Finite Group Databasebeta

← #24530

Subgroups — #24530

Order 256.

599 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 143 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×#450C211yesmaximal, characteristic
2C22×#102C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic
5#2071C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1877C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20#450—24no—
21C2×C4⋊C8C2212yescomplemented
22C2×#103C2212yescomplemented
23C2×#102—24no—
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#398—22no—
30#391C2212yescomplemented
31#390C2212yes—
32#386—22no—
33#385—21no—
34#385—21no—
35#384C2212yescomplemented
36#380C2211yescharacteristic
37#379—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yescomplemented
40#340C2212yescomplemented
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4D414yes—
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—22no—
54C23×C4C2311yescharacteristic
55C2×SD16—22no—
56C2×SD16—22no—
57C2×SD16—24no—
58C2×SD16—42no—
59C22×C8—22no—
60C22×C8—42no—
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C4×D4—24no—
67C4×D4—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4D411yescharacteristic
84C2×C22⋊C4D411yescharacteristic
85C2×C42C2312yes—
86C4⋊C8—42no—
87#103—42no—
88#102—24no—
89#102—24no—
90#102—22no—
91#102—22no—
92C22⋊C8—22no—
93#95—21no—
94#95—22no—
95#95—22no—
96#95D412yes—
97C24—21no—
98C24—21no—
99C24—24no—
100C24—44no—
101C24—42no—
102C24—42no—
103C24—82no—
104C24C2×D411yescharacteristic
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×D4—28no—
110C2×D4—24no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—44no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4C2×D412yes—
120C2×D4C2×D412yes—
121C2×D4#4114yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139SD16—44no—
140SD16—44no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—44no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C22⋊C4—22no—
160C22⋊C4—44no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—82no—
165C22⋊C4SD1614yescomplemented
166C42—22no—
167C42—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—24no—
172C23—22no—
173C23—48no—
174C23—41no—
175C23—41no—
176C23—42no—
177C23—42no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—48no—
182C23—84no—
183C23—88no—
184C23—88no—
185C23#12112yes—
186C23C2×SD1612yes—
187C23#13712yes—
188C23C22×D411yescharacteristic
189Q8—41no—
190Q8—41no—
191Q8—42no—
192D4—22no—
193D4—22no—
194D4—22no—
195D4—24no—
196D4—48no—
197D4—42no—
198D4#12314yes—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—88no—
219C2×C4—84no—
220C2×C4#12112yes—
221C2×C4C2×SD1614yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C4C2×#4111yescharacteristic
225C2×C4H(5,𝔽2)12yes—
226C8—41no—
227C8—42no—
228C8—41no—
229C22—22no—
230C22—24no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—816no—
242C22—82no—
243C22—82no—
244C22#46114yes—
245C22C2×#12111yescharacteristic
246C22#54612yes—
247C22C22×SD1611yescharacteristic
248C22C2×#13711yescharacteristic
249C22#57612yes—
250C4—22no—
251C4—24no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4—81no—
256C4—82no—
257C4C2×#12311yescharacteristic
258C4#53411yescharacteristic
259C4#55112yes—
260C2—24no—
261C2—44no—
262C2—82no—
263C2#241211yescharacteristic
264C2C2×#46111yescharacteristic
265C2#304111yescharacteristic
266C2#329512yes—
267C2#331412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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. #2502C2semidirect
5. #2071C2semidirect
6. #2055C2semidirect
7. #1892C2semidirect
8. #1877C2semidirect
9. #1850C2semidirect
10. #1824C2semidirect
11. #1802C2semidirect
21. C2×C4⋊C8C22semidirect
22. C2×#103C22semidirect
30. #391C22semidirect
35. #384C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #336C22semidirect
165. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C22×#1022
#25022
D4×C22⋊C42
#20712
#20552
#18922
#18772
#18502
#18242
#18022