Finite Group Databasebeta

← #23912

Subgroups — #23912

Order 256.

975 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 975 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#3484C211yesmaximal, characteristic, complemented
2C2×#535C211yesmaximal, characteristic, complemented
3C2×#455C211yesmaximal, characteristic, complemented
4C2×#453C211yesmaximal, characteristic, complemented
5#3005C212yesmaximal, complemented
6#2004C212yesmaximal, complemented
7#2002C212yesmaximal, complemented
8#1897C212yesmaximal, complemented
9#1887C211yesmaximal, characteristic, complemented
10#1885C211yesmaximal, characteristic, complemented
11#1783C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2211yescharacteristic
13C22×#41C2211yescharacteristic, complemented
14C23×D4C2211yescharacteristic
15C2×#137C2212yescomplemented
16C2×AGL(1,ℤ/8)C2212yescomplemented
17#546—24no—
18D42—24no—
19#535—24no—
20#535—24no—
21#535C2212yes—
22#534—22no—
23C2×#127C2211yescharacteristic
24C2×#127C2211yescharacteristic, complemented
25C2×#124C2211yescharacteristic
26C2×#124C2211yescharacteristic, complemented
27#525C2212yes—
28C2×#121—22no—
29C2×#120—22no—
30C2×#120—22no—
31#518C2211yescharacteristic
32C2×#117C2211yescharacteristic, complemented
33#455—24no—
34#453—24no—
35#452—24no—
36#449—24no—
37C2×#104C2212yescomplemented
38C2×#104C2212yescomplemented
39#419C2212yescomplemented
40#418C2212yescomplemented
41C2×#100C2211yescharacteristic
42C2×#100C2211yescharacteristic
43C2×#100C2212yescomplemented
44#407C2212yescomplemented
45#339C2212yescomplemented
46#339C2212yescomplemented
47C25D412yes—
48H(5,𝔽2)—22no—
49H(5,𝔽2)—22no—
50H(5,𝔽2)—42no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—24no—
55C2×#41—42no—
56C2×#41C2312yes—
57C22×Q8C2311yescharacteristic
58C22×D4—21no—
59C22×D4—22no—
60C22×D4—24no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—24no—
67C22×D4—42no—
68C22×D4D412yes—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74#137—24no—
75#137—42no—
76AGL(1,ℤ/8)—24no—
77AGL(1,ℤ/8)—42no—
78C2×Q16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×D8—22no—
82C2×M16C2312yes—
83C2×M16C2312yes—
84#127—22no—
85#127—22no—
86#127—22no—
87#127—22no—
88#127D414yescomplemented
89#124—22no—
90#124—22no—
91#124—22no—
92#124—22no—
93#124D414yescomplemented
94#123—24no—
95#123—22no—
96C22⋊Q8D414yescomplemented
97#121—216no—
98#121—28no—
99#121—24no—
100#121D414yescomplemented
101#120—216no—
102#120—28no—
103#120—24no—
104#120—22no—
105#120—24no—
106C4×D4—24no—
107C4×D4—24no—
108#117—22no—
109#117—22no—
110#117C2312yes—
111C2×C4⋊C4—21no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#104—24no—
120#104—24no—
121#103—24no—
122#102—24no—
123#100—22no—
124#100—22no—
125#100—42no—
126C22⋊C8—24no—
127C24—22no—
128C24—21no—
129C24—24no—
130C24—24no—
131C24—42no—
132C24—84no—
133C24C2×D412yes—
134C24C2×D412yes—
135C24C2×D411yescharacteristic
136#41—22no—
137#41—22no—
138#41—22no—
139#41—22no—
140#41—48no—
141#41—48no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8C2×D412yes—
148C2×Q8C2×D412yescomplemented
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—28no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—48no—
168C2×D4—48no—
169C2×D4—42no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—44no—
176C2×D4—84no—
177C2×D4—84no—
178C2×D4—88no—
179C2×D4C2×D412yes—
180C2×D4C2×D412yes—
181C2×D4C2×D412yes—
182C2×D4C2×D412yescomplemented
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D414yes—
197C22×C4C2411yescharacteristic
198Q16—44no—
199SD16—44no—
200SD16—44no—
201D8—44no—
202M16—22no—
203M16—22no—
204M16—22no—
205M16—22no—
206C2×C8—22no—
207C2×C8—22no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4C2×D414yes—
211C22⋊C4—24no—
212C22⋊C4—24no—
213C22⋊C4—28no—
214C22⋊C4—28no—
215C22⋊C4—24no—
216C22⋊C4—28no—
217C22⋊C4—22no—
218C22⋊C4—24no—
219C42—21no—
220C42—21no—
221C42—22no—
222C42C2×D412yescomplemented
223C42C2×D412yescomplemented
224C23—21no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—48no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—48no—
239C23—44no—
240C23—41no—
241C23—41no—
242C23—84no—
243C23—88no—
244C23—84no—
245C23—88no—
246C23#12012yes—
247C23#12014yes—
248C23C22×D411yescharacteristic
249Q8—22no—
250Q8—24no—
251Q8—22no—
252Q8—42no—
253D4—24no—
254D4—24no—
255D4—22no—
256D4—24no—
257D4—22no—
258D4—48no—
259D4—48no—
260D4—48no—
261D4—48no—
262D4—42no—
263D4—42no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—88no—
268D4—88no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4#12012yes—
289C2×C4C22×D411yescharacteristic
290C2×C4C22×D411yescharacteristic
291C2×C4C22×D412yes—
292C2×C4H(5,𝔽2)12yes—
293C8—42no—
294C8—42no—
295C22—22no—
296C22—22no—
297C22—24no—
298C22—22no—
299C22—22no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—44no—
310C22—44no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—816no—
315C22—82no—
316C22—164no—
317C22C2×#12011yescharacteristic
318C22C2×#12011yescharacteristic
319C22#53411yescharacteristic
320C4—22no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—44no—
326C4D4212yes—
327C4D4212yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—44no—
332C2—82no—
333C2—82no—
334C2—82no—
335C2#240311yescharacteristic
336C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3484C2semidirect
2. C2×#535C2semidirect
3. C2×#455C2semidirect
4. C2×#453C2semidirect
5. #3005C2semidirect
6. #2004C2semidirect
7. #2002C2semidirect
8. #1897C2semidirect
9. #1887C2semidirect
10. #1885C2semidirect
11. #1783C2semidirect
13. C22×#41C22semidirect
15. C2×#137C22semidirect
16. C2×AGL(1,ℤ/8)C22semidirect
24. C2×#127C22semidirect
26. C2×#124C22semidirect
32. C2×#117C22semidirect
37. C2×#104C22semidirect
38. C2×#104C22semidirect
39. #419C22semidirect
40. #418C22semidirect
43. C2×#100C22semidirect
44. #407C22semidirect
45. #339C22semidirect
46. #339C22semidirect
88. #127D4semidirect
93. #124D4semidirect
96. C22⋊Q8D4semidirect
100. #121D4semidirect
148. C2×Q8C2×D4semidirect
182. C2×D4C2×D4semidirect
222. C42C2×D4semidirect
223. C42C2×D4semidirect

Maximal subgroups

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

subgroupindex
#34842
C2×#5352
C2×#4552
C2×#4532
#30052
#20042
#20022
#18972
#18872
#18852
#17832