Finite Group Databasebeta

← #60456

Subgroups — #60456

Order 256.

1371 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C211yesmaximal, characteristic, complemented
2C2×#568C211yesmaximal, characteristic, complemented
3#3449C211yesmaximal, characteristic, complemented
4C2×#537C211yesmaximal, characteristic, complemented
5#3063C214yesmaximal, complemented
6#3062C214yesmaximal, complemented
7#3061C214yesmaximal, complemented
8#3060C214yesmaximal, complemented
9#3056C214yesmaximal, complemented
10#3055C214yesmaximal, complemented
11C2×#428C211yesmaximal, characteristic, complemented
12#2896C211yesmaximal, characteristic, complemented
13C2×#417C211yesmaximal, characteristic, complemented
14C22×#41C2211yescharacteristic
15C22×#41C2211yescharacteristic
16C22×#41C2211yescharacteristic
17#575—28no—
18C2×#137C2212yes—
19C2×#137C2212yescomplemented
20C2×AGL(1,ℤ/8)C2212yes—
21C2×AGL(1,ℤ/8)C2212yescomplemented
22C2×#135C2214yescomplemented
23#568—28no—
24C2×#131C2214yescomplemented
25C2×#131C2214yescomplemented
26C22×M16C2211yescharacteristic
27C22×M16C2211yescharacteristic
28C22×M16C2214yescomplemented
29#537—22no—
30#537—22no—
31#537C2214yes—
32#537C2214yescomplemented
33#537C2212yescomplemented
34#537C2212yescomplemented
35#536—21no—
36#536C2212yescomplemented
37#535—21no—
38#535C2212yescomplemented
39#534—22no—
40#532—22no—
41#525—22no—
42#525C2214yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2212yescomplemented
45C2×#121—21no—
46C2×#121—22no—
47C2×#121C2212yescomplemented
48C2×#117C2211yescharacteristic
49C22×C4⋊C4C2211yescharacteristic
50#470C2218yescomplemented
51#469C2218yescomplemented
52#468C2218yescomplemented
53#468C2218yescomplemented
54#467C2218yescomplemented
55#466C2218yescomplemented
56#465C2218yescomplemented
57#465C2218yescomplemented
58#428C2214yescomplemented
59#428C2214yescomplemented
60C2×#107C2212yescomplemented
61C2×#106C2212yescomplemented
62#419C2212yescomplemented
63#418C2212yescomplemented
64#417C2214yes—
65#417C2214yescomplemented
66#417C2212yescomplemented
67#417C2212yescomplemented
68#416C2214yescomplemented
69C2×#103C2212yescomplemented
70C2×#102C2212yescomplemented
71C2×#41—24no—
72C2×#41—21no—
73C2×#41—21no—
74C2×#41—21no—
75C2×#41—21no—
76C2×#41—24no—
77C2×#41—24no—
78C2×#41—24no—
79C2×#41—24no—
80C2×#41—24no—
81C2×#41D412yes—
82C2×#41D412yes—
83C2×#41C2312yes—
84C2×#41C2312yes—
85C2×#41C2312yes—
86C2×#41C2312yes—
87C2×#41C2312yes—
88C2×#41C2312yes—
89C22×Q8D411yescharacteristic
90C22×Q8C2311yescharacteristic
91C22×Q8C2311yescharacteristic
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—22no—
95C22×D4D412yes—
96C22×D4D411yescharacteristic
97C22×D4C2311yescharacteristic
98C22×D4C2311yescharacteristic
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2312yes—
103C23×C4C2311yescharacteristic
104#137—28no—
105#137—28no—
106AGL(1,ℤ/8)—28no—
107AGL(1,ℤ/8)—28no—
108#135—216no—
109C2×Q16C2314yes—
110C2×SD16C2314yes—
111C2×SD16C2314yes—
112C2×D8C2314yes—
113#131—216no—
114#131—216no—
115C2×M16—24no—
116C2×M16—216no—
117C2×M16D418yescomplemented
118C2×M16C2312yes—
119C2×M16C2312yes—
120C2×M16C2318yes—
121C2×M16C2312yes—
122C2×M16C2312yes—
123C22×C8C2312yes—
124C22×C8C2318yes—
125C22×C8C2312yes—
126C4⋊Q8—22no—
127#127—22no—
128#124—22no—
129#124—22no—
130#123—24no—
131#123—24no—
132C22⋊Q8—24no—
133C22⋊Q8—24no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—24no—
137C22⋊Q8C2318yes—
138C22⋊Q8C2314yes—
139C22⋊Q8C2314yes—
140#121—24no—
141#121—28no—
142#121—24no—
143#121—28no—
144#121—22no—
145#121—22no—
146#121—24no—
147#121—24no—
148#121—24no—
149#121C2318yes—
150#121C2314yes—
151#121C2314yes—
152#120—24no—
153#120—24no—
154C4×D4—24no—
155C4×D4—24no—
156#117—21no—
157#117—21no—
158#117C2312yes—
159#117C2312yes—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2312yes—
165C2×C4⋊C4C2312yes—
166C2×C4⋊C4C2312yes—
167C2×C4⋊C4C2311yescharacteristic
168C2×C4⋊C4C2311yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—21no—
173#107C2318yes—
174#106C2318yes—
175#103C2318yes—
176#103C2314yes—
177#103C2314yes—
178#102C2318yes—
179#102C2314yes—
180#102C2314yes—
181C24—21no—
182C24—21no—
183C24C2×D412yes—
184C24C2×D411yescharacteristic
185#41—28no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—24no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—24no—
194#41—24no—
195#41—24no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203C2×Q8—24no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8C2×D412yes—
211C2×Q8C2×D412yes—
212C2×Q8C2412yes—
213C2×Q8C2412yes—
214C2×D4—28no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—28no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—84no—
232C2×D4C2×D414yes—
233C2×D4C2×D414yes—
234C2×D4C2×D412yes—
235C2×D4C2×D412yes—
236C2×D4C2412yes—
237C2×D4C2412yes—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4C2×D412yes—
255C22×C4C2×D414yes—
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D414yes—
260C22×C4#4114yes—
261C22×C4C2414yes—
262C22×C4C2411yescharacteristic
263C22×C4C2411yescharacteristic
264C22×C4C2411yescharacteristic
265Q16—28no—
266SD16—28no—
267SD16—28no—
268D8—28no—
269M16—24no—
270M16—24no—
271M16—216no—
272M16C2×D418yes—
273M16C2×D418yescomplemented
274C2×C8—24no—
275C2×C8—216no—
276C2×C8C2×D418yescomplemented
277C2×C8C2414yes—
278C2×C8C2414yes—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—28no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4C2412yes—
288C4⋊C4C2414yes—
289C4⋊C4C2412yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—28no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—28no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C42—22no—
299C42—22no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23#12114yes—
314C23C22×D414yes—
315C23C22×D411yescharacteristic
316C23C22×D411yescharacteristic
317C23C22×D411yescharacteristic
318Q8—24no—
319Q8—24no—
320Q8—22no—
321Q8—22no—
322Q8—22no—
323Q8—22no—
324Q8—42no—
325Q8—42no—
326D4—28no—
327D4—28no—
328D4—24no—
329D4—24no—
330D4—22no—
331D4—22no—
332D4—22no—
333D4—22no—
334D4—44no—
335D4—44no—
336D4—44no—
337D4—44no—
338D4—42no—
339D4—42no—
340D4—84no—
341D4—84no—
342D4—88no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#12114yes—
364C2×C4#12114yes—
365C2×C4#12114yes—
366C2×C4C22×D414yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D412yes—
371C2×C4C22×D412yes—
372C2×C4C22×D412yes—
373C2×C4C2×#4112yes—
374C2×C4C2×#4112yes—
375C2×C4C2×#4112yes—
376C2×C4C2511yescharacteristic
377C8—24no—
378C8C22×D418yes—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22—84no—
397C22—84no—
398C22C2×#12112yes—
399C22C2×#12112yes—
400C22C2×#12112yes—
401C22C23×D411yescharacteristic
402C4—24no—
403C4—44no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4C2×#12112yes—
409C4C2×#12112yes—
410C4C2×#12112yes—
411C4C23×D411yescharacteristic
412C4C22×#4111yescharacteristic
413C2—22no—
414C2—22no—
415C2—22no—
416C2—44no—
417C2—82no—
418C2—82no—
419C2C22×#12111yescharacteristic
420C2#358511yescharacteristic
421C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#575C2semidirect
2. C2×#568C2semidirect
3. #3449C2semidirect
4. C2×#537C2semidirect
5. #3063C2semidirect
6. #3062C2semidirect
7. #3061C2semidirect
8. #3060C2semidirect
9. #3056C2semidirect
10. #3055C2semidirect
11. C2×#428C2semidirect
12. #2896C2semidirect
13. C2×#417C2semidirect
19. C2×#137C22semidirect
21. C2×AGL(1,ℤ/8)C22semidirect
22. C2×#135C22semidirect
24. C2×#131C22semidirect
25. C2×#131C22semidirect
28. C22×M16C22semidirect
32. #537C22semidirect
33. #537C22semidirect
34. #537C22semidirect
36. #536C22semidirect
38. #535C22semidirect
44. C2×C22⋊Q8C22semidirect
47. C2×#121C22semidirect
50. #470C22semidirect
51. #469C22semidirect
52. #468C22semidirect
53. #468C22semidirect
54. #467C22semidirect
55. #466C22semidirect
56. #465C22semidirect
57. #465C22semidirect
58. #428C22semidirect
59. #428C22semidirect
60. C2×#107C22semidirect
61. C2×#106C22semidirect
62. #419C22semidirect
63. #418C22semidirect
65. #417C22semidirect
66. #417C22semidirect
67. #417C22semidirect
68. #416C22semidirect
69. C2×#103C22semidirect
70. C2×#102C22semidirect
117. C2×M16D4semidirect
273. M16C2×D4semidirect
276. C2×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#5682
#34492
C2×#5372
#30632
#30622
#30612
#30602
#30562
#30552
C2×#4282
#28962
C2×#4172