Finite Group Databasebeta

← #31266

Subgroups — #31266

Order 256.

526 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#2562C211yesmaximal, characteristic, complemented
2#2408C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2076C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1893C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic, complemented
18C2×#125C2211yescharacteristic, complemented
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#120—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#107C2211yescharacteristic, complemented
24C2×#106C2211yescharacteristic, complemented
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic, complemented
36#400—21no—
37#394—21no—
38#394—21no—
39#392—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#388—21no—
43#388—21no—
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic, complemented
48#380C2211yescharacteristic, complemented
49#379—21no—
50C4×C4⋊C4C2211yescharacteristic, complemented
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—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4D411yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#125—41no—
82#125—41no—
83#121—22no—
84#121—22no—
85#121—41no—
86#121—41no—
87#120—44no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—41no—
115#106—41no—
116#102—22no—
117#102—22no—
118#102—41no—
119#102—41no—
120#102—41no—
121#102—41no—
122#102—41no—
123#102—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24—42no—
139C24—42no—
140C24—81no—
141C24—81no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4#4112yes—
171C2×D4#4112yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
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×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2411yescharacteristic
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—44no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C42—21no—
237C42—21no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—48no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—84no—
267C23—84no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23—84no—
272C23#13512yes—
273C23AGL(1,ℤ/8)12yes—
274C23C22×D411yescharacteristic
275D4—22no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—44no—
280D4—48no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—84no—
286D4—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
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×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C8—41no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—816no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#53411yescharacteristic
365C22#54611yescharacteristic
366C22#54611yescharacteristic
367C22C2×#13511yescharacteristic
368C22C2×AGL(1,ℤ/8)11yescharacteristic
369C22#57611yescharacteristic
370C22#57711yescharacteristic
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4#53811yescharacteristic
382C4#53811yescharacteristic
383C4#55111yescharacteristic
384C4H(3,𝔽4)11yescharacteristic
385C2—42no—
386C2—42no—
387C2—82no—
388C2—82no—
389C2#268411yescharacteristic
390C2#318811yescharacteristic
391C2#319811yescharacteristic
392C2#329611yescharacteristic
393C2#329711yescharacteristic
394C2#331011yescharacteristic
395C2#331811yescharacteristic
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. #2562C2semidirect
3. #2077C2semidirect
4. #2076C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #2025C2semidirect
8. #2025C2semidirect
9. #1926C2semidirect
10. #1925C2semidirect
11. #1893C2semidirect
12. #1893C2semidirect
13. #1869C2semidirect
14. #1786C2semidirect
15. #1786C2semidirect
17. C23×C8C22semidirect
18. C2×#125C22semidirect
23. C2×#107C22semidirect
24. C2×#106C22semidirect
35. C2×C22⋊C8C22semidirect
47. #380C22semidirect
48. #380C22semidirect
50. C4×C4⋊C4C22semidirect
51. #340C22semidirect
52. #340C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #336C22semidirect
58. #336C22semidirect

Maximal subgroups

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

subgroupindex
#25622
#24082
#20772
#20762
#20342
#20332
#20252
#19262
#19252
#18932
#18692
#17862