Finite Group Databasebeta

← #61728

Subgroups — #61728

Order 256.

1038 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3#3354C212yesmaximal, complemented
4#3345C214yesmaximal, complemented
5#3335C212yesmaximal, complemented
6#3332C212yesmaximal, complemented
7#3306C214yesmaximal, complemented
8#3294C212yesmaximal, complemented
9#3239C211yesmaximal, characteristic, complemented
10#3227C211yesmaximal, characteristic, complemented
11#3192C212yesmaximal, complemented
12#3188C212yesmaximal, complemented
13#3109C211yesmaximal, characteristic, complemented
14#3105C211yesmaximal, characteristic, complemented
15#3092C212yesmaximal, complemented
16C2×#481C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18C22×#41—21no—
19#555C2211yescharacteristic, complemented
20#554C2211yescharacteristic, complemented
21#553—21no—
22#553C2212yescomplemented
23#550C2211yescharacteristic, complemented
24#547—21no—
25#547—22no—
26#547—21no—
27#547C2212yes—
28#547C2212yescomplemented
29#547C2214yescomplemented
30#546—22no—
31#546C2212yescomplemented
32D42—21no—
33D42C2211yescharacteristic, complemented
34#540—21no—
35#538—22no—
36#532C2211yescharacteristic
37C2×C4⋊Q8C2211yescharacteristic, complemented
38#529C2211yescharacteristic
39#525C2212yes—
40C2×#123—22no—
41C2×C22⋊Q8—21no—
42C2×#121C2212yes—
43C2×#121C2212yescomplemented
44#518—21no—
45C2×C4×D4C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic, complemented
47C22×C4⋊C4C2212yescomplemented
48#501C2212yescomplemented
49#498C2212yescomplemented
50#489C2212yescomplemented
51#486C2212yescomplemented
52#484C2211yescharacteristic, complemented
53#484C2212yescomplemented
54#483C2214yescomplemented
55#482C2214yescomplemented
56#481C2211yescharacteristic
57#481C2212yes—
58#481C2214yescomplemented
59#481C2214yescomplemented
60#481C2212yescomplemented
61#481C2214yescomplemented
62#481C2212yescomplemented
63#481C2212yescomplemented
64#480C2214yescomplemented
65#478C2212yescomplemented
66#476C2212yescomplemented
67#476C2214yescomplemented
68#475C2212yescomplemented
69#474C2212yescomplemented
70#469C2214yescomplemented
71#469C2214yescomplemented
72#466C2214yescomplemented
73#465C2214yescomplemented
74#463C2212yescomplemented
75#459C2212yescomplemented
76#450C2211yescharacteristic, complemented
77#449C2214yescomplemented
78#447C2211yescharacteristic
79#447C2212yescomplemented
80#447C2214yescomplemented
81#442C2214yescomplemented
82C4×SD16C2212yescomplemented
83C4×D8C2212yescomplemented
84#436C2212yescomplemented
85#435C2214yescomplemented
86C8×D4C2212yescomplemented
87#432C2211yescharacteristic, complemented
88#431C2211yescharacteristic, complemented
89#428C2214yescomplemented
90C2×#106C2214yescomplemented
91#417C2214yescomplemented
92C2×#102C2214yescomplemented
93#407C2212yescomplemented
94C2×C22⋊C8C2212yescomplemented
95C2×#41—22no—
96C2×#41—24no—
97C2×#41—22no—
98C2×#41—21no—
99C2×#41—48no—
100C2×#41C2312yes—
101C22×Q8—21no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4D411yescharacteristic
106C22×D4C2311yescharacteristic
107C22×D4C2312yes—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C2312yes—
111C2×SD16C2312yes—
112C2×D8C2312yes—
113C2×D8C2314yes—
114C2×M16C2314yes—
115C22×C8C2314yes—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C2312yes—
119C4⋊Q8C2312yes—
120C4⋊Q8C2311yescharacteristic
121#127—21no—
122#127C2311yescharacteristic
123#126—22no—
124#126—21no—
125#125—21no—
126#125C2311yescharacteristic
127#124—22no—
128#124—21no—
129#124C2311yescharacteristic
130#123—22no—
131#123—21no—
132#123—22no—
133#123—24no—
134#123—24no—
135#123—24no—
136#123—24no—
137#123—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—24no—
141C22⋊Q8—22no—
142C22⋊Q8C2311yescharacteristic
143C22⋊Q8C2312yes—
144#121—22no—
145#121—21no—
146#121—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—21no—
151#121—21no—
152#121—22no—
153#121—22no—
154#121C2311yescharacteristic
155#121C2312yes—
156#121C2312yes—
157#121C2312yes—
158#121C2314yes—
159#121C2314yes—
160#121C2312yes—
161#120—22no—
162#120—22no—
163#120—22no—
164C4×Q8C2311yescharacteristic
165C4×D4—21no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4D411yescharacteristic
175C4×D4D412yes—
176C4×D4D411yescharacteristic
177C4×D4C2312yes—
178C4×D4C2312yes—
179C4×D4C2311yescharacteristic
180C4×D4C2312yes—
181C4×D4C2311yescharacteristic
182C4×D4C2312yes—
183C4×D4C2311yescharacteristic
184#117—21no—
185#117—21no—
186#117C2312yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4D411yescharacteristic
192C2×C4⋊C4C2311yescharacteristic
193C2×C4⋊C4C2312yes—
194C2×C4⋊C4C2314yes—
195C2×C4⋊C4C2314yes—
196C2×C4⋊C4C2312yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4D412yes—
202C2×C42C2311yescharacteristic
203#107C2312yes—
204#107C2314yes—
205#106C2312yes—
206#106C2314yes—
207#106C2314yes—
208#106C2314yes—
209#106C2314yes—
210C4⋊C8C2312yes—
211C4⋊C8C2312yes—
212#103C2312yes—
213#103C2314yes—
214#102C2312yes—
215#102C2314yes—
216#102C2314yes—
217#102C2314yes—
218#102C2314yes—
219#102C2312yes—
220#102C2312yes—
221#102C2314yes—
222C22⋊C8C2311yescharacteristic
223C22⋊C8C2312yes—
224C22⋊C8C2314yes—
225C22⋊C8C2311yescharacteristic
226C22⋊C8C2314yes—
227#97C2312yes—
228C4×C8C2312yes—
229C24—21no—
230C24—22no—
231C24C2×D412yes—
232#41—44no—
233#41—416no—
234#41—48no—
235#41—48no—
236#41—82no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—21no—
240C2×Q8—44no—
241C2×Q8C2411yescharacteristic
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—48no—
258C2×D4—44no—
259C2×D4—84no—
260C2×D4C2×D412yes—
261C2×D4C2×D411yescharacteristic
262C2×D4C2×D411yescharacteristic
263C2×D4C2411yescharacteristic
264C2×D4C2412yes—
265C2×D4C2412yes—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4C2×D411yescharacteristic
280C22×C4C2×D411yescharacteristic
281C22×C4C2×D412yes—
282C22×C4C2×D412yes—
283C22×C4C2×D412yes—
284C22×C4C2411yescharacteristic
285C22×C4C2412yes—
286C22×C4C2412yes—
287SD16—42no—
288D8—44no—
289D8—42no—
290M16—44no—
291C2×C8—44no—
292C2×C8C2412yes—
293C2×C8C2412yes—
294C2×C8C2414yes—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D412yes—
311C4⋊C4C2412yes—
312C4⋊C4C2414yes—
313C4⋊C4C2411yescharacteristic
314C4⋊C4C2412yes—
315C4⋊C4C2411yescharacteristic
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—21no—
329C22⋊C4—22no—
330C22⋊C4—21no—
331C22⋊C4—22no—
332C22⋊C4C2×D412yes—
333C22⋊C4C2×D412yes—
334C22⋊C4C2×D412yes—
335C22⋊C4C2×D412yes—
336C42—22no—
337C42—21no—
338C42C2×D411yescharacteristic
339C42C2×D411yescharacteristic
340C42C2411yescharacteristic
341C42C2411yescharacteristic
342C23—24no—
343C23—24no—
344C23—22no—
345C23—21no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—82no—
353C23—82no—
354C23C22×D412yes—
355C23C22×D412yes—
356C23C22×D411yescharacteristic
357Q8—22no—
358Q8—44no—
359Q8—44no—
360Q8—81no—
361D4—24no—
362D4—22no—
363D4—22no—
364D4—22no—
365D4—22no—
366D4—24no—
367D4—42no—
368D4—48no—
369D4—44no—
370D4—48no—
371D4—42no—
372D4—44no—
373D4—42no—
374D4—82no—
375D4—81no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4AGL(1,ℤ/8)14yes—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4C22×D412yes—
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D411yescharacteristic
408C2×C4C22×D411yescharacteristic
409C2×C4C22×D412yes—
410C2×C4C22×D411yescharacteristic
411C2×C4C2511yescharacteristic
412C8—22no—
413C8—24no—
414C8—22no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—48no—
423C22—42no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—42no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—84no—
433C22—164no—
434C22C2×AGL(1,ℤ/8)14yes—
435C22#57712yes—
436C22C23×D411yescharacteristic
437C4—21no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—42no—
445C4—44no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4C2×AGL(1,ℤ/8)12yes—
450C4#58512yes—
451C2—22no—
452C2—24no—
453C2—42no—
454C2—82no—
455C2—82no—
456C2—81no—
457C2#348511yescharacteristic
458C2C22×AGL(1,ℤ/8)11yescharacteristic
459C2C2×#57711yescharacteristic
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×#547C2semidirect
3. #3354C2semidirect
4. #3345C2semidirect
5. #3335C2semidirect
6. #3332C2semidirect
7. #3306C2semidirect
8. #3294C2semidirect
9. #3239C2semidirect
10. #3227C2semidirect
11. #3192C2semidirect
12. #3188C2semidirect
13. #3109C2semidirect
14. #3105C2semidirect
15. #3092C2semidirect
16. C2×#481C2semidirect
17. #2974C2semidirect
19. #555C22semidirect
20. #554C22semidirect
22. #553C22semidirect
23. #550C22semidirect
28. #547C22semidirect
29. #547C22semidirect
31. #546C22semidirect
33. D42C22semidirect
37. C2×C4⋊Q8C22semidirect
43. C2×#121C22semidirect
46. C2×C4×D4C22semidirect
47. C22×C4⋊C4C22semidirect
48. #501C22semidirect
49. #498C22semidirect
50. #489C22semidirect
51. #486C22semidirect
52. #484C22semidirect
53. #484C22semidirect
54. #483C22semidirect
55. #482C22semidirect
58. #481C22semidirect
59. #481C22semidirect
60. #481C22semidirect
61. #481C22semidirect
62. #481C22semidirect
63. #481C22semidirect
64. #480C22semidirect
65. #478C22semidirect
66. #476C22semidirect
67. #476C22semidirect
68. #475C22semidirect
69. #474C22semidirect
70. #469C22semidirect
71. #469C22semidirect
72. #466C22semidirect
73. #465C22semidirect
74. #463C22semidirect
75. #459C22semidirect
76. #450C22semidirect
77. #449C22semidirect
79. #447C22semidirect
80. #447C22semidirect
81. #442C22semidirect
82. C4×SD16C22semidirect
83. C4×D8C22semidirect
84. #436C22semidirect
85. #435C22semidirect
86. C8×D4C22semidirect
87. #432C22semidirect
88. #431C22semidirect
89. #428C22semidirect
90. C2×#106C22semidirect
91. #417C22semidirect
92. C2×#102C22semidirect
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×#5472
#33542
#33452
#33352
#33322
#33062
#32942
#32392
#32272
#31922
#31882
#31092
#31052
#30922
C2×#4812
#29742