Finite Group Databasebeta

← #18286

Subgroups — #18286

Order 256.

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

normal subgroups only

This group has 571 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
1C2×#543C211yesmaximal, characteristic, complemented
2#3187C212yesmaximal, complemented
3#2617C211yesmaximal, characteristic
4#1608C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1520C212yesmaximal, complemented
7C2×#323C211yesmaximal, characteristic, complemented
8#543—24no—
9#543—22no—
10#543C2214yes—
11C2×C22⋊Q8—22no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8C2212yes—
15C2×#121C2211yescharacteristic
16C2×#120—21no—
17C22×C22⋊C4—21no—
18C22×C22⋊C4C2211yescharacteristic
19#483C2214yescomplemented
20#481—22no—
21#465—24no—
22#450C2214yescomplemented
23#448—22no—
24C2×C22⋊C8C2212yes—
25#397C2212yes—
26#394—22no—
27#392C2212yes—
28#386—21no—
29#380C2211yescharacteristic
30#328C2218yescomplemented
31#323C2214yes—
32C25D411yescharacteristic
33C22×Q8—21no—
34C22×Q8D412yescomplemented
35C22×D4—21no—
36C22×D4—22no—
37C22×D4C2311yescharacteristic
38C23×C4—21no—
39C23×C4—21no—
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C2×SD16—24no—
43C22×C8—22no—
44C22⋊Q8—28no—
45C22⋊Q8—28no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48C22⋊Q8—24no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C22⋊Q8—28no—
52C22⋊Q8D414yescomplemented
53C22⋊Q8C2314yes—
54#121—22no—
55#121—22no—
56#120—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D412yescomplemented
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4D412yes—
74C2×C22⋊C4C2312yes—
75#106—24no—
76#103—24no—
77#102—24no—
78C22⋊C8—22no—
79C22⋊C8C2314yes—
80#95—22no—
81#95C2312yes—
82C24—21no—
83C24—21no—
84C24—22no—
85C24—21no—
86C24—81no—
87C24—81no—
88C24SD1614yescomplemented
89C24C2×D412yes—
90C24C2×D411yescharacteristic
91C2×Q8—24no—
92C2×Q8—22no—
93C2×Q8—82no—
94C2×Q8—82no—
95C2×Q8C2×D414yes—
96C2×D4—22no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—44no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—81no—
103C2×D4—82no—
104C2×D4—82no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—81no—
114C22×C4—81no—
115C22×C4—82no—
116C22×C4—82no—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122SD16—84no—
123C2×C8—24no—
124C2×C8—82no—
125C4⋊C4—24no—
126C4⋊C4—28no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—81no—
132C4⋊C4C2×D414yes—
133C22⋊C4—28no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—44no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—82no—
142C23—22no—
143C23—24no—
144C23—22no—
145C23—24no—
146C23—41no—
147C23—41no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—81no—
158C23—81no—
159C23—84no—
160C23—84no—
161C23#12011yescharacteristic
162C23#12011yescharacteristic
163C23C2×SD1614yes—
164C23C2×SD1612yes—
165C23#13712yes—
166C23C22×D411yescharacteristic
167Q8—44no—
168Q8—82no—
169Q8—84no—
170D4—44no—
171D4—88no—
172D4—82no—
173D4—82no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4#12012yes—
195C2×C4C22×D412yes—
196C8—44no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—24no—
204C22—42no—
205C22—42no—
206C22—44no—
207C22—48no—
208C22—42no—
209C22—42no—
210C22—82no—
211C22—84no—
212C22—82no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22#44812yes—
218C22#44812yes—
219C22C2×#12011yescharacteristic
220C22C22×SD1611yescharacteristic
221C22C2×#13711yescharacteristic
222C4—22no—
223C4—22no—
224C4—42no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—81no—
229C2—21no—
230C2—22no—
231C2—21no—
232C2—22no—
233C2—44no—
234C2—161no—
235C2C2×#44811yescharacteristic
236C2#301911yescharacteristic
237C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2617C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#543C2semidirect
2. #3187C2semidirect
4. #1608C2semidirect
5. #1596C2semidirect
6. #1520C2semidirect
7. C2×#323C2semidirect
19. #483C22semidirect
22. #450C22semidirect
30. #328C22semidirect
34. C22×Q8D4semidirect
52. C22⋊Q8D4semidirect
61. C2×C4⋊C4D4semidirect
88. C24SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#31872
#26172
#16082
#15962
#15202
C2×#3232