Finite Group Databasebeta

← #61802

Subgroups — #61802

Order 256.

930 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3506C211yesmaximal, characteristic, complemented
3#3354C212yesmaximal, complemented
4#3347C214yesmaximal, complemented
5#3341C212yesmaximal, complemented
6#3325C212yesmaximal, complemented
7#3318C214yesmaximal, complemented
8#3294C212yesmaximal, complemented
9#3235C211yesmaximal, characteristic, complemented
10#3228C211yesmaximal, characteristic, complemented
11#3199C214yesmaximal, complemented
12#3109C211yesmaximal, characteristic, complemented
13#3100C211yesmaximal, characteristic, complemented
14#3096C214yesmaximal, complemented
15#2981C211yesmaximal, characteristic, complemented
16H(3,𝔽4)—22no—
17#559—22no—
18#556—21no—
19#555—21no—
20#555C2212yescomplemented
21#555C2212yescomplemented
22#555C2212yescomplemented
23#553C2214yescomplemented
24#550—21no—
25#547C2212yescomplemented
26D42—21no—
27D42C2212yescomplemented
28D42C2212yescomplemented
29#538—22no—
30#535—22no—
31#535—22no—
32#535C2214yes—
33#535C2212yescomplemented
34#533C2211yescharacteristic, complemented
35#532C2211yescharacteristic
36#532C2211yescharacteristic
37#532C2211yescharacteristic
38#525—22no—
39#520C2212yescomplemented
40C2×C4×D4C2211yescharacteristic, complemented
41#501C2212yescomplemented
42#500C2211yescharacteristic, complemented
43#498C2211yescharacteristic, complemented
44#490C2212yescomplemented
45#486C2211yescharacteristic, complemented
46#486C2211yescharacteristic, complemented
47#482C2214yescomplemented
48#482C2214yescomplemented
49#482C2214yescomplemented
50#482C2214yescomplemented
51#482C2214yescomplemented
52#482C2214yescomplemented
53#481C2212yescomplemented
54#481C2214yescomplemented
55#480C2214yescomplemented
56#480C2212yescomplemented
57#478C2214yescomplemented
58#476C2212yescomplemented
59#476C2212yescomplemented
60#474C2212yescomplemented
61#474C2212yescomplemented
62#469C2214yescomplemented
63#469C2214yescomplemented
64#466C2214yescomplemented
65#466C2214yescomplemented
66#459C2212yes—
67#459C2212yes—
68#447C2212yescomplemented
69#447C2212yescomplemented
70#447C2214yescomplemented
71#447C2214yescomplemented
72#442C2214yescomplemented
73C4×D8C2212yescomplemented
74C4×D8C2212yescomplemented
75#436C2212yescomplemented
76#435C2214yescomplemented
77C8×D4C2212yescomplemented
78#432C2211yescharacteristic, complemented
79#431C2211yescharacteristic, complemented
80#428C2214yescomplemented
81#427C2214yescomplemented
82#418C2214yescomplemented
83C2×#102C2214yescomplemented
84#408C2214yescomplemented
85C2×#41—22no—
86C2×#41—21no—
87C2×#41—21no—
88C2×#41C2312yes—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4C2311yescharacteristic
92C22×D4C2314yes—
93C23×C4—21no—
94C2×D8C2312yes—
95C2×D8C2312yes—
96C2×D8C2314yes—
97C2×M16C2314yes—
98C22×C8C2314yes—
99C4⋊Q8D411yescharacteristic
100C4⋊Q8C2311yescharacteristic
101C4⋊Q8C2311yescharacteristic
102C4⋊Q8C2311yescharacteristic
103C4⋊Q8C2311yescharacteristic
104#127—24no—
105#127—22no—
106#127D411yescharacteristic
107#127C2311yescharacteristic
108#127C2311yescharacteristic
109#126—24no—
110#125C2312yes—
111#124—24no—
112#124—22no—
113#124—24no—
114#124—24no—
115#124—24no—
116#124—21no—
117#124—21no—
118#124D412yes—
119#123—24no—
120#123—24no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—22no—
124#121—22no—
125#121—22no—
126#121—22no—
127#121—24no—
128#121—21no—
129#121—21no—
130#121—22no—
131#121—22no—
132#121D412yes—
133#121D412yes—
134#121C2312yes—
135#121C2312yes—
136#121C2314yes—
137#121C2314yes—
138#121C2314yes—
139#121C2314yes—
140#120—24no—
141#120—24no—
142#120—24no—
143#120—24no—
144#120—24no—
145C4×Q8—22no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×D4—24no—
149C4×D4—22no—
150C4×D4—24no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4C2312yes—
156C4×D4C2312yes—
157C4×D4C2312yes—
158C4×D4C2312yes—
159C4×D4C2312yes—
160C4×D4C2312yes—
161#117—22no—
162#117—22no—
163#117—22no—
164#117C2312yes—
165#117C2314yes—
166#117C2314yes—
167C2×C4⋊C4C2311yescharacteristic
168C2×C4⋊C4C2314yes—
169C2×C22⋊C4—21no—
170C2×C42C2311yescharacteristic
171#107C2312yes—
172#107C2314yes—
173#107C2312yes—
174#107C2314yes—
175#106C2314yes—
176#106C2314yes—
177#106C2314yes—
178C4⋊C8C2312yes—
179C4⋊C8C2312yes—
180#102C2312yes—
181#102C2314yes—
182#102C2314yes—
183#102C2314yes—
184#102C2312yes—
185#102C2314yes—
186#102C2314yes—
187#102C2312yes—
188#102C2312yes—
189#102C2314yes—
190C22⋊C8C2312yes—
191C22⋊C8C2312yes—
192C22⋊C8C2314yes—
193C22⋊C8C2314yes—
194#97C2312yes—
195C4×C8C2311yescharacteristic
196C4×C8C2311yescharacteristic
197C24—21no—
198C24—24no—
199#41—44no—
200#41—82no—
201#41—82no—
202#41—84no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8C2×D412yes—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—24no—
220C2×D4—42no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4C2×D412yes—
226C2×D4C2×D414yes—
227C2×D4C2×D412yes—
228C2×D4C2×D412yes—
229C2×D4C2412yes—
230C2×D4C2412yes—
231C2×D4C2412yes—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—82no—
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2411yescharacteristic
242C22×C4C2414yes—
243D8—42no—
244D8—44no—
245D8—42no—
246M16—44no—
247C2×C8—44no—
248C2×C8C2412yes—
249C2×C8C2412yes—
250C2×C8C2414yes—
251C4⋊C4—21no—
252C4⋊C4—24no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4C2×D412yes—
260C4⋊C4C2×D412yes—
261C4⋊C4C2412yes—
262C4⋊C4C2412yes—
263C4⋊C4C2412yes—
264C4⋊C4C2412yes—
265C4⋊C4C2412yes—
266C22⋊C4—22no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—24no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4C2×D414yes—
280C22⋊C4C2×D414yes—
281C42—24no—
282C42—24no—
283C42—22no—
284C42C2×D411yescharacteristic
285C42C2×D411yescharacteristic
286C42C2411yescharacteristic
287C42C2411yescharacteristic
288C23—22no—
289C23—22no—
290C23—22no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—84no—
295C23—84no—
296C23—84no—
297C23C22×D414yes—
298C23C22×D411yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304D4—24no—
305D4—22no—
306D4—22no—
307D4—22no—
308D4—24no—
309D4—22no—
310D4—42no—
311D4—42no—
312D4—44no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—84no—
317D4—81no—
318D4—81no—
319D4—84no—
320D4—84no—
321D4—84no—
322D4—82no—
323D4—82no—
324D4—84no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4C22×D411yescharacteristic
342C2×C4C22×D414yes—
343C2×C4C22×D411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C2×C4C22×D411yescharacteristic
346C2×C4C22×D411yescharacteristic
347C2×C4C2511yescharacteristic
348C8—22no—
349C8—24no—
350C8—22no—
351C22—21no—
352C22—24no—
353C22—21no—
354C22—24no—
355C22—21no—
356C22—24no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—84no—
367C22—168no—
368C22C23×D411yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—24no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4#57612yes—
380C4#58512yes—
381C2—41no—
382C2—44no—
383C2—82no—
384C2—82no—
385C2—82no—
386C2#348511yescharacteristic
387C2C2×#57611yescharacteristic
388C2#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. #3549C2semidirect
2. #3506C2semidirect
3. #3354C2semidirect
4. #3347C2semidirect
5. #3341C2semidirect
6. #3325C2semidirect
7. #3318C2semidirect
8. #3294C2semidirect
9. #3235C2semidirect
10. #3228C2semidirect
11. #3199C2semidirect
12. #3109C2semidirect
13. #3100C2semidirect
14. #3096C2semidirect
15. #2981C2semidirect
20. #555C22semidirect
21. #555C22semidirect
22. #555C22semidirect
23. #553C22semidirect
25. #547C22semidirect
27. D42C22semidirect
28. D42C22semidirect
33. #535C22semidirect
34. #533C22semidirect
39. #520C22semidirect
40. C2×C4×D4C22semidirect
41. #501C22semidirect
42. #500C22semidirect
43. #498C22semidirect
44. #490C22semidirect
45. #486C22semidirect
46. #486C22semidirect
47. #482C22semidirect
48. #482C22semidirect
49. #482C22semidirect
50. #482C22semidirect
51. #482C22semidirect
52. #482C22semidirect
53. #481C22semidirect
54. #481C22semidirect
55. #480C22semidirect
56. #480C22semidirect
57. #478C22semidirect
58. #476C22semidirect
59. #476C22semidirect
60. #474C22semidirect
61. #474C22semidirect
62. #469C22semidirect
63. #469C22semidirect
64. #466C22semidirect
65. #466C22semidirect
68. #447C22semidirect
69. #447C22semidirect
70. #447C22semidirect
71. #447C22semidirect
72. #442C22semidirect
73. C4×D8C22semidirect
74. C4×D8C22semidirect
75. #436C22semidirect
76. #435C22semidirect
77. C8×D4C22semidirect
78. #432C22semidirect
79. #431C22semidirect
80. #428C22semidirect
81. #427C22semidirect
82. #418C22semidirect
83. C2×#102C22semidirect
84. #408C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#35062
#33542
#33472
#33412
#33252
#33182
#32942
#32352
#32282
#31992
#31092
#31002
#30962
#29812