Finite Group Databasebeta

← #61605

Subgroups — #61605

Order 256.

1116 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2C2×#550C211yesmaximal, characteristic, complemented
3#3333C212yesmaximal, complemented
4#3309C214yesmaximal, complemented
5#3305C212yesmaximal, complemented
6#3287C212yesmaximal, complemented
7#3274C212yesmaximal, complemented
8#3268C214yesmaximal, complemented
9#3217C211yesmaximal, characteristic, complemented
10#3205C211yesmaximal, characteristic, complemented
11#3194C212yesmaximal, complemented
12#3186C212yesmaximal, complemented
13#3112C211yesmaximal, characteristic, complemented
14#3108C211yesmaximal, characteristic, complemented
15#3006C212yesmaximal, complemented
16#3002C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18C22×#41C2212yescomplemented
19C2×AGL(1,ℤ/8)C2214yescomplemented
20C2×#135C2214yescomplemented
21#555C2211yescharacteristic, complemented
22#554C2211yescharacteristic, complemented
23#553—21no—
24#553C2212yescomplemented
25#550—21no—
26#550—21no—
27#550—22no—
28#550C2212yes—
29#550C2214yescomplemented
30#550C2212yescomplemented
31#550C2211yescharacteristic, complemented
32#546—22no—
33#546C2212yescomplemented
34D42—21no—
35D42C2211yescharacteristic, complemented
36#540—21no—
37#538—22no—
38#532C2211yescharacteristic
39C2×#127—21no—
40C2×#127C2211yescharacteristic, complemented
41#529C2211yescharacteristic
42#525C2212yes—
43C2×#121—22no—
44C2×#121C2212yes—
45C2×#121C2212yescomplemented
46#518—21no—
47C2×C4×Q8C2211yescharacteristic, complemented
48C2×C4×D4—21no—
49C2×C4×D4C2211yescharacteristic
50#496C2212yescomplemented
51#493C2212yescomplemented
52#490C2212yescomplemented
53#486C2212yescomplemented
54#483C2211yescharacteristic, complemented
55#482C2214yescomplemented
56#480C2211yescharacteristic
57#480C2214yescomplemented
58#480C2212yescomplemented
59#479C2212yescomplemented
60#477C2212yescomplemented
61#469C2214yescomplemented
62#468C2214yescomplemented
63#466C2214yescomplemented
64#466C2214yescomplemented
65#464C2212yescomplemented
66#460C2212yescomplemented
67#460C2212yescomplemented
68#459C2214yescomplemented
69#459C2212yescomplemented
70#452C2214yescomplemented
71#451C2211yescharacteristic, complemented
72#450C2212yescomplemented
73#449C2214yescomplemented
74#449C2214yescomplemented
75#449C2214yescomplemented
76#449C2214yescomplemented
77#448C2211yescharacteristic
78#448C2212yescomplemented
79#447C2212yes—
80#447C2212yescomplemented
81#447C2214yescomplemented
82#447C2212yescomplemented
83#440C2214yescomplemented
84C4×Q16C2212yescomplemented
85C4×SD16C2212yescomplemented
86#436C2212yescomplemented
87#435C2214yescomplemented
88C8×D4C2212yescomplemented
89#432C2211yescharacteristic, complemented
90#431C2211yescharacteristic, complemented
91#417C2214yescomplemented
92#416C2214yescomplemented
93#407C2212yescomplemented
94C2×C22⋊C8C2212yescomplemented
95C2×#41—22no—
96C2×#41—22no—
97C2×#41—21no—
98C2×#41—48no—
99C2×#41C2314yes—
100C2×#41C2314yes—
101C2×#41C2312yes—
102C22×Q8C2311yescharacteristic
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4D411yescharacteristic
110C22×D4C2312yes—
111C22×D4C2312yes—
112C23×C4—22no—
113C23×C4C2312yes—
114AGL(1,ℤ/8)—48no—
115#135—48no—
116C2×Q16C2312yes—
117C2×SD16C2314yes—
118C2×SD16C2312yes—
119C2×SD16C2314yes—
120C2×D8C2314yes—
121C2×D8C2314yes—
122C2×D8C2314yes—
123C2×M16C2314yes—
124C22×C8C2314yes—
125C4⋊Q8C2311yescharacteristic
126#127—22no—
127#127—22no—
128#127—24no—
129#127—21no—
130#127—21no—
131#127—21no—
132#127C2312yes—
133#127C2312yes—
134#127C2311yescharacteristic
135#126—22no—
136#126—21no—
137#125—21no—
138#125C2311yescharacteristic
139#124—22no—
140#124—21no—
141#124C2311yescharacteristic
142#123—22no—
143#123—21no—
144#123—22no—
145#123—22no—
146C22⋊Q8—22no—
147C22⋊Q8C2311yescharacteristic
148C22⋊Q8C2312yes—
149#121—22no—
150#121—21no—
151#121—22no—
152#121—22no—
153#121—22no—
154#121—22no—
155#121—21no—
156#121—21no—
157#121—24no—
158#121—24no—
159#121—24no—
160#121—24no—
161#121—22no—
162#121—22no—
163#121C2311yescharacteristic
164#121C2312yes—
165#121C2312yes—
166#121C2312yes—
167#121C2314yes—
168#121C2314yes—
169#121C2312yes—
170#120—22no—
171#120—22no—
172#120—22no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8C2312yes—
176C4×Q8C2312yes—
177C4×Q8C2311yescharacteristic
178C4×D4—21no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—24no—
185C4×D4—21no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4D411yescharacteristic
189C4×D4D412yes—
190C4×D4D411yescharacteristic
191C4×D4C2311yescharacteristic
192C4×D4C2312yes—
193C4×D4C2311yescharacteristic
194C4×D4C2312yes—
195C4×D4C2311yescharacteristic
196#117—21no—
197#117—21no—
198#117C2312yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4D411yescharacteristic
201C2×C4⋊C4C2311yescharacteristic
202C2×C4⋊C4C2312yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4D412yes—
207C2×C42—21no—
208C2×C42C2311yescharacteristic
209#107C2312yes—
210#107C2314yes—
211#106C2312yes—
212C4⋊C8C2312yes—
213C4⋊C8C2312yes—
214#103C2312yes—
215#103C2314yes—
216#103C2312yes—
217#103C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221#102C2314yes—
222#102C2312yes—
223#102C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2312yes—
226C22⋊C8C2314yes—
227C22⋊C8C2311yescharacteristic
228C22⋊C8C2314yes—
229#97C2312yes—
230C4×C8C2312yes—
231C24—22no—
232C24—22no—
233C24C2×D412yes—
234#41—28no—
235#41—28no—
236#41—44no—
237#41—44no—
238#41—44no—
239#41—48no—
240#41—48no—
241#41—82no—
242C2×Q8—24no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8C2411yescharacteristic
246C2×Q8C2412yes—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—82no—
272C2×D4—84no—
273C2×D4—84no—
274C2×D4C2×D411yescharacteristic
275C2×D4C2×D411yescharacteristic
276C2×D4C2×D412yes—
277C2×D4C2414yes—
278C2×D4C2411yescharacteristic
279C2×D4C2412yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—44no—
289C22×C4—48no—
290C22×C4C2×D411yescharacteristic
291C22×C4C2×D411yescharacteristic
292C22×C4C2×D412yes—
293C22×C4C2×D412yes—
294C22×C4C2×D412yes—
295C22×C4C2411yescharacteristic
296C22×C4C2412yes—
297C22×C4C2412yes—
298Q16—24no—
299SD16—28no—
300SD16—24no—
301SD16—44no—
302D8—44no—
303D8—44no—
304D8—44no—
305M16—44no—
306C2×C8—44no—
307C2×C8C2412yes—
308C2×C8C2412yes—
309C2×C8C2414yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4C2×D411yescharacteristic
321C4⋊C4C2×D411yescharacteristic
322C4⋊C4C2×D412yes—
323C4⋊C4C2411yescharacteristic
324C4⋊C4C2411yescharacteristic
325C4⋊C4C2412yes—
326C4⋊C4C2412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—22no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—21no—
338C22⋊C4—22no—
339C22⋊C4—21no—
340C22⋊C4—22no—
341C22⋊C4C2×D412yes—
342C22⋊C4C2×D412yes—
343C22⋊C4C2×D412yes—
344C22⋊C4C2×D412yes—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42C2×D411yescharacteristic
350C42C2×D411yescharacteristic
351C42C2411yescharacteristic
352C42C2411yescharacteristic
353C23—24no—
354C23—21no—
355C23—22no—
356C23—24no—
357C23—42no—
358C23—42no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—82no—
364C23—82no—
365C23C22×D412yes—
366C23C22×D412yes—
367C23C22×D411yescharacteristic
368Q8—24no—
369Q8—22no—
370Q8—22no—
371Q8—81no—
372Q8H(5,𝔽2)14yes—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—24no—
377D4—24no—
378D4—24no—
379D4—42no—
380D4—48no—
381D4—48no—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—42no—
386D4—44no—
387D4—42no—
388D4—44no—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—81no—
393D4—84no—
394D4—84no—
395D4—82no—
396D4—82no—
397D4—82no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4C22×D411yescharacteristic
421C2×C4C22×D412yes—
422C2×C4C22×D411yescharacteristic
423C2×C4C22×D411yescharacteristic
424C2×C4C22×D411yescharacteristic
425C2×C4C22×D412yes—
426C2×C4C22×D411yescharacteristic
427C2×C4C2511yescharacteristic
428C8—22no—
429C8—24no—
430C8—22no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—44no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—44no—
448C22—84no—
449C22—88no—
450C22—164no—
451C22#57612yes—
452C22C23×D411yescharacteristic
453C4—24no—
454C4—22no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4#57512yes—
466C4C2×H(5,𝔽2)12yes—
467C2—22no—
468C2—42no—
469C2—42no—
470C2—84no—
471C2—82no—
472C2—81no—
473C2#348411yescharacteristic
474C2C2×#57511yescharacteristic
475C2C2×#57611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3519C2semidirect
2. C2×#550C2semidirect
3. #3333C2semidirect
4. #3309C2semidirect
5. #3305C2semidirect
6. #3287C2semidirect
7. #3274C2semidirect
8. #3268C2semidirect
9. #3217C2semidirect
10. #3205C2semidirect
11. #3194C2semidirect
12. #3186C2semidirect
13. #3112C2semidirect
14. #3108C2semidirect
15. #3006C2semidirect
16. #3002C2semidirect
17. #2974C2semidirect
18. C22×#41C22semidirect
19. C2×AGL(1,ℤ/8)C22semidirect
20. C2×#135C22semidirect
21. #555C22semidirect
22. #554C22semidirect
24. #553C22semidirect
29. #550C22semidirect
30. #550C22semidirect
31. #550C22semidirect
33. #546C22semidirect
35. D42C22semidirect
40. C2×#127C22semidirect
45. C2×#121C22semidirect
47. C2×C4×Q8C22semidirect
50. #496C22semidirect
51. #493C22semidirect
52. #490C22semidirect
53. #486C22semidirect
54. #483C22semidirect
55. #482C22semidirect
57. #480C22semidirect
58. #480C22semidirect
59. #479C22semidirect
60. #477C22semidirect
61. #469C22semidirect
62. #468C22semidirect
63. #466C22semidirect
64. #466C22semidirect
65. #464C22semidirect
66. #460C22semidirect
67. #460C22semidirect
68. #459C22semidirect
69. #459C22semidirect
70. #452C22semidirect
71. #451C22semidirect
72. #450C22semidirect
73. #449C22semidirect
74. #449C22semidirect
75. #449C22semidirect
76. #449C22semidirect
78. #448C22semidirect
80. #447C22semidirect
81. #447C22semidirect
82. #447C22semidirect
83. #440C22semidirect
84. C4×Q16C22semidirect
85. C4×SD16C22semidirect
86. #436C22semidirect
87. #435C22semidirect
88. C8×D4C22semidirect
89. #432C22semidirect
90. #431C22semidirect
91. #417C22semidirect
92. #416C22semidirect
93. #407C22semidirect
94. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35192
C2×#5502
#33332
#33092
#33052
#32872
#32742
#32682
#32172
#32052
#31942
#31862
#31122
#31082
#30062
#30022
#29742