Finite Group Databasebeta

← #24865

Subgroups — #24865

Order 256.

619 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×#481C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2408C211yesmaximal, characteristic
4D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic, complemented
25#481—24no—
26C2×#106C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29C2×#106C2211yescharacteristic, complemented
30C2×#102—24no—
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic, complemented
41C2×C22⋊C8C2211yescharacteristic, complemented
42#394—21no—
43#392—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389—21no—
47#385—21no—
48#385—21no—
49#384C2211yescharacteristic, complemented
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic, complemented
53#379—21no—
54C4×C4⋊C4C2211yescharacteristic, complemented
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×C4—21no—
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#121—22no—
86#121—22no—
87#121—41no—
88#121—41no—
89#120—44no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—44no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121#106—22no—
122#106—22no—
123#106—41no—
124#106—41no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132#102D414yescomplemented
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140C24—21no—
141C24—21no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24—81no—
149C24C2×D411yescharacteristic
150C24C2×D411yescharacteristic
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—28no—
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×D4SD1614yes—
177C2×D4C2×D412yes—
178C2×D4#4112yes—
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—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4#4112yes—
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
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—44no—
220C2×C8C2×D412yes—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—21no—
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⋊C4C2×D412yes—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—21no—
250C42—21no—
251C42—42no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—48no—
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#12312yes—
284C23C2×SD1612yes—
285C23C2×SD1612yes—
286C23AGL(1,ℤ/8)12yes—
287C23C22×D411yescharacteristic
288D4—28no—
289D4—22no—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—48no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—84no—
299D4—84no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
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—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#12312yes—
339C2×C4C2×SD1612yes—
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—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—816no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#48114yescomplemented
379C22C2×#12311yescharacteristic
380C22D4211yescharacteristic
381C22#54611yescharacteristic
382C22C22×SD1611yescharacteristic
383C22C2×AGL(1,ℤ/8)11yescharacteristic
384C22#57611yescharacteristic
385C22#57711yescharacteristic
386C4—22no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4#52511yescharacteristic
397C4#53811yescharacteristic
398C4#54611yescharacteristic
399C4#55111yescharacteristic
400C2—24no—
401C2—42no—
402C2—82no—
403C2—82no—
404C2#241811yescharacteristic
405C2C2×#48111yescharacteristic
406C2#309611yescharacteristic
407C2#326411yescharacteristic
408C2D4×SD1611yescharacteristic
409C2#329511yescharacteristic
410C2#330611yescharacteristic
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×#481C2semidirect
2. C22×#102C2semidirect
4. D4×C4⋊C4C2semidirect
5. #2094C2semidirect
6. #2071C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #1926C2semidirect
10. #1926C2semidirect
11. #1874C2semidirect
12. #1869C2semidirect
13. #1853C2semidirect
14. #1807C2semidirect
15. #1786C2semidirect
24. C22×C4⋊C4C22semidirect
26. C2×#106C22semidirect
27. C2×#106C22semidirect
28. C2×#106C22semidirect
29. C2×#106C22semidirect
40. C2×C22⋊C8C22semidirect
41. C2×C22⋊C8C22semidirect
49. #384C22semidirect
52. #380C22semidirect
54. C4×C4⋊C4C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
132. #102D4semidirect
378. C22#481semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C22×#1022
#24082
D4×C4⋊C42
#20942
#20712
#20342
#19262
#18742
#18692
#18532
#18072
#17862