Finite Group Databasebeta

← #28137

Subgroups — #28137

Order 256.

494 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 115 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#3093C211yesmaximal, characteristic, complemented
2#3003C211yesmaximal, characteristic
3#2930C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5#2063C212yesmaximal, complemented
6#2051C212yesmaximal, complemented
7#1931C212yesmaximal, complemented
8#1907C212yesmaximal, complemented
9#1822C211yesmaximal, characteristic, complemented
10#1811C211yesmaximal, characteristic, complemented
11#1783C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2212yes—
14C2×#131C2211yescharacteristic, complemented
15C22×M16C2211yescharacteristic
16#525C2211yescharacteristic
17C2×#117C2211yescharacteristic
18#485—22no—
19#482—22no—
20#450—22no—
21#447—22no—
22#436—22no—
23#435—22no—
24C2×#108C2212yescomplemented
25#428C2212yescomplemented
26#424C2211yescharacteristic, complemented
27#418—22no—
28#418—22no—
29#418C2212yes—
30#417C2212yescomplemented
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#100C2212yes—
34C2×#100C2212yes—
35#408C2211yescharacteristic, complemented
36#407C2211yescharacteristic
37#407C2211yescharacteristic
38#407C2211yescharacteristic
39#405C2211yescharacteristic, complemented
40#343C2212yescomplemented
41#343C2212yescomplemented
42#341C2212yescomplemented
43C25—21no—
44C2×#41C2311yescharacteristic
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—21no—
49C22×D4—42no—
50C22×D4C2312yes—
51C23×C4C2311yescharacteristic
52AGL(1,ℤ/8)—24no—
53AGL(1,ℤ/8)—42no—
54C2×SD16—22no—
55C2×D8—22no—
56#131—42no—
57C2×M16—21no—
58C2×M16—42no—
59C2×M16D412yes—
60C2×M16D412yes—
61C2×M16C2311yescharacteristic
62C2×M16C2312yes—
63C2×M16C2312yes—
64C2×M16C2311yescharacteristic
65C22×C8—21no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#123—21no—
69C22⋊Q8D412yes—
70#121D412yes—
71#120—21no—
72C4×D4—22no—
73#117—22no—
74#117—22no—
75#117C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79#108—42no—
80#107—22no—
81#106—22no—
82C4⋊C8—22no—
83#103—22no—
84#102—22no—
85#102—22no—
86#102—22no—
87#102—24no—
88#102—24no—
89#100—24no—
90#100—42no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95#97—21no—
96C4×C8—21no—
97C24—22no—
98C24—44no—
99C24—42no—
100C24—84no—
101C24#4111yescharacteristic
102#41—22no—
103#41—22no—
104C2×Q8C2×D411yescharacteristic
105C2×D4—28no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—48no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—42no—
117C2×D4C2×D411yescharacteristic
118C2×D4#4114yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4#4111yescharacteristic
126C22×C4C2411yescharacteristic
127SD16—44no—
128D8—44no—
129M16—22no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16—22no—
135M16—42no—
136M16—42no—
137M16—42no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—21no—
144C2×C8—42no—
145C2×C8C2×D411yescharacteristic
146C2×C8C2×D411yescharacteristic
147C2×C8C2×D412yes—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4C2×D412yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—22no—
155C22⋊C4—42no—
156C42—21no—
157C42—42no—
158C23—22no—
159C23—21no—
160C23—24no—
161C23—42no—
162C23—48no—
163C23—42no—
164C23—42no—
165C23—84no—
166C23—84no—
167C23—84no—
168C23—816no—
169C23#12112yes—
170C23C2×#4111yescharacteristic
171Q8—22no—
172D4—24no—
173D4—22no—
174D4—24no—
175D4—48no—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—82no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4#12112yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4112yes—
194C2×C4H(5,𝔽2)12yes—
195C8—21no—
196C8—22no—
197C8—21no—
198C8—41no—
199C8—41no—
200C8—42no—
201C8—42no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—44no—
207C22—41no—
208C22—44no—
209C22—42no—
210C22—84no—
211C22—84no—
212C22—816no—
213C22—84no—
214C22C2×#12111yescharacteristic
215C22#55112yes—
216C4—22no—
217C4—41no—
218C4—42no—
219C4—82no—
220C4#52511yescharacteristic
221C4#53411yescharacteristic
222C4#54612yes—
223C2—22no—
224C2—42no—
225C2—81no—
226C2—84no—
227C2#253411yescharacteristic
228C2#306812yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3003C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3093C2semidirect
3. #2930C2semidirect
4. C2×#418C2semidirect
5. #2063C2semidirect
6. #2051C2semidirect
7. #1931C2semidirect
8. #1907C2semidirect
9. #1822C2semidirect
10. #1811C2semidirect
11. #1783C2semidirect
14. C2×#131C22semidirect
24. C2×#108C22semidirect
25. #428C22semidirect
26. #424C22semidirect
30. #417C22semidirect
35. #408C22semidirect
39. #405C22semidirect
40. #343C22semidirect
41. #343C22semidirect
42. #341C22semidirect

Maximal subgroups

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

subgroupindex
#30932
#30032
#29302
C2×#4182
#20632
#20512
#19312
#19072
#18222
#18112
#17832