Finite Group Databasebeta

← #24252

Subgroups — #24252

Order 256.

938 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 147 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×#460C211yesmaximal, characteristic, complemented
3C2×#450C212yesmaximal, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5#2536C211yesmaximal, characteristic, complemented
6#2033C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1961C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1786C212yesmaximal, complemented
12C23×D4—21no—
13C23×D4C2212yes—
14C22×SD16C2211yescharacteristic
15C22×SD16C2212yescomplemented
16D42—24no—
17D42—24no—
18C2×#127C2211yescharacteristic
19C2×C22⋊Q8C2212yescomplemented
20C2×#121—21no—
21C2×#121C2212yes—
22C2×#120—22no—
23C2×C4×Q8C2211yescharacteristic, complemented
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#460—24no—
27#450—28no—
28#440—24no—
29C2×#107C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yescomplemented
35C2×#102C2212yescomplemented
36C2×#102C2212yescomplemented
37C2×C22⋊C8C2212yescomplemented
38C2×C22⋊C8C2212yescomplemented
39C2×#97C2211yescharacteristic, complemented
40#398—21no—
41#394—21no—
42#388—22no—
43#382C2211yescharacteristic, complemented
44#380C2212yescomplemented
45#340C2212yescomplemented
46C25—22no—
47C22×Q8C2311yescharacteristic
48C22×D4—28no—
49C22×D4—22no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—24no—
53C22×D4—24no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—42no—
58C22×D4—42no—
59C22×D4—48no—
60C22×D4D411yescharacteristic
61C22×D4C2312yes—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2312yes—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—42no—
70C2×SD16D414yescomplemented
71C22×C8C2312yes—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#127—22no—
75#127D414yes—
76C22⋊Q8—24no—
77C22⋊Q8—42no—
78#121—28no—
79#121—24no—
80#121—24no—
81#120—216no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4D414yes—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D412yes—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101#107—22no—
102C4⋊C8—22no—
103#103—22no—
104#102—22no—
105#102—24no—
106#102—24no—
107#102—24no—
108#102—42no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111#97—22no—
112#95—22no—
113#95—21no—
114#95—21no—
115C24—21no—
116C24—28no—
117C24—22no—
118C24—44no—
119C24—44no—
120C24—44no—
121C24—44no—
122C24—82no—
123C24—82no—
124C24C2×D412yes—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×D412yescomplemented
131C2×D4—24no—
132C2×D4—28no—
133C2×D4—28no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—48no—
149C2×D4—48no—
150C2×D4—432no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—48no—
155C2×D4—82no—
156C2×D4C2×D414yes—
157C2×D4C2×D412yes—
158C2×D4C2×D412yes—
159C2×D4#4112yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—44no—
172C22×C4C2×D412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2411yescharacteristic
176SD16—28no—
177SD16—48no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—22no—
185C2×C8—21no—
186C2×C8C2×D412yescomplemented
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4C2×D412yes—
196C22⋊C4—28no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C42—42no—
202C42C2×D412yes—
203C42C2×D412yes—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—416no—
215C23—416no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—88no—
225C23—88no—
226C23—88no—
227C23—88no—
228C23—88no—
229C23AGL(1,ℤ/8)14yes—
230C23C22×D411yescharacteristic
231Q8—22no—
232Q8—22no—
233Q8—41no—
234Q8—41no—
235D4—28no—
236D4—22no—
237D4—22no—
238D4—24no—
239D4—22no—
240D4—24no—
241D4—48no—
242D4—416no—
243D4—416no—
244D4—42no—
245D4—42no—
246D4—84no—
247D4#12114yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—48no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4#12012yes—
269C2×C4#12012yes—
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C8—22no—
277C8—41no—
278C8—42no—
279C22—24no—
280C22—24no—
281C22—24no—
282C22—24no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—832no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22C2×#12011yescharacteristic
300C22D4211yescharacteristic
301C22#54611yescharacteristic
302C22C2×AGL(1,ℤ/8)12yes—
303C22#57712yes—
304C4—22no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—81no—
310C4—82no—
311C4C2×#12111yescharacteristic
312C4#53511yescharacteristic
313C4D4211yescharacteristic
314C4#54611yescharacteristic
315C2—44no—
316C2—44no—
317C2—84no—
318C2#240811yescharacteristic
319C2#300311yescharacteristic
320C2#300911yescharacteristic
321C2#326512yes—
322C2#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×#460C2semidirect
3. C2×#450C2semidirect
4. C2×#440C2semidirect
5. #2536C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
8. #1961C2semidirect
9. #1955C2semidirect
10. #1804C2semidirect
11. #1786C2semidirect
15. C22×SD16C22semidirect
19. C2×C22⋊Q8C22semidirect
23. C2×C4×Q8C22semidirect
29. C2×#107C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
38. C2×C22⋊C8C22semidirect
39. C2×#97C22semidirect
43. #382C22semidirect
44. #380C22semidirect
45. #340C22semidirect
70. C2×SD16D4semidirect
130. C2×Q8C2×D4semidirect
186. C2×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4602
C2×#4502
C2×#4402
#25362
#20332
#20002
#19612
#19552
#18042
#17862