Finite Group Databasebeta

← #10096

Subgroups — #10096

Order 256.

342 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 51 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#3215C211yesmaximal, characteristic, complemented
2#3138C211yesmaximal, characteristic, complemented
3#2657C211yesmaximal, characteristic
4#1555C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1506C211yesmaximal, characteristic, complemented
7#1300C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127—21no—
10C2×#127C2211yescharacteristic
11C2×#124—21no—
12C2×#124C2211yescharacteristic
13#497—21no—
14#490—22no—
15#489—21no—
16#486—21no—
17#486—21no—
18#464—22no—
19#451—21no—
20#450—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#418—22no—
24C4×M16C2211yescharacteristic
25#392—22no—
26#392—22no—
27#376C2211yescharacteristic
28#331—22no—
29#329—22no—
30#328—22no—
31C22×Q8D411yescharacteristic
32C22×D4—24no—
33C22×D4—21no—
34C22×D4—21no—
35C22×D4D411yescharacteristic
36C22×D4D412yes—
37C2×Q16—22no—
38C2×SD16—22no—
39C2×M16—21no—
40C4⋊Q8D411yescharacteristic, complemented
41C4⋊Q8D411yescharacteristic, complemented
42#127—21no—
43#127—21no—
44#127—21no—
45#127—21no—
46#127—44no—
47#125—21no—
48#124—21no—
49#124—21no—
50#124—21no—
51#124—21no—
52#124—44no—
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55C4×Q8—21no—
56C4×Q8—21no—
57#117—21no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—21no—
64C4⋊C8D412yescomplemented
65#103—22no—
66#102—24no—
67#102—24no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71#97—21no—
72#97—21no—
73#97D412yescomplemented
74C4×C8—21no—
75#95—21no—
76#95D412yes—
77C24—22no—
78C24—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—42no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—22no—
88C2×D4—44no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—48no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4—82no—
98C2×D4—84no—
99C2×D4—84no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105Q16—82no—
106SD16—82no—
107M16—42no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C22⋊C4—48no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C42—24no—
124C42—41no—
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42#4111yescharacteristic
129C23—42no—
130C23—42no—
131C23—42no—
132C23—42no—
133C23—44no—
134C23—41no—
135C23—42no—
136C23—44no—
137C23—42no—
138C23#12011yescharacteristic
139Q8—42no—
140Q8—42no—
141Q8—81no—
142Q8—81no—
143D4—44no—
144D4—42no—
145D4—81no—
146D4—82no—
147D4—82no—
148D4—82no—
149D4—82no—
150D4—816no—
151D4—82no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4#12011yescharacteristic
163C2×C4#12011yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12111yescharacteristic
167C2×C4#12711yescharacteristic
168C8—42no—
169C8—42no—
170C8—81no—
171C22—21no—
172C22—21no—
173C22—84no—
174C22—84no—
175C22—84no—
176C22—81no—
177C22—82no—
178C22—82no—
179C22—81no—
180C22—82no—
181C22—81no—
182C22#39211yescharacteristic
183C22#45512yes—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—44no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4#45312yes—
194C2—22no—
195C2—161no—
196C2—162no—
197C2#200211yescharacteristic
198C2#200411yescharacteristic
199C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#2657C2
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
1. #3215C2semidirect
2. #3138C2semidirect
4. #1555C2semidirect
5. #1527C2semidirect
6. #1506C2semidirect
7. #1300C2semidirect
40. C4⋊Q8D4semidirect
41. C4⋊Q8D4semidirect
64. C4⋊C8D4semidirect
73. #97D4semidirect

Maximal subgroups

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

subgroupindex
#32152
#31382
#26572
#15552
#15272
#15062
#13002