Finite Group Databasebeta

← #18554

Subgroups — #18554

Order 256.

622 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#41C211yesmaximal, characteristic, complemented
2#3184C211yesmaximal, characteristic, complemented
3#3115C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1651C211yesmaximal, characteristic, complemented
7#1642C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1627C211yesmaximal, characteristic, complemented
10#1625C211yesmaximal, characteristic, complemented
11#1621C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1557C211yesmaximal, characteristic, complemented
14#1490C211yesmaximal, characteristic, complemented
15#1489C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C2×AGL(1,ℤ/8)—21no—
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#548—21no—
21#547—21no—
22#547C2211yescharacteristic
23#546—22no—
24#546—21no—
25D42—21no—
26D42C2211yescharacteristic
27#532—21no—
28#532C2211yescharacteristic
29#532C2211yescharacteristic, complemented
30#525—22no—
31#525—21no—
32C4×#41C2211yescharacteristic
33C2×C4×D4—21no—
34C2×C4×D4C2211yescharacteristic
35#498C2211yescharacteristic, complemented
36#493C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#486C2211yescharacteristic, complemented
39#482—21no—
40#475C2211yescharacteristic, complemented
41#474C2211yescharacteristic, complemented
42#469—22no—
43#463—21no—
44#463—21no—
45#461C2211yescharacteristic
46#461C2211yescharacteristic, complemented
47#460C2211yescharacteristic, complemented
48#459C2211yescharacteristic
49#459C2211yescharacteristic, complemented
50#459C2211yescharacteristic, complemented
51#449—21no—
52#442—21no—
53#442—22no—
54#440—21no—
55#432C2211yescharacteristic, complemented
56#423C2211yescharacteristic, complemented
57#417—21no—
58C4×M16C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62#331C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—21no—
74C2×#41—21no—
75C2×#41—21no—
76C2×#41—21no—
77C2×#41—44no—
78C2×#41—44no—
79C2×#41D411yescharacteristic
80C22×Q8—21no—
81C22×D4—22no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4D411yescharacteristic
85C23×C4—22no—
86C23×C4—21no—
87AGL(1,ℤ/8)—44no—
88C2×SD16—21no—
89C2×SD16—21no—
90C2×SD16—21no—
91C2×D8—21no—
92C2×D8—21no—
93C2×D8—21no—
94C2×D8—22no—
95C2×M16—21no—
96C2×M16—21no—
97C4⋊Q8—21no—
98C4⋊Q8C2311yescharacteristic
99C4⋊Q8C2311yescharacteristic
100#127—21no—
101#127C2311yescharacteristic
102#127C2311yescharacteristic
103#124—22no—
104#124—21no—
105#124—21no—
106#123—22no—
107#123—22no—
108#123—22no—
109C22⋊Q8—22no—
110C22⋊Q8—21no—
111C22⋊Q8—22no—
112C22⋊Q8—21no—
113#121—22no—
114#121—22no—
115#121—22no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121D412yes—
124#120—22no—
125#120—22no—
126#120—22no—
127C4×Q8D411yescharacteristic
128C4×Q8C2311yescharacteristic
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4D411yescharacteristic
142C4×D4D411yescharacteristic
143C4×D4D411yescharacteristic
144C4×D4D412yes—
145C4×D4C2311yescharacteristic
146C4×D4C2311yescharacteristic
147C4×D4C2311yescharacteristic
148#117—21no—
149#117D411yescharacteristic
150C2×C4⋊C4—21no—
151C2×C4⋊C4D411yescharacteristic
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C42—21no—
155C2×C42C2311yescharacteristic
156#107—21no—
157#106—22no—
158#106—21no—
159C4⋊C8—21no—
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163C4⋊C8C2311yescharacteristic
164#103—21no—
165#103—21no—
166#103—21no—
167#102—21no—
168#102—22no—
169#102—21no—
170#102—21no—
171#102—21no—
172#102—21no—
173#102—22no—
174C22⋊C8—21no—
175#97—21no—
176#97—21no—
177C4×C8C2311yescharacteristic
178C4×C8C2311yescharacteristic
179C24—22no—
180C24—21no—
181#41—22no—
182#41—22no—
183#41—48no—
184#41—42no—
185#41—42no—
186#41—42no—
187#41—42no—
188#41—48no—
189#41—42no—
190#41—42no—
191#41—42no—
192#41—42no—
193C2×Q8—22no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8C2×D411yescharacteristic
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—48no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4C2×D411yescharacteristic
221C2×D4C2×D411yescharacteristic
222C2×D4C2×D411yescharacteristic
223C2×D4C2×D412yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243SD16—42no—
244SD16—42no—
245SD16—42no—
246D8—44no—
247D8—42no—
248D8—42no—
249D8—42no—
250M16—42no—
251M16—42no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4C2×D411yescharacteristic
267C4⋊C4C2×D411yescharacteristic
268C4⋊C4C2×D411yescharacteristic
269C4⋊C4C2×D412yes—
270C4⋊C4C2×D411yescharacteristic
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—22no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—22no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42C2×D411yescharacteristic
290C42C2×D411yescharacteristic
291C42C2×D411yescharacteristic
292C42C2411yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—42no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—41no—
302C23—81no—
303C23—82no—
304C23#12011yescharacteristic
305Q8—21no—
306Q8—44no—
307Q8—41no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311Q8AGL(1,ℤ/8)12yescomplemented
312D4—24no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—44no—
320D4—42no—
321D4—44no—
322D4—44no—
323D4—44no—
324D4—44no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—82no—
334D4—82no—
335D4AGL(1,ℤ/8)12yes—
336D4AGL(1,ℤ/8)12yescomplemented
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4#12011yescharacteristic
366C2×C4#12011yescharacteristic
367C2×C4#12011yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D411yescharacteristic
371C8—22no—
372C8—41no—
373C8—41no—
374C8—41no—
375C8—41no—
376C8—41no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—84no—
393C22—88no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×#12011yescharacteristic
398C4—21no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4C2×AGL(1,ℤ/8)11yescharacteristic
410C4C2×AGL(1,ℤ/8)11yescharacteristic
411C4C2×AGL(1,ℤ/8)11yescharacteristic
412C4#57511yescharacteristic
413C2—41no—
414C2—42no—
415C2—42no—
416C2—81no—
417C2—82no—
418C2—161no—
419C2#300311yescharacteristic
420C2#300611yescharacteristic
421C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. D4×#41C2semidirect
2. #3184C2semidirect
3. #3115C2semidirect
4. #3036C2semidirect
5. #1654C2semidirect
6. #1651C2semidirect
7. #1642C2semidirect
8. #1639C2semidirect
9. #1627C2semidirect
10. #1625C2semidirect
11. #1621C2semidirect
12. #1619C2semidirect
13. #1557C2semidirect
14. #1490C2semidirect
15. #1489C2semidirect
29. #532C22semidirect
35. #498C22semidirect
36. #493C22semidirect
37. #486C22semidirect
38. #486C22semidirect
40. #475C22semidirect
41. #474C22semidirect
46. #461C22semidirect
47. #460C22semidirect
49. #459C22semidirect
50. #459C22semidirect
55. #432C22semidirect
56. #423C22semidirect
58. C4×M16C22semidirect
59. #332C22semidirect
60. #332C22semidirect
61. #331C22semidirect
62. #331C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect
311. Q8AGL(1,ℤ/8)semidirect
336. D4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
D4×#412
#31842
#31152
#30362
#16542
#16512
#16422
#16392
#16272
#16252
#16212
#16192
#15572
#14902
#14892