Finite Group Databasebeta

← #24356

Subgroups — #24356

Order 256.

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

normal subgroups only

This group has 828 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, complemented
2C2×#460C212yesmaximal, complemented
3C2×#448C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6#2055C212yesmaximal, complemented
7#2014C212yesmaximal, complemented
8#1893C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×SD16—24no—
15C22×SD16—28no—
16C22×SD16C2212yes—
17C22×SD16C2212yes—
18C22×SD16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic, complemented
20C23×C8C2211yescharacteristic
21C2×#127C2212yes—
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×#120—21no—
25C2×C4×Q8C2212yescomplemented
26#460—28no—
27#448—24no—
28C2×C4⋊C8C2212yescomplemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yescomplemented
34C2×C22⋊C8C2211yescharacteristic
35#392—22no—
36#391—22no—
37#391C2212yescomplemented
38#390C2212yes—
39#380—21no—
40#380C2211yescharacteristic
41C4×C22⋊C4—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—24no—
49C22×Q8—42no—
50C22×Q8D414yescomplemented
51C22×Q8C2311yescharacteristic
52C22×Q8C2311yescharacteristic
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4—42no—
61C22×D4C2311yescharacteristic
62C22×D4C2312yes—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×SD16—232no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—24no—
73C2×SD16—24no—
74C2×SD16—42no—
75C2×SD16D418yescomplemented
76C22×C8—22no—
77C22×C8—24no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2312yes—
81#127—24no—
82#127—42no—
83#121—22no—
84#121—22no—
85#120—44no—
86#120—44no—
87C4×Q8—24no—
88C4×Q8—42no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—22no—
99C2×C42C2312yes—
100C4⋊C8—24no—
101#103—22no—
102#103—22no—
103#102—24no—
104#102—24no—
105#102—42no—
106C22⋊C8—22no—
107#95—22no—
108#95—22no—
109#95—21no—
110#95D412yes—
111C24—22no—
112C24—21no—
113C24—44no—
114C24—42no—
115C24—42no—
116C24—42no—
117C24—82no—
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—28no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—48no—
129C2×Q8C2×D412yes—
130C2×Q8C2×D412yes—
131C2×Q8C2×D414yes—
132C2×D4—28no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—48no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4—82no—
154C2×D4—82no—
155C2×D4C2×D414yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—44no—
164C22×C4C2×D412yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4#4112yes—
169C22×C4C2411yescharacteristic
170SD16—216no—
171SD16—216no—
172SD16—44no—
173SD16—44no—
174C2×C8—28no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8C2×D414yes—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—44no—
187C22⋊C4—22no—
188C22⋊C4—44no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4SD1614yescomplemented
193C42—22no—
194C42—22no—
195C42—44no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—48no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—42no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—88no—
212C23—88no—
213C23—84no—
214C23—84no—
215C23#12112yes—
216C23C2×SD1612yes—
217C23AGL(1,ℤ/8)12yes—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—24no—
221Q8—22no—
222Q8—22no—
223Q8—48no—
224Q8—42no—
225Q8#12014yescomplemented
226D4—24no—
227D4—24no—
228D4—48no—
229D4—41no—
230D4—42no—
231D4—41no—
232D4—42no—
233D4—42no—
234D4—88no—
235D4—88no—
236D4—84no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—48no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4#12112yes—
258C2×C4C2×SD1614yescomplemented
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D412yes—
261C2×C4C2×#4111yescharacteristic
262C2×C4H(5,𝔽2)12yes—
263C8—24no—
264C8—41no—
265C8—41no—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—816no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—81no—
279C22—82no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#46014yescomplemented
284C22C2×#12111yescharacteristic
285C22D4212yes—
286C22C22×SD1611yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22#57712yes—
289C4—22no—
290C4—24no—
291C4—24no—
292C4—42no—
293C4—42no—
294C4—82no—
295C4—81no—
296C4C2×#12011yescharacteristic
297C4#53411yescharacteristic
298C4#54612yes—
299C2—24no—
300C2—84no—
301C2—161no—
302C2#241011yescharacteristic
303C2C2×#46011yescharacteristic
304C2#304311yescharacteristic
305C2#326512yes—
306C2D4×SD1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
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. C23×SD16C2semidirect
2. C2×#460C2semidirect
3. C2×#448C2semidirect
5. Q8×C22⋊C4C2semidirect
6. #2055C2semidirect
7. #2014C2semidirect
8. #1893C2semidirect
9. #1880C2semidirect
10. #1850C2semidirect
11. #1786C2semidirect
18. C22×SD16C22semidirect
19. C22×SD16C22semidirect
25. C2×C4×Q8C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
33. C2×#102C22semidirect
37. #391C22semidirect
43. #340C22semidirect
44. #336C22semidirect
50. C22×Q8D4semidirect
75. C2×SD16D4semidirect
192. C22⋊C4SD16semidirect
225. Q8#120semidirect
258. C2×C4C2×SD16semidirect
283. C22#460semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4602
C2×#4482
#24332
Q8×C22⋊C42
#20552
#20142
#18932
#18802
#18502
#17862