Finite Group Databasebeta

← #24277

Subgroups — #24277

Order 256.

804 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4C2×#436C211yesmaximal, characteristic, complemented
5#2408C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C2×AGL(1,ℤ/8)—24no—
19C2×AGL(1,ℤ/8)—24no—
20C2×AGL(1,ℤ/8)—24no—
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×D8C2211yescharacteristic
24C22×D8C2211yescharacteristic
25C22×M16C2211yescharacteristic
26C22×M16C2211yescharacteristic, complemented
27C2×C22⋊Q8C2211yescharacteristic, complemented
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic, complemented
31C2×#120—21no—
32C2×C4×D4C2211yescharacteristic
33#469—24no—
34#468—24no—
35#436—24no—
36C2×#107C2211yescharacteristic, complemented
37C2×#106C2211yescharacteristic, complemented
38C2×C4⋊C8C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic, complemented
49C2×C4×C8C2211yescharacteristic, complemented
50#394—21no—
51#394—21no—
52#392—21no—
53#392—21no—
54#391C2211yescharacteristic, complemented
55#390C2211yescharacteristic
56#390C2211yescharacteristic
57#390C2211yescharacteristic, complemented
58#386—21no—
59#385—21no—
60#380C2211yescharacteristic
61#380C2211yescharacteristic
62#379—21no—
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—22no—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91AGL(1,ℤ/8)—232no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16D414yescomplemented
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—22no—
101C2×D8—22no—
102C2×D8—22no—
103C2×D8D414yescomplemented
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16D414yescomplemented
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114C22⋊Q8—22no—
115C22⋊Q8—41no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—41no—
120#121—41no—
121#121—41no—
122#120—44no—
123C4×D4—22no—
124C4×D4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4D411yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
144C2×C22⋊C4D411yescharacteristic
145C2×C42C2311yescharacteristic
146#107—22no—
147#106—22no—
148C4⋊C8—22no—
149#103—22no—
150#103—41no—
151#102—22no—
152#102—22no—
153#102—22no—
154#102—41no—
155#102—41no—
156#102—41no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C4×C8—22no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—42no—
168C24—42no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—81no—
173C24—81no—
174C24C2×D411yescharacteristic
175C24C2×D411yescharacteristic
176#41—28no—
177#41—44no—
178#41—44no—
179#41—44no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8C2×D412yes—
186C2×D4—28no—
187C2×D4—22no—
188C2×D4—22no—
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—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4C2×D412yes—
224C2×D4C2×D412yes—
225C2×D4C2×D412yes—
226C22×C4—21no—
227C22×C4—21no—
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—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D412yes—
248C22×C4C2×D411yescharacteristic
249C22×C4#4112yes—
250C22×C4C2411yescharacteristic
251SD16—28no—
252SD16—28no—
253D8—28no—
254D8—28no—
255M16—28no—
256M16—44no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C2×D412yes—
266C2×C8C2×D412yes—
267C2×C8C2×D412yes—
268C2×C8C2×D412yes—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—21no—
294C42—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—48no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23—84no—
334C23—84no—
335C23—84no—
336C23#12112yes—
337C23C22×D411yescharacteristic
338Q8—22no—
339Q8—22no—
340Q8—41no—
341Q8—41no—
342D4—22no—
343D4—22no—
344D4—22no—
345D4—22no—
346D4—22no—
347D4—22no—
348D4—44no—
349D4—48no—
350D4—48no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—84no—
358D4—84no—
359D4—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—48no—
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—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4#12112yes—
395C2×C4AGL(1,ℤ/8)12yescomplemented
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C2×C4C2×#4111yescharacteristic
400C2×C4H(5,𝔽2)11yescharacteristic
401C2×C4H(5,𝔽2)11yescharacteristic
402C8—22no—
403C8—41no—
404C8—41no—
405C8#12014yescomplemented
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—42no—
416C22—42no—
417C22—41no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—816no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C2×#12111yescharacteristic
443C22D4211yescharacteristic
444C22D4211yescharacteristic
445C22C2×AGL(1,ℤ/8)11yescharacteristic
446C22#57511yescharacteristic
447C22#57611yescharacteristic
448C22#57711yescharacteristic
449C4—22no—
450C4—22no—
451C4—41no—
452C4—42no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4C2×#12011yescharacteristic
459C4#53511yescharacteristic
460C4#54611yescharacteristic
461C4#54611yescharacteristic
462C2—42no—
463C2—81no—
464C2—82no—
465C2—82no—
466C2—82no—
467C2#240811yescharacteristic
468C2#305511yescharacteristic
469C2#306211yescharacteristic
470C2#327211yescharacteristic
471C2#327311yescharacteristic
472C2#327411yescharacteristic
473C2#327511yescharacteristic
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. C22×AGL(1,ℤ/8)C2semidirect
2. C2×#469C2semidirect
3. C2×#468C2semidirect
4. C2×#436C2semidirect
5. #2408C2semidirect
6. #2407C2semidirect
7. #2055C2semidirect
8. #2054C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1970C2semidirect
12. #1968C2semidirect
13. #1809C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
26. C22×M16C22semidirect
27. C2×C22⋊Q8C22semidirect
30. C2×#121C22semidirect
36. C2×#107C22semidirect
37. C2×#106C22semidirect
38. C2×C4⋊C8C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
48. C2×C22⋊C8C22semidirect
49. C2×C4×C8C22semidirect
54. #391C22semidirect
57. #390C22semidirect
63. #340C22semidirect
64. #340C22semidirect
97. C2×SD16D4semidirect
103. C2×D8D4semidirect
109. C2×M16D4semidirect
395. C2×C4AGL(1,ℤ/8)semidirect
405. C8#120semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#4692
C2×#4682
C2×#4362
#24082
#24072
#20552
#20542
#20002
#19992
#19702
#19682
#18092
#17882
#17862