Finite Group Databasebeta

← #24257

Subgroups — #24257

Order 256.

825 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C2×#442C211yesmaximal, characteristic, complemented
6#2537C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1955C211yesmaximal, characteristic, complemented
13#1802C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21#546—24no—
22#546—24no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121—21no—
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic, complemented
31C2×#120—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic, complemented
35C22×C22⋊C4—21no—
36#463—24no—
37#449—24no—
38#447—24no—
39#442—24no—
40C2×#106C2211yescharacteristic, complemented
41C2×C4⋊C8C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic, complemented
55C2×C22⋊C8C2211yescharacteristic, complemented
56C2×#97C2211yescharacteristic, complemented
57#394—21no—
58#394—21no—
59#384C2211yescharacteristic, complemented
60#380—21no—
61#380C2211yescharacteristic
62#380C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64#340C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—41no—
96C2×D8—22no—
97C2×D8—22no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—41no—
101C2×D8D414yescomplemented
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106#124—22no—
107#124D414yescomplemented
108#123—28no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111#121—28no—
112#121—22no—
113#121—22no—
114#121—22no—
115#121—22no—
116#121—41no—
117#121—41no—
118#121—44no—
119#120—28no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4D414yescomplemented
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4D411yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C22⋊C4—28no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4D411yescharacteristic
146C2×C22⋊C4D411yescharacteristic
147C2×C42C2311yescharacteristic
148#106—22no—
149C4⋊C8—22no—
150#103—22no—
151#103—22no—
152#103—41no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—41no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—22no—
165#95—21no—
166#95—21no—
167C24—21no—
168C24—24no—
169C24—21no—
170C24—21no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24—42no—
175C24—81no—
176C24—81no—
177C24C2×D411yescharacteristic
178C24C2×D411yescharacteristic
179#41—44no—
180#41—44no—
181#41—44no—
182#41—44no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8C2×D412yes—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
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—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4C2×D412yes—
225C2×D4C2×D412yes—
226C2×D4C2×D412yes—
227C2×D4C2×D412yes—
228C2×D4#4112yes—
229C22×C4—24no—
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—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2411yescharacteristic
260SD16—44no—
261D8—28no—
262D8—44no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8C2×D412yes—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C4⋊C4C2×D412yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—22no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—22no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42C2×D412yes—
300C42C2×D412yes—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—22no—
305C23—21no—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—48no—
320C23—48no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23AGL(1,ℤ/8)12yes—
343C23C22×D411yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348D4—22no—
349D4—22no—
350D4—22no—
351D4—22no—
352D4—22no—
353D4—22no—
354D4—22no—
355D4—44no—
356D4—48no—
357D4—48no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—84no—
363D4—84no—
364D4—84no—
365D4#12114yescomplemented
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—48no—
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—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4#12012yes—
403C2×C4#12012yes—
404C2×C4C22×D411yescharacteristic
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D411yescharacteristic
407C2×C4C2×#4111yescharacteristic
408C2×C4H(5,𝔽2)11yescharacteristic
409C2×C4H(5,𝔽2)11yescharacteristic
410C8—22no—
411C8—41no—
412C8—41no—
413C8—41no—
414C22—24no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—41no—
425C22—42no—
426C22—41no—
427C22—42no—
428C22—41no—
429C22—42no—
430C22—42no—
431C22—41no—
432C22—42no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—816no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C2×#12011yescharacteristic
451C22D4211yescharacteristic
452C22#54611yescharacteristic
453C22C2×AGL(1,ℤ/8)11yescharacteristic
454C22#57511yescharacteristic
455C22#57611yescharacteristic
456C22#57711yescharacteristic
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—42no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4C2×#12111yescharacteristic
467C4#53511yescharacteristic
468C4D4211yescharacteristic
469C4#54611yescharacteristic
470C2—42no—
471C2—44no—
472C2—81no—
473C2—82no—
474C2—82no—
475C2#240811yescharacteristic
476C2#300611yescharacteristic
477C2#301211yescharacteristic
478C2#326411yescharacteristic
479C2#326711yescharacteristic
480C2#330611yescharacteristic
481C2#331911yescharacteristic
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×#463C2semidirect
3. C2×#449C2semidirect
4. C2×#447C2semidirect
5. C2×#442C2semidirect
6. #2537C2semidirect
7. #2035C2semidirect
8. #2034C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1959C2semidirect
12. #1955C2semidirect
13. #1802C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
18. C22×SD16C22semidirect
30. C2×#121C22semidirect
34. C2×C4×D4C22semidirect
40. C2×#106C22semidirect
41. C2×C4⋊C8C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
51. C2×#102C22semidirect
54. C2×C22⋊C8C22semidirect
55. C2×C22⋊C8C22semidirect
56. C2×#97C22semidirect
59. #384C22semidirect
62. #380C22semidirect
63. #340C22semidirect
64. #340C22semidirect
101. C2×D8D4semidirect
107. #124D4semidirect
125. C4×D4D4semidirect
365. D4#121semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4632
C2×#4492
C2×#4472
C2×#4422
#25372
#20352
#20342
#20002
#19992
#19592
#19552
#18022
#17882
#17862