Finite Group Databasebeta

← #28347

Subgroups — #28347

Order 256.

542 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 139 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×#481C211yesmaximal, characteristic
2C2×#465C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic, complemented
4C22×#103C211yesmaximal, characteristic, complemented
5#2502C211yesmaximal, characteristic, complemented
6#2436C211yesmaximal, characteristic
7#2096C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1971C212yesmaximal, complemented
10#1959C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×#124C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#481—24no—
21#465—24no—
22#451—24no—
23C2×#106C2212yes—
24C2×#106C2211yescharacteristic
25C2×#103—24no—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2212yes—
34#398—22no—
35#393—22no—
36#391C2212yes—
37#391C2212yes—
38#386—22no—
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×Q16—24no—
56C2×Q16—24no—
57C2×Q16—42no—
58C2×SD16—22no—
59C2×SD16—42no—
60C22×C8—42no—
61C22×C8D414yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2312yes—
65#124—24no—
66#124—42no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69#121—22no—
70#121—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4D412yes—
83C2×C42C2312yes—
84#106—22no—
85#106—24no—
86#103—24no—
87#103—24no—
88#103—24no—
89#103—22no—
90#102—22no—
91#102—24no—
92C22⋊C8D414yes—
93#97—42no—
94#95—24no—
95#95—24no—
96#95D412yes—
97C24—21no—
98C24C2×D411yescharacteristic
99C2×Q8—22no—
100C2×Q8—28no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—44no—
110C2×Q8—48no—
111C2×Q8—44no—
112C2×Q8—42no—
113C2×Q8#4114yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—44no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4C2×D414yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136Q16—48no—
137SD16—44no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C2×C8C2×D414yes—
144C2×C8C2×D412yes—
145C2×C8C2×D412yes—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—84no—
155C22⋊C4—24no—
156C22⋊C4—48no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C23—22no—
162C23—22no—
163C23—41no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23#12712yes—
168C23#13714yes—
169C23C22×D411yescharacteristic
170Q8—24no—
171Q8—24no—
172Q8—48no—
173Q8—42no—
174Q8—42no—
175Q8—41no—
176Q8—41no—
177Q8—42no—
178D4—41no—
179D4—41no—
180D4—42no—
181D4—84no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—48no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—84no—
202C2×C4#12712yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C2×#4112yes—
206C2×C4H(5,𝔽2)12yes—
207C8—22no—
208C8—22no—
209C8—24no—
210C22—22no—
211C22—22no—
212C22—28no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—84no—
217C22#49714yes—
218C22C2×#12711yescharacteristic
219C22#54612yes—
220C22C2×#13712yes—
221C22#57712yes—
222C4—22no—
223C4—44no—
224C4—44no—
225C4—81no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4#52511yescharacteristic
230C4#53411yescharacteristic
231C4#54612yes—
232C2—24no—
233C2—161no—
234C2#253911yescharacteristic
235C2C2×#49711yescharacteristic
236C2#315411yescharacteristic
237C2#331614yes—
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
3. C2×#451C2semidirect
4. C22×#103C2semidirect
5. #2502C2semidirect
7. #2096C2semidirect
8. #2015C2semidirect
9. #1971C2semidirect
10. #1959C2semidirect
11. #1752C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4652
C2×#4512
C22×#1032
#25022
#24362
#20962
#20152
#19712
#19592
#17522