Finite Group Databasebeta

← #27279

Subgroups — #27279

Order 256.

598 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic, complemented
3#2441C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic
5#2071C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1957C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic, complemented
21C2×#121C2211yescharacteristic
22C2×#120—21no—
23C2×#120—21no—
24C2×#117C2211yescharacteristic
25#490—24no—
26#418—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×C4×C8C2211yescharacteristic, complemented
42#399—21no—
43#396—21no—
44#392—21no—
45#392—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#389—21no—
49#389C2211yescharacteristic, complemented
50#389C2211yescharacteristic, complemented
51#387—21no—
52#384C2211yescharacteristic, complemented
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×C4C2311yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#127—22no—
86#127—22no—
87#127—41no—
88#127—41no—
89#125—22no—
90#125—41no—
91#121—22no—
92#121—22no—
93#120—44no—
94#120—44no—
95#117—22no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129#102—41no—
130#102—41no—
131#102D414yescomplemented
132C22⋊C8—22no—
133#97—22no—
134C4×C8—41no—
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×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4#4112yes—
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208M16—44no—
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—21no—
220C2×C8C2×D412yes—
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—21no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4C2×D412yes—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—48no—
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—41no—
278C23—41no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23#12412yes—
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—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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#12412yes—
337C2×C4AGL(1,ℤ/8)12yes—
338C2×C4AGL(1,ℤ/8)12yes—
339C2×C4C22×D411yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4C2×#4111yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C8—22no—
346C8—41no—
347C8—41no—
348C8—41no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
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—
378C22C2×#12411yescharacteristic
379C22D4211yescharacteristic
380C22#54611yescharacteristic
381C22C2×AGL(1,ℤ/8)11yescharacteristic
382C22C2×AGL(1,ℤ/8)11yescharacteristic
383C22#57611yescharacteristic
384C22#57711yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4#52511yescharacteristic
396C4#53811yescharacteristic
397C4#54611yescharacteristic
398C4#55111yescharacteristic
399C2—42no—
400C2—82no—
401C2—82no—
402C2—161no—
403C2#247211yescharacteristic
404C2#313611yescharacteristic
405C2#313811yescharacteristic
406C2#327211yescharacteristic
407C2#327411yescharacteristic
408C2#331011yescharacteristic
409C2#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. C2×#418C2semidirect
3. #2441C2semidirect
5. #2071C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2025C2semidirect
9. #1957C2semidirect
10. #1936C2semidirect
11. #1934C2semidirect
12. #1895C2semidirect
13. #1875C2semidirect
14. #1786C2semidirect
15. #1753C2semidirect
20. C2×#125C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
40. C2×#97C22semidirect
41. C2×C4×C8C22semidirect
49. #389C22semidirect
50. #389C22semidirect
52. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
131. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4182
#24412
#24332
#20712
#20412
#20252
#19572
#19362
#19342
#18952
#18752
#17862
#17532