Finite Group Databasebeta

← #18318

Subgroups — #18318

Order 256.

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

normal subgroups only

This group has 586 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
1C2×#543C211yesmaximal, characteristic, complemented
2#3193C212yesmaximal, complemented
3#2539C211yesmaximal, characteristic
4#1614C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1508C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic, complemented
8#543—24no—
9#543—22no—
10#543C2214yes—
11C2×#123—21no—
12C2×C22⋊Q8—22no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8—22no—
15C2×C22⋊Q8C2212yes—
16C2×#121C2211yescharacteristic
17C2×#120—21no—
18C22×C22⋊C4—21no—
19C22×C22⋊C4C2211yescharacteristic
20#484C2214yescomplemented
21#480—22no—
22#468—24no—
23#450C2214yescomplemented
24#448—22no—
25#407C2212yes—
26#399C2212yes—
27#392C2212yes—
28#390—22no—
29C4×C22⋊C4C2211yescharacteristic
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—
43C2×M16—22no—
44#123—44no—
45C22⋊Q8—28no—
46C22⋊Q8—28no—
47C22⋊Q8—24no—
48C22⋊Q8—28no—
49C22⋊Q8—24no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8—28no—
53C22⋊Q8D414yescomplemented
54C22⋊Q8C2314yes—
55#121—22no—
56#121—22no—
57#120—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D412yescomplemented
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C22⋊C4C2312yes—
76C2×C42—21no—
77#107—24no—
78#103—24no—
79#102—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82#95C2312yes—
83C24—21no—
84C24—21no—
85C24—24no—
86C24—21no—
87C24—42no—
88C24—81no—
89C24C2×D411yescharacteristic
90C24C2×D412yes—
91C2×Q8—24no—
92C2×Q8—22no—
93C2×Q8—82no—
94C2×Q8—82no—
95C2×Q8C2×D414yes—
96C2×D4—22no—
97C2×D4—44no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—44no—
103C2×D4—81no—
104C2×D4—82no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
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—
123M16—82no—
124C2×C8—24no—
125C4⋊C4—24no—
126C4⋊C4—28no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—44no—
132C4⋊C4C2×D414yes—
133C22⋊C4—28no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C42—81no—
143C23—24no—
144C23—24no—
145C23—22no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—81no—
160C23—81no—
161C23—88no—
162C23#12011yescharacteristic
163C23#12011yescharacteristic
164C23#13714yes—
165C23C22×D411yescharacteristic
166Q8—44no—
167Q8—82no—
168Q8—84no—
169D4—44no—
170D4—88no—
171D4—82no—
172D4—82no—
173C2×C4—24no—
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—42no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—84no—
192C2×C4—84no—
193C2×C4#12012yes—
194C2×C4C22×D412yes—
195C8—44no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—21no—
203C22—21no—
204C22—44no—
205C22—44no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—84no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22#45312yes—
217C22C2×#12011yescharacteristic
218C22C2×#13712yes—
219C4—22no—
220C4—22no—
221C4—44no—
222C4—42no—
223C4—44no—
224C4—44no—
225C4—81no—
226C2—22no—
227C2—22no—
228C2—41no—
229C2—44no—
230C2—161no—
231C2#300411yescharacteristic
232C2C2×#45311yescharacteristic
233C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2539C2
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. #3193C2semidirect
4. #1614C2semidirect
5. #1596C2semidirect
6. #1508C2semidirect
7. #1462C2semidirect
20. #484C22semidirect
23. #450C22semidirect
30. #328C22semidirect
34. C22×Q8D4semidirect
53. C22⋊Q8D4semidirect
62. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#31932
#25392
#16142
#15962
#15082
#14622