Finite Group Databasebeta

← #60755

Subgroups — #60755

Order 256.

1097 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3449C211yesmaximal, characteristic, complemented
2C2×#537C211yesmaximal, characteristic, complemented
3#3204C214yesmaximal, complemented
4#3203C214yesmaximal, complemented
5#3201C214yesmaximal, complemented
6#3198C214yesmaximal, complemented
7#3095C211yesmaximal, characteristic, complemented
8#3094C211yesmaximal, characteristic, complemented
9#3092C212yesmaximal, complemented
10#3056C212yesmaximal, complemented
11#3055C212yesmaximal, complemented
12#3007C211yesmaximal, characteristic, complemented
13#3006C211yesmaximal, characteristic, complemented
14#3005C212yesmaximal, complemented
15#2879C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×#41C2211yescharacteristic, complemented
19C2×#137C2212yes—
20C2×AGL(1,ℤ/8)C2212yes—
21C22×M16C2212yescomplemented
22#537—22no—
23#537—22no—
24#537C2214yes—
25#537C2214yescomplemented
26#537C2212yescomplemented
27#537C2212yescomplemented
28#536—21no—
29#536C2212yescomplemented
30#535—21no—
31#535C2212yescomplemented
32#534—21no—
33#534—21no—
34#532—22no—
35#525—22no—
36#525C2212yes—
37#525C2212yes—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8C2211yescharacteristic
40C2×C22⋊Q8C2211yescharacteristic, complemented
41C2×#121—21no—
42C2×#121—21no—
43C2×#121—21no—
44C2×#121C2211yescharacteristic
45C2×#121C2211yescharacteristic, complemented
46C2×#117C2211yescharacteristic
47C22×C4⋊C4C2211yescharacteristic, complemented
48#485C2214yescomplemented
49#484C2214yescomplemented
50#484C2212yescomplemented
51#483C2214yescomplemented
52#483C2212yescomplemented
53#482C2214yescomplemented
54#481C2214yescomplemented
55#481C2212yescomplemented
56#480C2214yescomplemented
57#480C2212yescomplemented
58#470C2214yescomplemented
59#469C2214yescomplemented
60#468C2214yescomplemented
61#468C2214yescomplemented
62#467C2214yescomplemented
63#466C2214yescomplemented
64#465C2214yescomplemented
65#465C2214yescomplemented
66#452C2214yescomplemented
67#452C2214yescomplemented
68#452C2214yescomplemented
69#451C2212yescomplemented
70#450C2212yescomplemented
71#449C2214yescomplemented
72#449C2214yescomplemented
73#449C2214yescomplemented
74#448C2212yescomplemented
75#447C2212yescomplemented
76#428C2212yescomplemented
77#428C2212yescomplemented
78#419C2212yescomplemented
79#418C2212yescomplemented
80#417C2212yes—
81#417C2212yes—
82#408C2214yescomplemented
83#408C2214yescomplemented
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86#407C2212yescomplemented
87C2×#41—21no—
88C2×#41—21no—
89C2×#41—22no—
90C2×#41—44no—
91C2×#41—44no—
92C2×#41—44no—
93C2×#41D412yes—
94C2×#41D412yes—
95C2×#41C2312yes—
96C2×#41C2314yes—
97C2×#41C2312yes—
98C2×#41C2312yes—
99C2×#41C2312yes—
100C22×Q8D411yescharacteristic
101C22×Q8C2311yescharacteristic
102C22×Q8C2311yescharacteristic
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4D411yescharacteristic
108C22×D4D411yescharacteristic
109C22×D4D411yescharacteristic
110C22×D4C2311yescharacteristic
111C22×D4C2311yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118#137—44no—
119AGL(1,ℤ/8)—44no—
120C2×Q16C2314yes—
121C2×SD16C2314yes—
122C2×SD16C2314yes—
123C2×D8C2314yes—
124C2×M16—44no—
125C2×M16C2312yes—
126C2×M16C2312yes—
127C2×M16C2314yes—
128C22×C8C2314yes—
129C4⋊Q8—22no—
130#127—22no—
131#124—22no—
132#124—22no—
133#123—22no—
134#123—22no—
135#123—22no—
136#123—22no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C22⋊Q8—22no—
144C22⋊Q8C2312yes—
145C22⋊Q8C2312yes—
146C22⋊Q8C2314yes—
147C22⋊Q8C2312yes—
148C22⋊Q8C2312yes—
149C22⋊Q8C2312yes—
150C22⋊Q8C2312yes—
151#121—24no—
152#121—24no—
153#121—24no—
154#121—24no—
155#121—24no—
156#121—24no—
157#121—22no—
158#121—22no—
159#121—22no—
160#121—22no—
161#121—22no—
162#121—22no—
163#121—21no—
164#121—21no—
165#121—22no—
166#121C2312yes—
167#121C2312yes—
168#121C2314yes—
169#121C2312yes—
170#121C2312yes—
171#121C2312yes—
172#121C2312yes—
173#120—22no—
174#120—22no—
175#120—22no—
176#120—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181#117—21no—
182#117—21no—
183#117C2312yes—
184#117C2312yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2311yescharacteristic
190C2×C4⋊C4C2311yescharacteristic
191C2×C4⋊C4C2314yes—
192C2×C4⋊C4C2311yescharacteristic
193C2×C4⋊C4C2311yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200#107C2314yes—
201#107C2314yes—
202#106C2314yes—
203#106C2314yes—
204#103C2314yes—
205#103C2314yes—
206#103C2314yes—
207#103C2314yes—
208#102C2314yes—
209#102C2314yes—
210#102C2314yes—
211#102C2314yes—
212C22⋊C8C2312yes—
213C22⋊C8C2312yes—
214C22⋊C8C2312yes—
215C22⋊C8C2312yes—
216C22⋊C8C2314yes—
217C22⋊C8C2314yes—
218C24—21no—
219C24—21no—
220C24C2×D411yescharacteristic
221C24C2×D411yescharacteristic
222C24C2×D411yescharacteristic
223#41—24no—
224#41—24no—
225#41—44no—
226#41—44no—
227#41—42no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232#41—44no—
233#41—44no—
234#41—42no—
235#41—44no—
236#41—48no—
237#41—44no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C2×D412yes—
244C2×Q8C2×D412yes—
245C2×Q8C2412yes—
246C2×Q8C2412yes—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—24no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4C2×D412yes—
269C2×D4C2×D412yes—
270C2×D4C2×D412yes—
271C2×D4C2×D412yes—
272C2×D4C2×D412yes—
273C2×D4C2×D412yes—
274C2×D4C2412yes—
275C2×D4C2412yes—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—82no—
292C22×C4C2×D411yescharacteristic
293C22×C4C2×D411yescharacteristic
294C22×C4C2×D411yescharacteristic
295C22×C4C2×D412yes—
296C22×C4C2×D411yescharacteristic
297C22×C4C2×D411yescharacteristic
298C22×C4C2×D412yes—
299C22×C4C2411yescharacteristic
300C22×C4C2412yes—
301C22×C4C2411yescharacteristic
302C22×C4C2412yes—
303C22×C4C2411yescharacteristic
304Q16—44no—
305SD16—44no—
306SD16—44no—
307D8—44no—
308M16—24no—
309M16—24no—
310M16—44no—
311C2×C8—44no—
312C2×C8C2414yes—
313C2×C8C2414yes—
314C4⋊C4—24no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4C2412yes—
327C4⋊C4C2412yes—
328C4⋊C4C2412yes—
329C4⋊C4C2412yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—24no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—24no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C42—22no—
347C42—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23C22×D411yescharacteristic
366C23C22×D412yes—
367C23C22×D412yes—
368C23C22×D411yescharacteristic
369C23C22×D411yescharacteristic
370Q8—22no—
371Q8—24no—
372Q8—22no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377D4—22no—
378D4—24no—
379D4—22no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—48no—
388D4—44no—
389D4—44no—
390D4—42no—
391D4—82no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—84no—
397D4—82no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—84no—
427C2×C4C22×D411yescharacteristic
428C2×C4C22×D412yes—
429C2×C4C22×D411yescharacteristic
430C2×C4C22×D412yes—
431C2×C4C22×D411yescharacteristic
432C2×C4H(5,𝔽2)14yes—
433C2×C4C2511yescharacteristic
434C8—24no—
435C8—24no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—22no—
450C22—44no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—84no—
461C22#53412yes—
462C22C23×D411yescharacteristic
463C4—22no—
464C4—21no—
465C4—21no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4#53412yes—
474C4C2×H(5,𝔽2)12yes—
475C2—22no—
476C2—42no—
477C2—41no—
478C2—42no—
479C2—41no—
480C2—82no—
481C2—82no—
482C2C2×#53411yescharacteristic
483C2#358511yescharacteristic
484C2#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. #3449C2semidirect
2. C2×#537C2semidirect
3. #3204C2semidirect
4. #3203C2semidirect
5. #3201C2semidirect
6. #3198C2semidirect
7. #3095C2semidirect
8. #3094C2semidirect
9. #3092C2semidirect
10. #3056C2semidirect
11. #3055C2semidirect
12. #3007C2semidirect
13. #3006C2semidirect
14. #3005C2semidirect
15. #2879C2semidirect
18. C22×#41C22semidirect
21. C22×M16C22semidirect
25. #537C22semidirect
26. #537C22semidirect
27. #537C22semidirect
29. #536C22semidirect
31. #535C22semidirect
40. C2×C22⋊Q8C22semidirect
45. C2×#121C22semidirect
47. C22×C4⋊C4C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #484C22semidirect
51. #483C22semidirect
52. #483C22semidirect
53. #482C22semidirect
54. #481C22semidirect
55. #481C22semidirect
56. #480C22semidirect
57. #480C22semidirect
58. #470C22semidirect
59. #469C22semidirect
60. #468C22semidirect
61. #468C22semidirect
62. #467C22semidirect
63. #466C22semidirect
64. #465C22semidirect
65. #465C22semidirect
66. #452C22semidirect
67. #452C22semidirect
68. #452C22semidirect
69. #451C22semidirect
70. #450C22semidirect
71. #449C22semidirect
72. #449C22semidirect
73. #449C22semidirect
74. #448C22semidirect
75. #447C22semidirect
76. #428C22semidirect
77. #428C22semidirect
78. #419C22semidirect
79. #418C22semidirect
82. #408C22semidirect
83. #408C22semidirect
86. #407C22semidirect

Maximal subgroups

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

subgroupindex
#34492
C2×#5372
#32042
#32032
#32012
#31982
#30952
#30942
#30922
#30562
#30552
#30072
#30062
#30052
#28792