Finite Group Databasebeta

← #29384

Subgroups — #29384

Order 256.

354 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 101 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#2956C211yesmaximal, characteristic, complemented
2#2920C211yesmaximal, characteristic, complemented
3#2102C211yesmaximal, characteristic, complemented
4#2091C211yesmaximal, characteristic, complemented
5#2050C211yesmaximal, characteristic, complemented
6#2010C211yesmaximal, characteristic
7#1932C212yesmaximal, complemented
8#1896C212yesmaximal, complemented
9#1891C212yesmaximal, complemented
10#1861C212yesmaximal, complemented
11#1841C211yesmaximal, characteristic, complemented
12#575C2211yescharacteristic
13#568C2212yes—
14#525C2212yes—
15C22×C42C2211yescharacteristic
16#444—22no—
17#443—22no—
18#432—22no—
19#431—22no—
20#430C2211yescharacteristic
21#430C2212yes—
22#423C2211yescharacteristic
23#423C2211yescharacteristic
24#421C2212yes—
25C2×#104C2212yes—
26C2×#104C2212yes—
27#413C2212yes—
28#407C2211yescharacteristic
29#407C2211yescharacteristic
30#407C2212yes—
31C4×M16C2211yescharacteristic
32C4×M16C2211yescharacteristic
33#344C2212yes—
34#344C2212yes—
35#339C2212yes—
36#339C2212yes—
37#338C2211yescharacteristic
38H(3,ℤ/4)C2211yescharacteristic
39C2×#41C2312yes—
40C23×C4—21no—
41C23×C4C2311yescharacteristic
42#137—22no—
43AGL(1,ℤ/8)—22no—
44#135—22no—
45#131—24no—
46#131—42no—
47C2×M16—42no—
48C2×M16C2312yes—
49C2×M16C2312yes—
50C2×M16C2311yescharacteristic
51C2×M16C2311yescharacteristic
52C2×M16C2311yescharacteristic
53C2×M16C2311yescharacteristic
54#123—22no—
55C22⋊Q8—22no—
56#121—22no—
57#120—22no—
58C4×D4—42no—
59#117C2312yes—
60C2×C42—21no—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#108—22no—
68#108—42no—
69C4⋊C8—22no—
70C4⋊C8—22no—
71#104—22no—
72#104—22no—
73#104—22no—
74#104—22no—
75#104—42no—
76#101—22no—
77#100—22no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80C22⋊C8—42no—
81#97—21no—
82#97D412yescomplemented
83C4×C8—21no—
84C4×C8D412yescomplemented
85C24#4111yescharacteristic
86#41—22no—
87#41—22no—
88#41—22no—
89#41—22no—
90C2×Q8#4112yes—
91C2×D4—22no—
92C2×D4—42no—
93C2×D4#4112yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—44no—
106C22×C4#4111yescharacteristic
107C22×C4C2411yescharacteristic
108Q16—22no—
109SD16—24no—
110D8—22no—
111M16—22no—
112M16—22no—
113M16—22no—
114M16—22no—
115M16—22no—
116M16—22no—
117M16—22no—
118M16—22no—
119M16—44no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—44no—
125C2×C8C2×D412yes—
126C2×C8C2×D412yes—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C22⋊C4—22no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—22no—
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C42#4111yescharacteristic
142C42#4111yescharacteristic
143C23—21no—
144C23—22no—
145C23—22no—
146C23—42no—
147C23C2×#4111yescharacteristic
148Q8—22no—
149Q8—22no—
150D4—22no—
151D4—22no—
152D4—42no—
153D4—42no—
154D4—84no—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4C22×D411yescharacteristic
175C2×C4C2×#4112yes—
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)12yes—
178C8—22no—
179C8—22no—
180C8—41no—
181C8—42no—
182C8—41no—
183C8—42no—
184C22—22no—
185C22—24no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—41no—
190C22—42no—
191C22—82no—
192C22#52511yescharacteristic
193C22#55112yes—
194C4—22no—
195C4—22no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—42no—
200C4—82no—
201C4#52511yescharacteristic
202C4#53511yescharacteristic
203C4#55312yes—
204C2—21no—
205C2—22no—
206C2—42no—
207C2—82no—
208C2#259811yescharacteristic
subgroupG/H
centerC4#535
commutator, Frattini, cosocleC22×C4C24
socleC2#2598
prime radical#2010C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2956C2semidirect
2. #2920C2semidirect
3. #2102C2semidirect
4. #2091C2semidirect
5. #2050C2semidirect
7. #1932C2semidirect
8. #1896C2semidirect
9. #1891C2semidirect
10. #1861C2semidirect
11. #1841C2semidirect
82. #97D4semidirect
84. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#29562
#29202
#21022
#20912
#20502
#20102
#19322
#18962
#18912
#18612
#18412