Finite Group Databasebeta

← #61740

Subgroups — #61740

Order 256.

940 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2C2×#555C211yesmaximal, characteristic, complemented
3#3357C212yesmaximal, complemented
4#3347C214yesmaximal, complemented
5#3341C212yesmaximal, complemented
6#3325C212yesmaximal, complemented
7#3319C214yesmaximal, complemented
8#3305C212yesmaximal, complemented
9#3238C211yesmaximal, characteristic, complemented
10#3226C211yesmaximal, characteristic, complemented
11#3194C212yesmaximal, complemented
12#3186C212yesmaximal, complemented
13#3109C211yesmaximal, characteristic, complemented
14#3105C211yesmaximal, characteristic, complemented
15#3094C212yesmaximal, complemented
16#3091C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18#555—21no—
19#555—21no—
20#555—22no—
21#555C2212yes—
22#555C2211yescharacteristic, complemented
23#555C2212yescomplemented
24#555C2214yescomplemented
25#554C2211yescharacteristic, complemented
26#553—21no—
27#553C2212yescomplemented
28#550C2211yescharacteristic, complemented
29#546—22no—
30#546C2212yescomplemented
31D42—21no—
32D42C2211yescharacteristic, complemented
33#540—21no—
34#538—22no—
35#532C2211yescharacteristic
36C2×C4⋊Q8C2211yescharacteristic, complemented
37#529C2211yescharacteristic
38C2×#124—22no—
39#525C2212yes—
40C2×#121—21no—
41C2×#121C2212yes—
42C2×#121C2212yescomplemented
43#518—21no—
44C2×C4×D4C2211yescharacteristic
45C2×C4×D4C2211yescharacteristic, complemented
46C2×#117—21no—
47C2×#117C2212yescomplemented
48#501C2212yescomplemented
49#500C2212yescomplemented
50#489C2212yescomplemented
51#486C2212yescomplemented
52#485C2214yescomplemented
53#484C2211yescharacteristic, complemented
54#483C2212yescomplemented
55#482C2214yescomplemented
56#482C2214yescomplemented
57#482C2214yescomplemented
58#482C2214yescomplemented
59#481C2211yescharacteristic
60#481C2212yescomplemented
61#480C2212yes—
62#480C2212yescomplemented
63#480C2214yescomplemented
64#480C2212yescomplemented
65#478C2212yescomplemented
66#477C2212yescomplemented
67#476C2212yescomplemented
68#476C2212yescomplemented
69#474C2214yescomplemented
70#469C2214yescomplemented
71#468C2214yescomplemented
72#466C2214yescomplemented
73#466C2214yescomplemented
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
90#427C2214yescomplemented
91#418C2214yescomplemented
92#416C2214yescomplemented
93#407C2212yescomplemented
94C2×C22⋊C8C2212yescomplemented
95C2×#41—21no—
96C2×#41C2312yes—
97C22×Q8—21no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—21no—
101C22×D4D411yescharacteristic
102C22×D4C2311yescharacteristic
103C22×D4C2312yes—
104C23×C4—21no—
105C23×C4C2312yes—
106C2×SD16C2312yes—
107C2×D8C2312yes—
108C2×D8C2314yes—
109C2×M16C2314yes—
110C22×C8C2314yes—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8C2312yes—
114C4⋊Q8C2312yes—
115C4⋊Q8C2311yescharacteristic
116#127—21no—
117#127C2311yescharacteristic
118#126—22no—
119#126—21no—
120#125—21no—
121#125C2311yescharacteristic
122#124—24no—
123#124—24no—
124#124—24no—
125#124—24no—
126#124—22no—
127#124—21no—
128#124C2311yescharacteristic
129#123—22no—
130#123—21no—
131#123—22no—
132#123—22no—
133C22⋊Q8—22no—
134C22⋊Q8C2311yescharacteristic
135C22⋊Q8C2312yes—
136#121—22no—
137#121—22no—
138#121—24no—
139#121—22no—
140#121—21no—
141#121—22no—
142#121—22no—
143#121—22no—
144#121—22no—
145#121—21no—
146#121—21no—
147#121—22no—
148#121—22no—
149#121C2311yescharacteristic
150#121C2312yes—
151#121C2312yes—
152#121C2312yes—
153#121C2314yes—
154#121C2314yes—
155#121C2312yes—
156#120—22no—
157#120—22no—
158#120—22no—
159C4×Q8C2311yescharacteristic
160C4×D4—21no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—22no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4D411yescharacteristic
170C4×D4D412yes—
171C4×D4D411yescharacteristic
172C4×D4C2312yes—
173C4×D4C2312yes—
174C4×D4C2311yescharacteristic
175C4×D4C2312yes—
176C4×D4C2311yescharacteristic
177C4×D4C2312yes—
178C4×D4C2311yescharacteristic
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—24no—
184#117—21no—
185#117—21no—
186#117C2314yes—
187#117C2314yes—
188#117C2312yes—
189C2×C4⋊C4—21no—
190C2×C4⋊C4D411yescharacteristic
191C2×C4⋊C4C2311yescharacteristic
192C2×C4⋊C4C2312yes—
193C2×C4⋊C4C2312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D412yes—
200C2×C42—22no—
201C2×C42C2311yescharacteristic
202#107C2312yes—
203#107C2314yes—
204#107C2314yes—
205#107C2314yes—
206#106C2314yes—
207#106C2314yes—
208#106C2312yes—
209C4⋊C8C2312yes—
210C4⋊C8C2312yes—
211#103C2312yes—
212#103C2314yes—
213#102C2314yes—
214#102C2314yes—
215#102C2312yes—
216#102C2314yes—
217#102C2314yes—
218#102C2312yes—
219#102C2312yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2312yes—
223C22⋊C8C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2314yes—
226#97C2312yes—
227C4×C8C2312yes—
228C24—21no—
229C24—22no—
230C24C2×D412yes—
231#41—44no—
232#41—82no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—82no—
237C2×Q8C2411yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—84no—
255C2×D4—84no—
256C2×D4C2×D411yescharacteristic
257C2×D4C2×D411yescharacteristic
258C2×D4C2×D412yes—
259C2×D4C2411yescharacteristic
260C2×D4C2412yes—
261C2×D4C2412yes—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4C2×D411yescharacteristic
273C22×C4C2×D411yescharacteristic
274C22×C4C2×D412yes—
275C22×C4C2×D412yes—
276C22×C4C2×D412yes—
277C22×C4C2411yescharacteristic
278C22×C4C2412yes—
279C22×C4C2412yes—
280SD16—42no—
281D8—42no—
282D8—44no—
283M16—44no—
284C2×C8—44no—
285C2×C8C2412yes—
286C2×C8C2412yes—
287C2×C8C2414yes—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2×D411yescharacteristic
301C4⋊C4C2×D412yes—
302C4⋊C4C2412yes—
303C4⋊C4C2414yes—
304C4⋊C4C2411yescharacteristic
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2412yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4—22no—
323C22⋊C4—21no—
324C22⋊C4—22no—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C22⋊C4C2×D412yes—
328C22⋊C4C2×D412yes—
329C42—24no—
330C42—24no—
331C42—22no—
332C42—21no—
333C42C2×D411yescharacteristic
334C42C2×D411yescharacteristic
335C42C2411yescharacteristic
336C42C2411yescharacteristic
337C23—24no—
338C23—22no—
339C23—21no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—44no—
346C23—42no—
347C23—82no—
348C23—82no—
349C23C22×D412yes—
350C23C22×D412yes—
351C23C22×D411yescharacteristic
352Q8—22no—
353Q8—82no—
354Q8—82no—
355Q8—81no—
356D4—22no—
357D4—22no—
358D4—22no—
359D4—22no—
360D4—22no—
361D4—24no—
362D4—42no—
363D4—42no—
364D4—44no—
365D4—42no—
366D4—44no—
367D4—42no—
368D4—82no—
369D4—81no—
370D4—84no—
371D4—84no—
372D4—82no—
373D4—82no—
374D4—82no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4C22×D412yes—
398C2×C4C22×D412yes—
399C2×C4C22×D411yescharacteristic
400C2×C4C22×D411yescharacteristic
401C2×C4C22×D411yescharacteristic
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D411yescharacteristic
404C2×C4C2511yescharacteristic
405C8—22no—
406C8—24no—
407C8—22no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—41no—
419C22—42no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—42no—
425C22—84no—
426C22—84no—
427C22—164no—
428C22#57612yes—
429C22C23×D411yescharacteristic
430C4—21no—
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—44no—
440C4—41no—
441C4—42no—
442C4#57512yes—
443C4#58512yes—
444C2—22no—
445C2—42no—
446C2—42no—
447C2—82no—
448C2—82no—
449C2—81no—
450C2#348511yescharacteristic
451C2C2×#57511yescharacteristic
452C2C2×#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×#555C2semidirect
3. #3357C2semidirect
4. #3347C2semidirect
5. #3341C2semidirect
6. #3325C2semidirect
7. #3319C2semidirect
8. #3305C2semidirect
9. #3238C2semidirect
10. #3226C2semidirect
11. #3194C2semidirect
12. #3186C2semidirect
13. #3109C2semidirect
14. #3105C2semidirect
15. #3094C2semidirect
16. #3091C2semidirect
17. #2974C2semidirect
22. #555C22semidirect
23. #555C22semidirect
24. #555C22semidirect
25. #554C22semidirect
27. #553C22semidirect
28. #550C22semidirect
30. #546C22semidirect
32. D42C22semidirect
36. C2×C4⋊Q8C22semidirect
42. C2×#121C22semidirect
45. C2×C4×D4C22semidirect
47. C2×#117C22semidirect
48. #501C22semidirect
49. #500C22semidirect
50. #489C22semidirect
51. #486C22semidirect
52. #485C22semidirect
53. #484C22semidirect
54. #483C22semidirect
55. #482C22semidirect
56. #482C22semidirect
57. #482C22semidirect
58. #482C22semidirect
60. #481C22semidirect
62. #480C22semidirect
63. #480C22semidirect
64. #480C22semidirect
65. #478C22semidirect
66. #477C22semidirect
67. #476C22semidirect
68. #476C22semidirect
69. #474C22semidirect
70. #469C22semidirect
71. #468C22semidirect
72. #466C22semidirect
73. #466C22semidirect
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. #427C22semidirect
91. #418C22semidirect
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×#5552
#33572
#33472
#33412
#33252
#33192
#33052
#32382
#32262
#31942
#31862
#31092
#31052
#30942
#30912
#29742