Finite Group Databasebeta

← #60922

Subgroups — #60922

Order 256.

966 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508C211yesmaximal, characteristic, complemented
2#3460C211yesmaximal, characteristic, complemented
3#3259C214yesmaximal, complemented
4#3257C214yesmaximal, complemented
5#3253C212yesmaximal, complemented
6#3251C212yesmaximal, complemented
7#3247C212yesmaximal, complemented
8#3245C212yesmaximal, complemented
9#3081C212yesmaximal, complemented
10#3077C211yesmaximal, characteristic, complemented
11#3076C211yesmaximal, characteristic, complemented
12#3062C212yesmaximal, complemented
13#3058C212yesmaximal, complemented
14#3041C212yesmaximal, complemented
15#3038C211yesmaximal, characteristic, complemented
16C2×#459C211yesmaximal, characteristic, complemented
17#2906C211yesmaximal, characteristic, complemented
18C2×AGL(1,ℤ/8)C2212yes—
19C22×D8C2212yes—
20C2×#131C2212yes—
21#559—21no—
22#559—21no—
23#550—21no—
24#550—21no—
25#546—22no—
26#546—22no—
27#544C2211yescharacteristic
28#542—21no—
29#542C2212yescomplemented
30#540—21no—
31#540C2212yescomplemented
32#540C2214yescomplemented
33#538—22no—
34#537C2211yescharacteristic
35#535C2211yescharacteristic
36#535C2211yescharacteristic
37#535C2212yes—
38#533C2211yescharacteristic
39#532—21no—
40#532C2212yescomplemented
41#532C2212yescomplemented
42#532C2212yescomplemented
43C2×#127C2211yescharacteristic
44#529—21no—
45#529C2212yescomplemented
46#525—22no—
47#518C2212yes—
48C4×#41C2211yescharacteristic
49C4×#41C2211yescharacteristic
50C2×C4×D4C2211yescharacteristic
51#479C2212yescomplemented
52#478C2212yescomplemented
53#478C2214yescomplemented
54#478C2214yescomplemented
55#478C2214yescomplemented
56#477C2211yescharacteristic, complemented
57#476C2211yescharacteristic, complemented
58#476C2212yescomplemented
59#476C2214yescomplemented
60#475C2211yescharacteristic, complemented
61#474C2211yescharacteristic, complemented
62#474C2212yescomplemented
63#474C2214yescomplemented
64#469C2214yescomplemented
65#469C2214yescomplemented
66#469C2214yescomplemented
67#468C2214yescomplemented
68#466C2214yescomplemented
69#466C2214yescomplemented
70#466C2214yescomplemented
71#465C2214yescomplemented
72#463C2214yescomplemented
73#463C2214yescomplemented
74#460C2212yescomplemented
75#460C2212yescomplemented
76#459C2212yes—
77#459C2212yes—
78#459C2214yes—
79#459C2212yescomplemented
80#459C2214yescomplemented
81#459C2214yescomplemented
82#459C2212yescomplemented
83#428C2212yes—
84#427C2212yes—
85#424C2212yes—
86#424C2214yescomplemented
87#423C2211yescharacteristic
88#423C2212yescomplemented
89C2×C4⋊C8C2211yescharacteristic
90C2×C4⋊C8C2212yescomplemented
91#418C2212yes—
92#418C2212yes—
93#416C2212yes—
94C2×#102C2212yes—
95C2×#41—22no—
96C2×#41—22no—
97C2×#41—22no—
98C2×#41C2311yescharacteristic
99C2×#41C2311yescharacteristic
100C22×D4—21no—
101C22×D4—21no—
102C22×D4C2311yescharacteristic
103C22×D4C2312yes—
104C23×C4—21no—
105AGL(1,ℤ/8)—44no—
106C2×SD16C2314yes—
107C2×D8—44no—
108C2×D8C2314yes—
109C2×D8C2314yes—
110C2×D8C2314yes—
111#131—44no—
112C2×M16C2312yes—
113C2×M16C2314yes—
114C22×C8C2312yes—
115C22×C8C2314yes—
116C4⋊Q8—21no—
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2312yes—
120#127—21no—
121#127—22no—
122#127—21no—
123#127—21no—
124#127C2312yes—
125#127C2312yes—
126#127C2312yes—
127#126—22no—
128#126—22no—
129#125—21no—
130#125C2312yes—
131#125C2312yes—
132#124—21no—
133#124—22no—
134#124—22no—
135#124—22no—
136#124—22no—
137#124—22no—
138#124C2312yes—
139#123—22no—
140#123—22no—
141#123—22no—
142#123—24no—
143#123—24no—
144#123—22no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8C2312yes—
151#121—22no—
152#121—21no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—21no—
157#121—21no—
158#121—22no—
159#121—22no—
160#121—24no—
161#121—24no—
162#121C2312yes—
163#121C2314yes—
164#121C2314yes—
165#121C2312yes—
166#121C2312yes—
167#120—22no—
168#120—24no—
169#120—22no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8D411yescharacteristic
173C4×Q8D411yescharacteristic
174C4×Q8C2311yescharacteristic
175C4×Q8C2311yescharacteristic
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4D412yes—
187C4×D4D411yescharacteristic
188C4×D4D412yes—
189C4×D4D411yescharacteristic
190C4×D4C2312yes—
191C4×D4C2312yes—
192C4×D4C2311yescharacteristic
193C4×D4C2311yescharacteristic
194C4×D4C2314yes—
195C4×D4C2314yes—
196#117—21no—
197#117—22no—
198#117C2311yescharacteristic
199#117C2312yes—
200#117C2311yescharacteristic
201#117C2312yes—
202C2×C4⋊C4C2311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—24no—
205C2×C42—21no—
206C2×C42C2311yescharacteristic
207C2×C42C2312yes—
208#107C2312yes—
209#107C2312yes—
210#107C2314yes—
211#106C2312yes—
212#106C2312yes—
213#106C2314yes—
214C4⋊C8C2311yescharacteristic
215C4⋊C8C2311yescharacteristic
216C4⋊C8C2312yes—
217C4⋊C8C2312yes—
218C4⋊C8C2314yes—
219C4⋊C8C2312yes—
220C4⋊C8C2314yes—
221#103C2312yes—
222#103C2312yes—
223#102C2314yes—
224#102C2314yes—
225#102C2312yes—
226#102C2312yes—
227#102C2314yes—
228#102C2314yes—
229#102C2314yes—
230#102C2314yes—
231C24—22no—
232C24—21no—
233#41—22no—
234#41—22no—
235#41—22no—
236#41—22no—
237#41—84no—
238#41—84no—
239#41—84no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8C2×D411yescharacteristic
244C2×Q8C2411yescharacteristic
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4C2×D412yes—
266C2×D4C2×D411yescharacteristic
267C2×D4C2412yes—
268C2×D4C2411yescharacteristic
269C2×D4C2414yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—82no—
279C22×C4C2×D412yes—
280C22×C4C2×D411yescharacteristic
281C22×C4C2×D411yescharacteristic
282C22×C4C2×D412yes—
283C22×C4C2412yes—
284C22×C4C2411yescharacteristic
285SD16—44no—
286D8—44no—
287D8—44no—
288D8—44no—
289M16—24no—
290M16—44no—
291C2×C8—24no—
292C2×C8—44no—
293C2×C8C2414yes—
294C2×C8C2412yes—
295C2×C8C2412yes—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4C2×D411yescharacteristic
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D412yes—
310C4⋊C4C2×D412yes—
311C4⋊C4C2412yes—
312C4⋊C4C2411yescharacteristic
313C4⋊C4C2411yescharacteristic
314C4⋊C4C2412yes—
315C4⋊C4C2412yes—
316C22⋊C4—22no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—22no—
332C22⋊C4C2×D412yes—
333C22⋊C4C2×D412yes—
334C22⋊C4C2×D412yes—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—22no—
339C42C2×D411yescharacteristic
340C42C2×D412yes—
341C42C2×D411yescharacteristic
342C42C2×D412yes—
343C42C2411yescharacteristic
344C42C2412yes—
345C42C2411yescharacteristic
346C23—22no—
347C23—21no—
348C23—24no—
349C23—42no—
350C23—44no—
351C23—81no—
352C23—81no—
353C23—82no—
354C23—82no—
355C23C22×D412yes—
356C23C22×D411yescharacteristic
357Q8—22no—
358Q8—21no—
359Q8—21no—
360Q8—42no—
361Q8—42no—
362Q8—82no—
363D4—24no—
364D4—22no—
365D4—22no—
366D4—21no—
367D4—24no—
368D4—22no—
369D4—24no—
370D4—21no—
371D4—44no—
372D4—42no—
373D4—42no—
374D4—44no—
375D4—44no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—84no—
382D4—82no—
383D4—82no—
384D4—84no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—81no—
405C2×C4C22×D411yescharacteristic
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D412yes—
408C2×C4C22×D412yes—
409C2×C4C22×D412yes—
410C2×C4C22×D411yescharacteristic
411C2×C4C22×D411yescharacteristic
412C2×C4C22×D411yescharacteristic
413C2×C4H(5,𝔽2)12yes—
414C2×C42−1+412yes—
415C2×C4C2511yescharacteristic
416C8—22no—
417C8—22no—
418C8—24no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—44no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—42no—
429C22—44no—
430C22—42no—
431C22—41no—
432C22—44no—
433C22—84no—
434C22—164no—
435C22#53712yes—
436C22C23×D411yescharacteristic
437C4—21no—
438C4—22no—
439C4—21no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—42no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—41no—
448C4—42no—
449C4#53712yes—
450C4#57612yes—
451C4C2×H(5,𝔽2)11yescharacteristic
452C4C2×2−1+411yescharacteristic
453C2—22no—
454C2—42no—
455C2—81no—
456C2—82no—
457C2—84no—
458C2C2×#53711yescharacteristic
459C2C2×#57611yescharacteristic
460C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3508C2semidirect
2. #3460C2semidirect
3. #3259C2semidirect
4. #3257C2semidirect
5. #3253C2semidirect
6. #3251C2semidirect
7. #3247C2semidirect
8. #3245C2semidirect
9. #3081C2semidirect
10. #3077C2semidirect
11. #3076C2semidirect
12. #3062C2semidirect
13. #3058C2semidirect
14. #3041C2semidirect
15. #3038C2semidirect
16. C2×#459C2semidirect
17. #2906C2semidirect
29. #542C22semidirect
31. #540C22semidirect
32. #540C22semidirect
40. #532C22semidirect
41. #532C22semidirect
42. #532C22semidirect
45. #529C22semidirect
51. #479C22semidirect
52. #478C22semidirect
53. #478C22semidirect
54. #478C22semidirect
55. #478C22semidirect
56. #477C22semidirect
57. #476C22semidirect
58. #476C22semidirect
59. #476C22semidirect
60. #475C22semidirect
61. #474C22semidirect
62. #474C22semidirect
63. #474C22semidirect
64. #469C22semidirect
65. #469C22semidirect
66. #469C22semidirect
67. #468C22semidirect
68. #466C22semidirect
69. #466C22semidirect
70. #466C22semidirect
71. #465C22semidirect
72. #463C22semidirect
73. #463C22semidirect
74. #460C22semidirect
75. #460C22semidirect
79. #459C22semidirect
80. #459C22semidirect
81. #459C22semidirect
82. #459C22semidirect
86. #424C22semidirect
88. #423C22semidirect
90. C2×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#34602
#32592
#32572
#32532
#32512
#32472
#32452
#30812
#30772
#30762
#30622
#30582
#30412
#30382
C2×#4592
#29062