Finite Group Databasebeta

← #25594

Subgroups — #25594

Order 256.

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

normal subgroups only

This group has 525 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#3509C211yesmaximal, characteristic, complemented
2#3140C211yesmaximal, characteristic, complemented
3#3040C211yesmaximal, characteristic
4#2658C211yesmaximal, characteristic, complemented
5#2005C212yesmaximal, complemented
6#2003C212yesmaximal
7#1995C211yesmaximal, characteristic, complemented
8#1966C211yesmaximal, characteristic, complemented
9#1888C212yesmaximal, complemented
10#1886C212yesmaximal, complemented
11#1759C211yesmaximal, characteristic, complemented
12C2×#137C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2211yescharacteristic
14#557—22no—
15#556—22no—
16#552—24no—
17#541—22no—
18#536C2212yes—
19#533—22no—
20#533C2211yescharacteristic
21C2×C4⋊Q8C2212yes—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2212yes—
24C2×#124C2211yescharacteristic
25#519C2212yes—
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×#101C2212yes—
38C2×#101C2212yes—
39C4×M16C2211yescharacteristic
40#399—24no—
41#376C2211yescharacteristic
42#339C2212yes—
43#339C2212yes—
44#339C2212yes—
45C2×#41C2311yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×Q8C2312yes—
48C22×D4C2311yescharacteristic
49#137—42no—
50AGL(1,ℤ/8)—42no—
51C2×Q16—21no—
52C2×SD16—21no—
53C2×SD16—21no—
54C2×D8—21no—
55C2×M16C2311yescharacteristic
56C2×M16C2311yescharacteristic
57C2×M16C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—22no—
62C4⋊Q8—24no—
63C4⋊Q8—42no—
64C4⋊Q8D412yes—
65#126—28no—
66#125—24no—
67#125—28no—
68#125D412yes—
69#124—22no—
70#124—22no—
71#124—21no—
72#124—21no—
73#124—24no—
74#124—42no—
75#123—24no—
76C22⋊Q8—24no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C4×Q8—24no—
80C4×Q8—24no—
81C4×Q8—24no—
82C4×Q8—22no—
83C4×Q8D412yes—
84C4×D4—22no—
85C4×D4D412yes—
86#117—24no—
87#117—24no—
88#117—21no—
89#117C2311yescharacteristic
90#117C2312yes—
91C2×C4⋊C4—24no—
92C2×C4⋊C4D414yes—
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#101—42no—
105#101—42no—
106#97—21no—
107C4×C8—21no—
108#95—22no—
109C24—21no—
110#41—22no—
111#41—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8C2×D411yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4C2×D411yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4C2×D414yes—
135C22×C4C2×D412yes—
136C22×C4C2411yescharacteristic
137Q16—42no—
138SD16—42no—
139SD16—42no—
140D8—42no—
141M16—22no—
142M16—22no—
143M16—22no—
144M16—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C4⋊C4—28no—
149C4⋊C4—28no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—48no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D414yes—
158C22⋊C4—28no—
159C22⋊C4—21no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—48no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42—28no—
166C42—22no—
167C42—21no—
168C42—22no—
169C42—22no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D412yes—
173C42#4114yes—
174C23—21no—
175C23—44no—
176C23—41no—
177C23—42no—
178C23C22×D411yescharacteristic
179Q8—21no—
180Q8—21no—
181Q8—44no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—41no—
187Q8—41no—
188Q8—42no—
189D4—21no—
190D4—21no—
191D4—42no—
192D4—41no—
193D4—41no—
194D4—42no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4#12012yes—
209C2×C4#12012yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4112yes—
213C2×C4H(5,𝔽2)12yes—
214C8—41no—
215C8—41no—
216C8—42no—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—84no—
221C22—81no—
222C22—82no—
223C22C2×#12011yescharacteristic
224C22#54612yes—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—41no—
230C4—42no—
231C4—44no—
232C4—81no—
233C4#53211yescharacteristic
234C4#53511yescharacteristic
235C4#55012yes—
236C2—22no—
237C2—81no—
238C2—161no—
239C2#243111yescharacteristic
240C2#302012yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×AGL(1,ℤ/8)C22
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. #3509C2semidirect
2. #3140C2semidirect
4. #2658C2semidirect
5. #2005C2semidirect
7. #1995C2semidirect
8. #1966C2semidirect
9. #1888C2semidirect
10. #1886C2semidirect
11. #1759C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#31402
#30402
#26582
#20052
#20032
#19952
#19662
#18882
#18862
#17592