Finite Group Databasebeta

← #24377

Subgroups — #24377

Order 256.

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

normal subgroups only

This group has 725 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
1C22×#137C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3C2×#449C211yesmaximal, characteristic
4#2436C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic, complemented
6#2056C212yesmaximal, complemented
7#2015C212yesmaximal, complemented
8#1895C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal, complemented
10#1851C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C2×#137—24no—
15C2×#137—28no—
16C22×Q16C2212yes—
17C22×SD16C2212yes—
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2212yes—
22C2×#121C2211yescharacteristic
23C2×C4×D4C2212yes—
24#463—28no—
25#449—24no—
26C2×C4⋊C8C2212yes—
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33#393—21no—
34#393—21no—
35#391C2212yes—
36#391C2212yes—
37#386—22no—
38#386—22no—
39#381—21no—
40#380C2211yescharacteristic
41C4×C22⋊C4C2211yescharacteristic
42C2×#95—21no—
43#340C2212yes—
44#336C2212yes—
45C2×#41—22no—
46C2×#41—24no—
47C2×#41—42no—
48C2×#41D414yes—
49C22×Q8—21no—
50C22×Q8—24no—
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—42no—
54C22×Q8C2312yes—
55C22×Q8C2311yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4C2311yescharacteristic
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4C2311yescharacteristic
64#137—232no—
65C2×Q16—24no—
66C2×Q16—24no—
67C2×Q16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—22no—
71C2×SD16—42no—
72C2×SD16D418yescomplemented
73C2×D8—22no—
74C2×D8—24no—
75C2×M16—22no—
76C2×M16—24no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2312yes—
80#124—24no—
81#124—42no—
82#121—22no—
83#121—22no—
84C4×D4—24no—
85C4×D4—42no—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D412yes—
94C2×C42C2312yes—
95C4⋊C8—24no—
96#103—22no—
97#103—24no—
98#103—42no—
99#102—22no—
100#102—24no—
101C22⋊C8—22no—
102#95—21no—
103#95—22no—
104#95—22no—
105#95—22no—
106#95—22no—
107#95D411yescharacteristic
108#95D411yescharacteristic
109C24—21no—
110C24—21no—
111C24C2×D411yescharacteristic
112#41—28no—
113#41—44no—
114#41—48no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—28no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—48no—
126C2×Q8—44no—
127C2×Q8—42no—
128C2×Q8C2×D414yes—
129C2×Q8C2×D412yes—
130C2×D4—24no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—42no—
140C2×D4C2×D412yes—
141C2×D4C2×D414yes—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—82no—
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D412yes—
159C22×C4#4112yes—
160C22×C4C2411yescharacteristic
161Q16—216no—
162SD16—216no—
163SD16—44no—
164D8—44no—
165M16—28no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8C2×D414yes—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C22⋊C4—42no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—81no—
183C42—22no—
184C42—22no—
185C23—22no—
186C23—24no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—41no—
198C23—41no—
199C23#12112yes—
200C23C22×D411yescharacteristic
201Q8—24no—
202Q8—22no—
203Q8—22no—
204Q8—24no—
205Q8—48no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209D4—22no—
210D4—24no—
211D4—48no—
212D4—41no—
213D4—41no—
214D4—42no—
215D4—84no—
216D4#12014yes—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—88no—
240C2×C4#12112yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D412yes—
243C2×C4C2×#4111yescharacteristic
244C2×C4H(5,𝔽2)12yes—
245C8—24no—
246C8—41no—
247C8—41no—
248C22—22no—
249C22—24no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—81no—
262C22—81no—
263C22C2×#12111yescharacteristic
264C22D4212yes—
265C22#57512yes—
266C22#57711yescharacteristic
267C22#57811yescharacteristic
268C4—22no—
269C4—42no—
270C4—44no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4C2×#12011yescharacteristic
276C4#53411yescharacteristic
277C4#54612yes—
278C2—42no—
279C2—44no—
280C2—161no—
281C2#241011yescharacteristic
282C2#304011yescharacteristic
283C2#304611yescharacteristic
284C2#326912yes—
285C2#327512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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. C22×#137C2semidirect
2. C2×#463C2semidirect
4. #2436C2semidirect
5. #2339C2semidirect
6. #2056C2semidirect
7. #2015C2semidirect
8. #1895C2semidirect
9. #1877C2semidirect
10. #1851C2semidirect
11. #1787C2semidirect
72. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#4632
C2×#4492
#24362
#23392
#20562
#20152
#18952
#18772
#18512
#17872