Finite Group Databasebeta

← #61788

Subgroups — #61788

Order 256.

923 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3511C211yesmaximal, characteristic, complemented
3#3357C212yesmaximal, complemented
4#3354C212yesmaximal, complemented
5#3341C212yesmaximal, complemented
6#3340C212yesmaximal, complemented
7#3325C212yesmaximal, complemented
8#3324C212yesmaximal, complemented
9#3295C212yesmaximal, complemented
10#3294C212yesmaximal, complemented
11#3227C211yesmaximal, characteristic, complemented
12#3226C211yesmaximal, characteristic, complemented
13#3198C214yesmaximal, complemented
14#3106C211yesmaximal, characteristic, complemented
15#3106C211yesmaximal, characteristic, complemented
16#3096C214yesmaximal, complemented
17#2979C211yesmaximal, characteristic, complemented
18H(3,𝔽4)—22no—
19#556C2212yescomplemented
20#555—21no—
21#555C2212yes—
22#555C2212yes—
23#555C2212yescomplemented
24#554C2212yescomplemented
25#553—22no—
26#548—22no—
27#547C2212yescomplemented
28D42—21no—
29D42C2212yes—
30D42C2212yes—
31#542—22no—
32#537C2212yescomplemented
33#535—22no—
34#535—22no—
35#535C2214yes—
36#533—21no—
37#532—21no—
38#532C2211yescharacteristic
39#532C2211yescharacteristic
40#532C2211yescharacteristic
41C2×C4⋊Q8C2211yescharacteristic, complemented
42#520C2212yescomplemented
43C4×#41C2211yescharacteristic, complemented
44#500C2212yescomplemented
45#498C2212yescomplemented
46#486C2212yes—
47#486C2212yes—
48#485C2214yescomplemented
49#482C2214yescomplemented
50#482C2214yescomplemented
51#482C2214yescomplemented
52#482C2212yescomplemented
53#482C2212yescomplemented
54#481C2214yescomplemented
55#481C2214yescomplemented
56#480C2214yescomplemented
57#480C2214yescomplemented
58#477C2211yescharacteristic, complemented
59#477C2211yescharacteristic, complemented
60#476C2211yescharacteristic, complemented
61#476C2212yescomplemented
62#476C2211yescharacteristic, complemented
63#476C2212yescomplemented
64#474C2212yescomplemented
65#474C2212yescomplemented
66#469C2214yescomplemented
67#468C2214yescomplemented
68#466C2214yescomplemented
69#465C2214yescomplemented
70#460C2211yescharacteristic
71#460C2211yescharacteristic
72#459C2211yescharacteristic
73#459C2211yescharacteristic
74#450C2214yescomplemented
75#449C2212yescomplemented
76#449C2212yescomplemented
77#447C2214yescomplemented
78C4×SD16C2212yescomplemented
79C4×SD16C2212yescomplemented
80C4×D8C2212yescomplemented
81C4×D8C2212yescomplemented
82#436C2212yescomplemented
83#436C2212yescomplemented
84C8×D4C2212yescomplemented
85C8×D4C2212yescomplemented
86#431C2211yescharacteristic, complemented
87#431C2211yescharacteristic, complemented
88#428C2214yescomplemented
89#427C2214yescomplemented
90#418C2214yescomplemented
91C2×#102C2214yescomplemented
92#408C2214yescomplemented
93C2×#41—22no—
94C2×#41—21no—
95C2×#41—21no—
96C2×#41C2311yescharacteristic
97C2×#41C2312yes—
98C22×Q8—21no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4C2314yes—
102C2×SD16C2312yes—
103C2×SD16C2312yes—
104C2×D8C2312yes—
105C2×D8C2312yes—
106C2×M16C2314yes—
107C22×C8C2314yes—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8D411yescharacteristic
111C4⋊Q8C2312yes—
112C4⋊Q8C2312yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115#127—24no—
116#127D411yescharacteristic
117#127C2311yescharacteristic
118#127C2311yescharacteristic
119#126—24no—
120#125—22no—
121#124—24no—
122#124—22no—
123#124—24no—
124#124—24no—
125#124—24no—
126#124—21no—
127#124—21no—
128#124D412yes—
129#123—24no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—24no—
133C22⋊Q8C2314yes—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—24no—
139#121—21no—
140#121—21no—
141#121—22no—
142#121—22no—
143#121D412yes—
144#121D412yes—
145#121C2312yes—
146#121C2312yes—
147#121C2314yes—
148#121C2314yes—
149#121C2314yes—
150#120—24no—
151#120—24no—
152#120—24no—
153#120—24no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8C2311yescharacteristic
157C4×Q8C2311yescharacteristic
158C4×D4—22no—
159C4×D4—24no—
160C4×D4—22no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4C2311yescharacteristic
166C4×D4C2311yescharacteristic
167C4×D4C2312yes—
168C4×D4C2312yes—
169C4×D4C2312yes—
170C4×D4C2312yes—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—21no—
175#117C2311yescharacteristic
176#117C2314yes—
177#117C2314yes—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C2312yes—
180C2×C4⋊C4C2314yes—
181C2×C42—21no—
182C2×C42C2311yescharacteristic
183#107C2312yes—
184#107C2314yes—
185#107C2312yes—
186#107C2314yes—
187#106C2312yes—
188#106C2314yes—
189#106C2312yes—
190#106C2314yes—
191C4⋊C8C2311yescharacteristic
192C4⋊C8C2311yescharacteristic
193C4⋊C8C2311yescharacteristic
194C4⋊C8C2311yescharacteristic
195#103C2312yes—
196#103C2312yes—
197#102C2312yes—
198#102C2312yes—
199#102C2314yes—
200#102C2314yes—
201#102C2312yes—
202#102C2312yes—
203#102C2314yes—
204#102C2314yes—
205#102C2312yes—
206#102C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2312yes—
209C22⋊C8C2314yes—
210C22⋊C8C2314yes—
211C4×C8C2312yes—
212C4×C8C2312yes—
213C24—24no—
214#41—42no—
215#41—42no—
216#41—41no—
217#41—41no—
218#41—82no—
219#41—82no—
220#41—84no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—82no—
226C2×Q8C2×D412yes—
227C2×Q8C2411yescharacteristic
228C2×D4—21no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—24no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—84no—
242C2×D4—84no—
243C2×D4C2×D412yes—
244C2×D4C2×D412yes—
245C2×D4C2×D412yes—
246C2×D4C2×D414yes—
247C2×D4C2412yes—
248C2×D4C2412yes—
249C2×D4C2411yescharacteristic
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2411yescharacteristic
260C22×C4C2414yes—
261SD16—42no—
262SD16—42no—
263D8—42no—
264D8—42no—
265M16—44no—
266C2×C8—44no—
267C2×C8C2412yes—
268C2×C8C2412yes—
269C2×C8C2412yes—
270C2×C8C2412yes—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—24no—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C4⋊C4C2411yescharacteristic
283C4⋊C4C2414yes—
284C4⋊C4C2411yescharacteristic
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C22⋊C4—21no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4C2×D414yes—
301C22⋊C4C2×D414yes—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—24no—
306C42—24no—
307C42C2×D411yescharacteristic
308C42C2×D411yescharacteristic
309C42C2411yescharacteristic
310C42C2411yescharacteristic
311C23—21no—
312C23—22no—
313C23—22no—
314C23—84no—
315C23—84no—
316C23—84no—
317C23C22×D414yes—
318C23C22×D411yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—42no—
322Q8—44no—
323Q8—82no—
324Q8—81no—
325Q8—81no—
326D4—24no—
327D4—24no—
328D4—21no—
329D4—21no—
330D4—22no—
331D4—22no—
332D4—22no—
333D4—22no—
334D4—42no—
335D4—44no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—84no—
340D4—81no—
341D4—81no—
342D4—84no—
343D4—84no—
344D4—82no—
345D4—84no—
346D4—82no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4C22×D414yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C2511yescharacteristic
373C8—22no—
374C8—22no—
375C8—22no—
376C8—22no—
377C22—21no—
378C22—24no—
379C22—21no—
380C22—24no—
381C22—21no—
382C22—24no—
383C22—41no—
384C22—42no—
385C22—42no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—168no—
393C22C23×D411yescharacteristic
394C4—21no—
395C4—24no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—42no—
401C4—44no—
402C4—41no—
403C4—42no—
404C4#57612yes—
405C4#57712yes—
406C4#58511yescharacteristic
407C4#58511yescharacteristic
408C2—41no—
409C2—44no—
410C2—81no—
411C2—82no—
412C2—82no—
413C2#348511yescharacteristic
414C2C2×#57611yescharacteristic
415C2C2×#57711yescharacteristic
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. #3511C2semidirect
3. #3357C2semidirect
4. #3354C2semidirect
5. #3341C2semidirect
6. #3340C2semidirect
7. #3325C2semidirect
8. #3324C2semidirect
9. #3295C2semidirect
10. #3294C2semidirect
11. #3227C2semidirect
12. #3226C2semidirect
13. #3198C2semidirect
14. #3106C2semidirect
15. #3106C2semidirect
16. #3096C2semidirect
17. #2979C2semidirect
19. #556C22semidirect
23. #555C22semidirect
24. #554C22semidirect
27. #547C22semidirect
32. #537C22semidirect
41. C2×C4⋊Q8C22semidirect
42. #520C22semidirect
43. C4×#41C22semidirect
44. #500C22semidirect
45. #498C22semidirect
48. #485C22semidirect
49. #482C22semidirect
50. #482C22semidirect
51. #482C22semidirect
52. #482C22semidirect
53. #482C22semidirect
54. #481C22semidirect
55. #481C22semidirect
56. #480C22semidirect
57. #480C22semidirect
58. #477C22semidirect
59. #477C22semidirect
60. #476C22semidirect
61. #476C22semidirect
62. #476C22semidirect
63. #476C22semidirect
64. #474C22semidirect
65. #474C22semidirect
66. #469C22semidirect
67. #468C22semidirect
68. #466C22semidirect
69. #465C22semidirect
74. #450C22semidirect
75. #449C22semidirect
76. #449C22semidirect
77. #447C22semidirect
78. C4×SD16C22semidirect
79. C4×SD16C22semidirect
80. C4×D8C22semidirect
81. C4×D8C22semidirect
82. #436C22semidirect
83. #436C22semidirect
84. C8×D4C22semidirect
85. C8×D4C22semidirect
86. #431C22semidirect
87. #431C22semidirect
88. #428C22semidirect
89. #427C22semidirect
90. #418C22semidirect
91. C2×#102C22semidirect
92. #408C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#35112
#33572
#33542
#33412
#33402
#33252
#33242
#32952
#32942
#32272
#32262
#31982
#31062
#30962
#29792