Finite Group Databasebeta

← #10250

Subgroups — #10250

Order 256.

302 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 61 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#3208C211yesmaximal, characteristic
2#3078C211yesmaximal, characteristic, complemented
3#2548C211yesmaximal, characteristic
4#1545C211yesmaximal, characteristic
5#1503C211yesmaximal, characteristic, complemented
6#1479C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×#124—21no—
11C2×C4×Q8C2211yescharacteristic
12#492—21no—
13#486—21no—
14#479—22no—
15#477—21no—
16#475—21no—
17#460—22no—
18#450—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#428—21no—
22#423C2211yescharacteristic
23#419—21no—
24#394—22no—
25#390—22no—
26#390—21no—
27C4×C4⋊C4C2211yescharacteristic
28#331—22no—
29#328—22no—
30Q8⋊C8—22no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4—21no—
34C22×D4D412yes—
35C2×SD16—24no—
36C2×M16—21no—
37C4⋊Q8D411yescharacteristic
38C4⋊Q8D411yescharacteristic
39#127—21no—
40#127—21no—
41#127—21no—
42#127—21no—
43#125—21no—
44#124—42no—
45#124—42no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54#117—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—24no—
59C2×C22⋊C4—24no—
60C2×C42—21no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63#107—22no—
64#106—22no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D412yes—
68C4⋊C8D412yes—
69#103—22no—
70#103—22no—
71#102—24no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C4×C8—21no—
75C4×C8—21no—
76#95D412yes—
77C24—22no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8SD1612yescomplemented
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—44no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4—82no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104SD16—84no—
105M16—41no—
106M16—41no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—41no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—42no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—48no—
122C22⋊C4—41no—
123C22⋊C4—84no—
124C42—41no—
125C42—41no—
126C42—41no—
127C42—41no—
128C42—41no—
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—44no—
134C23—44no—
135C23—42no—
136C23—42no—
137C23—42no—
138C23#12011yescharacteristic
139Q8—22no—
140Q8—22no—
141Q8—41no—
142Q8—41no—
143Q8—42no—
144Q8—81no—
145D4—44no—
146D4—82no—
147D4—82no—
148D4—82no—
149D4—81no—
150D4—81no—
151D4—81no—
152D4—81no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12711yescharacteristic
171C2×C4C2×SD1611yescharacteristic
172C2×C4#13511yescharacteristic
173C2×C4AGL(1,ℤ/8)11yescharacteristic
174C2×C4#13711yescharacteristic
175C8—42no—
176C8—42no—
177C8—81no—
178C22—21no—
179C22—21no—
180C22—84no—
181C22—84no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22#39211yescharacteristic
186C22#45011yescharacteristic
187C22#45211yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—82no—
193C4—81no—
194C4—81no—
195C4—81no—
196C4#44811yescharacteristic
197C4#44911yescharacteristic
198C4#45512yes—
199C4#46511yescharacteristic
200C4#46811yescharacteristic
201C2—21no—
202C2—21no—
203C2—162no—
204C2#200011yescharacteristic
205C2#200411yescharacteristic
206C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3078C2semidirect
5. #1503C2semidirect
6. #1479C2semidirect
7. #1303C2semidirect
85. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#32082
#30782
#25482
#15452
#15032
#14792
#13032