Finite Group Databasebeta

← #60588

Subgroups — #60588

Order 256.

1067 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3494C211yesmaximal, characteristic, complemented
2C2×#520C211yesmaximal, characteristic, complemented
3#3125C214yesmaximal, complemented
4#3119C214yesmaximal, complemented
5#3117C212yesmaximal, complemented
6#3109C212yesmaximal, complemented
7#3105C212yesmaximal, complemented
8#3103C212yesmaximal, complemented
9#3092C211yesmaximal, characteristic, complemented
10C2×#481C211yesmaximal, characteristic, complemented
11#3084C212yesmaximal, complemented
12#3045C212yesmaximal, complemented
13#3006C211yesmaximal, characteristic, complemented
14#3002C211yesmaximal, characteristic, complemented
15#2952C212yesmaximal, complemented
16#2946C212yesmaximal, complemented
17#2924C211yesmaximal, characteristic, complemented
18C22×#41C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2212yes—
20C2×#135C2212yes—
21#555—21no—
22#553—21no—
23#553—21no—
24#551—22no—
25#547—21no—
26#546—22no—
27#544C2211yescharacteristic, complemented
28#542C2212yescomplemented
29#540C2212yescomplemented
30#535C2211yescharacteristic, complemented
31#532C2212yes—
32#529C2212yes—
33#525C2211yescharacteristic
34C2×#121C2211yescharacteristic
35#520—21no—
36#520—21no—
37#520—22no—
38#520C2212yes—
39#520C2212yes—
40#520C2214yescomplemented
41#520C2211yescharacteristic, complemented
42#518C2211yescharacteristic, complemented
43C2×C4×Q8C2212yescomplemented
44C2×C4×D4—22no—
45C2×C4×D4C2212yescomplemented
46C2×#117—21no—
47C2×#117C2211yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2211yescharacteristic
50#484C2212yescomplemented
51#483C2212yescomplemented
52#481C2212yes—
53#481C2212yes—
54#481C2214yescomplemented
55#481C2212yescomplemented
56#480C2212yescomplemented
57#479C2214yescomplemented
58#478C2214yescomplemented
59#476C2214yescomplemented
60#475C2214yescomplemented
61#464C2214yescomplemented
62#463C2214yescomplemented
63#460C2214yescomplemented
64#459C2214yescomplemented
65#452C2212yescomplemented
66#451C2211yescharacteristic, complemented
67#450C2211yescharacteristic, complemented
68#449C2214yes—
69#449C2212yescomplemented
70#448C2212yes—
71#448C2211yescharacteristic, complemented
72#447C2212yes—
73#447C2211yescharacteristic, complemented
74#442C2214yescomplemented
75#442C2214yescomplemented
76#441C2214yescomplemented
77#440C2214yescomplemented
78C4×SD16C2214yescomplemented
79C4×SD16C2214yescomplemented
80C4×SD16C2214yescomplemented
81C4×D8C2214yescomplemented
82#433C2214yescomplemented
83#432C2212yescomplemented
84#431C2212yescomplemented
85#428C2212yescomplemented
86C2×#106C2212yescomplemented
87#424C2212yescomplemented
88#417C2212yes—
89#417C2212yescomplemented
90#416C2212yes—
91C2×#102C2212yescomplemented
92#407C2211yescharacteristic
93C2×C22⋊C8C2211yescharacteristic
94#405C2212yescomplemented
95C2×#41—28no—
96C2×#41—21no—
97C2×#41—21no—
98C2×#41—22no—
99C2×#41C2312yes—
100C2×#41C2311yescharacteristic
101C2×#41C2312yes—
102C22×Q8C2311yescharacteristic
103C22×D4—21no—
104C22×D4—21no—
105C22×D4C2311yescharacteristic
106C22×D4C2311yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C2311yescharacteristic
111AGL(1,ℤ/8)—28no—
112#135—28no—
113C2×Q16C2312yes—
114C2×SD16C2312yes—
115C2×SD16C2312yes—
116C2×SD16C2314yes—
117C2×D8C2312yes—
118C2×D8C2314yes—
119C2×M16C2312yes—
120C22×C8C2312yes—
121C4⋊Q8—21no—
122C4⋊Q8C2312yes—
123#127C2312yes—
124#126—22no—
125#126—22no—
126#125—21no—
127#125C2312yes—
128#124—22no—
129#124—22no—
130#124C2312yes—
131#123—21no—
132#123—22no—
133#123—22no—
134#123—22no—
135#123—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8C2311yescharacteristic
139C22⋊Q8C2311yescharacteristic
140#121—21no—
141#121—21no—
142#121—22no—
143#121—21no—
144#121—21no—
145#121—22no—
146#121—22no—
147#121C2312yes—
148#121C2312yes—
149#121C2311yescharacteristic
150#121C2311yescharacteristic
151#120—21no—
152#120—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8C2314yes—
156C4×Q8C2314yes—
157C4×Q8C2312yes—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—22no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4C2314yes—
172C4×D4C2314yes—
173C4×D4C2312yes—
174C4×D4C2312yes—
175C4×D4C2312yes—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—21no—
180#117—22no—
181#117D412yes—
182#117D411yescharacteristic
183#117D411yescharacteristic
184#117C2311yescharacteristic
185#117C2311yescharacteristic
186#117C2311yescharacteristic
187#117C2312yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4D411yescharacteristic
197C2×C4⋊C4D411yescharacteristic
198C2×C4⋊C4C2312yes—
199C2×C4⋊C4C2312yes—
200C2×C4⋊C4C2312yes—
201C2×C4⋊C4C2311yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4D412yes—
206C2×C42—22no—
207C2×C42C2312yes—
208#107C2314yes—
209#106C2314yes—
210#106C2314yes—
211#106C2314yes—
212C4⋊C8C2314yes—
213C4⋊C8C2314yes—
214#103C2312yes—
215#103C2312yes—
216#103C2314yes—
217#103C2314yes—
218#102C2314yes—
219#102C2314yes—
220#102C2312yes—
221#102C2312yes—
222#102C2314yes—
223#102C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2311yescharacteristic
226C22⋊C8C2312yes—
227C22⋊C8C2312yes—
228C22⋊C8C2312yes—
229#97C2314yes—
230C4×C8C2314yes—
231C24—21no—
232C24—21no—
233C24C2×D411yescharacteristic
234#41—28no—
235#41—28no—
236#41—24no—
237#41—24no—
238#41—24no—
239#41—24no—
240#41—84no—
241#41#4118yescomplemented
242C2×Q8—24no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8C2411yescharacteristic
246C2×Q8C2412yes—
247C2×D4—28no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—82no—
262C2×D4C2412yes—
263C2×D4C2411yescharacteristic
264C2×D4C2412yes—
265C22×C4—28no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—84no—
281C22×C4—82no—
282C22×C4C2×D412yes—
283C22×C4C2×D411yescharacteristic
284C22×C4C2×D411yescharacteristic
285C22×C4C2×D411yescharacteristic
286C22×C4C2×D412yes—
287C22×C4C2411yescharacteristic
288C22×C4C2411yescharacteristic
289C22×C4C2411yescharacteristic
290Q16—24no—
291SD16—24no—
292SD16—28no—
293SD16—24no—
294D8—24no—
295D8—28no—
296M16—24no—
297C2×C8—24no—
298C2×C8C2412yes—
299C2×C8C2414yes—
300C2×C8C2412yes—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4C2×D411yescharacteristic
314C4⋊C4C2×D411yescharacteristic
315C4⋊C4C2×D412yes—
316C4⋊C4C2×D411yescharacteristic
317C4⋊C4C2×D411yescharacteristic
318C4⋊C4C2×D412yes—
319C4⋊C4C2414yes—
320C4⋊C4C2412yes—
321C4⋊C4C2411yescharacteristic
322C4⋊C4C2411yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—21no—
335C22⋊C4—22no—
336C22⋊C4—21no—
337C22⋊C4C2×D412yes—
338C22⋊C4C2×D412yes—
339C22⋊C4C2×D412yes—
340C22⋊C4C2×D412yes—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—21no—
345C42C2×D412yes—
346C42C2×D412yes—
347C42C2412yes—
348C42C2412yes—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23C22×D411yescharacteristic
362C23C22×D411yescharacteristic
363C23C22×D411yescharacteristic
364Q8—24no—
365Q8—22no—
366Q8—22no—
367Q8—82no—
368Q8C2×#4114yes—
369D4—28no—
370D4—24no—
371D4—28no—
372D4—22no—
373D4—24no—
374D4—22no—
375D4—42no—
376D4—44no—
377D4—42no—
378D4—42no—
379D4—82no—
380D4—82no—
381D4—81no—
382D4—82no—
383D4C2×#4114yes—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4C22×D412yes—
411C2×C4C22×D412yes—
412C2×C4C22×D412yes—
413C2×C4C22×D411yescharacteristic
414C2×C4C22×D412yes—
415C2×C4C22×D411yescharacteristic
416C2×C4C22×D411yescharacteristic
417C2×C4C2×#4114yes—
418C2×C4C2511yescharacteristic
419C8—22no—
420C8—22no—
421C8—24no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—21no—
430C22—44no—
431C22—48no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—41no—
439C22—42no—
440C22—162no—
441C22#52512yes—
442C22#57712yes—
443C22C23×D411yescharacteristic
444C4—22no—
445C4—22no—
446C4—21no—
447C4—24no—
448C4—21no—
449C4—22no—
450C4—22no—
451C4—44no—
452C4—41no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4#52512yes—
457C4C22×#4112yes—
458C2—22no—
459C2—22no—
460C2—41no—
461C2—44no—
462C2—81no—
463C2—82no—
464C2C2×#52511yescharacteristic
465C2C2×#57711yescharacteristic
466C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3494C2semidirect
2. C2×#520C2semidirect
3. #3125C2semidirect
4. #3119C2semidirect
5. #3117C2semidirect
6. #3109C2semidirect
7. #3105C2semidirect
8. #3103C2semidirect
9. #3092C2semidirect
10. C2×#481C2semidirect
11. #3084C2semidirect
12. #3045C2semidirect
13. #3006C2semidirect
14. #3002C2semidirect
15. #2952C2semidirect
16. #2946C2semidirect
17. #2924C2semidirect
27. #544C22semidirect
28. #542C22semidirect
29. #540C22semidirect
30. #535C22semidirect
40. #520C22semidirect
41. #520C22semidirect
42. #518C22semidirect
43. C2×C4×Q8C22semidirect
45. C2×C4×D4C22semidirect
50. #484C22semidirect
51. #483C22semidirect
54. #481C22semidirect
55. #481C22semidirect
56. #480C22semidirect
57. #479C22semidirect
58. #478C22semidirect
59. #476C22semidirect
60. #475C22semidirect
61. #464C22semidirect
62. #463C22semidirect
63. #460C22semidirect
64. #459C22semidirect
65. #452C22semidirect
66. #451C22semidirect
67. #450C22semidirect
69. #449C22semidirect
71. #448C22semidirect
73. #447C22semidirect
74. #442C22semidirect
75. #442C22semidirect
76. #441C22semidirect
77. #440C22semidirect
78. C4×SD16C22semidirect
79. C4×SD16C22semidirect
80. C4×SD16C22semidirect
81. C4×D8C22semidirect
82. #433C22semidirect
83. #432C22semidirect
84. #431C22semidirect
85. #428C22semidirect
86. C2×#106C22semidirect
87. #424C22semidirect
89. #417C22semidirect
91. C2×#102C22semidirect
94. #405C22semidirect
241. #41#41semidirect

Maximal subgroups

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

subgroupindex
#34942
C2×#5202
#31252
#31192
#31172
#31092
#31052
#31032
#30922
C2×#4812
#30842
#30452
#30062
#30022
#29522
#29462
#29242