Finite Group Databasebeta

← #33007

Subgroups — #33007

Order 256.

873 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×C4×D4C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic, complemented
5#1877C214yesmaximal, complemented
6#1864C214yesmaximal, complemented
7#1823C211yesmaximal, characteristic, complemented
8#1817C211yesmaximal, characteristic, complemented
9#1786C211yesmaximal, characteristic
10C23×D4C2211yescharacteristic
11C24×C4—21no—
12C22×SD16C2212yes—
13C2×C22⋊Q8C2211yescharacteristic
14C2×C4×D4—24no—
15C2×C4×D4—24no—
16C2×C4×D4—22no—
17C2×C4×D4C2214yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20C22×C42C2211yescharacteristic
21#483—24no—
22#450—24no—
23C2×#106C2212yescomplemented
24C2×#103C2212yescomplemented
25C2×#102C2212yes—
26C2×#102C2214yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#391C2212yescomplemented
30#389C2212yescomplemented
31#387—22no—
32#385—21no—
33#381—21no—
34#380C2211yescharacteristic
35#340C2214yescomplemented
36#336C2214yescomplemented
37C25—21no—
38C22×Q8C2311yescharacteristic
39C22×D4—28no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4D414yes—
43C22×D4C2312yes—
44C23×C4—28no—
45C23×C4—21no—
46C23×C4—24no—
47C23×C4—44no—
48C23×C4—44no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×SD16—24no—
53C2×SD16—28no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C4×D4—216no—
59C4×D4—28no—
60C4×D4—28no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D414yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—28no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42D412yes—
76C2×C42C2312yes—
77#106—24no—
78#103—24no—
79#102—24no—
80#102—28no—
81C22⋊C8—22no—
82C22⋊C8—22no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95—22no—
87C24—28no—
88C24—24no—
89C24—22no—
90C24—44no—
91C24—42no—
92C24—42no—
93C24C2×D411yescharacteristic
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—24no—
97C2×D4—28no—
98C2×D4—216no—
99C2×D4—216no—
100C2×D4—24no—
101C2×D4—22no—
102C2×D4—24no—
103C2×D4C2×D414yes—
104C2×D4#4118yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—216no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—48no—
117C22×C4—432no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—48no—
121C22×C4SD1614yescomplemented
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128SD16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—24no—
132C2×C8—24no—
133C4⋊C4—24no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—24no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4C2×D414yes—
141C22⋊C4—216no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C42—24no—
145C42—22no—
146C42—22no—
147C23—22no—
148C23—22no—
149C23—216no—
150C23—24no—
151C23—28no—
152C23—416no—
153C23—416no—
154C23—48no—
155C23—42no—
156C23—42no—
157C23—44no—
158C23—48no—
159C23#12012yes—
160C23C2×SD1612yes—
161C23AGL(1,ℤ/8)12yes—
162C23C22×D411yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—216no—
166D4—24no—
167D4—28no—
168D4#12318yes—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—28no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—432no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—48no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4#12012yes—
192C2×C4C2×SD1614yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4114yes—
196C8—42no—
197C8—42no—
198C22—24no—
199C22—22no—
200C22—24no—
201C22—22no—
202C22—28no—
203C22—432no—
204C22—48no—
205C22—48no—
206C22—44no—
207C22—44no—
208C22#45014yes—
209C22C2×#12011yescharacteristic
210C22#52512yes—
211C22C22×SD1611yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22#57512yes—
214C4—22no—
215C4—24no—
216C4—44no—
217C4—42no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4C2×#12312yes—
222C4#52512yes—
223C2—24no—
224C2—48no—
225C2#284911yescharacteristic
226C2C2×#45011yescharacteristic
227C2#300511yescharacteristic
228C2#310112yes—
229C2#311412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. #2855C2semidirect
5. #1877C2semidirect
6. #1864C2semidirect
7. #1823C2semidirect
8. #1817C2semidirect
23. C2×#106C22semidirect
24. C2×#103C22semidirect
29. #391C22semidirect
30. #389C22semidirect
35. #340C22semidirect
36. #336C22semidirect
121. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4832
C2×#4502
#28552
#18772
#18642
#18232
#18172
#17862