Finite Group Databasebeta

← #27264

Subgroups — #27264

Order 256.

608 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2442C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic
5#2073C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1961C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1893C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic, complemented
21C2×#121C2211yescharacteristic
22C2×#120—21no—
23C2×#120—21no—
24C22×C4⋊C4C2211yescharacteristic
25#490—24no—
26C2×#102—24no—
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic, complemented
41C2×#97C2211yescharacteristic, complemented
42#400—21no—
43#398—21no—
44#392—21no—
45#392—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#389C2211yescharacteristic, complemented
49#389C2211yescharacteristic, complemented
50#385—21no—
51#382C2211yescharacteristic, complemented
52#380—21no—
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
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—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#127—22no—
87#127—22no—
88#127—41no—
89#127—41no—
90#125—22no—
91#125—41no—
92#121—22no—
93#121—22no—
94#120—44no—
95#120—44no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102—41no—
128#102—41no—
129#102D414yescomplemented
130C22⋊C8—22no—
131#97—22no—
132#97—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95D411yescharacteristic
139#95D411yescharacteristic
140C24—21no—
141C24—21no—
142C24—21no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24—81no—
150C24C2×D411yescharacteristic
151C2×D4—28no—
152C2×D4—21no—
153C2×D4—21no—
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—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4C2×D412yes—
190C2×D4#4112yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—81no—
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4#4112yes—
209C22×C4#4112yes—
210C22×C4C2411yescharacteristic
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—44no—
223C2×C8C2×D412yes—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4C2×D412yes—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—48no—
258C23—41no—
259C23—41no—
260C23—41no—
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—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23#12412yes—
284C23AGL(1,ℤ/8)12yes—
285C23AGL(1,ℤ/8)12yes—
286C23C22×D411yescharacteristic
287D4—22no—
288D4—22no—
289D4—22no—
290D4—22no—
291D4—48no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—84no—
301D4—84no—
302D4—84no—
303D4—84no—
304D4—84no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
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—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—84no—
339C2×C4#12412yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C2×#4111yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—22no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—816no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#49014yescomplemented
379C22C2×#12411yescharacteristic
380C22D4211yescharacteristic
381C22#54611yescharacteristic
382C22C2×AGL(1,ℤ/8)11yescharacteristic
383C22C2×AGL(1,ℤ/8)11yescharacteristic
384C22#57611yescharacteristic
385C22#57711yescharacteristic
386C4—22no—
387C4—41no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4#52511yescharacteristic
397C4#53811yescharacteristic
398C4#54611yescharacteristic
399C4#55111yescharacteristic
400C2—24no—
401C2—82no—
402C2—82no—
403C2—161no—
404C2#247211yescharacteristic
405C2C2×#49011yescharacteristic
406C2#313611yescharacteristic
407C2#326411yescharacteristic
408C2#326511yescharacteristic
409C2#331011yescharacteristic
410C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#490C2semidirect
2. C22×#102C2semidirect
3. #2442C2semidirect
5. #2073C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2025C2semidirect
9. #1961C2semidirect
10. #1936C2semidirect
11. #1936C2semidirect
12. #1893C2semidirect
13. #1874C2semidirect
14. #1786C2semidirect
15. #1752C2semidirect
20. C2×#125C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
40. C2×#97C22semidirect
41. C2×#97C22semidirect
48. #389C22semidirect
49. #389C22semidirect
51. #382C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
129. #102D4semidirect
378. C22#490semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C22×#1022
#24422
#24332
#20732
#20412
#20252
#19612
#19362
#18932
#18742
#17862
#17522