Finite Group Databasebeta

← #24385

Subgroups — #24385

Order 256.

946 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 153 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×#468C212yesmaximal, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#407C211yesmaximal, characteristic, complemented
5#2848C211yesmaximal, characteristic, complemented
6#2033C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1895C212yesmaximal, complemented
9#1872C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4—21no—
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C22×SD16C2212yescomplemented
16C22×M16C2212yescomplemented
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic, complemented
19C2×C22⋊Q8C2211yescharacteristic, complemented
20C2×#121—21no—
21C2×#121—21no—
22C2×#121—28no—
23C2×#121C2212yes—
24C2×#121C2212yes—
25C22×C4⋊C4C2211yescharacteristic
26C22×C22⋊C4—21no—
27#468—28no—
28#450—24no—
29C2×#107C2212yescomplemented
30C2×#103C2212yescomplemented
31C2×#102C2212yescomplemented
32C2×#102C2212yescomplemented
33C2×#102C2212yescomplemented
34#407—24no—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yescomplemented
38#385—21no—
39#385—21no—
40#385—21no—
41#380C2211yescharacteristic
42#380C2211yescharacteristic, complemented
43#380C2211yescharacteristic, complemented
44#379—21no—
45#340C2212yescomplemented
46#340C2212yescomplemented
47C25—21no—
48C25D411yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—28no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—44no—
59C22×D4—44no—
60C22×D4C2312yes—
61C23×C4—21no—
62C23×C4—24no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—42no—
66C23×C4D411yescharacteristic
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—24no—
73C2×SD16—24no—
74C2×SD16—42no—
75C2×M16—24no—
76C2×M16—24no—
77C2×M16—42no—
78C22×C8C2312yes—
79C22×C8C2312yes—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—22no—
83C22⋊Q8—42no—
84#121—28no—
85#121—28no—
86#121—24no—
87#121—24no—
88#121—24no—
89#121D418yes—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2312yes—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108#107—24no—
109#103—24no—
110#102—24no—
111#102—24no—
112#102—42no—
113C22⋊C8—22no—
114C22⋊C8—24no—
115C22⋊C8D414yescomplemented
116#95—21no—
117#95—21no—
118#95—21no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—24no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—41no—
127C24—41no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24C2×D411yescharacteristic
135C24C2×D411yescharacteristic
136C24C2×D411yescharacteristic
137C2×Q8—22no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×D4—24no—
142C2×D4—28no—
143C2×D4—28no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—42no—
152C2×D4—432no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—42no—
158C2×D4C2×D414yes—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—48no—
184C22×C4—48no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D414yes—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4#4112yes—
193C22×C4C2411yescharacteristic
194SD16—48no—
195M16—48no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8C2×D414yescomplemented
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—21no—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4C2×D414yes—
212C22⋊C4—28no—
213C22⋊C4—28no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—28no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—24no—
238C23—44no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—44no—
243C23—48no—
244C23—48no—
245C23—48no—
246C23—44no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—84no—
254C23—84no—
255C23—84no—
256C23—84no—
257C23#12012yes—
258C23#12112yes—
259C23AGL(1,ℤ/8)12yes—
260C23AGL(1,ℤ/8)12yes—
261C23#13712yes—
262C23C22×D411yescharacteristic
263Q8—42no—
264Q8—42no—
265D4—24no—
266D4—24no—
267D4—48no—
268D4—416no—
269D4—416no—
270D4—42no—
271D4—42no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4#12012yes—
305C2×C4#12112yes—
306C2×C4C22×D411yescharacteristic
307C2×C4C22×D412yes—
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)12yes—
310C8—24no—
311C8—42no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—416no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—816no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22#46814yescomplemented
342C22C2×#12011yescharacteristic
343C22C2×#12111yescharacteristic
344C22#53411yescharacteristic
345C22C2×AGL(1,ℤ/8)11yescharacteristic
346C22C2×AGL(1,ℤ/8)11yescharacteristic
347C22C2×#13711yescharacteristic
348C22#57511yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—44no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4D4212yes—
358C4#54612yes—
359C2—24no—
360C2—42no—
361C2—42no—
362C2—84no—
363C2#241011yescharacteristic
364C2#300311yescharacteristic
365C2#300711yescharacteristic
366C2C2×#46811yescharacteristic
367C2#305511yescharacteristic
368C2#319311yescharacteristic
369C2#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×#468C2semidirect
3. C2×#450C2semidirect
4. C2×#407C2semidirect
5. #2848C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
8. #1895C2semidirect
9. #1872C2semidirect
10. #1852C2semidirect
11. #1786C2semidirect
15. C22×SD16C22semidirect
16. C22×M16C22semidirect
18. C2×C22⋊Q8C22semidirect
19. C2×C22⋊Q8C22semidirect
29. C2×#107C22semidirect
30. C2×#103C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
33. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
42. #380C22semidirect
43. #380C22semidirect
45. #340C22semidirect
46. #340C22semidirect
115. C22⋊C8D4semidirect
201. C2×C8C2×D4semidirect
341. C22#468semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#4682
C2×#4502
C2×#4072
#28482
#20332
#20002
#18952
#18722
#18522
#17862