Finite Group Databasebeta

← #25762

Subgroups — #25762

Order 256.

861 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic, complemented
2C2×#495C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#449C211yesmaximal, characteristic, complemented
6C2×#450C211yesmaximal, characteristic, complemented
7C2×#447C211yesmaximal, characteristic, complemented
8#2540C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic, complemented
13#1983C211yesmaximal, characteristic, complemented
14#1954C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22C22×D8C2211yescharacteristic
23C22×D8C2211yescharacteristic
24#546—24no—
25#546—24no—
26C2×#124C2211yescharacteristic
27C2×#124C2211yescharacteristic, complemented
28C2×#123—21no—
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×C22⋊Q8C2211yescharacteristic, complemented
32C2×#121—21no—
33C2×#121C2211yescharacteristic
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic, complemented
36C2×#120—21no—
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C22×C22⋊C4—21no—
40#495—24no—
41#463—24no—
42#452—24no—
43#450—24no—
44#449—24no—
45#447—24no—
46C2×C4⋊C8C2211yescharacteristic, complemented
47C2×#103C2211yescharacteristic, complemented
48C2×#103C2211yescharacteristic, complemented
49C2×#103C2211yescharacteristic, complemented
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic
54C2×#102C2211yescharacteristic, complemented
55C2×C22⋊C8C2211yescharacteristic
56C2×C22⋊C8C2211yescharacteristic
57C2×C22⋊C8C2211yescharacteristic, complemented
58C2×C22⋊C8C2211yescharacteristic, complemented
59C2×C4×C8C2211yescharacteristic, complemented
60#399—21no—
61#392—21no—
62#390—21no—
63#390—21no—
64C4×C4⋊C4C2211yescharacteristic, complemented
65#340C2211yescharacteristic, complemented
66#340C2211yescharacteristic, complemented
67C25—21no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—41no—
73C2×#41—41no—
74C22×Q8C2311yescharacteristic
75C22×Q8C2311yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—41no—
86C22×D4—41no—
87C22×D4D411yescharacteristic
88C22×D4C2311yescharacteristic
89C22×D4C2311yescharacteristic
90C22×D4C2311yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—41no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—22no—
101C2×SD16—22no—
102C2×SD16—22no—
103C2×SD16—22no—
104C2×SD16—22no—
105C2×SD16—41no—
106C2×SD16—41no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—41no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117#124—22no—
118#124—22no—
119#124—22no—
120#124D414yescomplemented
121#123—28no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128#121—28no—
129#121—22no—
130#121—22no—
131#121—22no—
132#121—22no—
133#121—41no—
134#121—41no—
135#120—28no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4D414yescomplemented
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4D411yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C22⋊C4—28no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4D411yescharacteristic
160C2×C22⋊C4D411yescharacteristic
161C2×C42—21no—
162C2×C42C2311yescharacteristic
163C4⋊C8—22no—
164#103—22no—
165#103—22no—
166#103—22no—
167#102—22no—
168#102—22no—
169#102—22no—
170#102D414yescomplemented
171#102D414yescomplemented
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C4×C8—22no—
177#95—21no—
178C24—21no—
179C24—24no—
180C24—21no—
181C24—21no—
182C24—42no—
183C24—42no—
184C24—42no—
185C24—42no—
186C24—81no—
187C24—81no—
188C24C2×D411yescharacteristic
189C24C2×D411yescharacteristic
190#41—44no—
191#41—44no—
192#41—44no—
193#41—44no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8C2×D412yes—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—81no—
241C2×D4—81no—
242C2×D4C2×D412yes—
243C2×D4C2×D412yes—
244C2×D4C2×D412yes—
245C2×D4C2×D412yes—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2411yescharacteristic
274Q16—44no—
275SD16—44no—
276SD16—44no—
277SD16—44no—
278D8—44no—
279D8—44no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C2×D412yes—
291C2×C8C2×D412yes—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4C2×D412yes—
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—81no—
316C42C2×D412yes—
317C42C2×D412yes—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—48no—
336C23—48no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—84no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23—84no—
359C23#13512yes—
360C23C22×D411yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—41no—
364Q8—41no—
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369D4—22no—
370D4—22no—
371D4—22no—
372D4—22no—
373D4—22no—
374D4—44no—
375D4—48no—
376D4—48no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382D4—41no—
383D4—84no—
384D4—84no—
385D4—84no—
386D4#12714yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—48no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—84no—
422C2×C4#12012yes—
423C2×C4#12012yes—
424C2×C4C22×D411yescharacteristic
425C2×C4C22×D411yescharacteristic
426C2×C4C22×D411yescharacteristic
427C2×C4C22×D411yescharacteristic
428C2×C4H(5,𝔽2)11yescharacteristic
429C2×C4H(5,𝔽2)11yescharacteristic
430C8—22no—
431C8—22no—
432C8—41no—
433C8—41no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—41no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—816no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22C2×#12011yescharacteristic
471C22D4211yescharacteristic
472C22D4211yescharacteristic
473C22C2×#13511yescharacteristic
474C22#57511yescharacteristic
475C22#57611yescharacteristic
476C22#57711yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4C2×#12711yescharacteristic
487C4#53511yescharacteristic
488C4D4211yescharacteristic
489C4D4211yescharacteristic
490C2—42no—
491C2—44no—
492C2—81no—
493C2—82no—
494C2—161no—
495C2#243311yescharacteristic
496C2#300211yescharacteristic
497C2#301211yescharacteristic
498C2#327311yescharacteristic
499C2#327511yescharacteristic
500C2#328011yescharacteristic
501C2#328211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#546C2semidirect
2. C2×#495C2semidirect
3. C2×#463C2semidirect
4. C2×#452C2semidirect
5. C2×#449C2semidirect
6. C2×#450C2semidirect
7. C2×#447C2semidirect
8. #2540C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #2000C2semidirect
12. #1999C2semidirect
13. #1983C2semidirect
14. #1954C2semidirect
15. C4×#102C2semidirect
18. C22×Q16C22semidirect
21. C22×SD16C22semidirect
27. C2×#124C22semidirect
31. C2×C22⋊Q8C22semidirect
35. C2×#121C22semidirect
46. C2×C4⋊C8C22semidirect
47. C2×#103C22semidirect
48. C2×#103C22semidirect
49. C2×#103C22semidirect
54. C2×#102C22semidirect
57. C2×C22⋊C8C22semidirect
58. C2×C22⋊C8C22semidirect
59. C2×C4×C8C22semidirect
64. C4×C4⋊C4C22semidirect
65. #340C22semidirect
66. #340C22semidirect
120. #124D4semidirect
139. C4×D4D4semidirect
170. #102D4semidirect
171. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4952
C2×#4632
C2×#4522
C2×#4492
C2×#4502
C2×#4472
#25402
#20012
#20002
#19992
#19832
#19542
C4×#1022