Finite Group Databasebeta

← #28229

Subgroups — #28229

Order 256.

499 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 135 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×#468C212yesmaximal, complemented
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#436C211yesmaximal, characteristic, complemented
4#2536C211yesmaximal, characteristic, complemented
5#2535C211yesmaximal, characteristic
6#2035C212yesmaximal, complemented
7#2034C212yesmaximal, complemented
8#1846C211yesmaximal, characteristic, complemented
9#1809C212yesmaximal, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×M16C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×#121C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#468—28no—
20C4×SD16—24no—
21#436—24no—
22C2×#107C2212yes—
23C2×#106C2211yescharacteristic
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2211yescharacteristic
28C2×#102C2212yes—
29C2×#102C2211yescharacteristic
30C2×C22⋊C8C2212yes—
31C2×C4×C8C2211yescharacteristic
32#398—21no—
33#397—21no—
34#394—21no—
35#392—21no—
36#388—22no—
37#386—22no—
38#382C2211yescharacteristic
39#382C2211yescharacteristic
40#380C2212yes—
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44C22×Q8C2311yescharacteristic
45C22×D4—22no—
46C22×D4D411yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4C2312yes—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16D414yescomplemented
53C2×M16—24no—
54C2×M16—24no—
55C2×M16—24no—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22⋊Q8—24no—
60C22⋊Q8—42no—
61#121—24no—
62#121—42no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4D412yes—
82C2×C42—21no—
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—22no—
86#106—42no—
87C4⋊C8—22no—
88#103—24no—
89#103—22no—
90#102—24no—
91#102—22no—
92C22⋊C8—24no—
93C4×C8D414yescomplemented
94#95—22no—
95#95—22no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100C24—21no—
101C24C2×D412yes—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8C2×D412yes—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—42no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—82no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—82no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—28no—
137M16—48no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C2×C8C2×D412yes—
146C2×C8#4112yes—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C22⋊C4—24no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—82no—
164C42—42no—
165C42C2×D412yes—
166C42#4112yes—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—42no—
178C23—42no—
179C23C22×D411yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184D4—22no—
185D4—22no—
186D4—44no—
187D4—41no—
188D4—41no—
189D4—88no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4#12112yes—
214C2×C4#12112yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4H(5,𝔽2)11yescharacteristic
220C2×C42−1+411yescharacteristic
221C8—22no—
222C8—42no—
223C8#12114yes—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—44no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22C2×#12111yescharacteristic
237C22D4211yescharacteristic
238C22#54711yescharacteristic
239C22#57512yes—
240C22#57712yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—42no—
245C4—82no—
246C4—82no—
247C4—81no—
248C4—81no—
249C4C2×#12111yescharacteristic
250C4#53711yescharacteristic
251C4#54711yescharacteristic
252C4#55511yescharacteristic
253C2—82no—
254C2—82no—
255C2#253711yescharacteristic
256C2#305411yescharacteristic
257C2#305711yescharacteristic
258C2#327512yes—
259C2#335112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
2. C2×C4×SD16C2semidirect
3. C2×#436C2semidirect
4. #2536C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #1846C2semidirect
9. #1809C2semidirect
10. #1805C2semidirect
11. #1804C2semidirect
52. C2×SD16D4semidirect
93. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×C4×SD162
C2×#4362
#25362
#25352
#20352
#20342
#18462
#18092
#18052
#18042