Finite Group Databasebeta

← #24468

Subgroups — #24468

Order 256.

751 conjugacy classes of subgroups.

The subgroup lattice has 545 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
5C2×#421C211yesmaximal, characteristic, complemented
6#2619C211yesmaximal, characteristic, complemented
7#2040C211yesmaximal, characteristic, complemented
8#2037C211yesmaximal, characteristic, complemented
9#2006C211yesmaximal, characteristic, complemented
10#2002C211yesmaximal, characteristic, complemented
11#1900C211yesmaximal, characteristic, complemented
12#1897C211yesmaximal, characteristic, complemented
13#1883C211yesmaximal, characteristic, complemented
14#1860C211yesmaximal, 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×#121C2211yescharacteristic
37C2×#121C2211yescharacteristic, complemented
38#520C2211yescharacteristic
39C2×#117C2211yescharacteristic, complemented
40#469—21no—
41#468—21no—
42#466—21no—
43#465—21no—
44#453—24no—
45#449—21no—
46#449—21no—
47#448—21no—
48#447—21no—
49#428C2211yescharacteristic, complemented
50#421—24no—
51C2×#104C2211yescharacteristic
52C2×#104C2211yescharacteristic
53C2×#104C2211yescharacteristic, complemented
54C2×#104C2211yescharacteristic, complemented
55C2×#104C2211yescharacteristic, complemented
56C2×#104C2211yescharacteristic, complemented
57#418C2211yescharacteristic, complemented
58#417C2211yescharacteristic
59#417C2211yescharacteristic, complemented
60#417C2211yescharacteristic, complemented
61C2×#100C2211yescharacteristic, complemented
62C2×#100C2211yescharacteristic, complemented
63#407C2211yescharacteristic, complemented
64#394—22no—
65#392—22no—
66#390C2211yescharacteristic, complemented
67#390C2211yescharacteristic, complemented
68#382C2211yescharacteristic, 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×C4C2311yescharacteristic
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—22no—
106AGL(1,ℤ/8)—22no—
107AGL(1,ℤ/8)—41no—
108AGL(1,ℤ/8)—41no—
109AGL(1,ℤ/8)—41no—
110C2×SD16—21no—
111C2×SD16—21no—
112C2×SD16—21no—
113C2×D8—21no—
114C2×D8—21no—
115C2×D8—21no—
116C2×M16—21no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16C2311yescharacteristic
120C2×M16C2311yescharacteristic
121C2×M16C2311yescharacteristic
122C2×M16C2311yescharacteristic
123C22×C8—21no—
124#127—22no—
125#127—21no—
126#127—22no—
127#127—41no—
128#124—22no—
129#124—21no—
130#123—22no—
131#123—21no—
132C22⋊Q8—22no—
133C22⋊Q8—21no—
134C22⋊Q8—21no—
135C22⋊Q8D411yescharacteristic, complemented
136C22⋊Q8D411yescharacteristic, complemented
137#121—22no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—22no—
142#121—21no—
143#121—21no—
144#121—21no—
145#121—21no—
146#121—21no—
147#121—21no—
148#121—22no—
149#121—21no—
150#121—21no—
151#121—21no—
152#121—21no—
153#121—21no—
154#121—21no—
155#121D411yescharacteristic, complemented
156#121D411yescharacteristic, complemented
157#120—22no—
158#120—22no—
159#120—22no—
160#120—21no—
161#120—22no—
162C4×Q8—21no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—21no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—21no—
169#117—21no—
170#117—22no—
171#117—41no—
172#117C2311yescharacteristic
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4D412yes—
183C2×C42C2311yescharacteristic
184C2×C42C2311yescharacteristic
185#107—21no—
186#106—21no—
187#104—22no—
188#104—22no—
189#104—22no—
190#104—22no—
191#104—41no—
192#104D414yescomplemented
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#102—21no—
198#102—21no—
199#102—21no—
200#102—21no—
201#100—22no—
202#100—41no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205#95—22no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24C2×D412yes—
210C24C2×D411yescharacteristic
211#41—21no—
212#41—21no—
213#41—22no—
214#41—22no—
215#41—21no—
216#41—21no—
217#41—21no—
218#41—21no—
219#41—21no—
220#41—21no—
221#41—21no—
222#41—21no—
223#41—42no—
224#41—42no—
225#41—41no—
226#41—41no—
227#41—42no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232#41—42no—
233#41C2×D412yes—
234#41#4112yes—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8C2×D411yescharacteristic
239C2×Q8C2×D411yescharacteristic
240C2×Q8C2×D411yescharacteristic
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
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—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—84no—
286C2×D4C2×D411yescharacteristic
287C2×D4C2×D411yescharacteristic
288C2×D4C2×D411yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C2×D411yescharacteristic
307C22×C4C2×D411yescharacteristic
308C22×C4C2×D411yescharacteristic
309C22×C4C2×D411yescharacteristic
310C22×C4C2×D411yescharacteristic
311C22×C4C2×D412yes—
312C22×C4C2411yescharacteristic
313SD16—42no—
314SD16—42no—
315SD16—42no—
316D8—42no—
317D8—42no—
318D8—42no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—21no—
323M16—21no—
324M16—21no—
325M16—21no—
326M16—42no—
327M16C2×D412yescomplemented
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—42no—
342C4⋊C4C2×D411yescharacteristic
343C4⋊C4C2×D411yescharacteristic
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—21no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—22no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×D412yescomplemented
367C23—21no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23#12011yescharacteristic
391C23#12011yescharacteristic
392C23C22×D411yescharacteristic
393Q8—21no—
394Q8—21no—
395Q8—21no—
396Q8—21no—
397Q8—21no—
398Q8—42no—
399Q8#12112yes—
400D4—22no—
401D4—22no—
402D4—21no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—21no—
407D4—42no—
408D4—42no—
409D4—44no—
410D4—42no—
411D4—44no—
412D4—44no—
413D4—42no—
414D4—41no—
415D4—41no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—81no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—82no—
431D4—88no—
432D4—82no—
433D4—82no—
434D4#12112yes—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4#12011yescharacteristic
472C2×C4#12011yescharacteristic
473C2×C4C22×D411yescharacteristic
474C2×C4C22×D411yescharacteristic
475C2×C4C22×D411yescharacteristic
476C2×C4C2×#4111yescharacteristic
477C2×C4H(5,𝔽2)11yescharacteristic
478C2×C4H(5,𝔽2)11yescharacteristic
479C8—22no—
480C8—41no—
481C8—41no—
482C8—41no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—84no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—164no—
514C22C2×#12011yescharacteristic
515C22D4211yescharacteristic
516C22#54611yescharacteristic
517C4—21no—
518C4—21no—
519C4—22no—
520C4—42no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—81no—
527C4—81no—
528C4C2×#12111yescharacteristic
529C4#53411yescharacteristic
530C4D4211yescharacteristic
531C4#54611yescharacteristic
532C2—21no—
533C2—21no—
534C2—41no—
535C2—41no—
536C2—42no—
537C2—81no—
538C2—82no—
539C2—81no—
540C2—161no—
541C2#241011yescharacteristic
542C2#301911yescharacteristic
543C2#302111yescharacteristic
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. C2×#421C2semidirect
6. #2619C2semidirect
7. #2040C2semidirect
8. #2037C2semidirect
9. #2006C2semidirect
10. #2002C2semidirect
11. #1900C2semidirect
12. #1897C2semidirect
13. #1883C2semidirect
14. #1860C2semidirect
15. #1792C2semidirect
19. C2×AGL(1,ℤ/8)C22semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
21. C22×M16C22semidirect
32. C2×#127C22semidirect
37. C2×#121C22semidirect
39. C2×#117C22semidirect
49. #428C22semidirect
53. C2×#104C22semidirect
54. C2×#104C22semidirect
55. C2×#104C22semidirect
56. C2×#104C22semidirect
57. #418C22semidirect
59. #417C22semidirect
60. #417C22semidirect
61. C2×#100C22semidirect
62. C2×#100C22semidirect
63. #407C22semidirect
66. #390C22semidirect
67. #390C22semidirect
68. #382C22semidirect
69. #343C22semidirect
70. #343C22semidirect
71. #339C22semidirect
72. #339C22semidirect
85. C2×#41D4semidirect
86. C2×#41D4semidirect
90. C22×Q8D4semidirect
99. C22×D4D4semidirect
135. C22⋊Q8D4semidirect
136. C22⋊Q8D4semidirect
155. #121D4semidirect
156. #121D4semidirect
192. #104D4semidirect
327. M16C2×D4semidirect
366. C42C2×D4semidirect

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
C2×#4212
#26192
#20402
#20372
#20062
#20022
#19002
#18972
#18832
#18602
#17922