Finite Group Databasebeta

← #24424

Subgroups — #24424

Order 256.

996 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C22×#102C211yesmaximal, characteristic, complemented
6#2416C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4—21no—
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic, complemented
21C22×D8C2211yescharacteristic, complemented
22C23×C8C2211yescharacteristic, complemented
23D42—24no—
24D42—24no—
25C2×#127C2211yescharacteristic
26C2×#121—21no—
27C2×#121—21no—
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic, complemented
31C2×#121C2211yescharacteristic, complemented
32C2×#120—21no—
33C2×#120—21no—
34C2×C4×D4C2211yescharacteristic
35C2×C4×D4C2211yescharacteristic
36C22×C4⋊C4C2211yescharacteristic, complemented
37#466—24no—
38#447—24no—
39#447—24no—
40C2×#107C2211yescharacteristic, 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×#102C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic, complemented
54C2×C22⋊C8C2211yescharacteristic, complemented
55#396—21no—
56#394—21no—
57#390C2211yescharacteristic, complemented
58#389C2211yescharacteristic, complemented
59#386—21no—
60#380C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C25—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—28no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—21no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4—41no—
86C22×D4—41no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4D411yescharacteristic
90C22×D4D414yescomplemented
91C22×D4C2311yescharacteristic
92C22×D4C2311yescharacteristic
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C2×D8—22no—
100C2×D8—22no—
101C2×D8—22no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—41no—
106C2×D8—41no—
107C2×D8—41no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115#127—22no—
116#127—22no—
117#121—28no—
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121—22no—
124#121—41no—
125#121—44no—
126#121D414yescomplemented
127#120—28no—
128#120—28no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4D411yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4D411yescharacteristic
151C2×C22⋊C4D411yescharacteristic
152C2×C42C2311yescharacteristic
153#107—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—22no—
160#102—22no—
161#102—22no—
162#102—41no—
163#102D414yescomplemented
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—41no—
167#95—21no—
168#95—21no—
169C24—21no—
170C24—24no—
171C24—21no—
172C24—21no—
173C24—24no—
174C24—21no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24—81no—
184C24—81no—
185C24—81no—
186C24—81no—
187C24C2×D411yescharacteristic
188C24C2×D411yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—28no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
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—21no—
216C2×D4—21no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—432no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—81no—
241C2×D4—81no—
242C2×D4—81no—
243C2×D4D814yescomplemented
244C2×D4C2×D412yes—
245C2×D4C2×D412yes—
246C2×D4C2×D412yes—
247C2×D4C2×D412yes—
248C2×D4#4112yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
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—21no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D412yes—
273C22×C4C2×D412yes—
274C22×C4C2×D411yescharacteristic
275C22×C4C2411yescharacteristic
276D8—44no—
277D8—44no—
278D8—44no—
279C2×C8—21no—
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×C8—44no—
291C2×C8C2×D412yescomplemented
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4C2×D412yes—
302C4⋊C4C2×D412yescomplemented
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—81no—
313C42—21no—
314C42—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—22no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—48no—
331C23—48no—
332C23—48no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
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—48no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—84no—
358C23—84no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23—84no—
364C23—84no—
365C23—84no—
366C23—84no—
367C23#12012yes—
368C23C2×D812yes—
369C23C2×D812yes—
370C23AGL(1,ℤ/8)12yes—
371C23C22×D411yescharacteristic
372D4—28no—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—22no—
377D4—22no—
378D4—22no—
379D4—22no—
380D4—48no—
381D4—48no—
382D4—48no—
383D4—48no—
384D4—48no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—84no—
394D4—84no—
395D4—84no—
396D4#12114yescomplemented
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—48no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4#12012yes—
430C2×C4C2×D812yes—
431C2×C4C22×D411yescharacteristic
432C2×C4C22×D411yescharacteristic
433C2×C4C22×D411yescharacteristic
434C2×C4C2×#4111yescharacteristic
435C2×C4H(5,𝔽2)11yescharacteristic
436C2×C4H(5,𝔽2)11yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8—41no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—42no—
453C22—41no—
454C22—42no—
455C22—41no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—41no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—816no—
468C22—816no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#44714yescomplemented
485C22C2×#12011yescharacteristic
486C22D4211yescharacteristic
487C22#54611yescharacteristic
488C22C22×D811yescharacteristic
489C22C2×AGL(1,ℤ/8)11yescharacteristic
490C22#57611yescharacteristic
491C22#57711yescharacteristic
492C4—22no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4C2×#12111yescharacteristic
501C4#53411yescharacteristic
502C4D4211yescharacteristic
503C4#54611yescharacteristic
504C2—24no—
505C2—42no—
506C2—44no—
507C2—82no—
508C2—82no—
509C2—161no—
510C2#241011yescharacteristic
511C2C2×#44711yescharacteristic
512C2#301211yescharacteristic
513C2#326511yescharacteristic
514C2D4×D811yescharacteristic
515C2#329411yescharacteristic
516C2#331811yescharacteristic
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×D42C2semidirect
2. C2×#466C2semidirect
3. C2×#447C2semidirect
4. C2×#447C2semidirect
5. C22×#102C2semidirect
6. #2416C2semidirect
7. #2041C2semidirect
8. #2033C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1879C2semidirect
12. #1874C2semidirect
13. #1869C2semidirect
14. #1862C2semidirect
15. #1786C2semidirect
20. C22×D8C22semidirect
21. C22×D8C22semidirect
22. C23×C8C22semidirect
30. C2×#121C22semidirect
31. C2×#121C22semidirect
36. C22×C4⋊C4C22semidirect
40. C2×#107C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
49. C2×#102C22semidirect
50. C2×#102C22semidirect
51. C2×#102C22semidirect
52. C2×C22⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
54. C2×C22⋊C8C22semidirect
57. #390C22semidirect
58. #389C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
90. C22×D4D4semidirect
126. #121D4semidirect
163. #102D4semidirect
243. C2×D4D8semidirect
291. C2×C8C2×D4semidirect
302. C4⋊C4C2×D4semidirect
396. D4#121semidirect
484. C22#447semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4662
C2×#4472
C22×#1022
#24162
#20412
#20332
#20112
#19992
#18792
#18742
#18692
#18622
#17862