes—
161C22⋊C4—28no—
162C22⋊C4—28no—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—48no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—22no—
170C42—21no—
171C42—22no—
172C42—22no—
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C42C2×D412yes—
176C42#4114yescomplemented
177C23—24no—
178C23—21no—
179C23—24no—
180C23—24no—
181C23—48no—
182C23—44no—
183C23—41no—
184C23—42no—
185C23C22×D411yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—44no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192D4—21no—
193D4—21no—
194D4—48no—
195D4—42no—
196D4—42no—
197D4—41no—
198D4—42no—
199D4—41no—
200D4—42no—
201D4—42no—
202D4—42no—
203D4—84no—
204D4—88no—
205D4—816no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—88no—
217C2×C4#12012yes—
218C2×C4#12012yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C2×#4112yes—
222C2×C4H(5,𝔽2)12yes—
223C8—41no—
224C8—41no—
225C8—42no—
226C22—22no—
227C22—41no—
228C22—44no—
229C22—42no—
230C22—48no—
231C22—84no—
232C22—84no—
233C22—84no—
234C22—81no—
235C22—82no—
236C22C2×#12011yescharacteristic
237C22#54612yes—
238C4—22no—
239C4—22no—
240C4—44no—
241C4—41no—
242C4—42no—
243C4—44no—
244C4—81no—
245C4#53211yescharacteristic
246C4#53511yescharacteristic
247C4#55012yes—
248C2—22no—
249C2—81no—
250C2—84no—
251C2—161no—
252C2#243111yescharacteristic
253C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3508C2semidirect
2. #3140C2semidirect
3. #3040C2semidirect
4. #2657C2semidirect
5. #2004C2semidirect
6. #2002C2semidirect
7. #1995C2semidirect
8. #1965C2semidirect
9. #1887C2semidirect
10. #1885C2semidirect
11. #1759C2semidirect
61. C4⋊Q8D4semidirect
68. #125D4semidirect
83. C4×Q8D4semidirect
88. C4×D4D4semidirect
176. C42#41semidirect

Maximal subgroups

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

subgroupindex
#35082
#31402
#30402
#26572
#20042
#20022
#19952
#19652
#18872
#18852
#17592