Finite Group Databasebeta

← #18178

Subgroups — #18178

Order 256.

693 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 117 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#3109C212yesmaximal, complemented
3#2397C211yesmaximal, characteristic, complemented
4#1613C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1505C212yesmaximal, 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×#121—22no—
16C2×C4×D4—22no—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic, complemented
19#481C2214yescomplemented
20#476—22no—
21#459—22no—
22#447C2214yescomplemented
23#442—24no—
24#432C2212yescomplemented
25#399—21no—
26#399C2212yescomplemented
27#392—21no—
28#392C2212yescomplemented
29#376C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32H(5,𝔽2)—48no—
33C2×#41—24no—
34C2×#41—22no—
35C22×D4—24no—
36C22×D4—24no—
37C22×D4—22no—
38C22×D4—22no—
39C22×D4D411yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4D412yes—
43C2×D8—24no—
44C4⋊Q8D412yescomplemented
45#127D412yescomplemented
46#124—22no—
47#123—28no—
48C22⋊Q8—24no—
49#121—24no—
50#121—24no—
51#121—24no—
52#121—22no—
53#121—24no—
54#121C2314yes—
55#120—28no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—48no—
62C4×D4D414yescomplemented
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4C2312yes—
72C2×C42—21no—
73C2×C42C2311yescharacteristic
74#106—24no—
75C4⋊C8—22no—
76#102—24no—
77#102—24no—
78C22⋊C8C2314yes—
79#97—22no—
80#95—21no—
81#95C2312yes—
82C24—24no—
83C24—21no—
84C24C2×D412yes—
85#41—48no—
86#41—416no—
87C2×Q8—22no—
88C2×D4—24no—
89C2×D4—28no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—44no—
97C2×D4—42no—
98C2×D4—416no—
99C2×D4—44no—
100C2×D4—48no—
101C2×D4—44no—
102C2×D4—48no—
103C2×D4—84no—
104C2×D4C2×D414yes—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119D8—48no—
120C2×C8—24no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—44no—
126C4⋊C4C2×D414yes—
127C22⋊C4—28no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—24no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C22⋊C4—42no—
134C42—24no—
135C42C2×D412yes—
136C42C2×D412yes—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—24no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—41no—
145C23—48no—
146C23—41no—
147C23—42no—
148C23—84no—
149C23—84no—
150C23AGL(1,ℤ/8)14yes—
151C23C22×D411yescharacteristic
152Q8—44no—
153D4—28no—
154D4—24no—
155D4—44no—
156D4—42no—
157D4—44no—
158D4—416no—
159D4—416no—
160D4—48no—
161D4—44no—
162D4—84no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—41no—
169C2×C4—48no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4#12011yescharacteristic
181C2×C4#12012yes—
182C2×C4#12011yescharacteristic
183C2×C4C22×D412yes—
184C8—44no—
185C22—21no—
186C22—21no—
187C22—24no—
188C22—44no—
189C22—44no—
190C22—42no—
191C22—42no—
192C22—44no—
193C22—44no—
194C22—42no—
195C22—44no—
196C22—82no—
197C22—82no—
198C22—81no—
199C22—81no—
200C22—168no—
201C22C2×#12011yescharacteristic
202C22C2×AGL(1,ℤ/8)12yes—
203C4—22no—
204C4—21no—
205C4—22no—
206C4—44no—
207C4—44no—
208C4—42no—
209C4—81no—
210C4#45312yes—
211C2—22no—
212C2—44no—
213C2—82no—
214C2—84no—
215C2#300311yescharacteristic
216C2C2×#45311yescharacteristic
217C2#302511yescharacteristic
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. #3109C2semidirect
3. #2397C2semidirect
4. #1613C2semidirect
5. #1595C2semidirect
6. #1505C2semidirect
7. #1466C2semidirect
18. C2×C4×D4C22semidirect
19. #481C22semidirect
22. #447C22semidirect
24. #432C22semidirect
26. #399C22semidirect
28. #392C22semidirect
29. #376C22semidirect
30. #327C22semidirect
31. #323C22semidirect
44. C4⋊Q8D4semidirect
45. #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
#31092
#23972
#16132
#15952
#15052
#14662