Finite Group Databasebeta

← #18133

Subgroups — #18133

Order 256.

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

normal subgroups only

This group has 655 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#3007C212yesmaximal, complemented
3#2404C211yesmaximal, characteristic
4#1602C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1511C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C2×#137—22no—
10#543—24no—
11#543—22no—
12#543C2214yes—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—22no—
16C2×C22⋊Q8C2212yes—
17C2×#120—21no—
18C22×C22⋊C4—21no—
19C22×C22⋊C4C2211yescharacteristic
20#452—24no—
21#451C2214yescomplemented
22#450C2214yescomplemented
23#417—22no—
24#407C2212yes—
25#393C2212yes—
26#392C2212yes—
27#386—21no—
28#386—22no—
29#381C2211yescharacteristic
30#328C2218yescomplemented
31#323C2214yes—
32C25D411yescharacteristic
33C2×#41—22no—
34C2×#41—22no—
35C2×#41—44no—
36C22×Q8—21no—
37C22×Q8D412yescomplemented
38C22×Q8C2311yescharacteristic
39C22×D4—21no—
40C22×D4—22no—
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47#137—48no—
48C2×Q16—24no—
49C2×SD16—24no—
50C2×M16—22no—
51C22⋊Q8—28no—
52C22⋊Q8—28no—
53C22⋊Q8—24no—
54C22⋊Q8—28no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—28no—
59C22⋊Q8D414yescomplemented
60C22⋊Q8C2314yes—
61#120—44no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D412yescomplemented
66C2×C22⋊C4—22no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4D412yes—
76C2×C22⋊C4C2312yes—
77#103—24no—
78#102—24no—
79C22⋊C8—22no—
80C22⋊C8C2314yes—
81#95—21no—
82#95—22no—
83#95C2312yes—
84C24—21no—
85C24—21no—
86C24—24no—
87C24—21no—
88C24—42no—
89C24—81no—
90C24C2×D411yescharacteristic
91C24C2×D412yes—
92#41—28no—
93#41—44no—
94#41—44no—
95#41—44no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—44no—
101C2×Q8—82no—
102C2×Q8C2×D414yes—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—82no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—81no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2411yescharacteristic
132Q16—48no—
133SD16—48no—
134M16—44no—
135C2×C8—24no—
136C4⋊C4—24no—
137C4⋊C4—28no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—24no—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—44no—
148C22⋊C4—42no—
149C22⋊C4—82no—
150C23—24no—
151C23—24no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—41no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—41no—
163C23—42no—
164C23—41no—
165C23—42no—
166C23—81no—
167C23—81no—
168C23—88no—
169C23#12011yescharacteristic
170C23#12011yescharacteristic
171C23#13714yescomplemented
172C23C22×D411yescharacteristic
173Q8—24no—
174Q8—44no—
175Q8—42no—
176Q8—82no—
177Q8—84no—
178D4—24no—
179D4—42no—
180D4—44no—
181D4—44no—
182D4—88no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—48no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4—84no—
203C2×C4—84no—
204C2×C4#12012yes—
205C2×C4C22×D412yes—
206C8—44no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—24no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22—44no—
217C22—48no—
218C22—44no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—82no—
223C22—82no—
224C22—84no—
225C22—81no—
226C22—81no—
227C22B(4,𝔽2)12yes—
228C22C2×#12011yescharacteristic
229C22C2×#13712yes—
230C4—22no—
231C4—22no—
232C4—42no—
233C4—44no—
234C4—42no—
235C4—44no—
236C4—82no—
237C2—22no—
238C2—22no—
239C2—41no—
240C2—44no—
241C2—82no—
242C2#300411yescharacteristic
243C2#301911yescharacteristic
244C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2404C2
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. #3007C2semidirect
4. #1602C2semidirect
5. #1596C2semidirect
6. #1511C2semidirect
7. #1462C2semidirect
21. #451C22semidirect
22. #450C22semidirect
30. #328C22semidirect
37. C22×Q8D4semidirect
59. C22⋊Q8D4semidirect
65. C2×C4⋊C4D4semidirect
171. C23#137semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#30072
#24042
#16022
#15962
#15112
#14622