Finite Group Databasebeta

← #18358

Subgroups — #18358

Order 256.

642 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 115 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#3496C211yesmaximal, characteristic, complemented
2#3213C212yesmaximal, complemented
3#2631C211yesmaximal, characteristic, complemented
4#1601C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1514C212yesmaximal, complemented
7#1466C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9#547—22no—
10#546—24no—
11D42—22no—
12#537—21no—
13#534C2214yes—
14#532C2212yes—
15C2×#124C2211yescharacteristic, complemented
16C2×#121—22no—
17C2×C4×D4C2211yescharacteristic
18#496—22no—
19#489—22no—
20#463—24no—
21#448C2214yescomplemented
22#447C2214yescomplemented
23#432C2212yescomplemented
24#397—21no—
25#393C2212yescomplemented
26#392—21no—
27#392C2212yescomplemented
28#386—22no—
29#383C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32H(5,𝔽2)—48no—
33C2×#41—24no—
34C2×#41—22no—
35C22×Q8C2311yescharacteristic
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—22no—
39C22×D4—22no—
40C22×D4D411yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4D412yes—
43C2×SD16—24no—
44C2×D8—24no—
45C4⋊Q8D412yescomplemented
46#127D412yescomplemented
47#124—22no—
48#124—22no—
49#124—22no—
50#123—28no—
51C22⋊Q8—24no—
52#121—24no—
53#121—24no—
54#121—24no—
55#121—22no—
56#121—24no—
57#121C2314yes—
58#120—28no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4D414yescomplemented
63C2×C4⋊C4—22no—
64C2×C4⋊C4D411yescharacteristic
65C2×C22⋊C4—24no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4C2312yes—
71C2×C42C2311yescharacteristic
72C4⋊C8—22no—
73#103—24no—
74#102—24no—
75C22⋊C8C2314yes—
76#97—22no—
77#95—21no—
78#95—21no—
79#95—22no—
80#95C2312yes—
81C24—24no—
82C24—21no—
83C24C2×D412yes—
84#41—48no—
85#41—416no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—81no—
89C2×D4—24no—
90C2×D4—28no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—416no—
98C2×D4—48no—
99C2×D4—48no—
100C2×D4—44no—
101C2×D4—81no—
102C2×D4—84no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4C2×D414yes—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—82no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117SD16—84no—
118D8—84no—
119C2×C8—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C2×D414yes—
124C22⋊C4—28no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—42no—
131C22⋊C4—82no—
132C22⋊C4—82no—
133C42C2×D412yes—
134C42C2×D412yes—
135C23—22no—
136C23—22no—
137C23—24no—
138C23—42no—
139C23—42no—
140C23—48no—
141C23—41no—
142C23—41no—
143C23—42no—
144C23—41no—
145C23—42no—
146C23—84no—
147C23—84no—
148C23AGL(1,ℤ/8)14yes—
149C23C22×D411yescharacteristic
150Q8—44no—
151Q8—44no—
152D4—28no—
153D4—44no—
154D4—44no—
155D4—44no—
156D4—44no—
157D4—416no—
158D4—48no—
159D4—84no—
160D4—88no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—48no—
168C2×C4—48no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12012yes—
179C2×C4C22×D412yes—
180C8—44no—
181C22—21no—
182C22—21no—
183C22—24no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—44no—
188C22—44no—
189C22—44no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—168no—
197C22C2×#12011yescharacteristic
198C22C2×AGL(1,ℤ/8)12yes—
199C4—22no—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—44no—
204C4—42no—
205C4—82no—
206C4—81no—
207C2—22no—
208C2—44no—
209C2—84no—
210C2—161no—
211C2#300311yescharacteristic
212C2#301911yescharacteristic
213C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. #3213C2semidirect
3. #2631C2semidirect
4. #1601C2semidirect
5. #1595C2semidirect
6. #1514C2semidirect
7. #1466C2semidirect
15. C2×#124C22semidirect
21. #448C22semidirect
22. #447C22semidirect
23. #432C22semidirect
25. #393C22semidirect
27. #392C22semidirect
29. #383C22semidirect
30. #327C22semidirect
31. #323C22semidirect
45. C4⋊Q8D4semidirect
46. #127D4semidirect
62. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
#34962
#32132
#26312
#16012
#15952
#15142
#14662