Finite Group Databasebeta

← #23862

Subgroups — #23862

Order 256.

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

normal subgroups only

This group has 944 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×#137C211yesmaximal, characteristic, complemented
2C2×#546C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic, complemented
4C2×#450C212yesmaximal, complemented
5#2409C211yesmaximal, characteristic, complemented
6#2015C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1895C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic, complemented
14C23×D4C2211yescharacteristic
15C2×#137—24no—
16C2×#137—28no—
17C22×Q16C2212yescomplemented
18C22×SD16C2212yes—
19C22×SD16C2212yes—
20C22×M16C2211yescharacteristic, complemented
21#546—24no—
22#546—24no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2212yes—
27C2×#121—21no—
28C2×#121C2211yescharacteristic
29C2×#120—21no—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C22×C22⋊C4—21no—
33#450—28no—
34#448—24no—
35C2×#103C2212yescomplemented
36C2×#103C2212yescomplemented
37C2×#102C2212yes—
38C2×#102C2212yes—
39C2×C22⋊C8C2212yes—
40C2×C22⋊C8C2211yescharacteristic, complemented
41#395—21no—
42#393—21no—
43#391C2212yescomplemented
44#386—21no—
45#385—21no—
46#380C2211yescharacteristic, complemented
47#340C2212yescomplemented
48#336C2212yescomplemented
49C25—21no—
50C2×#41—22no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41D414yes—
54C22×Q8—21no—
55C22×Q8—24no—
56C22×Q8—22no—
57C22×Q8—22no—
58C22×Q8—42no—
59C22×Q8C2311yescharacteristic
60C22×Q8C2312yes—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—24no—
66C22×D4—21no—
67C22×D4—42no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76#137—232no—
77C2×Q16—24no—
78C2×Q16—24no—
79C2×Q16—24no—
80C2×SD16—24no—
81C2×SD16—24no—
82C2×SD16—24no—
83C2×SD16—24no—
84C2×SD16—42no—
85C2×SD16D418yescomplemented
86C2×M16—22no—
87C2×M16—24no—
88C22×C8C2312yes—
89C22×C8C2312yes—
90#124—22no—
91#124—22no—
92#123—28no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—24no—
96C22⋊Q8—42no—
97#121—28no—
98#121—22no—
99#121D414yes—
100#120—28no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2312yes—
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—28no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42C2311yescharacteristic
121#103—24no—
122#103—42no—
123#102—24no—
124#102—24no—
125C22⋊C8—22no—
126C22⋊C8—24no—
127#95—21no—
128#95—22no—
129#95—22no—
130C24—21no—
131C24—24no—
132C24—21no—
133C24—44no—
134C24—42no—
135C24—42no—
136C24—82no—
137C24C2×D411yescharacteristic
138C24C2×D411yescharacteristic
139#41—28no—
140#41—44no—
141#41—48no—
142C2×Q8—21no—
143C2×Q8—22no—
144C2×Q8—28no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—44no—
152C2×Q8—48no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8C2×D412yes—
156C2×Q8C2×D414yes—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—44no—
169C2×D4—48no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—82no—
174C2×D4C2×D412yes—
175C2×D4C2×D414yes—
176C2×D4C2×D412yes—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D412yes—
198C22×C4C2×D412yes—
199C22×C4C2411yescharacteristic
200Q16—216no—
201SD16—216no—
202SD16—48no—
203M16—28no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8C2×D414yes—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—44no—
216C4⋊C4C2×D412yes—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—24no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C42—21no—
227C42—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—48no—
240C23—48no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—88no—
249C23—84no—
250C23—88no—
251C23#12012yes—
252C23#13712yescomplemented
253C23C22×D411yescharacteristic
254Q8—24no—
255Q8—22no—
256Q8—22no—
257Q8—24no—
258Q8—48no—
259Q8—42no—
260Q8—42no—
261Q8—42no—
262D4—22no—
263D4—24no—
264D4—24no—
265D4—44no—
266D4—48no—
267D4—48no—
268D4—41no—
269D4—41no—
270D4—84no—
271D4#12014yes—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4#12012yes—
300C2×C4C22×D411yescharacteristic
301C2×C4C22×D412yes—
302C2×C4C22×D411yescharacteristic
303C2×C4H(5,𝔽2)12yes—
304C8—24no—
305C8—42no—
306C22—22no—
307C22—24no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—44no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—816no—
325C22—82no—
326C22—82no—
327C22C2×#12011yescharacteristic
328C22D4212yes—
329C22C2×#13711yescharacteristic
330C22#57511yescharacteristic
331C22#57712yes—
332C4—22no—
333C4—41no—
334C4—42no—
335C4—44no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—82no—
340C4C2×#12011yescharacteristic
341C4#53411yescharacteristic
342C4D4212yes—
343C2—42no—
344C2—42no—
345C2—44no—
346C2—82no—
347C2#240311yescharacteristic
348C2#300711yescharacteristic
349C2#300911yescharacteristic
350C2#326612yes—
351C2#327512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
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
1. C22×#137C2semidirect
3. C2×#448C2semidirect
4. C2×#450C2semidirect
5. #2409C2semidirect
6. #2015C2semidirect
7. #2001C2semidirect
8. #1895C2semidirect
9. #1877C2semidirect
10. #1870C2semidirect
11. #1788C2semidirect
13. C23×Q8C22semidirect
17. C22×Q16C22semidirect
20. C22×M16C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
40. C2×C22⋊C8C22semidirect
43. #391C22semidirect
46. #380C22semidirect
47. #340C22semidirect
48. #336C22semidirect
85. C2×SD16D4semidirect
252. C23#137semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#5462
C2×#4482
C2×#4502
#24092
#20152
#20012
#18952
#18772
#18702
#17882