Finite Group Databasebeta

← #24009

Subgroups — #24009

Order 256.

886 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 145 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×#121C211yesmaximal, characteristic
2C2×#482C212yesmaximal, complemented
3#2280C211yesmaximal, characteristic, complemented
4#2034C212yesmaximal, complemented
5#2033C212yesmaximal, complemented
6#1875C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic, complemented
8#1852C211yesmaximal, characteristic, complemented
9#1786C211yesmaximal, characteristic, complemented
10#1786C211yesmaximal, characteristic, complemented
11#1779C211yesmaximal, characteristic, complemented
12C23×D4—21no—
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×#121—21no—
16C2×#121—21no—
17C2×#121—28no—
18C2×#121C2212yes—
19C2×#121C2212yes—
20C2×#117C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic, complemented
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4—21no—
24#482—28no—
25C2×#107C2212yescomplemented
26C2×#106C2212yescomplemented
27C2×#102C2212yescomplemented
28C2×#102C2212yescomplemented
29C2×#102C2212yescomplemented
30C2×#102C2212yescomplemented
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2212yescomplemented
34C2×C22⋊C8C2212yescomplemented
35#381—21no—
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38#380C2211yescharacteristic, complemented
39#380C2211yescharacteristic, complemented
40#379—21no—
41C4×C22⋊C4—21no—
42C4×C22⋊C4—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25—21no—
46C25D411yescharacteristic
47C22×D4—28no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—24no—
51C22×D4—21no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4—42no—
55C22×D4—44no—
56C22×D4—44no—
57C22×D4C2312yes—
58C23×C4—21no—
59C23×C4—24no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—42no—
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C22×C8C2312yes—
70C22×C8C2312yes—
71#121—28no—
72#121—28no—
73#121—24no—
74#121—24no—
75#121—24no—
76#121D418yes—
77#117—24no—
78#117—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—28no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97#107—24no—
98#106—24no—
99#102—24no—
100#102—24no—
101#102—42no—
102#102—42no—
103C22⋊C8—24no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106C22⋊C8—42no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—24no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—41no—
119C24—41no—
120C24—42no—
121C24—42no—
122C24—42no—
123C24—42no—
124C24—42no—
125C24—42no—
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C24C2×D411yescharacteristic
129C2×D4—24no—
130C2×D4—28no—
131C2×D4—28no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—48no—
138C2×D4—48no—
139C2×D4—42no—
140C2×D4—432no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—42no—
146C2×D4C2×D414yes—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D414yes—
178C22×C4C2×D411yescharacteristic
179C22×C4#4112yes—
180C22×C4#4112yes—
181C22×C4C2411yescharacteristic
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C4⋊C4—24no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4C2×D414yes—
195C22⋊C4—28no—
196C22⋊C4—28no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—42no—
204C42—42no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—28no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—24no—
223C23—44no—
224C23—44no—
225C23—44no—
226C23—44no—
227C23—44no—
228C23—48no—
229C23—48no—
230C23—48no—
231C23—44no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—84no—
239C23—84no—
240C23—84no—
241C23—84no—
242C23#12012yes—
243C23#12312yes—
244C23#13512yes—
245C23AGL(1,ℤ/8)12yes—
246C23AGL(1,ℤ/8)12yes—
247C23C22×D411yescharacteristic
248D4—24no—
249D4—24no—
250D4—48no—
251D4—416no—
252D4—416no—
253D4—42no—
254D4—42no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4#12012yes—
288C2×C4#12312yes—
289C2×C4C22×D412yes—
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4H(5,𝔽2)12yes—
293C8—42no—
294C8—42no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—416no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—816no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22#48214yescomplemented
325C22C2×#12011yescharacteristic
326C22C2×#12311yescharacteristic
327C22#53411yescharacteristic
328C22C2×#13511yescharacteristic
329C22C2×AGL(1,ℤ/8)11yescharacteristic
330C22C2×AGL(1,ℤ/8)11yescharacteristic
331C22#57511yescharacteristic
332C4—22no—
333C4—22no—
334C4—41no—
335C4—44no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#54612yes—
341C4#54612yes—
342C2—24no—
343C2—42no—
344C2—42no—
345C2—84no—
346C2#240511yescharacteristic
347C2#300211yescharacteristic
348C2#300311yescharacteristic
349C2C2×#48211yescharacteristic
350C2#309411yescharacteristic
351C2#318811yescharacteristic
352C2#319411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#482C2semidirect
3. #2280C2semidirect
4. #2034C2semidirect
5. #2033C2semidirect
6. #1875C2semidirect
7. #1853C2semidirect
8. #1852C2semidirect
9. #1786C2semidirect
10. #1786C2semidirect
11. #1779C2semidirect
20. C2×#117C22semidirect
21. C2×#117C22semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×#102C22semidirect
28. C2×#102C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
33. C2×C22⋊C8C22semidirect
34. C2×C22⋊C8C22semidirect
38. #380C22semidirect
39. #380C22semidirect
43. #340C22semidirect
44. #340C22semidirect
324. C22#482semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#4822
#22802
#20342
#20332
#18752
#18532
#18522
#17862
#17792