Finite Group Databasebeta

← #18794

Subgroups — #18794

Order 256.

607 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#41C211yesmaximal, characteristic, complemented
2#3207C211yesmaximal, characteristic, complemented
3#3109C211yesmaximal, characteristic, complemented
4#3106C211yesmaximal, characteristic, complemented
5#1637C211yesmaximal, characteristic, complemented
6#1635C211yesmaximal, characteristic, complemented
7#1631C211yesmaximal, characteristic, complemented
8#1629C211yesmaximal, characteristic, complemented
9#1627C211yesmaximal, characteristic, complemented
10#1625C211yesmaximal, characteristic, complemented
11#1621C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1553C211yesmaximal, characteristic, complemented
14#1488C211yesmaximal, characteristic, complemented
15#1478C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17#550C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#548—21no—
20#547—21no—
21#547C2211yescharacteristic
22#546—22no—
23#546—21no—
24D42—21no—
25D42C2211yescharacteristic
26#532—21no—
27#532C2211yescharacteristic
28#532C2211yescharacteristic, complemented
29#525—22no—
30#525—21no—
31C4×#41C2211yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4C2211yescharacteristic
34#493C2211yescharacteristic, complemented
35#492C2211yescharacteristic, complemented
36#487C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#482—21no—
39#481—21no—
40#477C2211yescharacteristic, complemented
41#476C2211yescharacteristic
42#476C2211yescharacteristic, complemented
43#476C2211yescharacteristic, complemented
44#463—22no—
45#460C2211yescharacteristic, complemented
46#459C2211yescharacteristic
47#459C2211yescharacteristic, complemented
48#459C2211yescharacteristic, complemented
49#449—21no—
50#449—22no—
51#447—21no—
52#442—21no—
53#442—21no—
54C4×SD16—21no—
55C4×D8—21no—
56#432C2211yescharacteristic, complemented
57#431C2211yescharacteristic, complemented
58#431C2211yescharacteristic, 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—
87C2×SD16—21no—
88C2×SD16—22no—
89C2×D8—21no—
90C2×D8—21no—
91C2×D8—21no—
92C2×D8—22no—
93C4⋊Q8—21no—
94C4⋊Q8C2311yescharacteristic
95C4⋊Q8C2311yescharacteristic
96#127—21no—
97#127C2311yescharacteristic
98#127C2311yescharacteristic
99#124—22no—
100#124—21no—
101#124—21no—
102#123—22no—
103#123—22no—
104#123—22no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C22⋊Q8—21no—
109#121—22no—
110#121—22no—
111#121—22no—
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121D412yes—
120#120—22no—
121#120—22no—
122#120—22no—
123C4×Q8D411yescharacteristic
124C4×Q8C2311yescharacteristic
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4D411yescharacteristic
138C4×D4D411yescharacteristic
139C4×D4D411yescharacteristic
140C4×D4D412yes—
141C4×D4C2311yescharacteristic
142C4×D4C2311yescharacteristic
143C4×D4C2311yescharacteristic
144#117—21no—
145#117D411yescharacteristic
146C2×C4⋊C4—21no—
147C2×C4⋊C4D411yescharacteristic
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C42—21no—
151C2×C42C2311yescharacteristic
152#107—21no—
153#106—21no—
154#106—21no—
155#106—21no—
156C4⋊C8—21no—
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161#103—21no—
162#103—22no—
163#102—21no—
164#102—21no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—22no—
171C22⋊C8—21no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174#97—21no—
175C4×C8—21no—
176C4×C8C2311yescharacteristic
177C4×C8C2311yescharacteristic
178C24—22no—
179C24—21no—
180#41—22no—
181#41—22no—
182#41—48no—
183#41—42no—
184#41—42no—
185#41—48no—
186#41—42no—
187#41—42no—
188#41—42no—
189#41—42no—
190#41—82no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8C2×D411yescharacteristic
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—48no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4C2×D411yescharacteristic
219C2×D4C2×D411yescharacteristic
220C2×D4C2×D411yescharacteristic
221C2×D4C2×D412yes—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241SD16—42no—
242SD16—82no—
243D8—42no—
244D8—42no—
245D8—42no—
246D8—82no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4C2×D412yes—
262C4⋊C4C2×D411yescharacteristic
263C4⋊C4C2×D411yescharacteristic
264C4⋊C4C2×D411yescharacteristic
265C4⋊C4C2×D411yescharacteristic
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—22no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—22no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C2×D411yescharacteristic
285C42C2×D411yescharacteristic
286C42C2×D411yescharacteristic
287C42C2411yescharacteristic
288C23—21no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—44no—
296C23—41no—
297C23—81no—
298C23—82no—
299C23#12011yescharacteristic
300Q8—21no—
301Q8—44no—
302Q8—42no—
303Q8—41no—
304Q8—41no—
305Q8AGL(1,ℤ/8)12yes—
306D4—24no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—42no—
313D4—44no—
314D4—42no—
315D4—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—82no—
328D4—82no—
329D4#13512yes—
330D4AGL(1,ℤ/8)12yes—
331C2×C4—24no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4#12011yescharacteristic
360C2×C4#12011yescharacteristic
361C2×C4#12011yescharacteristic
362C2×C4C22×D411yescharacteristic
363C2×C4C22×D411yescharacteristic
364C2×C4C22×D411yescharacteristic
365C8—41no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—41no—
370C8—41no—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—84no—
387C22—88no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C2×#12011yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4C2×#13511yescharacteristic
404C4C2×AGL(1,ℤ/8)11yescharacteristic
405C4C2×AGL(1,ℤ/8)11yescharacteristic
406C4#57511yescharacteristic
407C2—41no—
408C2—42no—
409C2—42no—
410C2—81no—
411C2—82no—
412C2—161no—
413C2#300211yescharacteristic
414C2#300311yescharacteristic
415C2#302111yescharacteristic
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. #3207C2semidirect
3. #3109C2semidirect
4. #3106C2semidirect
5. #1637C2semidirect
6. #1635C2semidirect
7. #1631C2semidirect
8. #1629C2semidirect
9. #1627C2semidirect
10. #1625C2semidirect
11. #1621C2semidirect
12. #1619C2semidirect
13. #1553C2semidirect
14. #1488C2semidirect
15. #1478C2semidirect
28. #532C22semidirect
34. #493C22semidirect
35. #492C22semidirect
36. #487C22semidirect
37. #486C22semidirect
40. #477C22semidirect
42. #476C22semidirect
43. #476C22semidirect
45. #460C22semidirect
47. #459C22semidirect
48. #459C22semidirect
56. #432C22semidirect
57. #431C22semidirect
58. #431C22semidirect
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

Maximal subgroups

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

subgroupindex
D4×#412
#32072
#31092
#31062
#16372
#16352
#16312
#16292
#16272
#16252
#16212
#16192
#15532
#14882
#14782