Finite Group Databasebeta

← #24432

Subgroups — #24432

Order 256.

961 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42C211yesmaximal, characteristic
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5C22×#102C211yesmaximal, characteristic, complemented
6#2417C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4—21no—
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic, complemented
21C22×SD16C2211yescharacteristic, complemented
22C23×C8C2211yescharacteristic, complemented
23D42—24no—
24D42—24no—
25C2×#127C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic, complemented
27C2×C22⋊Q8C2211yescharacteristic, complemented
28C2×#121—21no—
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#120—21no—
32C2×#120—21no—
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic, complemented
36#465—24no—
37#450—24no—
38#450—24no—
39C2×#106C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#102—24no—
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#102C2211yescharacteristic, complemented
49C2×#102C2211yescharacteristic, complemented
50C2×#102C2211yescharacteristic, complemented
51C2×C22⋊C8C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic, complemented
54#398—21no—
55#396—21no—
56#391C2211yescharacteristic, complemented
57#389C2211yescharacteristic, complemented
58#386—21no—
59#380—21no—
60#380C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C25—21no—
67C22×Q8C2311yescharacteristic
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—28no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—21no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4—41no—
86C22×D4—44no—
87C22×D4—44no—
88C22×D4D411yescharacteristic
89C22×D4D414yes—
90C22×D4C2311yescharacteristic
91C22×D4C2311yescharacteristic
92C22×D4C2311yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—22no—
101C2×SD16—22no—
102C2×SD16—22no—
103C2×SD16—41no—
104C2×SD16—41no—
105C2×SD16—41no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—41no—
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113#127—22no—
114#127—22no—
115C22⋊Q8—22no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—41no—
119#121—28no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121D414yes—
124#120—28no—
125#120—28no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4D411yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4D411yescharacteristic
148C2×C22⋊C4D411yescharacteristic
149C2×C42C2311yescharacteristic
150#106—22no—
151#103—22no—
152#102—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—41no—
160#102D414yescomplemented
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—41no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168C24—21no—
169C24—24no—
170C24—21no—
171C24—21no—
172C24—24no—
173C24—42no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—81no—
182C24—81no—
183C24—81no—
184C24—81no—
185C24C2×D411yescharacteristic
186C24C2×D411yescharacteristic
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—432no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—81no—
234C2×D4—81no—
235C2×D4SD1614yescomplemented
236C2×D4C2×D412yes—
237C2×D4C2×D412yes—
238C2×D4C2×D412yes—
239C2×D4C2×D412yes—
240C2×D4#4112yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—21no—
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—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2×D411yescharacteristic
268C22×C4C2411yescharacteristic
269SD16—44no—
270SD16—44no—
271SD16—44no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—44no—
284C2×C8C2×D412yescomplemented
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—81no—
296C4⋊C4C2×D412yes—
297C4⋊C4C2×D412yescomplemented
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—22no—
301C22⋊C4—24no—
302C22⋊C4—22no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—81no—
306C42—21no—
307C42—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—21no—
315C23—22no—
316C23—21no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—48no—
324C23—48no—
325C23—48no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—48no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23—84no—
352C23—84no—
353C23#12012yes—
354C23C2×SD1612yes—
355C23C2×SD1612yes—
356C23AGL(1,ℤ/8)12yes—
357C23C22×D411yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—41no—
361Q8—41no—
362D4—28no—
363D4—22no—
364D4—22no—
365D4—22no—
366D4—22no—
367D4—22no—
368D4—22no—
369D4—22no—
370D4—48no—
371D4—48no—
372D4—48no—
373D4—48no—
374D4—48no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—84no—
380D4#12114yes—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—48no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
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—84no—
416C2×C4#12012yes—
417C2×C4C2×SD1612yes—
418C2×C4C22×D411yescharacteristic
419C2×C4C22×D411yescharacteristic
420C2×C4C22×D411yescharacteristic
421C2×C4C2×#4111yescharacteristic
422C2×C4H(5,𝔽2)11yescharacteristic
423C2×C4H(5,𝔽2)11yescharacteristic
424C8—22no—
425C8—41no—
426C8—41no—
427C8—41no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—42no—
440C22—41no—
441C22—42no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—816no—
455C22—816no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#45014yescomplemented
465C22C2×#12011yescharacteristic
466C22D4211yescharacteristic
467C22#54611yescharacteristic
468C22C22×SD1611yescharacteristic
469C22C2×AGL(1,ℤ/8)11yescharacteristic
470C22#57611yescharacteristic
471C22#57711yescharacteristic
472C4—22no—
473C4—22no—
474C4—41no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4C2×#12111yescharacteristic
482C4#53411yescharacteristic
483C4D4211yescharacteristic
484C4#54611yescharacteristic
485C2—24no—
486C2—42no—
487C2—44no—
488C2—82no—
489C2—82no—
490C2#241011yescharacteristic
491C2C2×#45011yescharacteristic
492C2#301211yescharacteristic
493C2#326411yescharacteristic
494C2D4×SD1611yescharacteristic
495C2#329511yescharacteristic
496C2#331011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×D42C2
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
2. C2×#465C2semidirect
3. C2×#450C2semidirect
4. C2×#450C2semidirect
5. C22×#102C2semidirect
6. #2417C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2014C2semidirect
10. #2000C2semidirect
11. #1877C2semidirect
12. #1874C2semidirect
13. #1869C2semidirect
14. #1864C2semidirect
15. #1786C2semidirect
20. C22×SD16C22semidirect
21. C22×SD16C22semidirect
22. C23×C8C22semidirect
26. C2×C22⋊Q8C22semidirect
27. C2×C22⋊Q8C22semidirect
35. C22×C4⋊C4C22semidirect
39. C2×#106C22semidirect
40. C2×#103C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
49. C2×#102C22semidirect
50. C2×#102C22semidirect
51. C2×C22⋊C8C22semidirect
52. C2×C22⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
56. #391C22semidirect
57. #389C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
160. #102D4semidirect
235. C2×D4SD16semidirect
284. C2×C8C2×D4semidirect
297. C4⋊C4C2×D4semidirect
464. C22#450semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4652
C2×#4502
C22×#1022
#24172
#20412
#20342
#20142
#20002
#18772
#18742
#18692
#18642
#17862