Finite Group Databasebeta

← #24358

Subgroups — #24358

Order 256.

799 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×SD16C211yesmaximal, characteristic
2C2×#461C212yesmaximal, complemented
3C2×#450C211yesmaximal, characteristic
4#2434C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic
6#2055C212yesmaximal, complemented
7#2013C212yesmaximal, complemented
8#1892C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×SD16—24no—
15C22×SD16—28no—
16C22×SD16C2212yes—
17C22×SD16C2211yescharacteristic
18C22×SD16C2212yes—
19C22×SD16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C4⋊Q8C2212yescomplemented
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×D4C2212yes—
24C22×C22⋊C4—21no—
25#461—28no—
26#450—24no—
27C2×C4⋊C8C2212yescomplemented
28C2×#103C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×#103C2212yescomplemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34#395—22no—
35#391C2212yescomplemented
36#390C2212yes—
37#386—22no—
38#385—21no—
39#385—21no—
40#380C2211yescharacteristic
41#379—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—42no—
51C22×Q8C2311yescharacteristic
52C22×Q8C2312yes—
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—42no—
57C22×D4D414yes—
58C22×D4C2311yescharacteristic
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C2311yescharacteristic
63C2×SD16—232no—
64C2×SD16—24no—
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16—42no—
73C2×SD16D418yes—
74C22×C8—22no—
75C22×C8—24no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2312yes—
79C4⋊Q8—24no—
80C4⋊Q8—42no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C4×D4—24no—
84C4×D4—42no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42C2312yes—
99C4⋊C8—24no—
100#103—24no—
101#103—24no—
102#103—42no—
103#102—22no—
104#102—22no—
105C22⋊C8—22no—
106#95—21no—
107#95—22no—
108#95—22no—
109#95D412yes—
110C24—21no—
111C24—21no—
112C24—24no—
113C24—44no—
114C24—42no—
115C24—42no—
116C24—82no—
117C24C2×D411yescharacteristic
118C2×Q8—22no—
119C2×Q8—28no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—44no—
129C2×Q8—48no—
130C2×Q8—44no—
131C2×Q8—42no—
132C2×Q8C2×D414yes—
133C2×D4—28no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—44no—
141C2×D4—48no—
142C2×D4—44no—
143C2×D4C2×D412yes—
144C2×D4C2×D412yes—
145C2×D4C2×D414yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—82no—
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4#4112yes—
162C22×C4C2411yescharacteristic
163SD16—216no—
164SD16—216no—
165SD16—44no—
166SD16—44no—
167C2×C8—28no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8C2×D414yes—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C22⋊C4—22no—
183C22⋊C4—44no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—44no—
187C22⋊C4—82no—
188C22⋊C4SD1614yescomplemented
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—22no—
196C23—48no—
197C23—41no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—48no—
205C23—84no—
206C23—88no—
207C23—88no—
208C23#12112yes—
209C23C2×SD1612yes—
210C23#13712yes—
211C23C22×D411yescharacteristic
212Q8—24no—
213Q8—24no—
214Q8—48no—
215Q8—42no—
216Q8—42no—
217Q8—41no—
218Q8—41no—
219Q8—42no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—24no—
224D4—48no—
225D4—42no—
226D4#12014yes—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—88no—
248C2×C4#12112yes—
249C2×C4C2×SD1614yescomplemented
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D412yes—
252C2×C4C2×#4111yescharacteristic
253C2×C4H(5,𝔽2)12yes—
254C8—24no—
255C8—41no—
256C8—41no—
257C22—22no—
258C22—24no—
259C22—24no—
260C22—22no—
261C22—24no—
262C22—44no—
263C22—44no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—816no—
270C22—82no—
271C22—82no—
272C22#46114yescomplemented
273C22C2×#12111yescharacteristic
274C22D4212yes—
275C22C22×SD1611yescharacteristic
276C22C2×#13711yescharacteristic
277C22#57712yes—
278C4—22no—
279C4—24no—
280C4—44no—
281C4—42no—
282C4—81no—
283C4—82no—
284C4—81no—
285C4C2×#12011yescharacteristic
286C4#53411yescharacteristic
287C4#54612yes—
288C2—24no—
289C2—44no—
290C2—82no—
291C2#241011yescharacteristic
292C2C2×#46111yescharacteristic
293C2#304211yescharacteristic
294C2#326612yes—
295C2D4×SD1612yes—
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
2. C2×#461C2semidirect
4. #2434C2semidirect
6. #2055C2semidirect
7. #2013C2semidirect
8. #1892C2semidirect
9. #1877C2semidirect
10. #1850C2semidirect
11. #1787C2semidirect
21. C2×C4⋊Q8C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
35. #391C22semidirect
43. #340C22semidirect
44. #336C22semidirect
188. C22⋊C4SD16semidirect
249. C2×C4C2×SD16semidirect
272. C22#461semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4612
C2×#4502
#24342
D4×C22⋊C42
#20552
#20132
#18922
#18772
#18502
#17872