Finite Group Databasebeta

← C2×#20261375 #27155181

Subgroups — C2×#20261375 #27155181

Order 12416.

1900 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095511C211yesmaximal, characteristic, complemented
2C23×C22⋊D97C211yesmaximal, characteristic, complemented
3#20261375C2116yesmaximal, direct product factor
4C2×#15690796C211yesmaximal, characteristic, complemented
5C2×#15690793C211yesmaximal, characteristic, complemented
6C2×#15690792C211yesmaximal, characteristic, complemented
7C2×#15690787C211yesmaximal, characteristic, complemented
8C2×#15690784C211yesmaximal, characteristic, complemented
9C2×#15690783C211yesmaximal, characteristic, complemented
10C2×#15690783C211yesmaximal, characteristic, complemented
11C2×#15690780C211yesmaximal, characteristic, complemented
12C2×#15690779C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
14C2×#15690770C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
16C22×C22⋊C4×C97C211yesmaximal, characteristic, complemented
17C24×D97C2211yescharacteristic
18C2×#13095511—28no—
19C2×#13095511C2218yescomplemented
20C22×Dic194C2211yescharacteristic, complemented
21C22×D388C2211yescharacteristic, complemented
22C22×D388C2211yescharacteristic, complemented
23C22×C22⋊D97—28no—
24C22×C22⋊D97C2211yescharacteristic
25C22×C22⋊D97C2211yescharacteristic
26C22×C22⋊D97C2211yescharacteristic
27C22×C22⋊D97C2211yescharacteristic
28C22×C22⋊D97C2211yescharacteristic
29C22×C22⋊D97C2218yescomplemented
30C22×C22⋊D97C2211yescharacteristic, complemented
31C22×C22⋊D97C2211yescharacteristic, complemented
32C23×Dic97C2211yescharacteristic
33C22×C4×D97C2211yescharacteristic, complemented
34C22×C4×D97C2211yescharacteristic, complemented
35C22×C4×D97C2211yescharacteristic, complemented
36#15690796C2218yescomplemented
37#15690793C2218yescomplemented
38#15690792C2218yescomplemented
39#15690787C2218yescomplemented
40#15690784C2218yescomplemented
41#15690783C2218yescomplemented
42#15690783C2218yescomplemented
43#15690780C2218yescomplemented
44#15690779C2218yescomplemented
45C4×C22⋊D97C2218yescomplemented
46#15690770C2218yescomplemented
47C2×C4⋊Dic97C2211yescharacteristic, complemented
48C2×#13095490C2211yescharacteristic, complemented
49C2×#13095490C2211yescharacteristic, complemented
50C2×#13095490C2211yescharacteristic, complemented
51C2×C22⋊Dic97C2211yescharacteristic
52C2×C22⋊Dic97C2211yescharacteristic, complemented
53C2×C22⋊Dic97C2211yescharacteristic, complemented
54C2×#13095488C2211yescharacteristic, complemented
55C2×#13095488C2211yescharacteristic, complemented
56C2×#13095488C2211yescharacteristic, complemented
57C2×#13095488C2211yescharacteristic, complemented
58C2×#13095488C2211yescharacteristic, complemented
59C22⋊C4×D97C2218yescomplemented
60C2×C4×Dic97C2211yescharacteristic, complemented
61C25×C97C2211yescharacteristic
62C23×C4×C97C2211yescharacteristic, complemented
63C23×C4×C97C2211yescharacteristic, complemented
64C2×C22⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2218yescomplemented
67C2×C22⋊C4×C97C2211yescharacteristic, complemented
68C2×C22⋊C4×C97C2211yescharacteristic, complemented
69C23×D97—24no—
70C23×D97—24no—
71C23×D97—42no—
72C23×D97C2311yescharacteristic
73C23×D97C2311yescharacteristic
74C23×D97C2314yes—
75C23×D97C2311yescharacteristic
76#13095511—232no—
77C2×Dic194—24no—
78C2×Dic194C2314yescomplemented
79C2×D388—24no—
80C2×D388—42no—
81C2×D388C2314yescomplemented
82C2×D388C2314yescomplemented
83C2×C22⋊D97—24no—
84C2×C22⋊D97—24no—
85C2×C22⋊D97—24no—
86C2×C22⋊D97—24no—
87C2×C22⋊D97—232no—
88C2×C22⋊D97—24no—
89C2×C22⋊D97—42no—
90C2×C22⋊D97D418yescomplemented
91C2×C22⋊D97C2314yes—
92C2×C22⋊D97C2314yes—
93C2×C22⋊D97C2314yes—
94C2×C22⋊D97C2314yes—
95C2×C22⋊D97C2314yes—
96C2×C22⋊D97C2314yescomplemented
97C2×C22⋊D97C2314yescomplemented
98C22×Dic97—24no—
99C22×Dic97C2311yescharacteristic
100C22×Dic97C2311yescharacteristic
101C22×Dic97C2311yescharacteristic
102C22×Dic97C2311yescharacteristic
103C22×Dic97C2314yes—
104C22×Dic97C2311yescharacteristic
105C2×C4×D97—24no—
106C2×C4×D97—24no—
107C2×C4×D97—42no—
108C2×C4×D97C2314yescomplemented
109C2×C4×D97C2314yescomplemented
110C2×C4×D97C2314yescomplemented
111C4⋊Dic97C2314yescomplemented
112#13095490C2314yescomplemented
113#13095490C2314yescomplemented
114#13095490C2314yescomplemented
115C22⋊Dic97C2314yes—
116C22⋊Dic97C2314yescomplemented
117C22⋊Dic97C2314yescomplemented
118#13095488C2314yescomplemented
119#13095488C2314yescomplemented
120#13095488C2314yescomplemented
121#13095488C2314yescomplemented
122#13095488C2314yescomplemented
123C4×Dic97C2314yescomplemented
124C24×C97—24no—
125C24×C97—24no—
126C24×C97—42no—
127C24×C97C2311yescharacteristic
128C24×C97C2314yes—
129C24×C97C2311yescharacteristic
130C24×C97C2311yescharacteristic
131C22×C4×C97—24no—
132C22×C4×C97—42no—
133C22×C4×C97C2311yescharacteristic
134C22×C4×C97C2311yescharacteristic
135C22×C4×C97C2311yescharacteristic
136C22×C4×C97C2311yescharacteristic
137C22×C4×C97C2314yescomplemented
138C22×C4×C97C2314yescomplemented
139C22⋊C4×C97C2314yes—
140C22⋊C4×C97C2314yes—
141C22⋊C4×C97C2314yescomplemented
142C22⋊C4×C97C2314yescomplemented
143C23×C97—22no—
144C23×C97—22no—
145C23×C97—22no—
146C23×C97—22no—
147C23×C97—28no—
148C23×C97—22no—
149C23×C97—22no—
150C23×C97—28no—
151C23×C97—22no—
152C23×C97—44no—
153C23×C97—416no—
154C23×C97C2×D414yes—
155C23×C97#4114yescomplemented
156C23×C97C2412yes—
157C23×C97C2412yes—
158C23×C97C2412yes—
159C23×C97C2411yescharacteristic
160C22×D97—22no—
161C22×D97—22no—
162C22×D97—22no—
163C22×D97—22no—
164C22×D97—22no—
165C22×D97—22no—
166C22×D97—28no—
167C22×D97—22no—
168C22×D97—22no—
169C22×D97—28no—
170C22×D97—44no—
171C22×D97—416no—
172C22×D97C2×D414yes—
173C22×D97C2412yes—
174C22×D97C2412yes—
175C22×D97C2412yes—
176C2×C4×C97—22no—
177C2×C4×C97—22no—
178C2×C4×C97—28no—
179C2×C4×C97—22no—
180C2×C4×C97—22no—
181C2×C4×C97—44no—
182C2×C4×C97C2412yes—
183C2×C4×C97C2412yes—
184C2×C4×C97C2412yes—
185C2×C4×C97C2412yes—
186C22⋊D97—28no—
187C22⋊D97—28no—
188C22⋊D97—28no—
189C22⋊D97—28no—
190C22⋊D97—28no—
191C22⋊D97—44no—
192C22⋊D97C2×D4116yescomplemented
193C2×Dic97—22no—
194C2×Dic97—22no—
195C2×Dic97—22no—
196C2×Dic97—22no—
197C2×Dic97—28no—
198C2×Dic97C2×D414yes—
199C2×Dic97C2412yes—
200C2×Dic97C2412yes—
201C2×Dic97C2412yes—
202C2×Dic97C2412yes—
203C2×Dic97C2412yes—
204D388—28no—
205D388—44no—
206C4×D97—28no—
207C4×D97—28no—
208C4×D97—44no—
209Dic194—28no—
210C22×C97—28no—
211C22×C97—28no—
212C22×C97—22no—
213C22×C97—22no—
214C22×C97—22no—
215C22×C97—22no—
216C22×C97—22no—
217C22×C97—22no—
218C22×C97—22no—
219C22×C97—416no—
220C22×C97—44no—
221C22×C97C22×D414yes—
222C22×C97C22×D412yes—
223C22×C97C2×#4112yes—
224C22×C97C2×#4114yescomplemented
225C22×C97H(5,𝔽2)12yes—
226C22×C97C2511yescharacteristic
227C2×D97—28no—
228C2×D97—22no—
229C2×D97—22no—
230C2×D97—22no—
231C2×D97—22no—
232C2×D97—22no—
233C2×D97—22no—
234C2×D97—22no—
235C2×D97—22no—
236C2×D97—416no—
237C2×D97—44no—
238C2×D97—44no—
239C2×D97C22×D414yes—
240C4×C97—22no—
241C4×C97—22no—
242C4×C97—22no—
243C4×C97—22no—
244Dic97—22no—
245Dic97—22no—
246Dic97—22no—
247Dic97—22no—
248Dic97C22×D414yes—
249C2×C97—24no—
250C2×C97—24no—
251C2×C97—42no—
252C2×C97#54614yescomplemented
253C2×C97C23×D411yescharacteristic
254C2×C97C22×#4111yescharacteristic
255C2×C97C2×H(5,𝔽2)11yescharacteristic
256D97—24no—
257D97—42no—
258D97—42no—
259C2×#546—971nomaximal, Sylow
260C97C2×#54611yescharacteristic, complemented, Sylow
261C22×#41—971no—
262C23×D4—971no—
263#546—9716no—
264C2×#124—971no—
265C2×#123—971no—
266C2×#123—971no—
267C2×C22⋊Q8—971no—
268C2×#121—971no—
269C2×#121—971no—
270C2×#121—971no—
271C2×#120—971no—
272C2×#120—971no—
273C2×C4×D4—971no—
274C2×C4×D4—971no—
275C22×C22⋊C4—971no—
276C22×C22⋊C4D9711yescharacteristic, complemented
277C25—971no—
278C25C2×D9711yescharacteristic
279C2×#41—978no—
280C2×#41—1948no—
281C22×Q8—971no—
282C22×D4—971no—
283C22×D4—971no—
284C22×D4—971no—
285C22×D4—971no—
286C22×D4—971no—
287C22×D4—971no—
288C22×D4—971no—
289C22×D4—978no—
290C22×D4—971no—
291C22×D4—971no—
292C22×D4—1948no—
293C23×C4—971no—
294C23×C4—971no—
295C23×C4—971no—
296C23×C4—971no—
297C23×C4C2×D9711yescharacteristic, complemented
298C23×C4C2×D9711yescharacteristic, complemented
299#124—978no—
300#123—978no—
301#123—978no—
302C22⋊Q8—978no—
303#121—978no—
304#121—978no—
305#121—978no—
306#120—978no—
307#120—978no—
308C4×D4—978no—
309C4×D4—978no—
310C2×C4⋊C4—971no—
311C2×C4⋊C4—971no—
312C2×C4⋊C4—971no—
313C2×C4⋊C4—971no—
314C2×C22⋊C4—971no—
315C2×C22⋊C4—971no—
316C2×C22⋊C4—971no—
317C2×C22⋊C4—971no—
318C2×C22⋊C4—971no—
319C2×C22⋊C4—971no—
320C2×C22⋊C4—971no—
321C2×C22⋊C4—971no—
322C2×C22⋊C4—978no—
323C2×C22⋊C4C2×D9711yescharacteristic
324C2×C22⋊C4C2×D9711yescharacteristic
325C2×C22⋊C4C2×D9718yescomplemented
326C2×C22⋊C4C2×D9711yescharacteristic, complemented
327C2×C22⋊C4C2×D9711yescharacteristic, complemented
328C2×C42—971no—
329C24—24no—
330C24—24no—
331C24—42no—
332C24—971no—
333C24—974no—
334C24—971no—
335C24—971no—
336C24—3882no—
337C24—1944no—
338C24—1944no—
339C24C22×D9711yescharacteristic
340C24C22×D9714yes—
341C24C22×D9711yescharacteristic
342C24C22×D9711yescharacteristic
343#41—19432no—
344C2×Q8—974no—
345C2×Q8—1944no—
346C2×D4—978no—
347C2×D4—974no—
348C2×D4—974no—
349C2×D4—974no—
350C2×D4—974no—
351C2×D4—974no—
352C2×D4—974no—
353C2×D4—974no—
354C2×D4—974no—
355C2×D4—974no—
356C2×D4—3882no—
357C2×D4—3882no—
358C2×D4—1944no—
359C2×D4—1944no—
360C2×D4—19432no—
361C2×D4—1944no—
362C2×D4—1944no—
363C2×D4—1944no—
364C2×D4—1944no—
365C22×C4—24no—
366C22×C4—42no—
367C22×C4—971no—
368C22×C4—974no—
369C22×C4—971no—
370C22×C4—971no—
371C22×C4—971no—
372C22×C4—971no—
373C22×C4—974no—
374C22×C4—974no—
375C22×C4—974no—
376C22×C4—3882no—
377C22×C4—1944no—
378C22×C4—1944no—
379C22×C4—1944no—
380C22×C4C22×D9711yescharacteristic
381C22×C4C22×D9711yescharacteristic
382C22×C4C22×D9711yescharacteristic
383C22×C4C22×D9711yescharacteristic
384C22×C4C22×D9714yescomplemented
385C22×C4C22×D9714yescomplemented
386C4⋊C4—974no—
387C4⋊C4—974no—
388C4⋊C4—974no—
389C4⋊C4—974no—
390C22⋊C4—974no—
391C22⋊C4—974no—
392C22⋊C4—974no—
393C22⋊C4—974no—
394C22⋊C4—974no—
395C22⋊C4—974no—
396C22⋊C4—974no—
397C22⋊C4—974no—
398C22⋊C4C22×D9714yes—
399C22⋊C4C22×D9714yes—
400C22⋊C4C22×D9714yescomplemented
401C22⋊C4C22×D9714yescomplemented
402C42—974no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—28no—
408C23—22no—
409C23—22no—
410C23—28no—
411C23—22no—
412C23—44no—
413C23—416no—
414C23—974no—
415C23—972no—
416C23—972no—
417C23—972no—
418C23—38816no—
419C23—3884no—
420C23—1948no—
421C23—1942no—
422C23—1942no—
423C23—1942no—
424C23—1942no—
425C23—1948no—
426C23—1942no—
427C23—1942no—
428C23—1942no—
429C23—1942no—
430C23D4×D9714yes—
431C23#1309551114yescomplemented
432C23C23×D9712yes—
433C23C23×D9712yes—
434C23C23×D9712yes—
435C23C23×D9711yescharacteristic
436Q8—1948no—
437D4—9716no—
438D4—3884no—
439D4—3884no—
440D4—1948no—
441D4—1948no—
442D4—1948no—
443D4—1948no—
444D4—1948no—
445D4—1948no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—44no—
452C2×C4—974no—
453C2×C4—972no—
454C2×C4—972no—
455C2×C4—972no—
456C2×C4—972no—
457C2×C4—972no—
458C2×C4—3884no—
459C2×C4—1948no—
460C2×C4—1948no—
461C2×C4—1948no—
462C2×C4—1942no—
463C2×C4—1942no—
464C2×C4—1942no—
465C2×C4—1942no—
466C2×C4C23×D9712yes—
467C2×C4C23×D9712yes—
468C2×C4C23×D9712yes—
469C2×C4C23×D9712yes—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—28no—
475C22—28no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—416no—
480C22—44no—
481C22—974no—
482C22—38816no—
483C22—3884no—
484C22—3884no—
485C22—1942no—
486C22—1942no—
487C22—1942no—
488C22—1942no—
489C22—1942no—
490C22—1942no—
491C22—1942no—
492C22—1942no—
493C22—1948no—
494C22C2×D4×D9714yes—
495C22C2×D4×D9712yes—
496C22C2×#1309551112yes—
497C22C2×#1309551114yescomplemented
498C22#1569085512yes—
499C22C24×D9711yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—974no—
505C4—1942no—
506C4—1942no—
507C4—1942no—
508C4—1942no—
509C2—24no—
510C2—24no—
511C2—42no—
512C2—3882no—
513C2—3882no—
514C2—1944no—
515C2#2026137514yesdirect product factor
516C2C22×D4×D9711yescharacteristic
517C2C22×#1309551111yescharacteristic
518C2C2×#1569085511yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×C22⋊C4×C97C2
socleC23×C97C24
hypercenterC22×C22⋊C4D97
hypocenterC97C2×#546
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C23×C22⋊D97C2semidirect
3. #20261375C2direct (N × H)
4. C2×#15690796C2semidirect
5. C2×#15690793C2semidirect
6. C2×#15690792C2semidirect
7. C2×#15690787C2semidirect
8. C2×#15690784C2semidirect
9. C2×#15690783C2semidirect
10. C2×#15690783C2semidirect
11. C2×#15690780C2semidirect
12. C2×#15690779C2semidirect
13. C2×C4×C22⋊D97C2semidirect
14. C2×#15690770C2semidirect
15. C2×C22⋊C4×D97C2semidirect
16. C22×C22⋊C4×C97C2semidirect
19. C2×#13095511C22semidirect
20. C22×Dic194C22semidirect
21. C22×D388C22semidirect
22. C22×D388C22semidirect
29. C22×C22⋊D97C22semidirect
30. C22×C22⋊D97C22semidirect
31. C22×C22⋊D97C22semidirect
33. C22×C4×D97C22semidirect
34. C22×C4×D97C22semidirect
35. C22×C4×D97C22semidirect
36. #15690796C22semidirect
37. #15690793C22semidirect
38. #15690792C22semidirect
39. #15690787C22semidirect
40. #15690784C22semidirect
41. #15690783C22semidirect
42. #15690783C22semidirect
43. #15690780C22semidirect
44. #15690779C22semidirect
45. C4×C22⋊D97C22semidirect
46. #15690770C22semidirect
47. C2×C4⋊Dic97C22semidirect
48. C2×#13095490C22semidirect
49. C2×#13095490C22semidirect
50. C2×#13095490C22semidirect
52. C2×C22⋊Dic97C22semidirect
53. C2×C22⋊Dic97C22semidirect
54. C2×#13095488C22semidirect
55. C2×#13095488C22semidirect
56. C2×#13095488C22semidirect
57. C2×#13095488C22semidirect
58. C2×#13095488C22semidirect
59. C22⋊C4×D97C22semidirect
60. C2×C4×Dic97C22semidirect
62. C23×C4×C97C22semidirect
63. C23×C4×C97C22semidirect
66. C2×C22⋊C4×C97C22semidirect
67. C2×C22⋊C4×C97C22semidirect
68. C2×C22⋊C4×C97C22semidirect
78. C2×Dic194C23semidirect
81. C2×D388C23semidirect
82. C2×D388C23semidirect
90. C2×C22⋊D97D4semidirect
96. C2×C22⋊D97C23semidirect
97. C2×C22⋊D97C23semidirect
108. C2×C4×D97C23semidirect
109. C2×C4×D97C23semidirect
110. C2×C4×D97C23semidirect
111. C4⋊Dic97C23semidirect
112. #13095490C23semidirect
113. #13095490C23semidirect
114. #13095490C23semidirect
116. C22⋊Dic97C23semidirect
117. C22⋊Dic97C23semidirect
118. #13095488C23semidirect
119. #13095488C23semidirect
120. #13095488C23semidirect
121. #13095488C23semidirect
122. #13095488C23semidirect
123. C4×Dic97C23semidirect
137. C22×C4×C97C23semidirect
138. C22×C4×C97C23semidirect
141. C22⋊C4×C97C23semidirect
142. C22⋊C4×C97C23semidirect
155. C23×C97#41semidirect
192. C22⋊D97C2×D4semidirect
224. C22×C97C2×#41semidirect
252. C2×C97#546semidirect
260. C97C2×#546semidirect
276. C22×C22⋊C4D97semidirect
297. C23×C4C2×D97semidirect
298. C23×C4C2×D97semidirect
325. C2×C22⋊C4C2×D97semidirect
326. C2×C22⋊C4C2×D97semidirect
327. C2×C22⋊C4C2×D97semidirect
384. C22×C4C22×D97semidirect
385. C22×C4C22×D97semidirect
400. C22⋊C4C22×D97semidirect
401. C22⋊C4C22×D97semidirect
431. C23#13095511semidirect
497. C22C2×#13095511semidirect
515. C2#20261375direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955112
C23×C22⋊D972
#202613752
C2×#156907962
C2×#156907932
C2×#156907922
C2×#156907872
C2×#156907842
C2×#156907832
C2×#156907802
C2×#156907792
C2×C4×C22⋊D972
C2×#156907702
C2×C22⋊C4×D972
C22×C22⋊C4×C972
C2×#54697