Finite Group Databasebeta

← #61218

Subgroups — #61218

Order 256.

1083 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic, complemented
2#3496C211yesmaximal, characteristic, complemented
3#3319C214yesmaximal, complemented
4#3318C214yesmaximal, complemented
5#3307C214yesmaximal, complemented
6#3306C214yesmaximal, complemented
7#3135C211yesmaximal, characteristic, complemented
8#3099C212yesmaximal, complemented
9#3077C211yesmaximal, characteristic, complemented
10#3062C212yesmaximal, complemented
11#3036C211yesmaximal, characteristic, complemented
12#3008C212yesmaximal, complemented
13C2×#442C211yesmaximal, characteristic, complemented
14#2932C211yesmaximal, characteristic, complemented
15#2897C212yesmaximal, complemented
16C2×H(5,𝔽2)C2212yes—
17C2×AGL(1,ℤ/8)C2212yescomplemented
18C22×D8C2211yescharacteristic
19C2×#131C2212yescomplemented
20#559—22no—
21#556—21no—
22#555C2212yescomplemented
23#553C2214yescomplemented
24#550—21no—
25#547—21no—
26#547C2212yescomplemented
27#547C2212yescomplemented
28#546—22no—
29#546C2214yescomplemented
30D42—21no—
31D42C2212yescomplemented
32#538—22no—
33#537C2212yes—
34#535C2212yescomplemented
35#534—22no—
36#533C2211yescharacteristic, complemented
37#532C2211yescharacteristic
38#532C2211yescharacteristic
39#525—22no—
40C2×#121—22no—
41#520C2212yescomplemented
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic, complemented
44#491C2212yescomplemented
45#490C2212yescomplemented
46#489C2212yescomplemented
47#489C2212yescomplemented
48#485C2214yescomplemented
49#483C2214yescomplemented
50#482C2214yescomplemented
51#481C2214yescomplemented
52#478C2214yescomplemented
53#476C2212yescomplemented
54#474C2212yescomplemented
55#469C2214yescomplemented
56#468C2214yescomplemented
57#466C2214yescomplemented
58#465C2214yescomplemented
59#463C2214yescomplemented
60#461C2212yescomplemented
61#459C2212yescomplemented
62#449C2214yescomplemented
63#449C2214yescomplemented
64#447C2214yescomplemented
65#447C2214yescomplemented
66#442C2214yescomplemented
67#442C2214yescomplemented
68#435C2214yescomplemented
69#435C2214yescomplemented
70#428C2212yescomplemented
71C2×#106C2211yescharacteristic, complemented
72#423C2211yescharacteristic, complemented
73#418C2212yes—
74#418C2214yescomplemented
75#417C2212yescomplemented
76#417C2214yescomplemented
77#416C2212yescomplemented
78#416C2212yescomplemented
79#416C2214yescomplemented
80C2×#102C2211yescharacteristic
81C2×#102C2214yescomplemented
82C2×#102C2211yescharacteristic, complemented
83#408C2212yescomplemented
84C2×#97C2211yescharacteristic, complemented
85H(5,𝔽2)—28no—
86H(5,𝔽2)—44no—
87C2×#41—22no—
88C2×#41—22no—
89C2×#41C2312yes—
90C2×#41C2314yes—
91C2×#41C2312yes—
92C22×D4—24no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4C2311yescharacteristic
96C22×D4C2314yes—
97C22×D4C2311yescharacteristic
98C23×C4—21no—
99C23×C4—21no—
100AGL(1,ℤ/8)—44no—
101C2×SD16C2314yes—
102C2×D8—24no—
103C2×D8C2314yes—
104C2×D8C2314yes—
105#131—44no—
106C2×M16C2312yes—
107C2×M16C2314yes—
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110C22×C8C2314yes—
111C4⋊Q8D411yescharacteristic
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115#127—22no—
116#127D411yescharacteristic
117#127C2311yescharacteristic
118#126—24no—
119#125C2312yes—
120#124—22no—
121#124—24no—
122#124D412yes—
123#124C2312yes—
124#123—24no—
125#123—24no—
126#123—24no—
127#123—24no—
128C22⋊Q8—22no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8—22no—
132C22⋊Q8C2314yes—
133#121—24no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—24no—
138#121—21no—
139#121—21no—
140#121—22no—
141#121—24no—
142#121D412yes—
143#121D412yes—
144#121C2314yes—
145#121C2314yes—
146#121C2314yes—
147#120—24no—
148#120—24no—
149#120—24no—
150C4×Q8—22no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×D4—24no—
154C4×D4—22no—
155C4×D4—24no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4C2312yes—
163C4×D4C2312yes—
164C4×D4C2312yes—
165C4×D4C2312yes—
166C4×D4C2312yes—
167C4×D4C2312yes—
168#117—22no—
169#117C2312yes—
170#117C2314yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2311yescharacteristic
173C2×C4⋊C4C2311yescharacteristic
174C2×C4⋊C4C2314yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—24no—
178C2×C42C2311yescharacteristic
179#107C2314yes—
180#106C2314yes—
181#106C2314yes—
182C4⋊C8C2312yes—
183C4⋊C8C2312yes—
184#103C2312yes—
185#103C2314yes—
186#103C2312yes—
187#103C2312yes—
188#103C2312yes—
189#102C2314yes—
190#102C2312yes—
191#102C2314yes—
192#102C2314yes—
193#102C2314yes—
194#102C2314yes—
195#102C2312yes—
196#102C2312yes—
197#102C2312yes—
198C22⋊C8C2314yes—
199C22⋊C8C2314yes—
200#97C2312yes—
201#97C2312yes—
202C24—21no—
203C24—21no—
204C24—24no—
205#41—24no—
206#41—24no—
207#41—28no—
208#41—24no—
209#41—24no—
210#41—44no—
211#41—48no—
212#41—84no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8C2×D412yes—
216C2×Q8C2412yes—
217C2×D4—28no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—44no—
232C2×D4—48no—
233C2×D4—416no—
234C2×D4—84no—
235C2×D4C2×D414yes—
236C2×D4C2×D412yes—
237C2×D4C2×D412yes—
238C2×D4C2×D412yes—
239C2×D4#4114yescomplemented
240C2×D4C2412yes—
241C2×D4C2412yes—
242C2×D4C2412yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2414yes—
257C22×C4C2411yescharacteristic
258SD16—44no—
259D8—28no—
260D8—44no—
261M16—24no—
262M16—44no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—44no—
266C2×C8C2412yes—
267C2×C8C2412yes—
268C2×C8C2414yes—
269C4⋊C4—21no—
270C4⋊C4—24no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4C2×D412yes—
280C4⋊C4C2×D412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—22no—
296C22⋊C4—24no—
297C22⋊C4C2×D414yes—
298C22⋊C4C2×D414yes—
299C42—24no—
300C42—22no—
301C42C2×D411yescharacteristic
302C42C2×D411yescharacteristic
303C42C2411yescharacteristic
304C42C2411yescharacteristic
305C23—22no—
306C23—24no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—84no—
316C23—84no—
317C23C22×D414yes—
318C23C22×D411yescharacteristic
319Q8—24no—
320Q8—22no—
321Q8—22no—
322Q8—42no—
323Q8—81no—
324D4—28no—
325D4—24no—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—22no—
330D4—22no—
331D4—22no—
332D4—42no—
333D4—42no—
334D4—48no—
335D4—42no—
336D4—48no—
337D4—44no—
338D4—42no—
339D4—81no—
340D4—84no—
341D4—84no—
342D4—84no—
343D4C2×#4114yes—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4C22×D414yes—
363C2×C4C22×D411yescharacteristic
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D411yescharacteristic
368C2×C4C2×#4112yes—
369C2×C4H(5,𝔽2)12yes—
370C2×C4C2511yescharacteristic
371C8—22no—
372C8—22no—
373C8—24no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—84no—
389C22—84no—
390C22—168no—
391C22#54612yes—
392C22C23×D411yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—24no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4#54612yes—
404C4C22×#4111yescharacteristic
405C4C2×H(5,𝔽2)11yescharacteristic
406C2—22no—
407C2—44no—
408C2—44no—
409C2—82no—
410C2—82no—
411C2C2×#54611yescharacteristic
412C2#358511yescharacteristic
413C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3506C2semidirect
2. #3496C2semidirect
3. #3319C2semidirect
4. #3318C2semidirect
5. #3307C2semidirect
6. #3306C2semidirect
7. #3135C2semidirect
8. #3099C2semidirect
9. #3077C2semidirect
10. #3062C2semidirect
11. #3036C2semidirect
12. #3008C2semidirect
13. C2×#442C2semidirect
14. #2932C2semidirect
15. #2897C2semidirect
17. C2×AGL(1,ℤ/8)C22semidirect
19. C2×#131C22semidirect
22. #555C22semidirect
23. #553C22semidirect
26. #547C22semidirect
27. #547C22semidirect
29. #546C22semidirect
31. D42C22semidirect
34. #535C22semidirect
36. #533C22semidirect
41. #520C22semidirect
43. C2×C4×D4C22semidirect
44. #491C22semidirect
45. #490C22semidirect
46. #489C22semidirect
47. #489C22semidirect
48. #485C22semidirect
49. #483C22semidirect
50. #482C22semidirect
51. #481C22semidirect
52. #478C22semidirect
53. #476C22semidirect
54. #474C22semidirect
55. #469C22semidirect
56. #468C22semidirect
57. #466C22semidirect
58. #465C22semidirect
59. #463C22semidirect
60. #461C22semidirect
61. #459C22semidirect
62. #449C22semidirect
63. #449C22semidirect
64. #447C22semidirect
65. #447C22semidirect
66. #442C22semidirect
67. #442C22semidirect
68. #435C22semidirect
69. #435C22semidirect
70. #428C22semidirect
71. C2×#106C22semidirect
72. #423C22semidirect
74. #418C22semidirect
75. #417C22semidirect
76. #417C22semidirect
77. #416C22semidirect
78. #416C22semidirect
79. #416C22semidirect
81. C2×#102C22semidirect
82. C2×#102C22semidirect
83. #408C22semidirect
84. C2×#97C22semidirect
239. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
#35062
#34962
#33192
#33182
#33072
#33062
#31352
#30992
#30772
#30622
#30362
#30082
C2×#4422
#29322
#28972