Finite Group Databasebeta

← #24464

Subgroups — #24464

Order 256.

800 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3488C211yesmaximal, characteristic, complemented
2#3055C211yesmaximal, characteristic, complemented
3C2×#453C211yesmaximal, characteristic, complemented
4#3006C211yesmaximal, characteristic, complemented
5C22×#104C211yesmaximal, characteristic, complemented
6#2856C211yesmaximal, characteristic, complemented
7#2039C211yesmaximal, characteristic, complemented
8#2037C211yesmaximal, characteristic, complemented
9#2006C211yesmaximal, characteristic, complemented
10#2002C211yesmaximal, characteristic, complemented
11#1900C211yesmaximal, characteristic, complemented
12#1896C211yesmaximal, characteristic, complemented
13#1883C211yesmaximal, characteristic, complemented
14#1859C211yesmaximal, characteristic, complemented
15#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
20C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C22×M16C2211yescharacteristic, complemented
22#550—21no—
23#548—21no—
24#546—21no—
25#546—22no—
26D42—22no—
27D42—21no—
28#537—21no—
29#537C2211yescharacteristic
30#535—21no—
31#535C2211yescharacteristic
32C2×#127C2211yescharacteristic, complemented
33#525—21no—
34#525C2211yescharacteristic
35C2×#121—21no—
36C2×#121—22no—
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic, complemented
39#520C2211yescharacteristic
40C22×C42C2211yescharacteristic, complemented
41#469—21no—
42#468—21no—
43#466—21no—
44#465—21no—
45#453—24no—
46#449—21no—
47#449—21no—
48#448—21no—
49#447—21no—
50#428C2211yescharacteristic, complemented
51C2×#104—24no—
52C2×#104C2211yescharacteristic
53C2×#104C2211yescharacteristic
54C2×#104C2211yescharacteristic, complemented
55C2×#104C2211yescharacteristic, complemented
56C2×#104C2211yescharacteristic, complemented
57C2×#104C2211yescharacteristic, complemented
58#418C2211yescharacteristic, complemented
59#417C2211yescharacteristic
60#417C2211yescharacteristic, complemented
61#417C2211yescharacteristic, complemented
62C2×#100C2211yescharacteristic, complemented
63C2×#100C2211yescharacteristic, complemented
64#407C2211yescharacteristic, complemented
65#390—22no—
66#390C2211yescharacteristic, complemented
67#390C2211yescharacteristic, complemented
68#384C2211yescharacteristic, complemented
69#343C2211yescharacteristic, complemented
70#343C2211yescharacteristic, complemented
71#339C2211yescharacteristic, complemented
72#339C2211yescharacteristic, complemented
73H(5,𝔽2)—22no—
74H(5,𝔽2)—22no—
75H(5,𝔽2)—41no—
76H(5,𝔽2)—41no—
77C2×#41—21no—
78C2×#41—21no—
79C2×#41—21no—
80C2×#41—21no—
81C2×#41—22no—
82C2×#41—22no—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41D411yescharacteristic, complemented
86C2×#41D411yescharacteristic, complemented
87C2×#41C2311yescharacteristic
88C2×#41C2311yescharacteristic
89C2×#41C2311yescharacteristic
90C22×Q8D411yescharacteristic, complemented
91C22×D4—21no—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4—21no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4D412yes—
99C22×D4D411yescharacteristic, complemented
100C22×D4C2311yescharacteristic
101C22×D4C2311yescharacteristic
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2311yescharacteristic
105AGL(1,ℤ/8)—22no—
106AGL(1,ℤ/8)—22no—
107AGL(1,ℤ/8)—22no—
108AGL(1,ℤ/8)—41no—
109AGL(1,ℤ/8)—41no—
110AGL(1,ℤ/8)—41no—
111C2×SD16—21no—
112C2×SD16—21no—
113C2×SD16—21no—
114C2×D8—21no—
115C2×D8—21no—
116C2×D8—21no—
117C2×M16—21no—
118C2×M16—22no—
119C2×M16—41no—
120C2×M16C2311yescharacteristic
121C2×M16C2311yescharacteristic
122C2×M16C2311yescharacteristic
123C2×M16C2311yescharacteristic
124C22×C8—21no—
125#127—22no—
126#127—21no—
127#127—22no—
128#127—41no—
129#124—22no—
130#124—21no—
131#123—22no—
132#123—21no—
133C22⋊Q8—22no—
134C22⋊Q8—21no—
135C22⋊Q8—21no—
136C22⋊Q8D411yescharacteristic, complemented
137C22⋊Q8D411yescharacteristic, complemented
138#121—22no—
139#121—22no—
140#121—22no—
141#121—22no—
142#121—22no—
143#121—21no—
144#121—21no—
145#121—21no—
146#121—21no—
147#121—21no—
148#121—21no—
149#121—22no—
150#121—21no—
151#121—21no—
152#121—21no—
153#121—21no—
154#121—21no—
155#121—21no—
156#121—48no—
157#121D411yescharacteristic, complemented
158#121D411yescharacteristic, complemented
159#120—22no—
160#120—22no—
161#120—22no—
162#120—21no—
163#120—22no—
164C4×Q8—21no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—21no—
171#117—21no—
172#117C2311yescharacteristic
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4D412yes—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—41no—
185C2×C42C2311yescharacteristic
186C2×C42C2311yescharacteristic
187#107—21no—
188#106—21no—
189#104—22no—
190#104—22no—
191#104—22no—
192#104—22no—
193#104—41no—
194#104D414yescomplemented
195#103—21no—
196#103—21no—
197#103—21no—
198#102—21no—
199#102—21no—
200#102—21no—
201#102—21no—
202#102—21no—
203#100—22no—
204#100—41no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C24—22no—
208C24—21no—
209C24—21no—
210C24C2×D411yescharacteristic
211C24C2×D412yes—
212#41—21no—
213#41—21no—
214#41—22no—
215#41—22no—
216#41—21no—
217#41—21no—
218#41—21no—
219#41—21no—
220#41—21no—
221#41—21no—
222#41—21no—
223#41—21no—
224#41—42no—
225#41—42no—
226#41—41no—
227#41—41no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232#41—42no—
233#41—42no—
234#41C2×D412yes—
235#41#4112yes—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8C2×D411yescharacteristic
240C2×Q8C2×D411yescharacteristic
241C2×Q8C2×D411yescharacteristic
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—84no—
287C2×D4C2×D411yescharacteristic
288C2×D4C2×D411yescharacteristic
289C2×D4C2×D411yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C2×D411yescharacteristic
311C22×C4C2×D411yescharacteristic
312C22×C4C2×D411yescharacteristic
313C22×C4C2×D411yescharacteristic
314C22×C4C2×D411yescharacteristic
315C22×C4C2×D412yes—
316C22×C4C2411yescharacteristic
317SD16—42no—
318SD16—42no—
319SD16—42no—
320D8—42no—
321D8—42no—
322D8—42no—
323M16—21no—
324M16—21no—
325M16—21no—
326M16—21no—
327M16—21no—
328M16—21no—
329M16—21no—
330M16—42no—
331M16C2×D412yescomplemented
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—44no—
346C4⋊C4C2×D411yescharacteristic
347C4⋊C4C2×D411yescharacteristic
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—21no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C42—22no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—42no—
370C42C2×D412yescomplemented
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—42no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—81no—
392C23—82no—
393C23—82no—
394C23#12011yescharacteristic
395C23#12011yescharacteristic
396C23C22×D411yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—21no—
400Q8—21no—
401Q8—21no—
402Q8—42no—
403Q8#12112yescomplemented
404D4—22no—
405D4—22no—
406D4—21no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—21no—
411D4—42no—
412D4—42no—
413D4—44no—
414D4—42no—
415D4—44no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—44no—
421D4—42no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—88no—
431D4—81no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—82no—
438D4#12112yescomplemented
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—48no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4#12011yescharacteristic
477C2×C4#12011yescharacteristic
478C2×C4C22×D411yescharacteristic
479C2×C4C22×D411yescharacteristic
480C2×C4C22×D411yescharacteristic
481C2×C4C2×#4111yescharacteristic
482C2×C4H(5,𝔽2)11yescharacteristic
483C2×C4H(5,𝔽2)11yescharacteristic
484C8—22no—
485C8—41no—
486C8—41no—
487C8—41no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—164no—
518C22#45312yescomplemented
519C22C2×#12011yescharacteristic
520C22D4211yescharacteristic
521C22#54611yescharacteristic
522C4—24no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—81no—
533C4C2×#12111yescharacteristic
534C4#53411yescharacteristic
535C4D4211yescharacteristic
536C4#54611yescharacteristic
537C2—21no—
538C2—21no—
539C2—22no—
540C2—41no—
541C2—42no—
542C2—81no—
543C2—82no—
544C2—81no—
545C2—161no—
546C2#241011yescharacteristic
547C2C2×#45311yescharacteristic
548C2#301811yescharacteristic
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. #3488C2semidirect
2. #3055C2semidirect
3. C2×#453C2semidirect
4. #3006C2semidirect
5. C22×#104C2semidirect
6. #2856C2semidirect
7. #2039C2semidirect
8. #2037C2semidirect
9. #2006C2semidirect
10. #2002C2semidirect
11. #1900C2semidirect
12. #1896C2semidirect
13. #1883C2semidirect
14. #1859C2semidirect
15. #1792C2semidirect
19. C2×AGL(1,ℤ/8)C22semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
21. C22×M16C22semidirect
32. C2×#127C22semidirect
38. C2×#121C22semidirect
40. C22×C42C22semidirect
50. #428C22semidirect
54. C2×#104C22semidirect
55. C2×#104C22semidirect
56. C2×#104C22semidirect
57. C2×#104C22semidirect
58. #418C22semidirect
60. #417C22semidirect
61. #417C22semidirect
62. C2×#100C22semidirect
63. C2×#100C22semidirect
64. #407C22semidirect
66. #390C22semidirect
67. #390C22semidirect
68. #384C22semidirect
69. #343C22semidirect
70. #343C22semidirect
71. #339C22semidirect
72. #339C22semidirect
85. C2×#41D4semidirect
86. C2×#41D4semidirect
90. C22×Q8D4semidirect
99. C22×D4D4semidirect
136. C22⋊Q8D4semidirect
137. C22⋊Q8D4semidirect
157. #121D4semidirect
158. #121D4semidirect
194. #104D4semidirect
331. M16C2×D4semidirect
370. C42C2×D4semidirect
403. Q8#121semidirect
438. D4#121semidirect
518. C22#453semidirect

Maximal subgroups

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

subgroupindex
#34882
#30552
C2×#4532
#30062
C22×#1042
#28562
#20392
#20372
#20062
#20022
#19002
#18962
#18832
#18592
#17922