Finite Group Databasebeta

← #20744

Subgroups — #20744

Order 256.

719 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#518C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2314C211yesmaximal, characteristic, complemented
4#1825C212yesmaximal, complemented
5#1804C212yesmaximal, complemented
6#1802C212yesmaximal, complemented
7#1786C212yesmaximal
8#1762C212yesmaximal, complemented
9C4×#102C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11#518—24no—
12#518C418yescomplemented
13C2×C4×D4—22no—
14C2×C4×D4C2212yes—
15C2×C4×D4C2212yes—
16C2×#117C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C22⋊C4—22no—
19#433—24no—
20C2×C4⋊C8C2212yescomplemented
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2212yes—
26C2×#97C2211yescharacteristic, complemented
27C2×C4×C8C2211yescharacteristic, complemented
28#388—22no—
29#387—22no—
30#384C2211yescharacteristic, complemented
31#382C2211yescharacteristic, complemented
32#380C2212yes—
33C4×C4⋊C4C2212yescomplemented
34#340C2212yescomplemented
35#340C2212yescomplemented
36#340C2212yescomplemented
37#340C2212yescomplemented
38C25—21no—
39C22×D4—24no—
40C22×D4—24no—
41C22×D4—21no—
42C22×D4—22no—
43C22×D4C2×C414yes—
44C22×D4C2312yes—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4×D4—216no—
51C4×D4—24no—
52C4×D4—24no—
53C4×D4C2×C418yes—
54C4×D4C2×C418yes—
55#117—22no—
56#117C2×C414yes—
57#117C2×C414yes—
58#117D414yes—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—216no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D412yes—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C4⋊C8—24no—
75#102—24no—
76#102—24no—
77#102—24no—
78#102—24no—
79C22⋊C8—24no—
80#97—22no—
81C4×C8—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85C24—24no—
86C24—24no—
87C24—22no—
88C24—24no—
89C24—82no—
90C24—82no—
91C24C2×D411yescharacteristic
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—216no—
95C2×D4—216no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4C22×C414yes—
101C2×D4#4114yes—
102C2×D4#4114yes—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—28no—
107C22×C4—28no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—82no—
111C22×C4—82no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4C22×C412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—82no—
132C4⋊C4C22⋊C418yes—
133C4⋊C4C22×C414yes—
134C22⋊C4—24no—
135C22⋊C4—28no—
136C22⋊C4—28no—
137C22⋊C4—82no—
138C22⋊C4C22⋊C418yes—
139C42—24no—
140C42—82no—
141C42C22×C412yes—
142C42C22×C412yes—
143C42C2×D412yes—
144C42C2×D412yes—
145C23—216no—
146C23—24no—
147C23—21no—
148C23—21no—
149C23—24no—
150C23—24no—
151C23—22no—
152C23—22no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—88no—
158C23—88no—
159C23—84no—
160C23—84no—
161C23C2×C22⋊C412yes—
162C23C22×D411yescharacteristic
163D4—216no—
164D4—24no—
165D4—24no—
166D4—24no—
167D4#11718yes—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—88no—
186C2×C4—88no—
187C2×C4C2×C22⋊C412yes—
188C2×C4C2×C22⋊C414yes—
189C2×C4C2×C22⋊C414yes—
190C2×C4C23×C411yescharacteristic
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4112yes—
193C2×C4C2×#4112yes—
194C8—42no—
195C8—42no—
196C22—24no—
197C22—22no—
198C22—28no—
199C22—42no—
200C22—44no—
201C22—44no—
202C22—41no—
203C22—44no—
204C22—41no—
205C22—44no—
206C22—44no—
207C22—44no—
208C22—816no—
209C22C22×C22⋊C411yescharacteristic
210C22#52511yescharacteristic
211C22#52511yescharacteristic
212C22#57612yes—
213C22#57712yes—
214C4—24no—
215C4—24no—
216C4—22no—
217C4—44no—
218C4—82no—
219C4—82no—
220C4C2×#11712yes—
221C4#52512yes—
222C2—42no—
223C2—48no—
224C2#228011yescharacteristic
225C2#289712yes—
226C2#311812yes—
227C2#312211yescharacteristic
228C2#312311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#433C2semidirect
3. #2314C2semidirect
4. #1825C2semidirect
5. #1804C2semidirect
6. #1802C2semidirect
8. #1762C2semidirect
9. C4×#102C2semidirect
12. #518C4semidirect
20. C2×C4⋊C8C22semidirect
26. C2×#97C22semidirect
27. C2×C4×C8C22semidirect
30. #384C22semidirect
31. #382C22semidirect
33. C4×C4⋊C4C22semidirect
34. #340C22semidirect
35. #340C22semidirect
36. #340C22semidirect
37. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4332
#23142
#18252
#18042
#18022
#17862
#17622
C4×#1022