Finite Group Databasebeta

← #25191

Subgroups — #25191

Order 256.

505 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3063C211yesmaximal, characteristic, complemented
2#3046C211yesmaximal, characteristic, complemented
3C2×#455C211yesmaximal, characteristic, complemented
4#3012C211yesmaximal, characteristic
5C2×#443C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic, complemented
7#2063C211yesmaximal, characteristic, complemented
8#2052C211yesmaximal, characteristic, complemented
9#2040C211yesmaximal, characteristic, complemented
10#2003C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1937C211yesmaximal, characteristic, complemented
13#1908C211yesmaximal, characteristic, complemented
14#1907C211yesmaximal, characteristic, complemented
15#1860C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×#135C2211yescharacteristic
21C2×#131C2211yescharacteristic
22C2×#131C2211yescharacteristic, complemented
23#537C2211yescharacteristic
24#536C2211yescharacteristic, complemented
25C2×#124C2211yescharacteristic
26#520C2211yescharacteristic
27#470—21no—
28#468—21no—
29#467—21no—
30#465—21no—
31#464—21no—
32#463—21no—
33#462—21no—
34#461—21no—
35#455—24no—
36#452—21no—
37#450—21no—
38#449—21no—
39#447—21no—
40#443—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×#103C2211yescharacteristic, complemented
56C2×#103C2211yescharacteristic, complemented
57C2×#102—21no—
58C2×#102C2211yescharacteristic
59C2×#102C2211yescharacteristic
60C2×#101C2211yescharacteristic, complemented
61C2×#100C2211yescharacteristic
62#408C2211yescharacteristic
63C2×C4×C8C2211yescharacteristic, 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×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4C2311yescharacteristic
83#137—22no—
84#137—22no—
85#137—22no—
86#137—41no—
87AGL(1,ℤ/8)—22no—
88AGL(1,ℤ/8)—41no—
89#135—22no—
90#135—22no—
91C2×Q16—21no—
92C2×Q16—21no—
93C2×Q16D411yescharacteristic
94C2×SD16—21no—
95C2×SD16—21no—
96C2×SD16—21no—
97C2×SD16D411yescharacteristic
98C2×SD16D411yescharacteristic
99C2×D8—21no—
100C2×D8D411yescharacteristic
101#131—22no—
102#131—22no—
103#131—41no—
104#131—41no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C4⋊Q8—21no—
116#124—21no—
117#124—22no—
118#124—22no—
119#123—41no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124#121—21no—
125#121—21no—
126#121—41no—
127C4×Q8—21no—
128C4×D4—21no—
129C4×D4—41no—
130#117—21no—
131#117C2311yescharacteristic
132C2×C4⋊C4—21no—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C22⋊C4—41no—
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#103—21no—
149#103—21no—
150#103—21no—
151#103—21no—
152#103D411yescharacteristic
153#103D411yescharacteristic
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
164#102D411yescharacteristic
165#101—41no—
166#100—22no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C4×C8—22no—
170#95—41no—
171C24—21no—
172C24—41no—
173#41—21no—
174#41—21no—
175#41—24no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—42no—
185#41—42no—
186#41—42no—
187#41C2×D412yes—
188#41#4112yes—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8C2×D411yescharacteristic
196C2×Q8C2×D411yescharacteristic
197C2×Q8C2×D411yescharacteristic
198C2×D4—21no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4C2×D411yescharacteristic
216C2×D4C2×D411yescharacteristic
217C2×D4C2×D411yescharacteristic
218C2×D4#4112yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4C2411yescharacteristic
229Q16—22no—
230Q16—42no—
231Q16—42no—
232SD16—22no—
233SD16—22no—
234SD16—42no—
235SD16—42no—
236SD16—42no—
237D8—22no—
238D8—42no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—21no—
243M16—21no—
244M16—21no—
245M16—21no—
246M16—42no—
247M16—42no—
248M16C2×D412yes—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8C2×D411yescharacteristic
260C2×C8C2×D411yescharacteristic
261C2×C8C2×D411yescharacteristic
262C2×C8C2×D411yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4C2×D411yescharacteristic
272C4⋊C4C2×D411yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—44no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42—21no—
281C42—21no—
282C42—41no—
283C42C2×D412yes—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—44no—
289C23—41no—
290C23—41no—
291C23—81no—
292C23—81no—
293C23—161no—
294C23—161no—
295C23H(5,𝔽2)11yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8#12112yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—21no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—22no—
315D4—44no—
316D4—44no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—161no—
324D4#12112yes—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—44no—
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—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—161no—
348C2×C4C22×D411yescharacteristic
349C2×C4C22×D411yescharacteristic
350C2×C4C22×D411yescharacteristic
351C2×C4C2×#4111yescharacteristic
352C2×C4C2×#4111yescharacteristic
353C2×C4H(5,𝔽2)11yescharacteristic
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—161no—
378C22—164no—
379C22#53811yescharacteristic
380C22D4211yescharacteristic
381C22#54611yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4C2×#12111yescharacteristic
392C4#52511yescharacteristic
393C4D4211yescharacteristic
394C4#54611yescharacteristic
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—81no—
400C2—82no—
401C2#242411yescharacteristic
402C2#329011yescharacteristic
403C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#3012C2
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. #3063C2semidirect
2. #3046C2semidirect
3. C2×#455C2semidirect
5. C2×#443C2semidirect
6. #2897C2semidirect
7. #2063C2semidirect
8. #2052C2semidirect
9. #2040C2semidirect
10. #2003C2semidirect
11. #1958C2semidirect
12. #1937C2semidirect
13. #1908C2semidirect
14. #1907C2semidirect
15. #1860C2semidirect
22. C2×#131C22semidirect
24. #536C22semidirect
41. C2×#108C22semidirect
42. #428C22semidirect
43. #424C22semidirect
46. C2×#104C22semidirect
52. #417C22semidirect
55. C2×#103C22semidirect
56. C2×#103C22semidirect
60. C2×#101C22semidirect
63. C2×C4×C8C22semidirect
64. #382C22semidirect
65. #343C22semidirect
66. #343C22semidirect
67. #341C22semidirect
68. #339C22semidirect
144. #104D4semidirect

Maximal subgroups

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

subgroupindex
#30632
#30462
C2×#4552
#30122
C2×#4432
#28972
#20632
#20522
#20402
#20032
#19582
#19372
#19082
#19072
#18602