Finite Group Databasebeta

← #25187

Subgroups — #25187

Order 256.

525 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#471C211yesmaximal, characteristic, complemented
2#3042C211yesmaximal, characteristic, complemented
3C2×#453C211yesmaximal, characteristic
4#3009C211yesmaximal, characteristic, complemented
5#2946C211yesmaximal, characteristic, complemented
6#2897C211yesmaximal, characteristic, complemented
7#2063C211yesmaximal, characteristic, complemented
8#2051C211yesmaximal, characteristic, complemented
9#2038C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1977C211yesmaximal, characteristic, complemented
12#1929C211yesmaximal, characteristic, complemented
13#1887C211yesmaximal, characteristic, complemented
14#1882C211yesmaximal, characteristic, complemented
15#1866C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22C2×#131C2211yescharacteristic
23#536C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#520C2211yescharacteristic
26#518C2211yescharacteristic, complemented
27#471—24no—
28#463—21no—
29#463—21no—
30#461—21no—
31#461—21no—
32#453—24no—
33#452—21no—
34#452—21no—
35#450—21no—
36#450—21no—
37#442—21no—
38#440—21no—
39C4×SD16—21no—
40C4×D8—21no—
41C2×#108C2211yescharacteristic, complemented
42C2×#108C2211yescharacteristic, complemented
43#428C2211yescharacteristic, complemented
44#424C2211yescharacteristic, complemented
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47#419C2211yescharacteristic, complemented
48#418—21no—
49#418C2211yescharacteristic
50#418C2211yescharacteristic
51#417—21no—
52#417C2211yescharacteristic
53#416—21no—
54#416C2211yescharacteristic
55#416C2211yescharacteristic
56C2×#102—21no—
57C2×#102C2211yescharacteristic
58C2×#102C2211yescharacteristic, complemented
59C2×#101C2211yescharacteristic, complemented
60C2×#100C2211yescharacteristic
61C2×#100C2211yescharacteristic, complemented
62#408C2211yescharacteristic
63#405C2211yescharacteristic, complemented
64#391C2211yescharacteristic, complemented
65#343C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67#339C2211yescharacteristic, complemented
68#336C2211yescharacteristic, 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
77C22×Q8C2311yescharacteristic
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84#137—22no—
85#137—22no—
86AGL(1,ℤ/8)—22no—
87AGL(1,ℤ/8)—22no—
88AGL(1,ℤ/8)—41no—
89AGL(1,ℤ/8)D414yes—
90C2×Q16—21no—
91C2×SD16—21no—
92C2×SD16—21no—
93C2×SD16—22no—
94C2×SD16—21no—
95C2×SD16—22no—
96C2×SD16—21no—
97C2×SD16—21no—
98C2×SD16—21no—
99C2×SD16—21no—
100C2×SD16—41no—
101C2×SD16—41no—
102C2×D8—21no—
103C2×D8—21no—
104#131—22no—
105#131—41no—
106C2×M16—21no—
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C22×C8—21no—
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114C4⋊Q8—21no—
115#127—22no—
116#127—22no—
117#124—21no—
118#123—41no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C4×Q8—21no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—41no—
128#117—21no—
129#117C2311yescharacteristic
130#117C2311yescharacteristic
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—41no—
134C2×C42C2311yescharacteristic
135#108—22no—
136#108—41no—
137#107—21no—
138#106—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141#104—22no—
142#104—22no—
143#103—21no—
144#103—21no—
145#103—21no—
146#103D411yescharacteristic, complemented
147#103D411yescharacteristic, complemented
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#102D411yescharacteristic, complemented
160#102D411yescharacteristic, complemented
161#102D411yescharacteristic, complemented
162#102D411yescharacteristic, complemented
163#101—22no—
164#100—22no—
165#100—22no—
166C22⋊C8—21no—
167C22⋊C8—21no—
168#97—21no—
169C4×C8—21no—
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—42no—
182#41—42no—
183#41—42no—
184#41C2×D412yes—
185#41#4112yes—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8C2×D411yescharacteristic
193C2×Q8C2×D411yescharacteristic
194C2×D4—21no—
195C2×D4—24no—
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—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—82no—
216C2×D4—82no—
217C2×D4C2×D412yes—
218C2×D4C2×D411yescharacteristic
219C2×D4C2×D411yescharacteristic
220C2×D4#4112yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4C2411yescharacteristic
231Q16—42no—
232SD16—24no—
233SD16—42no—
234SD16—42no—
235SD16—42no—
236SD16—42no—
237SD16—42no—
238SD16—42no—
239D8—24no—
240D8—42no—
241M16—21no—
242M16—21no—
243M16—21no—
244M16—21no—
245M16—21no—
246M16—21no—
247M16—21no—
248M16—42no—
249M16C2×D412yes—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—42no—
259C2×C8C2×D411yescharacteristic
260C2×C8C2×D411yescharacteristic
261C2×C8C2×D411yescharacteristic
262C2×C8C2×D411yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4C2×D411yescharacteristic
269C4⋊C4C2×D411yescharacteristic
270C4⋊C4C2×D411yescharacteristic
271C4⋊C4C2×D411yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—41no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—44no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—161no—
298C23—161no—
299C23H(5,𝔽2)11yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8#12112yes—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—22no—
313D4—21no—
314D4—22no—
315D4—21no—
316D4—22no—
317D4—44no—
318D4—44no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—88no—
326D4#12112yes—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—44no—
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—161no—
347C2×C4—161no—
348C2×C4—161no—
349C2×C4C22×D411yescharacteristic
350C2×C4C22×D411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C2×C4C2×#4111yescharacteristic
353C2×C4C2×#4111yescharacteristic
354C2×C4H(5,𝔽2)11yescharacteristic
355C8—22no—
356C8—21no—
357C8—21no—
358C8—21no—
359C8—21no—
360C8—41no—
361C8—41no—
362C8—41no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—82no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—161no—
380C22—161no—
381C22—164no—
382C22#53811yescharacteristic
383C22D4211yescharacteristic
384C22#54611yescharacteristic
385C4—22no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4C2×#12111yescharacteristic
395C4#52511yescharacteristic
396C4D4211yescharacteristic
397C4#54611yescharacteristic
398C2—21no—
399C2—21no—
400C2—42no—
401C2—81no—
402C2—82no—
403C2—82no—
404C2#242411yescharacteristic
405C2#328811yescharacteristic
406C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#453C2
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. C2×#471C2semidirect
2. #3042C2semidirect
4. #3009C2semidirect
5. #2946C2semidirect
6. #2897C2semidirect
7. #2063C2semidirect
8. #2051C2semidirect
9. #2038C2semidirect
10. #2014C2semidirect
11. #1977C2semidirect
12. #1929C2semidirect
13. #1887C2semidirect
14. #1882C2semidirect
15. #1866C2semidirect
21. C22×SD16C22semidirect
26. #518C22semidirect
41. C2×#108C22semidirect
42. C2×#108C22semidirect
43. #428C22semidirect
44. #424C22semidirect
47. #419C22semidirect
58. C2×#102C22semidirect
59. C2×#101C22semidirect
61. C2×#100C22semidirect
63. #405C22semidirect
64. #391C22semidirect
65. #343C22semidirect
66. #339C22semidirect
67. #339C22semidirect
68. #336C22semidirect
146. #103D4semidirect
147. #103D4semidirect
148. #103D4semidirect
149. #103D4semidirect
159. #102D4semidirect
160. #102D4semidirect
161. #102D4semidirect
162. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4712
#30422
C2×#4532
#30092
#29462
#28972
#20632
#20512
#20382
#20142
#19772
#19292
#18872
#18822
#18662