Finite Group Databasebeta

← #31050

Subgroups — #31050

Order 256.

447 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3099C211yesmaximal, characteristic, complemented
2#3046C211yesmaximal, characteristic, complemented
3#3012C211yesmaximal, characteristic
4#3009C211yesmaximal, characteristic, complemented
5#2079C211yesmaximal, characteristic, complemented
6#2063C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic, complemented
8#2038C211yesmaximal, characteristic, complemented
9#2019C211yesmaximal, characteristic, complemented
10#2007C211yesmaximal, characteristic, complemented
11#1974C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1937C211yesmaximal, characteristic, complemented
14#1883C211yesmaximal, characteristic, complemented
15#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic, complemented
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×#135C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22#537C2211yescharacteristic
23#536C2211yescharacteristic
24#520C2211yescharacteristic, complemented
25#485—21no—
26#483—21no—
27#482—21no—
28#481—21no—
29#464—21no—
30#463—21no—
31#462—21no—
32#461—21no—
33#452—21no—
34#452—21no—
35#452—21no—
36#450—21no—
37#450—21no—
38#450—21no—
39#449—21no—
40#447—21no—
41C2×#108C2211yescharacteristic, complemented
42#428C2211yescharacteristic, complemented
43C2×#106C2211yescharacteristic, complemented
44#424C2211yescharacteristic, complemented
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic, complemented
48#418C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#416C2211yescharacteristic
53C2×#103C2211yescharacteristic, complemented
54C2×#102C2211yescharacteristic
55C2×#101C2211yescharacteristic, complemented
56C2×#100C2211yescharacteristic
57#408C2211yescharacteristic
58#408C2211yescharacteristic
59#408C2211yescharacteristic, complemented
60C2×C4×C8C2211yescharacteristic, complemented
61#391C2211yescharacteristic, complemented
62#390C2211yescharacteristic
63#343C2211yescharacteristic, complemented
64#343C2211yescharacteristic, complemented
65#341C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67H(5,𝔽2)—22no—
68H(5,𝔽2)—41no—
69H(5,𝔽2)—41no—
70H(5,𝔽2)—41no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2311yescharacteristic
74C2×#41C2311yescharacteristic
75C2×#41C2311yescharacteristic
76C22×Q8C2311yescharacteristic
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4C2311yescharacteristic
81#137—41no—
82#137—41no—
83#137—41no—
84#137—41no—
85AGL(1,ℤ/8)—22no—
86AGL(1,ℤ/8)—41no—
87#135—41no—
88#135—41no—
89C2×Q16—21no—
90C2×Q16—21no—
91C2×Q16D411yescharacteristic
92C2×SD16—21no—
93C2×SD16—21no—
94C2×SD16—21no—
95C2×SD16—22no—
96C2×SD16—21no—
97C2×SD16—21no—
98C2×SD16—41no—
99C2×SD16D411yescharacteristic
100C2×SD16D411yescharacteristic
101C2×D8—21no—
102C2×D8D411yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2311yescharacteristic
105C2×M16C2311yescharacteristic
106C2×M16C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C4⋊Q8—21no—
110#124—21no—
111#123—41no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C22⋊Q8—21no—
116#121—21no—
117#121—21no—
118#121—41no—
119C4×Q8—21no—
120C4×D4—21no—
121C4×D4—41no—
122#117—21no—
123#117C2311yescharacteristic
124C2×C4⋊C4—21no—
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—41no—
128C2×C42C2311yescharacteristic
129#108—41no—
130#107—21no—
131#106—21no—
132#106—21no—
133#106—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136#104—41no—
137#104—41no—
138#104—41no—
139#103—21no—
140#103—21no—
141#103—21no—
142#103—21no—
143#103—21no—
144#103D411yescharacteristic
145#103D411yescharacteristic
146#102—21no—
147#102—21no—
148#102—21no—
149#102—21no—
150#102—21no—
151#102—21no—
152#102—21no—
153#102D411yescharacteristic
154#102D411yescharacteristic
155#101—41no—
156#100—41no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C4×C8—41no—
164#95—41no—
165C24—21no—
166C24—41no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—42no—
180#41—42no—
181#41—42no—
182#41—42no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8C2×D411yescharacteristic
190C2×Q8C2×D411yescharacteristic
191C2×Q8C2×D411yescharacteristic
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—41no—
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D411yescharacteristic
211C2×D4C2×D411yescharacteristic
212C2×D4#4112yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C2411yescharacteristic
223Q16—41no—
224Q16—42no—
225Q16—42no—
226SD16—41no—
227SD16—41no—
228SD16—42no—
229SD16—42no—
230SD16—42no—
231SD16—42no—
232SD16—42no—
233D8—41no—
234D8—42no—
235M16—21no—
236M16—21no—
237M16—21no—
238M16—21no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8C2×D411yescharacteristic
252C2×C8C2×D411yescharacteristic
253C2×C8C2×D411yescharacteristic
254C2×C8C2×D411yescharacteristic
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4C2×D411yescharacteristic
264C4⋊C4C2×D411yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—82no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—44no—
281C23—41no—
282C23—41no—
283C23—81no—
284C23—81no—
285C23—161no—
286C23—161no—
287C23H(5,𝔽2)11yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—22no—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—44no—
307D4—44no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—161no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—161no—
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—21no—
347C8—21no—
348C8—21no—
349C8—21no—
350C8—41no—
351C8—41no—
352C8—41no—
353C8—41no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22—161no—
370C22—164no—
371C22#53811yescharacteristic
372C22D4211yescharacteristic
373C22#56011yescharacteristic
374C4—42no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4#53411yescharacteristic
384C4#53511yescharacteristic
385C4#54611yescharacteristic
386C4#54611yescharacteristic
387C2—21no—
388C2—21no—
389C2—81no—
390C2—81no—
391C2—81no—
392C2—82no—
393C2#267811yescharacteristic
394C2#329011yescharacteristic
395C2#329111yescharacteristic
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. #3099C2semidirect
2. #3046C2semidirect
4. #3009C2semidirect
5. #2079C2semidirect
6. #2063C2semidirect
7. #2052C2semidirect
8. #2038C2semidirect
9. #2019C2semidirect
10. #2007C2semidirect
11. #1974C2semidirect
12. #1968C2semidirect
13. #1937C2semidirect
14. #1883C2semidirect
15. #1792C2semidirect
18. C2×#137C22semidirect
24. #520C22semidirect
41. C2×#108C22semidirect
42. #428C22semidirect
43. C2×#106C22semidirect
44. #424C22semidirect
47. C2×#104C22semidirect
53. C2×#103C22semidirect
55. C2×#101C22semidirect
59. #408C22semidirect
60. C2×C4×C8C22semidirect
61. #391C22semidirect
63. #343C22semidirect
64. #343C22semidirect
65. #341C22semidirect
66. #339C22semidirect

Maximal subgroups

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

subgroupindex
#30992
#30462
#30122
#30092
#20792
#20632
#20522
#20382
#20192
#20072
#19742
#19682
#19372
#18832
#17922