Finite Group Databasebeta

← #24032

Subgroups — #24032

Order 256.

911 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 151 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×D42C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic, complemented
3C22×#102C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2041C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic, complemented
9#1862C212yesmaximal, complemented
10#1786C212yesmaximal, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4—21no—
13C23×D4C2211yescharacteristic
14C23×D4C2211yescharacteristic
15C23×C8C2211yescharacteristic, complemented
16D42—24no—
17D42—24no—
18C2×#127C2211yescharacteristic
19C2×#123—21no—
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×#120—21no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic, complemented
28#480—24no—
29C2×#107C2212yescomplemented
30C2×#102—24no—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yescomplemented
34C2×#102C2212yescomplemented
35C2×#102C2212yescomplemented
36C2×#102C2212yescomplemented
37C2×C22⋊C8C2211yescharacteristic, complemented
38C2×C22⋊C8C2212yescomplemented
39#398—21no—
40#397—21no—
41#389C2212yescomplemented
42#381—21no—
43#380C2211yescharacteristic, complemented
44#380C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C25—21no—
49C22×D4—24no—
50C22×D4—28no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—24no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—22no—
60C22×D4—21no—
61C22×D4—42no—
62C22×D4—42no—
63C22×D4—44no—
64C22×D4—44no—
65C22×D4D411yescharacteristic
66C22×D4D414yes—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C22×C8—22no—
75C22×C8—42no—
76C22×C8C2312yes—
77C22×C8C2312yes—
78#127—22no—
79#127—22no—
80#123—44no—
81#121—28no—
82#121—22no—
83#121—22no—
84#121—22no—
85#121D414yes—
86#120—28no—
87#120—28no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2312yes—
98C2×C4⋊C4C2312yes—
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42C2311yescharacteristic
108#107—24no—
109#102—24no—
110#102—24no—
111#102—24no—
112#102—24no—
113#102—42no—
114#102—42no—
115C22⋊C8—22no—
116C22⋊C8—42no—
117#95—22no—
118#95—21no—
119#95—22no—
120C24—21no—
121C24—24no—
122C24—21no—
123C24—21no—
124C24—24no—
125C24—44no—
126C24—44no—
127C24—42no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—82no—
132C24—82no—
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—28no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—42no—
157C2×D4—48no—
158C2×D4—432no—
159C2×D4—44no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—44no—
165C2×D4—42no—
166C2×D4—82no—
167C2×D4D814yes—
168C2×D4C2×D412yes—
169C2×D4C2×D412yes—
170C2×D4C2×D412yes—
171C2×D4#4114yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2×D411yescharacteristic
193C22×C4C2411yescharacteristic
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—44no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4C2×D412yes—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—44no—
217C42—21no—
218C42—21no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—48no—
230C23—48no—
231C23—48no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—48no—
244C23—84no—
245C23—88no—
246C23—88no—
247C23—88no—
248C23—84no—
249C23—88no—
250C23#12012yes—
251C23C2×D812yes—
252C23C2×D812yes—
253C23AGL(1,ℤ/8)12yes—
254C23C22×D411yescharacteristic
255D4—28no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—24no—
260D4—24no—
261D4—48no—
262D4—48no—
263D4—48no—
264D4—48no—
265D4—48no—
266D4—41no—
267D4—41no—
268D4—42no—
269D4—84no—
270D4#12314yescomplemented
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—48no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—84no—
295C2×C4#12012yes—
296C2×C4C2×D812yes—
297C2×C4C22×D411yescharacteristic
298C2×C4C22×D411yescharacteristic
299C2×C4C2×#4112yes—
300C2×C4H(5,𝔽2)12yes—
301C8—42no—
302C8—42no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—22no—
309C22—24no—
310C22—41no—
311C22—42no—
312C22—41no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—44no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—816no—
324C22—816no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22#44714yes—
330C22C2×#12011yescharacteristic
331C22#54612yes—
332C22C22×D811yescharacteristic
333C22C2×AGL(1,ℤ/8)11yescharacteristic
334C22#57712yes—
335C4—22no—
336C4—22no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—82no—
341C4—82no—
342C4C2×#12311yescharacteristic
343C4#53411yescharacteristic
344C4#54612yes—
345C2—24no—
346C2—42no—
347C2—44no—
348C2—82no—
349C2—82no—
350C2#240511yescharacteristic
351C2C2×#44711yescharacteristic
352C2#300911yescharacteristic
353C2#329412yes—
354C2#331012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×D42C2
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
2. C2×#480C2semidirect
3. C22×#102C2semidirect
4. #2486C2semidirect
5. #2041C2semidirect
6. #2033C2semidirect
7. #1893C2semidirect
8. #1869C2semidirect
9. #1862C2semidirect
10. #1786C2semidirect
11. #1786C2semidirect
15. C23×C8C22semidirect
27. C22×C4⋊C4C22semidirect
29. C2×#107C22semidirect
33. C2×#102C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
38. C2×C22⋊C8C22semidirect
41. #389C22semidirect
43. #380C22semidirect
44. #380C22semidirect
45. #340C22semidirect
46. #336C22semidirect
270. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4802
C22×#1022
#24862
#20412
#20332
#18932
#18692
#18622
#17862