Finite Group Databasebeta

← #24456

Subgroups — #24456

Order 256.

705 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#576C211yesmaximal, characteristic, complemented
2C2×#472C211yesmaximal, characteristic, complemented
3#3045C211yesmaximal, characteristic, complemented
4C2×#453C211yesmaximal, characteristic, complemented
5#3008C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic, complemented
7#2051C211yesmaximal, characteristic, complemented
8#2045C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#2002C211yesmaximal, characteristic, complemented
11#1907C211yesmaximal, characteristic, complemented
12#1883C211yesmaximal, characteristic, complemented
13#1876C211yesmaximal, characteristic, complemented
14#1866C211yesmaximal, characteristic, complemented
15#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×H(5,𝔽2)C2211yescharacteristic
18#576—24no—
19#576—24no—
20C2×AGL(1,ℤ/8)—21no—
21C2×AGL(1,ℤ/8)—21no—
22C2×AGL(1,ℤ/8)C2211yescharacteristic
23C2×AGL(1,ℤ/8)C2211yescharacteristic
24C2×AGL(1,ℤ/8)C2211yescharacteristic
25C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
26C2×#135—21no—
27C2×#135C2211yescharacteristic
28C22×D8—21no—
29C22×D8C2211yescharacteristic
30C22×D8C2211yescharacteristic, complemented
31C2×#131C2211yescharacteristic
32C2×#131C2211yescharacteristic, complemented
33#535C2211yescharacteristic, complemented
34C2×#127C2211yescharacteristic, complemented
35#520C2211yescharacteristic, complemented
36#472—24no—
37#464—21no—
38#463—21no—
39#460—21no—
40#459—21no—
41#453—24no—
42#449—21no—
43#449—21no—
44#447—21no—
45#447—21no—
46C2×#108C2211yescharacteristic, complemented
47#424C2211yescharacteristic, complemented
48C2×#104C2211yescharacteristic
49C2×#104C2211yescharacteristic, complemented
50C2×#104C2211yescharacteristic, complemented
51#418—21no—
52#418C2211yescharacteristic, complemented
53#418C2211yescharacteristic, complemented
54#417—21no—
55#417C2211yescharacteristic
56#417C2211yescharacteristic, complemented
57#416—21no—
58#416C2211yescharacteristic
59C2×#102—21no—
60C2×#102C2211yescharacteristic, complemented
61C2×#102C2211yescharacteristic, complemented
62C2×#100C2211yescharacteristic, complemented
63C2×#100C2211yescharacteristic, complemented
64C2×#100C2211yescharacteristic, complemented
65#408C2211yescharacteristic, complemented
66#390C2211yescharacteristic, complemented
67#343C2211yescharacteristic, complemented
68#341C2211yescharacteristic, complemented
69#339C2211yescharacteristic, complemented
70#336C2211yescharacteristic, complemented
71H(5,𝔽2)—22no—
72H(5,𝔽2)—22no—
73H(5,𝔽2)—22no—
74H(5,𝔽2)—22no—
75H(5,𝔽2)—22no—
76H(5,𝔽2)—41no—
77H(5,𝔽2)—41no—
78H(5,𝔽2)D414yescomplemented
79C2×#41—21no—
80C2×#41—21no—
81C2×#41—21no—
82C2×#41—21no—
83C2×#41C2311yescharacteristic
84C2×#41C2311yescharacteristic
85C2×#41C2311yescharacteristic
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4—41no—
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C22×D4C2311yescharacteristic
96AGL(1,ℤ/8)—28no—
97AGL(1,ℤ/8)—28no—
98AGL(1,ℤ/8)—22no—
99AGL(1,ℤ/8)—22no—
100AGL(1,ℤ/8)—22no—
101AGL(1,ℤ/8)—22no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—22no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)D414yescomplemented
106#135—28no—
107#135—22no—
108#135—22no—
109C2×Q16D411yescharacteristic, complemented
110C2×SD16—21no—
111C2×SD16—21no—
112C2×SD16—21no—
113C2×SD16—21no—
114C2×SD16D411yescharacteristic, complemented
115C2×SD16D411yescharacteristic, complemented
116C2×D8—28no—
117C2×D8—21no—
118C2×D8—21no—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—22no—
122C2×D8—21no—
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—41no—
129C2×D8D411yescharacteristic, complemented
130#131—22no—
131#131—22no—
132#131—22no—
133#131—41no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16C2311yescharacteristic
137C2×M16C2311yescharacteristic
138C2×M16C2311yescharacteristic
139C2×M16C2311yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2311yescharacteristic
143C22×C8C2311yescharacteristic
144#127—22no—
145#127—21no—
146#127—41no—
147#124—21no—
148#121—21no—
149#121—21no—
150#120—41no—
151C4×Q8—21no—
152C4×D4—21no—
153C4×D4—41no—
154#117—21no—
155#117C2311yescharacteristic
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2311yescharacteristic
158C2×C22⋊C4—41no—
159C2×C42C2311yescharacteristic
160#108—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163#104—22no—
164#104—22no—
165#104—41no—
166#103—21no—
167#103—21no—
168#103D411yescharacteristic, complemented
169#103D411yescharacteristic, complemented
170#102—21no—
171#102—21no—
172#102—21no—
173#102—21no—
174#102—21no—
175#102—21no—
176#102—21no—
177#102—21no—
178#102—21no—
179#102—21no—
180#102D411yescharacteristic, complemented
181#102D411yescharacteristic, complemented
182#100—22no—
183#100—22no—
184#100—22no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—41no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41—21no—
200#41—21no—
201#41—21no—
202#41—21no—
203#41—42no—
204#41—42no—
205#41—42no—
206#41—42no—
207#41—42no—
208#41C2×D412yes—
209#41C2×D412yes—
210#41C2×D412yes—
211#41#4112yes—
212C2×Q8C2×D411yescharacteristic
213C2×Q8C2×D411yescharacteristic
214C2×Q8C2×D411yescharacteristic
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4C2×D412yes—
256C2×D4C2×D412yes—
257C2×D4C2×D411yescharacteristic
258C2×D4C2×D411yescharacteristic
259C2×D4C2×D411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C2411yescharacteristic
269Q16—22no—
270SD16—24no—
271SD16—24no—
272SD16—22no—
273SD16—22no—
274SD16—42no—
275SD16—42no—
276D8—24no—
277D8—24no—
278D8—24no—
279D8—22no—
280D8—24no—
281D8—42no—
282D8—42no—
283D8—42no—
284D8—42no—
285M16—24no—
286M16—21no—
287M16—21no—
288M16—21no—
289M16—21no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16—42no—
294M16C2×D412yes—
295C2×C8—24no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—42no—
305C2×C8C2×D411yescharacteristic
306C2×C8C2×D411yescharacteristic
307C2×C8C2×D411yescharacteristic
308C2×C8C2×D411yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4C2×D411yescharacteristic
313C4⋊C4C2×D411yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—82no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—44no—
329C23—44no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—161no—
344C23—161no—
345C23—161no—
346C23—161no—
347C23H(5,𝔽2)11yescharacteristic
348Q8—21no—
349Q8—21no—
350Q8—21no—
351Q8—21no—
352Q8#12012yes—
353Q8#12112yes—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—22no—
358D4—22no—
359D4—22no—
360D4—21no—
361D4—22no—
362D4—21no—
363D4—22no—
364D4—21no—
365D4—21no—
366D4—44no—
367D4—44no—
368D4—44no—
369D4—44no—
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—88no—
381D4—161no—
382D4—161no—
383D4#12012yes—
384D4#12112yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—161no—
404C2×C4C22×D411yescharacteristic
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D411yescharacteristic
408C2×C4C2×#4111yescharacteristic
409C2×C4H(5,𝔽2)11yescharacteristic
410C8—21no—
411C8—22no—
412C8—21no—
413C8—21no—
414C8—21no—
415C8—41no—
416C8—41no—
417C8—41no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—164no—
445C22—164no—
446C22#53411yescharacteristic
447C22D4211yescharacteristic
448C22#54611yescharacteristic
449C4—22no—
450C4—22no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—81no—
456C4C2×#12011yescharacteristic
457C4C2×#12111yescharacteristic
458C4D4211yescharacteristic
459C4#54611yescharacteristic
460C2—21no—
461C2—21no—
462C2—42no—
463C2—42no—
464C2—81no—
465C2—82no—
466C2—82no—
467C2—161no—
468C2#241011yescharacteristic
469C2#328811yescharacteristic
470C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#576C2semidirect
2. C2×#472C2semidirect
3. #3045C2semidirect
4. C2×#453C2semidirect
5. #3008C2semidirect
6. #2897C2semidirect
7. #2051C2semidirect
8. #2045C2semidirect
9. #2011C2semidirect
10. #2002C2semidirect
11. #1907C2semidirect
12. #1883C2semidirect
13. #1876C2semidirect
14. #1866C2semidirect
15. #1792C2semidirect
25. C2×AGL(1,ℤ/8)C22semidirect
30. C22×D8C22semidirect
32. C2×#131C22semidirect
33. #535C22semidirect
34. C2×#127C22semidirect
35. #520C22semidirect
46. C2×#108C22semidirect
47. #424C22semidirect
49. C2×#104C22semidirect
50. C2×#104C22semidirect
52. #418C22semidirect
53. #418C22semidirect
56. #417C22semidirect
60. C2×#102C22semidirect
61. C2×#102C22semidirect
62. C2×#100C22semidirect
63. C2×#100C22semidirect
64. C2×#100C22semidirect
65. #408C22semidirect
66. #390C22semidirect
67. #343C22semidirect
68. #341C22semidirect
69. #339C22semidirect
70. #336C22semidirect
78. H(5,𝔽2)D4semidirect
105. AGL(1,ℤ/8)D4semidirect
109. C2×Q16D4semidirect
114. C2×SD16D4semidirect
115. C2×SD16D4semidirect
129. C2×D8D4semidirect
168. #103D4semidirect
169. #103D4semidirect
180. #102D4semidirect
181. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
C2×#4722
#30452
C2×#4532
#30082
#28972
#20512
#20452
#20112
#20022
#19072
#18832
#18762
#18662
#17922