Finite Group Databasebeta

← #27539

Subgroups — #27539

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic, complemented
2D4×C22⋊C4C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#1936C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1924C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic, complemented
13#1786C211yesmaximal, characteristic
14#1786C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C2×#126—21no—
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C22⋊C4—21no—
21C2×#106C2211yescharacteristic, complemented
22C2×#106C2211yescharacteristic, complemented
23C2×C4⋊C8C2211yescharacteristic, complemented
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic, complemented
35#400—21no—
36#397—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic, complemented
40#389C2211yescharacteristic, complemented
41#387—21no—
42#386—21no—
43#386—21no—
44#384C2211yescharacteristic, complemented
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#379—21no—
49C4×C4⋊C4C2211yescharacteristic, complemented
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic, complemented
52#340C2211yescharacteristic, complemented
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#336C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59C25—21no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#126—44no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—41no—
80C4×D4—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#106—41no—
107#106—41no—
108C4⋊C8—41no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—41no—
116#102—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119#97—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24—21no—
129C24—24no—
130C24—21no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24—42no—
135C24—81no—
136C24—81no—
137C24C2×D411yescharacteristic
138C2×D4—24no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4#4112yes—
154C2×D4#4112yes—
155C2×D4#4112yes—
156C2×D4#4112yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—42no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—48no—
236C23—48no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—84no—
248C23—84no—
249C23—84no—
250C23—84no—
251C23—84no—
252C23AGL(1,ℤ/8)12yes—
253C23C22×D411yescharacteristic
254D4—22no—
255D4—22no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—48no—
260D4—41no—
261D4—41no—
262D4#12614yescomplemented
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4AGL(1,ℤ/8)12yes—
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C8—41no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—22no—
313C22—24no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—816no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#52511yescharacteristic
335C22#54611yescharacteristic
336C22#54611yescharacteristic
337C22C2×AGL(1,ℤ/8)11yescharacteristic
338C22C2×AGL(1,ℤ/8)11yescharacteristic
339C22#57711yescharacteristic
340C22#57711yescharacteristic
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4C2×#12611yescharacteristic
353C4#53811yescharacteristic
354C4#55111yescharacteristic
355C4#55111yescharacteristic
356C2—42no—
357C2—44no—
358C2—82no—
359C2#248011yescharacteristic
360C2#310911yescharacteristic
361C2#312311yescharacteristic
362C2#330611yescharacteristic
363C2#330611yescharacteristic
364C2#331011yescharacteristic
365C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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. #2496C2semidirect
3. #2077C2semidirect
4. #2077C2semidirect
5. #2071C2semidirect
6. #2065C2semidirect
7. #1936C2semidirect
8. #1926C2semidirect
9. #1924C2semidirect
10. #1864C2semidirect
11. #1864C2semidirect
12. #1802C2semidirect
15. #1762C2semidirect
21. C2×#106C22semidirect
22. C2×#106C22semidirect
23. C2×C4⋊C8C22semidirect
34. C2×#97C22semidirect
39. #389C22semidirect
40. #389C22semidirect
44. #384C22semidirect
49. C4×C4⋊C4C22semidirect
51. #340C22semidirect
52. #340C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #336C22semidirect
58. #336C22semidirect
262. D4#126semidirect

Maximal subgroups

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

subgroupindex
#24962
D4×C22⋊C42
#20772
#20712
#20652
#19362
#19262
#19242
#18642
#18022
#17862
#17622