Finite Group Databasebeta

← #24044

Subgroups — #24044

Order 256.

884 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42C211yesmaximal, characteristic
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1786C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4—21no—
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×M16C2211yescharacteristic, complemented
20D42—24no—
21D42—24no—
22C2×#127C2211yescharacteristic
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27C2×#120—21no—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic, complemented
31#482—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34#418—24no—
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic, complemented
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×C22⋊C8C2211yescharacteristic, complemented
48C2×C22⋊C8C2211yescharacteristic, complemented
49C2×C22⋊C8C2211yescharacteristic, complemented
50#397—21no—
51#394—21no—
52#389C2211yescharacteristic, complemented
53#389C2211yescharacteristic, complemented
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic, complemented
57#380C2211yescharacteristic, complemented
58#380C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C25—21no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—28no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—22no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—44no—
84C22×D4—44no—
85C22×D4D411yescharacteristic
86C22×D4D414yes—
87C22×D4C2311yescharacteristic
88C22×D4C2311yescharacteristic
89C22×D4C2311yescharacteristic
90C23×C4—21no—
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100#127—22no—
101#127—22no—
102#121—28no—
103#121—22no—
104#121—22no—
105#121—22no—
106#121D414yes—
107#120—28no—
108#120—28no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—22no—
113#117—22no—
114#117—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4D411yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—41no—
145#102—41no—
146#102—41no—
147#102—41no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156C24—21no—
157C24—24no—
158C24—21no—
159C24—21no—
160C24—24no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—42no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—81no—
170C24—81no—
171C24—81no—
172C24—81no—
173C24C2×D411yescharacteristic
174C24C2×D411yescharacteristic
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—28no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—432no—
209C2×D4—44no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—42no—
216C2×D4—81no—
217C2×D4—81no—
218C2×D4C2×D412yes—
219C2×D4C2×D412yes—
220C2×D4C2×D412yes—
221C2×D4#4112yes—
222C2×D4#4112yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D412yes—
245C22×C4C2×D412yes—
246C22×C4C2×D411yescharacteristic
247C22×C4C2411yescharacteristic
248M16—44no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4C2×D412yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—22no—
277C22⋊C4—24no—
278C22⋊C4—22no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C42—21no—
284C42—21no—
285C42—42no—
286C23—21no—
287C23—22no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—48no—
304C23—48no—
305C23—48no—
306C23—48no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23#12012yes—
334C23AGL(1,ℤ/8)12yes—
335C23C22×D411yescharacteristic
336D4—28no—
337D4—22no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—48no—
345D4—48no—
346D4—48no—
347D4—48no—
348D4—48no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—84no—
354D4#12314yes—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—48no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4#12012yes—
390C2×C4AGL(1,ℤ/8)12yes—
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C2×#4111yescharacteristic
394C2×C4C2×#4111yescharacteristic
395C2×C4H(5,𝔽2)11yescharacteristic
396C2×C4H(5,𝔽2)11yescharacteristic
397C8—41no—
398C8—41no—
399C8—41no—
400C8—41no—
401C22—22no—
402C22—24no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—42no—
410C22—41no—
411C22—42no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—816no—
429C22—816no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C2×#12011yescharacteristic
439C22#54611yescharacteristic
440C22#54611yescharacteristic
441C22C2×AGL(1,ℤ/8)11yescharacteristic
442C22C2×AGL(1,ℤ/8)11yescharacteristic
443C22#57611yescharacteristic
444C22#57711yescharacteristic
445C4—22no—
446C4—22no—
447C4—41no—
448C4—42no—
449C4—41no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4C2×#12311yescharacteristic
455C4#53411yescharacteristic
456C4#54611yescharacteristic
457C4#54611yescharacteristic
458C2—42no—
459C2—42no—
460C2—44no—
461C2—82no—
462C2—82no—
463C2#240511yescharacteristic
464C2#300311yescharacteristic
465C2#301211yescharacteristic
466C2#331011yescharacteristic
467C2#331811yescharacteristic
468C2#332411yescharacteristic
469C2#332511yescharacteristic
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×#482C2semidirect
3. C2×#418C2semidirect
4. #2485C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1895C2semidirect
10. #1870C2semidirect
11. #1864C2semidirect
12. #1862C2semidirect
13. #1786C2semidirect
14. #1786C2semidirect
15. #1786C2semidirect
19. C22×M16C22semidirect
30. C2×#117C22semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
39. C2×#102C22semidirect
40. C2×#102C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×C22⋊C8C22semidirect
48. C2×C22⋊C8C22semidirect
49. C2×C22⋊C8C22semidirect
52. #389C22semidirect
53. #389C22semidirect
56. #380C22semidirect
57. #380C22semidirect
58. #380C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4822
C2×#4182
#24852
#20412
#20342
#20332
#18952
#18702
#18642
#18622
#17862