Finite Group Databasebeta

← #18810

Subgroups — #18810

Order 256.

521 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2#3251C211yesmaximal, characteristic, complemented
3#3207C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1646C211yesmaximal, characteristic, complemented
6#1643C211yesmaximal, characteristic, complemented
7#1642C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1624C211yesmaximal, characteristic, complemented
10#1622C211yesmaximal, characteristic, complemented
11#1621C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1560C211yesmaximal, characteristic, complemented
14#1495C211yesmaximal, characteristic, complemented
15#1486C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#553—21no—
20#553—21no—
21#550C2211yescharacteristic
22#546—22no—
23#546—21no—
24D42—21no—
25D42C2211yescharacteristic
26#540—21no—
27#538—22no—
28#532C2211yescharacteristic
29#532C2211yescharacteristic, complemented
30#529C2211yescharacteristic
31#525—21no—
32C2×#121—21no—
33#518—21no—
34C2×C4×D4C2211yescharacteristic
35#493C2211yescharacteristic, complemented
36#492C2211yescharacteristic, complemented
37#487C2211yescharacteristic, complemented
38#486C2211yescharacteristic, complemented
39#478—21no—
40#475C2211yescharacteristic, complemented
41#474C2211yescharacteristic, complemented
42#469—21no—
43#468—21no—
44#463—22no—
45#463—21no—
46#463—21no—
47#463—21no—
48#461C2211yescharacteristic
49#461C2211yescharacteristic, complemented
50#460C2211yescharacteristic, complemented
51#459C2211yescharacteristic
52#459C2211yescharacteristic, complemented
53#459C2211yescharacteristic, complemented
54#449—22no—
55#431C2211yescharacteristic, complemented
56#423C2211yescharacteristic, complemented
57#423C2211yescharacteristic, complemented
58#417—21no—
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, 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—21no—
72C2×#41—21no—
73C2×#41D412yes—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4D411yescharacteristic
78C23×C4—21no—
79AGL(1,ℤ/8)—44no—
80C2×SD16—21no—
81C2×SD16—21no—
82C2×SD16—22no—
83C2×SD16—21no—
84C2×D8—21no—
85C2×D8—21no—
86C2×D8—22no—
87C2×D8—21no—
88C2×M16—21no—
89C2×M16—21no—
90C4⋊Q8C2311yescharacteristic
91C4⋊Q8C2311yescharacteristic
92#127—21no—
93#127C2311yescharacteristic
94#127C2311yescharacteristic
95#126—22no—
96#126—21no—
97#125—21no—
98#125D411yescharacteristic
99#124—22no—
100#124—21no—
101#124D411yescharacteristic
102#124D412yes—
103#123—22no—
104#123—22no—
105#123—22no—
106#123—21no—
107C22⋊Q8—21no—
108C22⋊Q8—22no—
109C22⋊Q8D411yescharacteristic
110#121—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#121—21no—
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#121—21no—
124#121D411yescharacteristic
125#120—22no—
126#120—22no—
127#120—22no—
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×D4C2311yescharacteristic
144C4×D4C2311yescharacteristic
145C4×D4C2311yescharacteristic
146#117—21no—
147#117—21no—
148#117—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4D411yescharacteristic
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C42C2311yescharacteristic
155#107—21no—
156#106—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163C4⋊C8C2311yescharacteristic
164#103—22no—
165#103—21no—
166#103—21no—
167#103—21no—
168#102—22no—
169#102—21no—
170#102—21no—
171#102—21no—
172#102—21no—
173#102—21no—
174C22⋊C8—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177C24—22no—
178C24—21no—
179#41—42no—
180#41—42no—
181#41—81no—
182#41—81no—
183#41—82no—
184C2×Q8—21no—
185C2×Q8C2×D411yescharacteristic
186C2×Q8C2×D412yes—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—84no—
208C2×D4C2×D411yescharacteristic
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D411yescharacteristic
211C2×D4C2×D412yes—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227SD16—42no—
228SD16—42no—
229SD16—81no—
230SD16—82no—
231D8—42no—
232D8—42no—
233D8—81no—
234D8—82no—
235M16—42no—
236M16—81no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4C2×D411yescharacteristic
255C4⋊C4C2×D411yescharacteristic
256C4⋊C4C2×D411yescharacteristic
257C4⋊C4C2×D411yescharacteristic
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—22no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—22no—
277C42—21no—
278C42C2×D411yescharacteristic
279C42C2×D411yescharacteristic
280C42C2×D411yescharacteristic
281C42C2411yescharacteristic
282C23—21no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—41no—
290C23—81no—
291C23—82no—
292C23—82no—
293C23#12011yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—41no—
298Q8—41no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—44no—
308D4—42no—
309D4—44no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4—82no—
322D4—82no—
323D4AGL(1,ℤ/8)12yes—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#12011yescharacteristic
350C2×C4#12011yescharacteristic
351C2×C4#12011yescharacteristic
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D411yescharacteristic
354C2×C4C22×D411yescharacteristic
355C8—41no—
356C8—41no—
357C8—41no—
358C8—41no—
359C8—41no—
360C8—41no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—41no—
371C22—42no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—84no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—164no—
381C22C2×#12011yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—81no—
393C4C2×AGL(1,ℤ/8)11yescharacteristic
394C4#57511yescharacteristic
395C4#57611yescharacteristic
396C4#57711yescharacteristic
397C2—41no—
398C2—42no—
399C2—81no—
400C2—82no—
401C2—81no—
402C2—161no—
403C2#300611yescharacteristic
404C2#301211yescharacteristic
405C2#301911yescharacteristic
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. #3519C2semidirect
2. #3251C2semidirect
3. #3207C2semidirect
4. #3036C2semidirect
5. #1646C2semidirect
6. #1643C2semidirect
7. #1642C2semidirect
8. #1639C2semidirect
9. #1624C2semidirect
10. #1622C2semidirect
11. #1621C2semidirect
12. #1619C2semidirect
13. #1560C2semidirect
14. #1495C2semidirect
15. #1486C2semidirect
29. #532C22semidirect
35. #493C22semidirect
36. #492C22semidirect
37. #487C22semidirect
38. #486C22semidirect
40. #475C22semidirect
41. #474C22semidirect
49. #461C22semidirect
50. #460C22semidirect
52. #459C22semidirect
53. #459C22semidirect
55. #431C22semidirect
56. #423C22semidirect
57. #423C22semidirect
59. #333C22semidirect
60. #333C22semidirect
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
#35192
#32512
#32072
#30362
#16462
#16432
#16422
#16392
#16242
#16222
#16212
#16192
#15602
#14952
#14862