Finite Group Databasebeta

← #24873

Subgroups — #24873

Order 256.

590 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2408C211yesmaximal, characteristic
4#2352C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic, complemented
23C22×C4⋊C4C2211yescharacteristic
24#482—24no—
25C2×#107C2211yescharacteristic, complemented
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29C2×#102—24no—
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic, complemented
41#394—21no—
42#392—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#388—21no—
46#387—21no—
47#386—21no—
48#382C2211yescharacteristic, complemented
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic, complemented
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54C4×C22⋊C4—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#121—22no—
85#121—22no—
86#121—41no—
87#121—41no—
88#120—44no—
89C4×D4—22no—
90C4×D4—22no—
91#117—22no—
92#117—41no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118#107—22no—
119#107—41no—
120#106—22no—
121#106—41no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129#102D414yescomplemented
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4C2×D412yes—
177C2×D4#4112yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4#4112yes—
202C22×C4#4112yes—
203C22×C4C2411yescharacteristic
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—44no—
216C2×C8C2×D412yes—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—42no—
230C4⋊C4C2×D412yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42—81no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—48no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23#12312yes—
280C23#13512yes—
281C23AGL(1,ℤ/8)12yes—
282C23C22×D411yescharacteristic
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—48no—
288D4—44no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294D4—84no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4#12312yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—816no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#48214yescomplemented
373C22C2×#12311yescharacteristic
374C22D4211yescharacteristic
375C22#54611yescharacteristic
376C22C2×#13511yescharacteristic
377C22C2×AGL(1,ℤ/8)11yescharacteristic
378C22#57611yescharacteristic
379C22#57711yescharacteristic
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—42no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4#52511yescharacteristic
391C4#53811yescharacteristic
392C4#54611yescharacteristic
393C4#55111yescharacteristic
394C2—24no—
395C2—81no—
396C2—82no—
397C2—82no—
398C2#241811yescharacteristic
399C2C2×#48211yescharacteristic
400C2#309611yescharacteristic
401C2#326411yescharacteristic
402C2#328211yescharacteristic
403C2#329711yescharacteristic
404C2#331011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C22×#102C2semidirect
4. #2352C2semidirect
5. #2092C2semidirect
6. #2073C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1926C2semidirect
10. #1925C2semidirect
11. #1875C2semidirect
12. #1869C2semidirect
13. #1852C2semidirect
14. #1807C2semidirect
15. #1786C2semidirect
22. C2×#117C22semidirect
25. C2×#107C22semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×#106C22semidirect
39. C2×C22⋊C8C22semidirect
40. C2×C22⋊C8C22semidirect
48. #382C22semidirect
51. #380C22semidirect
53. C4×C4⋊C4C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
129. #102D4semidirect
372. C22#482semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C22×#1022
#24082
#23522
#20922
#20732
#20342
#20332
#19262
#19252
#18752
#18692
#18522
#18072
#17862