Finite Group Databasebeta

← #24021

Subgroups — #24021

Order 256.

657 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 127 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#3485C211yesmaximal, characteristic
2#3093C212yesmaximal, complemented
3#2297C211yesmaximal, characteristic, complemented
4#2038C214yesmaximal, complemented
5#1896C212yesmaximal, complemented
6#1856C212yesmaximal, complemented
7#1792C212yesmaximal, complemented
8#1785C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic
10C22×#41C2211yescharacteristic
11#547—24no—
12#546—24no—
13#536C2212yes—
14#534—22no—
15#525C2212yes—
16C2×#123—22no—
17#518C2211yescharacteristic
18C2×#117C2211yescharacteristic, complemented
19C22×C42C2211yescharacteristic, complemented
20#485—24no—
21#482—24no—
22#428C2214yescomplemented
23C2×#104C2214yes—
24#417C2214yes—
25C2×#101C2212yescomplemented
26C2×#100C2212yes—
27#407C2212yescomplemented
28#391C2212yescomplemented
29#390C2212yes—
30#386—22no—
31C4×C22⋊C4—22no—
32#339C2214yescomplemented
33H(5,𝔽2)—24no—
34H(5,𝔽2)—42no—
35C2×#41—22no—
36C2×#41—24no—
37C2×#41—24no—
38C2×#41—42no—
39C2×#41C2312yes—
40C22×Q8C2311yescharacteristic
41C22×D4—22no—
42C22×D4—22no—
43C22×D4D412yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4D412yes—
47C23×C4C2311yescharacteristic
48C2×M16C2314yes—
49C4⋊Q8—22no—
50#124—22no—
51#123—28no—
52#123—24no—
53#123—22no—
54#123—24no—
55C22⋊Q8—24no—
56C22⋊Q8D414yes—
57#121—28no—
58#121D414yes—
59#120—24no—
60#120—22no—
61C4×D4—24no—
62C4×D4—24no—
63#117—24no—
64#117C2312yes—
65C2×C4⋊C4—24no—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—24no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—42no—
74C2×C42C2312yes—
75#107—24no—
76#106—24no—
77#104—44no—
78#103—24no—
79#102—24no—
80#101—42no—
81#100—24no—
82C22⋊C8—24no—
83#95—24no—
84C24—22no—
85C24C2×D411yescharacteristic
86C24C2×D412yes—
87#41—24no—
88#41—24no—
89#41—416no—
90#41—48no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8C2×D412yes—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—42no—
103C2×D4—48no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4C2×D412yes—
107C2×D4#4114yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D414yes—
123C22×C4C2411yescharacteristic
124M16—24no—
125M16—24no—
126C2×C8—24no—
127C4⋊C4—28no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4C2×D414yes—
131C22⋊C4—24no—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—22no—
135C22⋊C4—24no—
136C22⋊C4—48no—
137C22⋊C4—42no—
138C42—22no—
139C42—24no—
140C42—42no—
141C42—42no—
142C23—22no—
143C23—21no—
144C23—21no—
145C23—22no—
146C23—22no—
147C23—42no—
148C23—44no—
149C23—44no—
150C23—84no—
151C23#12012yes—
152C23#12314yes—
153C23C22×D411yescharacteristic
154Q8—24no—
155Q8—44no—
156D4—28no—
157D4—24no—
158D4—48no—
159D4—48no—
160D4—42no—
161D4—42no—
162D4—84no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—28no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—48no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—48no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4#12012yes—
182C2×C4C22×D412yes—
183C2×C4C2×#4112yes—
184C2×C4H(5,𝔽2)12yes—
185C8—44no—
186C22—22no—
187C22—22no—
188C22—24no—
189C22—24no—
190C22—44no—
191C22—42no—
192C22—42no—
193C22—44no—
194C22—42no—
195C22—44no—
196C22—82no—
197C22—164no—
198C22C2×#12011yescharacteristic
199C22C2×#12311yescharacteristic
200C22#53411yescharacteristic
201C4—22no—
202C4—44no—
203C4—42no—
204C4—42no—
205C4—44no—
206C4—82no—
207C4#54614yes—
208C2—21no—
209C2—21no—
210C2—42no—
211C2—44no—
212C2—82no—
213C2#240511yescharacteristic
214C2#301812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3093C2semidirect
3. #2297C2semidirect
4. #2038C2semidirect
5. #1896C2semidirect
6. #1856C2semidirect
7. #1792C2semidirect
8. #1785C2semidirect
18. C2×#117C22semidirect
19. C22×C42C22semidirect
22. #428C22semidirect
25. C2×#101C22semidirect
27. #407C22semidirect
28. #391C22semidirect
32. #339C22semidirect

Maximal subgroups

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

subgroupindex
#34852
#30932
#22972
#20382
#18962
#18562
#17922
#17852