Finite Group Databasebeta

← #23855

Subgroups — #23855

Order 256.

965 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 965 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C23×SD16C211yesmaximal, characteristic, complemented
2C2×D4×Q8C211yesmaximal, characteristic, complemented
3C2×#448C212yesmaximal, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5#2408C211yesmaximal, characteristic
6#2013C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1892C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×Q8C2211yescharacteristic, complemented
14C23×D4C2211yescharacteristic
15C22×SD16—24no—
16C22×SD16—28no—
17C22×SD16C2212yes—
18C22×SD16C2212yes—
19C22×SD16C2212yes—
20C23×C8C2211yescharacteristic
21D4×Q8—24no—
22D4×Q8—24no—
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic, complemented
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic, complemented
29C2×#121C2212yes—
30C2×#120—21no—
31C2×C4×Q8C2211yescharacteristic, complemented
32C2×C4×D4—21no—
33C2×C4×D4C2211yescharacteristic
34#450—24no—
35#448—28no—
36C2×#103C2212yescomplemented
37C2×#103C2212yescomplemented
38C2×#103C2212yescomplemented
39C2×#102C2212yes—
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2212yescomplemented
42#394—21no—
43#392—21no—
44#390C2212yes—
45#380C2211yescharacteristic
46#379—21no—
47#340C2212yescomplemented
48#336C2212yescomplemented
49C25—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—21no—
53C22×Q8—21no—
54C22×Q8—22no—
55C22×Q8—24no—
56C22×Q8—24no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8D414yescomplemented
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—24no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—42no—
69C22×D4D411yescharacteristic
70C22×D4C2312yes—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×SD16—232no—
76C2×SD16—24no—
77C2×SD16—24no—
78C2×SD16—24no—
79C2×SD16—24no—
80C2×SD16—24no—
81C2×SD16—24no—
82C2×SD16—24no—
83C2×SD16—42no—
84C2×SD16D418yescomplemented
85C22×C8—22no—
86C22×C8—24no—
87C22×C8C2312yes—
88C22×C8C2312yes—
89C4⋊Q8—28no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C22⋊Q8—28no—
93C22⋊Q8—28no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8D414yescomplemented
98#121—24no—
99#121—42no—
100#120—44no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×D4—28no—
104C4×D4—22no—
105C4×D4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2312yes—
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#103—24no—
125#103—24no—
126#103—42no—
127#102—24no—
128C22⋊C8—22no—
129C22⋊C8—24no—
130#95—21no—
131C24—22no—
132C24—44no—
133C24—42no—
134C24—42no—
135C24—42no—
136C24—82no—
137C24C2×D411yescharacteristic
138C24C2×D411yescharacteristic
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—28no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—48no—
154C2×Q8—48no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8C2×D412yes—
158C2×Q8C2×D412yes—
159C2×Q8C2×D414yes—
160C2×Q8C2×D412yes—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—28no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—48no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—82no—
176C2×D4SD1614yescomplemented
177C2×D4C2×D414yes—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D412yes—
197C22×C4C2×D412yes—
198C22×C4C2×D411yescharacteristic
199C22×C4C2411yescharacteristic
200SD16—216no—
201SD16—216no—
202SD16—48no—
203C2×C8—28no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8C2×D414yes—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4C2×D412yes—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C42—24no—
229C42—21no—
230C42—21no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—22no—
237C23—24no—
238C23—48no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—88no—
246C23—88no—
247C23—84no—
248C23—84no—
249C23#12012yes—
250C23C2×SD1612yes—
251C23C2×SD1612yescomplemented
252C23#13712yes—
253C23C22×D411yescharacteristic
254Q8—22no—
255Q8—24no—
256Q8—22no—
257Q8—22no—
258Q8—24no—
259Q8—48no—
260Q8—48no—
261Q8—41no—
262Q8—41no—
263Q8—42no—
264Q8#12014yescomplemented
265D4—28no—
266D4—24no—
267D4—24no—
268D4—48no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—48no—
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#12012yes—
299C2×C4C2×SD1612yescomplemented
300C2×C4C22×D411yescharacteristic
301C2×C4C22×D412yes—
302C2×C4C22×D411yescharacteristic
303C2×C4H(5,𝔽2)12yes—
304C8—24no—
305C8—42no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—816no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#44814yescomplemented
324C22C2×#12011yescharacteristic
325C22D4212yes—
326C22C22×SD1611yescharacteristic
327C22C2×#13711yescharacteristic
328C22#57712yes—
329C4—22no—
330C4—24no—
331C4—22no—
332C4—41no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—82no—
338C4C2×#12011yescharacteristic
339C4#53411yescharacteristic
340C4D4212yes—
341C2—24no—
342C2—42no—
343C2—84no—
344C2#240311yescharacteristic
345C2C2×#44811yescharacteristic
346C2#301011yescharacteristic
347C2#326612yes—
348C2D4×SD1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
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. C23×SD16C2semidirect
2. C2×D4×Q8C2semidirect
3. C2×#448C2semidirect
4. C2×#450C2semidirect
6. #2013C2semidirect
7. #2000C2semidirect
8. #1892C2semidirect
9. #1880C2semidirect
10. #1869C2semidirect
11. #1787C2semidirect
13. C23×Q8C22semidirect
24. C2×C4⋊Q8C22semidirect
28. C2×C22⋊Q8C22semidirect
31. C2×C4×Q8C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
41. C2×C22⋊C8C22semidirect
47. #340C22semidirect
48. #336C22semidirect
59. C22×Q8D4semidirect
84. C2×SD16D4semidirect
97. C22⋊Q8D4semidirect
176. C2×D4SD16semidirect
251. C23C2×SD16semidirect
264. Q8#120semidirect
299. C2×C4C2×SD16semidirect
323. C22#448semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×D4×Q82
C2×#4482
C2×#4502
#24082
#20132
#20002
#18922
#18802
#18692
#17872