Finite Group Databasebeta

← #28336

Subgroups — #28336

Order 256.

576 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 155 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×#483C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic
4C22×#103C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic
6#2434C211yesmaximal, characteristic, complemented
7#2096C212yesmaximal, complemented
8#2013C212yesmaximal, complemented
9#1968C212yesmaximal, complemented
10#1956C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2212yes—
14C22×SD16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C4⋊Q8C2212yes—
17C2×#123—22no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#483—24no—
22#465—24no—
23#448—24no—
24C2×#106C2212yes—
25C2×#106C2211yescharacteristic
26C2×#103—24no—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2212yes—
35#399—22no—
36#395—22no—
37#391C2212yes—
38#390C2212yes—
39#385—22no—
40#384C2212yes—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—22no—
58C2×SD16—42no—
59C2×SD16—42no—
60C22×C8—42no—
61C22×C8D414yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2312yes—
65C4⋊Q8—24no—
66C4⋊Q8—42no—
67#123—48no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#121—22no—
71#121—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4D412yes—
85C2×C42C2312yes—
86#106—22no—
87#106—24no—
88#103—24no—
89#103—24no—
90#103—24no—
91#103—22no—
92#103—24no—
93#102—22no—
94C22⋊C8D414yes—
95C4×C8—42no—
96#95—24no—
97#95D412yes—
98C24—21no—
99C24C2×D411yescharacteristic
100C2×Q8—22no—
101C2×Q8—28no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—44no—
111C2×Q8—48no—
112C2×Q8—44no—
113C2×Q8—42no—
114C2×Q8#4114yes—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—42no—
120C2×D4—44no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2×D414yes—
136C22×C4C2411yescharacteristic
137SD16—48no—
138SD16—44no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—44no—
144C2×C8C2×D414yes—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—48no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—48no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—48no—
160C22⋊C4SD1618yescomplemented
161C42—22no—
162C42—22no—
163C23—22no—
164C23—22no—
165C23—41no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23#12712yes—
170C23C2×SD1614yes—
171C23C22×D411yescharacteristic
172Q8—24no—
173Q8—24no—
174Q8—48no—
175Q8—42no—
176Q8—42no—
177Q8—41no—
178Q8—41no—
179Q8—42no—
180D4—41no—
181D4—41no—
182D4—42no—
183D4—84no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—84no—
203C2×C4#12712yes—
204C2×C4C2×SD1618yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4112yes—
208C2×C4H(5,𝔽2)12yes—
209C8—22no—
210C8—22no—
211C8—24no—
212C22—22no—
213C22—28no—
214C22—22no—
215C22—81no—
216C22—81no—
217C22—84no—
218C22—81no—
219C22#49214yes—
220C22C2×#12711yescharacteristic
221C22#54612yes—
222C22C22×SD1612yes—
223C22#57812yes—
224C4—28no—
225C4—22no—
226C4—44no—
227C4—82no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4#52511yescharacteristic
232C4#53411yescharacteristic
233C4#54612yes—
234C2—24no—
235C2—161no—
236C2#253911yescharacteristic
237C2C2×#49211yescharacteristic
238C2#315611yescharacteristic
239C2#329814yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#483C2semidirect
4. C22×#103C2semidirect
6. #2434C2semidirect
7. #2096C2semidirect
8. #2013C2semidirect
9. #1968C2semidirect
10. #1956C2semidirect
11. C8×C22⋊C4C2semidirect
160. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4652
C2×#4482
C22×#1032
#25012
#24342
#20962
#20132
#19682
#19562
C8×C22⋊C42