Finite Group Databasebeta

← #62847

Subgroups — #62847

Order 256.

3553 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3521C218yesmaximal, complemented
2#3520C218yesmaximal, complemented
3#3515C218yesmaximal, complemented
4#3513C218yesmaximal, complemented
5#3507C212yesmaximal, complemented
6#3505C214yesmaximal, complemented
7#3501C211yesmaximal, characteristic, complemented
8#3496C211yesmaximal, characteristic, complemented
9#3492C212yesmaximal, complemented
10#3491C214yesmaximal, complemented
11#3485C212yesmaximal, complemented
12#3461C212yesmaximal, complemented
13#3459C214yesmaximal, complemented
14#3455C212yesmaximal, complemented
15C2×#539C214yesmaximal, complemented
16C4×H(5,𝔽2)C212yesmaximal, complemented
17C2×#520C211yesmaximal, characteristic, complemented
18C2×H(5,𝔽2)C2212yes—
19C22×#41C2211yescharacteristic
20#559C2214yescomplemented
21#557C2214yescomplemented
22#556C2218yescomplemented
23#556C2214yescomplemented
24#554C2218yescomplemented
25#554C2218yescomplemented
26#554C2214yescomplemented
27#554C2214yescomplemented
28#553C2218yescomplemented
29#553C22116yescomplemented
30#553C2218yescomplemented
31#552C22116yescomplemented
32#552C22116yescomplemented
33#552C2218yescomplemented
34#552C2218yescomplemented
35#552C2218yescomplemented
36#551C22116yescomplemented
37#551C22116yescomplemented
38#551C22116yescomplemented
39D4×Q8C2214yescomplemented
40#548C2218yescomplemented
41#547C2218yescomplemented
42#547C2218yescomplemented
43#547C2218yescomplemented
44#547C2214yescomplemented
45#546C2218yescomplemented
46#546C2218yescomplemented
47#546C2218yescomplemented
48#546C2218yescomplemented
49D42C2214yescomplemented
50#544C2212yescomplemented
51#543C2214yescomplemented
52#542C2214yescomplemented
53#542C2214yescomplemented
54#542C2218yescomplemented
55#541C2214yescomplemented
56#541C2218yescomplemented
57#540C2214yescomplemented
58#539C2214yescomplemented
59#539C2218yescomplemented
60#539C2218yescomplemented
61#539C2218yescomplemented
62#539C22116yescomplemented
63#539C22116yescomplemented
64#539C2214yescomplemented
65#539C22116yescomplemented
66#538C2214yescomplemented
67#538C2214yescomplemented
68#538C22116yescomplemented
69#537C2212yes—
70#536C2214yescomplemented
71#534C2214yes—
72#534C2218yescomplemented
73#533C2212yescomplemented
74#533C2218yescomplemented
75#532C2212yes—
76#529C2218yescomplemented
77#529C2218yescomplemented
78#529C2218yescomplemented
79#529C22116yescomplemented
80#529C2218yescomplemented
81#529C2218yescomplemented
82C2×#126C2218yescomplemented
83C2×#125C2218yescomplemented
84#525C2214yescomplemented
85#525C2218yescomplemented
86#525C22116yescomplemented
87C2×#123C2218yescomplemented
88C2×#123C2218yescomplemented
89C2×C22⋊Q8C2214yescomplemented
90C2×C22⋊Q8C2218yescomplemented
91C2×#121C2214yes—
92#520C2218yescomplemented
93#520C2218yescomplemented
94#520C2214yescomplemented
95#520C2218yescomplemented
96#518C2212yes—
97#518C2218yescomplemented
98#518C2218yescomplemented
99C4×#41C2214yescomplemented
100C4×#41C2218yescomplemented
101C2×C4×Q8C2211yescharacteristic, complemented
102C2×C4×Q8C2211yescharacteristic, complemented
103C2×C4×D4C2211yescharacteristic
104C2×C4×D4C2211yescharacteristic
105C2×C4×D4C2212yescomplemented
106C2×C4×D4C2212yescomplemented
107C2×C4×D4C2218yescomplemented
108C2×C4×D4C2218yescomplemented
109C2×#117C2211yescharacteristic, complemented
110C2×#117C2212yescomplemented
111C22×C4⋊C4C2212yescomplemented
112C22×C4⋊C4C2211yescharacteristic, complemented
113H(5,𝔽2)—216no—
114C2×#41—28no—
115C2×#41C2314yes—
116C2×#41C2318yes—
117C2×#41C2318yes—
118C2×#41C2318yes—
119C22×Q8C2311yescharacteristic
120C22×Q8C2314yes—
121C22×D4C2312yes—
122C22×D4C2318yes—
123C22×D4C2318yes—
124C22×D4C2314yes—
125C22×D4C2311yescharacteristic
126C23×C4C2312yes—
127C23×C4C2311yescharacteristic
128C23×C4C2312yes—
129C23×C4C2312yes—
130C23×C4C2312yes—
131C23×C4C2318yes—
132C4⋊Q8C2312yes—
133C4⋊Q8C2318yes—
134C4⋊Q8C2312yes—
135C4⋊Q8C2312yes—
136C4⋊Q8C2318yes—
137#127C2312yes—
138#126C2318yes—
139#126C23116yes—
140#126C23116yes—
141#126C23116yes—
142#126C2318yes—
143#126C23116yes—
144#126C2318yes—
145#126C2318yes—
146#126C23116yes—
147#125C2318yes—
148#125C23116yes—
149#125C23116yes—
150#125C23116yes—
151#125C2318yes—
152#125C2318yes—
153#125C2318yes—
154#125C2318yes—
155#125C2314yes—
156#124C2318yes—
157#124C2318yes—
158#124C2318yes—
159#124C2318yes—
160#124C2318yes—
161#124C2314yes—
162#123C2318yes—
163#123C2318yes—
164#123C23116yes—
165#123C23116yes—
166#123C23116yes—
167#123C23116yes—
168#123C23116yes—
169#123C23116yes—
170#123C2318yes—
171#123C23116yes—
172#123C23116yes—
173#123C23116yes—
174#123C2318yes—
175#123C2318yes—
176C22⋊Q8C2318yes—
177C22⋊Q8C2314yes—
178C22⋊Q8C2318yes—
179C22⋊Q8C23116yes—
180C22⋊Q8C2318yes—
181C22⋊Q8C23116yes—
182C22⋊Q8C23116yes—
183C22⋊Q8C2318yes—
184C22⋊Q8C2318yes—
185C22⋊Q8C23116yes—
186C22⋊Q8C2314yes—
187C22⋊Q8C2314yes—
188C22⋊Q8C2314yes—
189C22⋊Q8C2318yes—
190C22⋊Q8C23116yes—
191C22⋊Q8C2318yes—
192C22⋊Q8C2318yes—
193C22⋊Q8C2318yes—
194#121C2314yes—
195#121C2318yes—
196#121C2318yes—
197#121C2318yes—
198#121C2318yes—
199#121C23116yes—
200#121C2318yes—
201#121C2314yes—
202#121C2314yes—
203#121C2314yes—
204#121C23116yes—
205#121C2318yes—
206#121C2318yes—
207#120C2318yes—
208#120C23116yes—
209#120C23116yes—
210C4×Q8C2318yes—
211C4×Q8C2314yes—
212C4×Q8C2314yes—
213C4×Q8C2314yes—
214C4×Q8C2314yes—
215C4×Q8C2318yes—
216C4×Q8C2318yes—
217C4×Q8C2314yes—
218C4×Q8C2314yes—
219C4×Q8C2318yes—
220C4×D4C2318yes—
221C4×D4C2314yes—
222C4×D4C2314yes—
223C4×D4C2318yes—
224C4×D4C23116yes—
225C4×D4C2314yes—
226C4×D4C2318yes—
227C4×D4C23116yes—
228C4×D4C23116yes—
229C4×D4C2318yes—
230C4×D4C23116yes—
231C4×D4C23116yes—
232C4×D4C2318yes—
233C4×D4C2318yes—
234C4×D4C2318yes—
235C4×D4C2318yes—
236C4×D4C2314yes—
237C4×D4C23116yes—
238C4×D4C2318yes—
239C4×D4C23116yes—
240C4×D4C2318yes—
241C4×D4C2318yes—
242C4×D4C2318yes—
243C4×D4C2314yes—
244C4×D4C2314yes—
245#117C2314yes—
246#117C2314yes—
247#117C23116yes—
248#117C2314yes—
249#117C2318yes—
250#117C23116yes—
251#117C2318yes—
252#117C23116yes—
253#117C2318yes—
254#117C23116yes—
255C2×C4⋊C4C2311yescharacteristic
256C2×C4⋊C4C2312yes—
257C2×C4⋊C4C2311yescharacteristic
258C2×C4⋊C4C2311yescharacteristic
259C2×C4⋊C4C2312yes—
260C2×C4⋊C4C2312yes—
261C2×C4⋊C4C2318yes—
262C2×C4⋊C4C2318yes—
263C2×C4⋊C4C2318yes—
264C2×C4⋊C4C2311yescharacteristic
265C2×C4⋊C4C2318yes—
266C2×C4⋊C4C2318yes—
267C2×C4⋊C4C2314yes—
268C2×C4⋊C4C2318yes—
269C2×C4⋊C4C2318yes—
270C2×C4⋊C4C2314yes—
271C2×C4⋊C4C2318yes—
272C2×C22⋊C4C2312yes—
273C2×C22⋊C4C2312yes—
274C2×C22⋊C4C2311yescharacteristic
275C2×C22⋊C4C2311yescharacteristic
276C2×C22⋊C4C2318yes—
277C2×C22⋊C4C2318yes—
278C2×C22⋊C4C23116yes—
279C2×C22⋊C4C2318yes—
280C2×C22⋊C4C2318yes—
281C2×C22⋊C4C2318yes—
282C2×C42C2311yescharacteristic
283C2×C42C2311yescharacteristic
284C2×C42C2312yes—
285C2×C42C2312yes—
286C2×C42C2318yes—
287C2×C42C2318yes—
288C24C2418yes—
289C24C2411yescharacteristic
290C24C2412yes—
291#41—232no—
292#41—232no—
293#41—216no—
294#41—88no—
295C2×Q8—24no—
296C2×Q8—84no—
297C2×Q8C2414yes—
298C2×Q8C2414yes—
299C2×Q8C2414yes—
300C2×Q8C2414yes—
301C2×Q8C2418yes—
302C2×D4—28no—
303C2×D4—232no—
304C2×D4—232no—
305C2×D4—84no—
306C2×D4#4118yescomplemented
307C2×D4C2418yes—
308C2×D4C2418yes—
309C2×D4C24116yes—
310C2×D4C2414yes—
311C2×D4C2414yes—
312C2×D4C2418yes—
313C2×D4C2418yes—
314C2×D4C2418yes—
315C2×D4C2414yes—
316C2×D4C2414yes—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—82no—
320C22×C4—88no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2412yes—
324C22×C4C2418yes—
325C22×C4C2418yes—
326C22×C4C2418yes—
327C22×C4C2411yescharacteristic
328C22×C4C2414yes—
329C22×C4C2411yescharacteristic
330C22×C4C2411yescharacteristic
331C22×C4C2418yes—
332C22×C4C2418yes—
333C22×C4C2411yescharacteristic
334C22×C4C2412yes—
335C22×C4C2418yes—
336C22×C4C2411yescharacteristic
337C22×C4C2412yes—
338C22×C4C2418yes—
339C22×C4C2414yes—
340C22×C4C2418yes—
341C22×C4C24116yes—
342C22×C4C2414yes—
343C22×C4C2418yes—
344C22×C4C2411yescharacteristic
345C4⋊C4C2414yes—
346C4⋊C4C2412yes—
347C4⋊C4C2418yes—
348C4⋊C4C2412yes—
349C4⋊C4C2414yes—
350C4⋊C4C2414yes—
351C4⋊C4C2418yes—
352C4⋊C4C2414yes—
353C4⋊C4C2414yes—
354C4⋊C4C2418yes—
355C4⋊C4C24116yes—
356C4⋊C4C2414yes—
357C4⋊C4C2418yes—
358C4⋊C4C2418yes—
359C4⋊C4C2418yes—
360C4⋊C4C2414yes—
361C4⋊C4C2414yes—
362C4⋊C4C2418yes—
363C4⋊C4C24116yes—
364C4⋊C4C2418yes—
365C4⋊C4C24116yes—
366C4⋊C4C24116yes—
367C4⋊C4C2414yes—
368C4⋊C4C2418yes—
369C4⋊C4C2418yes—
370C4⋊C4C2414yes—
371C4⋊C4C2414yes—
372C22⋊C4C2418yes—
373C22⋊C4C2418yes—
374C22⋊C4C24116yes—
375C22⋊C4C2418yes—
376C22⋊C4C24116yes—
377C22⋊C4C2418yes—
378C22⋊C4C2418yes—
379C22⋊C4C2418yes—
380C22⋊C4C2418yes—
381C22⋊C4C2418yes—
382C22⋊C4C2412yes—
383C22⋊C4C2414yes—
384C22⋊C4C2412yes—
385C22⋊C4C24116yes—
386C22⋊C4C24116yes—
387C22⋊C4C24116yes—
388C22⋊C4C2418yes—
389C22⋊C4C2418yes—
390C22⋊C4C2418yes—
391C42C2414yes—
392C42C2418yes—
393C42C24116yes—
394C42C2418yes—
395C42C2418yes—
396C42C2418yes—
397C42C2418yes—
398C42C2412yes—
399C42C2412yes—
400C23—22no—
401C23—216no—
402C23—42no—
403C23—41no—
404C23—42no—
405C23—41no—
406C23—48no—
407C23—48no—
408C23C2×#4118yes—
409C23C2518yes—
410C23C2512yes—
411C23C2514yes—
412C23C2511yescharacteristic
413Q8—28no—
414Q8—28no—
415Q8—44no—
416Q8—48no—
417Q8—44no—
418D4—232no—
419D4—216no—
420D4—216no—
421D4—28no—
422D4—44no—
423D4—48no—
424D4—48no—
425D4—44no—
426D4—48no—
427D4C2×#41116yes—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—216no—
431C2×C4—216no—
432C2×C4—28no—
433C2×C4—216no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—48no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—416no—
445C2×C4—41no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—41no—
449C2×C4C2×#4114yes—
450C2×C4C2512yes—
451C2×C4C2512yes—
452C2×C4C2512yes—
453C2×C4C2514yes—
454C2×C4C2518yes—
455C2×C4C2514yes—
456C2×C4C2518yes—
457C2×C4C2511yescharacteristic
458C2×C4C2518yes—
459C2×C4C2514yes—
460C2×C4C2511yescharacteristic
461C2×C4C2512yes—
462C2×C4C2512yes—
463C22—21no—
464C22—24no—
465C22—21no—
466C22—24no—
467C22—22no—
468C22—28no—
469C22—22no—
470C22—28no—
471C22—28no—
472C22—22no—
473C22—48no—
474C22—44no—
475C22—816no—
476C22C22×#4118yes—
477C22C22×#4112yes—
478C22C2611yescharacteristic
479C4—22no—
480C4—28no—
481C4—24no—
482C4—22no—
483C4—24no—
484C4—21no—
485C4—24no—
486C4—28no—
487C4—21no—
488C4—22no—
489C4—28no—
490C4—22no—
491C4C22×#4114yes—
492C2—22no—
493C2—28no—
494C2—42no—
495C2—48no—
496C2C23×#4111yescharacteristic
497C2H(7,𝔽2)11yescharacteristic
498C22−1+611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3521C2semidirect
2. #3520C2semidirect
3. #3515C2semidirect
4. #3513C2semidirect
5. #3507C2semidirect
6. #3505C2semidirect
7. #3501C2semidirect
8. #3496C2semidirect
9. #3492C2semidirect
10. #3491C2semidirect
11. #3485C2semidirect
12. #3461C2semidirect
13. #3459C2semidirect
14. #3455C2semidirect
15. C2×#539C2semidirect
16. C4×H(5,𝔽2)C2semidirect
17. C2×#520C2semidirect
20. #559C22semidirect
21. #557C22semidirect
22. #556C22semidirect
23. #556C22semidirect
24. #554C22semidirect
25. #554C22semidirect
26. #554C22semidirect
27. #554C22semidirect
28. #553C22semidirect
29. #553C22semidirect
30. #553C22semidirect
31. #552C22semidirect
32. #552C22semidirect
33. #552C22semidirect
34. #552C22semidirect
35. #552C22semidirect
36. #551C22semidirect
37. #551C22semidirect
38. #551C22semidirect
39. D4×Q8C22semidirect
40. #548C22semidirect
41. #547C22semidirect
42. #547C22semidirect
43. #547C22semidirect
44. #547C22semidirect
45. #546C22semidirect
46. #546C22semidirect
47. #546C22semidirect
48. #546C22semidirect
49. D42C22semidirect
50. #544C22semidirect
51. #543C22semidirect
52. #542C22semidirect
53. #542C22semidirect
54. #542C22semidirect
55. #541C22semidirect
56. #541C22semidirect
57. #540C22semidirect
58. #539C22semidirect
59. #539C22semidirect
60. #539C22semidirect
61. #539C22semidirect
62. #539C22semidirect
63. #539C22semidirect
64. #539C22semidirect
65. #539C22semidirect
66. #538C22semidirect
67. #538C22semidirect
68. #538C22semidirect
70. #536C22semidirect
72. #534C22semidirect
73. #533C22semidirect
74. #533C22semidirect
76. #529C22semidirect
77. #529C22semidirect
78. #529C22semidirect
79. #529C22semidirect
80. #529C22semidirect
81. #529C22semidirect
82. C2×#126C22semidirect
83. C2×#125C22semidirect
84. #525C22semidirect
85. #525C22semidirect
86. #525C22semidirect
87. C2×#123C22semidirect
88. C2×#123C22semidirect
89. C2×C22⋊Q8C22semidirect
90. C2×C22⋊Q8C22semidirect
92. #520C22semidirect
93. #520C22semidirect
94. #520C22semidirect
95. #520C22semidirect
97. #518C22semidirect
98. #518C22semidirect
99. C4×#41C22semidirect
100. C4×#41C22semidirect
101. C2×C4×Q8C22semidirect
102. C2×C4×Q8C22semidirect
105. C2×C4×D4C22semidirect
106. C2×C4×D4C22semidirect
107. C2×C4×D4C22semidirect
108. C2×C4×D4C22semidirect
109. C2×#117C22semidirect
110. C2×#117C22semidirect
111. C22×C4⋊C4C22semidirect
112. C22×C4⋊C4C22semidirect
306. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
#35212
#35202
#35152
#35132
#35072
#35052
#35012
#34962
#34922
#34912
#34852
#34612
#34592
#34552
C2×#5392
C4×H(5,𝔽2)2
C2×#5202