Finite Group Databasebeta

← #25192

Subgroups — #25192

Order 256.

527 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
1#3062C211yesmaximal, characteristic, complemented
2#3045C211yesmaximal, characteristic, complemented
3C2×#453C211yesmaximal, characteristic, complemented
4#3012C211yesmaximal, characteristic, complemented
5C2×#444C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic, complemented
7#2063C211yesmaximal, characteristic, complemented
8#2051C211yesmaximal, characteristic, complemented
9#2040C211yesmaximal, characteristic, complemented
10#2002C211yesmaximal, characteristic, complemented
11#1961C211yesmaximal, characteristic, complemented
12#1938C211yesmaximal, characteristic, complemented
13#1907C211yesmaximal, characteristic, complemented
14#1907C211yesmaximal, characteristic, complemented
15#1860C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×#135C2211yescharacteristic
21C2×#131C2211yescharacteristic
22C2×#131C2211yescharacteristic, complemented
23#537C2211yescharacteristic
24#535C2211yescharacteristic, complemented
25C2×#127C2211yescharacteristic
26#520C2211yescharacteristic
27#469—21no—
28#468—21no—
29#466—21no—
30#465—21no—
31#464—21no—
32#463—21no—
33#460—21no—
34#459—21no—
35#453—24no—
36#452—21no—
37#450—21no—
38#449—21no—
39#447—21no—
40#444—24no—
41C2×#108C2211yescharacteristic, complemented
42#428C2211yescharacteristic, complemented
43#424C2211yescharacteristic, complemented
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic, complemented
47#418—21no—
48#418C2211yescharacteristic
49#417—21no—
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#417C2211yescharacteristic, complemented
53#416—21no—
54#416C2211yescharacteristic
55C2×#102—21no—
56C2×#102C2211yescharacteristic
57C2×#102C2211yescharacteristic
58C2×#102C2211yescharacteristic, complemented
59C2×#102C2211yescharacteristic, complemented
60C2×#100C2211yescharacteristic
61C2×#100C2211yescharacteristic, complemented
62#408C2211yescharacteristic
63C2×#97C2211yescharacteristic, complemented
64#382C2211yescharacteristic, complemented
65#343C2211yescharacteristic, complemented
66#343C2211yescharacteristic, complemented
67#341C2211yescharacteristic, complemented
68#339C2211yescharacteristic, complemented
69H(5,𝔽2)—22no—
70H(5,𝔽2)—22no—
71H(5,𝔽2)—22no—
72H(5,𝔽2)—41no—
73C2×#41—21no—
74C2×#41—21no—
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C2×#41C2311yescharacteristic
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84AGL(1,ℤ/8)—22no—
85AGL(1,ℤ/8)—22no—
86AGL(1,ℤ/8)—22no—
87AGL(1,ℤ/8)—22no—
88AGL(1,ℤ/8)—41no—
89AGL(1,ℤ/8)—41no—
90#135—22no—
91#135—22no—
92C2×Q16D411yescharacteristic, complemented
93C2×SD16—21no—
94C2×SD16—21no—
95C2×SD16—21no—
96C2×SD16D411yescharacteristic, complemented
97C2×SD16D411yescharacteristic, complemented
98C2×D8—21no—
99C2×D8—21no—
100C2×D8—21no—
101C2×D8D411yescharacteristic, complemented
102#131—22no—
103#131—22no—
104#131—41no—
105#131—41no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C2×M16C2311yescharacteristic
112C22×C8—21no—
113C22×C8—21no—
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116#127—21no—
117#127—22no—
118#127—22no—
119#124—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122#121—21no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—41no—
127#120—41no—
128C4×Q8—21no—
129C4×D4—21no—
130C4×D4—41no—
131#117—21no—
132#117C2311yescharacteristic
133C2×C4⋊C4—21no—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C42C2311yescharacteristic
137#108—22no—
138#107—21no—
139#106—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142#104—22no—
143#104—22no—
144#104D414yescomplemented
145#103—21no—
146#103—21no—
147#103—21no—
148#103D411yescharacteristic, complemented
149#103D411yescharacteristic, complemented
150#102—21no—
151#102—21no—
152#102—21no—
153#102—21no—
154#102—21no—
155#102—21no—
156#102—21no—
157#102—21no—
158#102—21no—
159#102—21no—
160#102—21no—
161#102—21no—
162#102—21no—
163#102D411yescharacteristic, complemented
164#102D411yescharacteristic, complemented
165#100—22no—
166#100—41no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169#97—22no—
170#95—41no—
171C24—21no—
172C24—21no—
173C24—41no—
174#41—21no—
175#41—21no—
176#41—24no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—21no—
185#41—42no—
186#41—42no—
187#41—42no—
188#41C2×D412yes—
189#41#4112yes—
190C2×Q8C2×D411yescharacteristic
191C2×Q8C2×D411yescharacteristic
192C2×Q8C2×D411yescharacteristic
193C2×D4—21no—
194C2×D4—24no—
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—21no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—44no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—42no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4C2×D411yescharacteristic
222C2×D4C2×D411yescharacteristic
223C2×D4C2×D411yescharacteristic
224C2×D4#4112yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4C2411yescharacteristic
234Q16—22no—
235SD16—22no—
236SD16—22no—
237SD16—42no—
238SD16—42no—
239SD16—42no—
240D8—22no—
241D8—42no—
242D8—42no—
243D8—42no—
244M16—21no—
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—21no—
250M16—21no—
251M16—42no—
252M16—42no—
253M16C2×D412yes—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8C2×D411yescharacteristic
265C2×C8C2×D411yescharacteristic
266C2×C8C2×D411yescharacteristic
267C2×C8C2×D411yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4C2×D411yescharacteristic
275C4⋊C4C2×D411yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C42—21no—
283C42—21no—
284C42—41no—
285C42C2×D412yes—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—44no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—161no—
301C23—161no—
302C23H(5,𝔽2)11yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8#12112yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—22no—
318D4—44no—
319D4—44no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—88no—
331D4—161no—
332D4—161no—
333D4—161no—
334D4#12112yes—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—161no—
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D411yescharacteristic
357C2×C4C22×D411yescharacteristic
358C2×C4C2×#4111yescharacteristic
359C2×C4C2×#4111yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—22no—
366C8—41no—
367C8—41no—
368C8—41no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—82no—
382C22—82no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—161no—
389C22—161no—
390C22—164no—
391C22#53811yescharacteristic
392C22D4211yescharacteristic
393C22#54611yescharacteristic
394C4—22no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—81no—
402C4C2×#12111yescharacteristic
403C4#52511yescharacteristic
404C4D4211yescharacteristic
405C4#54611yescharacteristic
406C2—21no—
407C2—21no—
408C2—42no—
409C2—81no—
410C2—81no—
411C2—82no—
412C2—161no—
413C2#242411yescharacteristic
414C2#328811yescharacteristic
415C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3062C2semidirect
2. #3045C2semidirect
3. C2×#453C2semidirect
4. #3012C2semidirect
5. C2×#444C2semidirect
6. #2897C2semidirect
7. #2063C2semidirect
8. #2051C2semidirect
9. #2040C2semidirect
10. #2002C2semidirect
11. #1961C2semidirect
12. #1938C2semidirect
13. #1907C2semidirect
14. #1907C2semidirect
15. #1860C2semidirect
22. C2×#131C22semidirect
24. #535C22semidirect
41. C2×#108C22semidirect
42. #428C22semidirect
43. #424C22semidirect
46. C2×#104C22semidirect
52. #417C22semidirect
58. C2×#102C22semidirect
59. C2×#102C22semidirect
61. C2×#100C22semidirect
63. C2×#97C22semidirect
64. #382C22semidirect
65. #343C22semidirect
66. #343C22semidirect
67. #341C22semidirect
68. #339C22semidirect
92. C2×Q16D4semidirect
96. C2×SD16D4semidirect
97. C2×SD16D4semidirect
101. C2×D8D4semidirect
144. #104D4semidirect
148. #103D4semidirect
149. #103D4semidirect
163. #102D4semidirect
164. #102D4semidirect

Maximal subgroups

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

subgroupindex
#30622
#30452
C2×#4532
#30122
C2×#4442
#28972
#20632
#20512
#20402
#20022
#19612
#19382
#19072
#18602