Finite Group Databasebeta

← C2×#20261473 #27155851

Subgroups — C2×#20261473 #27155851

Order 12416.

1344 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261473C2116yesmaximal, direct product factor
2C2×#15690806C211yesmaximal, characteristic, complemented
3C2×#15690805C211yesmaximal, characteristic, complemented
4C2×#15690803C211yesmaximal, characteristic, complemented
5C2×#15690795C211yesmaximal, characteristic, complemented
6C2×#15690794C211yesmaximal, characteristic, complemented
7C2×#15690793C211yesmaximal, characteristic, complemented
8C2×#15690784C211yesmaximal, characteristic, complemented
9C2×#15690783C211yesmaximal, characteristic, complemented
10C2×#15690782C211yesmaximal, characteristic, complemented
11C2×#15690782C211yesmaximal, characteristic, complemented
12C2×#15690781C211yesmaximal, characteristic, complemented
13C2×#15690780C211yesmaximal, characteristic, complemented
14C2×#15690778C211yesmaximal, characteristic, complemented
15C2×#15690778C211yesmaximal, characteristic, complemented
16C2×#121×C97C211yesmaximal, characteristic, complemented
17C24×D97C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×D388C2211yescharacteristic
20C22×C22⋊D97C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C22⋊D97C2211yescharacteristic
23C22×C22⋊D97C2211yescharacteristic
24C22×C22⋊D97C2211yescharacteristic
25C22×C22⋊D97C2211yescharacteristic, complemented
26C22×C22⋊D97C2211yescharacteristic, complemented
27C23×Dic97C2211yescharacteristic, complemented
28C22×C4×D97C2211yescharacteristic, complemented
29#15690806C2218yescomplemented
30#15690805C2218yescomplemented
31#15690803C2218yescomplemented
32#15690795C2218yescomplemented
33#15690794C2218yescomplemented
34#15690793C2218yescomplemented
35#15690784C2218yescomplemented
36#15690783C2218yescomplemented
37#15690782C2218yescomplemented
38#15690782C2218yescomplemented
39#15690781C2218yescomplemented
40#15690780C2218yescomplemented
41#15690778C2218yescomplemented
42#15690778C2218yescomplemented
43C2×C4⋊Dic97C2211yescharacteristic, complemented
44C2×C4⋊Dic97C2211yescharacteristic, complemented
45C2×#13095490C2211yescharacteristic, complemented
46C2×#13095490C2211yescharacteristic, complemented
47C2×#13095490C2211yescharacteristic, complemented
48C2×C22⋊Dic97C2211yescharacteristic
49C2×C22⋊Dic97C2211yescharacteristic
50C2×C22⋊Dic97C2211yescharacteristic, complemented
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
57C2×#13095488C2211yescharacteristic, complemented
58C2×C4×Dic97C2211yescharacteristic, complemented
59C22×D4×C97C2211yescharacteristic
60C22×D4×C97C2211yescharacteristic
61C22×D4×C97C2211yescharacteristic, complemented
62C23×C4×C97C2211yescharacteristic, complemented
63#121×C97C2218yescomplemented
64C2×C4⋊C4×C97C2211yescharacteristic, complemented
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic, complemented
67C23×D97—42no—
68C23×D97—42no—
69C23×D97—42no—
70C23×D97C2311yescharacteristic
71C23×D97C2311yescharacteristic
72C23×D97C2311yescharacteristic
73C23×D97C2314yes—
74C2×D388—42no—
75C2×D388—42no—
76C2×D388C2314yes—
77C2×D388C2314yes—
78C2×C22⋊D97—42no—
79C2×C22⋊D97—42no—
80C2×C22⋊D97—42no—
81C2×C22⋊D97—42no—
82C2×C22⋊D97—42no—
83C2×C22⋊D97—42no—
84C2×C22⋊D97—42no—
85C2×C22⋊D97C2314yes—
86C2×C22⋊D97C2314yes—
87C2×C22⋊D97C2314yes—
88C2×C22⋊D97C2314yes—
89C2×C22⋊D97C2314yes—
90C2×C22⋊D97C2314yescomplemented
91C2×C22⋊D97C2314yescomplemented
92C22×Dic97—42no—
93C22×Dic97C2311yescharacteristic
94C22×Dic97C2311yescharacteristic
95C22×Dic97C2311yescharacteristic
96C22×Dic97C2311yescharacteristic
97C22×Dic97C2311yescharacteristic
98C22×Dic97C2314yescomplemented
99C2×C4×D97—42no—
100C2×C4×D97C2314yescomplemented
101C4⋊Dic97C2314yescomplemented
102C4⋊Dic97C2314yescomplemented
103#13095490C2314yescomplemented
104#13095490C2314yescomplemented
105#13095490C2314yescomplemented
106C22⋊Dic97C2314yes—
107C22⋊Dic97C2314yes—
108C22⋊Dic97C2314yescomplemented
109#13095488C2314yes—
110#13095488C2314yes—
111#13095488C2314yes—
112#13095488C2314yes—
113#13095488C2314yes—
114#13095488C2314yes—
115#13095488C2314yescomplemented
116C4×Dic97C2314yescomplemented
117C24×C97C2311yescharacteristic
118C24×C97C2311yescharacteristic
119C24×C97C2311yescharacteristic
120C2×D4×C97—42no—
121C2×D4×C97—42no—
122C2×D4×C97—42no—
123C2×D4×C97C2314yes—
124C2×D4×C97C2314yes—
125C2×D4×C97C2314yescomplemented
126C22×C4×C97—42no—
127C22×C4×C97C2311yescharacteristic
128C22×C4×C97C2311yescharacteristic
129C22×C4×C97C2311yescharacteristic
130C22×C4×C97C2311yescharacteristic
131C22×C4×C97C2314yescomplemented
132C4⋊C4×C97C2314yescomplemented
133C22⋊C4×C97C2314yes—
134C22⋊C4×C97C2314yescomplemented
135C23×C97—22no—
136C23×C97—22no—
137C23×C97—22no—
138C23×C97—22no—
139C23×C97—22no—
140C23×C97—22no—
141C23×C97—22no—
142C23×C97—22no—
143C23×C97—22no—
144C23×C97C2412yes—
145C23×C97C2412yes—
146C23×C97C2412yes—
147C23×C97C2411yescharacteristic
148C22×D97—22no—
149C22×D97—22no—
150C22×D97—22no—
151C22×D97—22no—
152C22×D97—22no—
153C22×D97—22no—
154C22×D97—22no—
155C22×D97—22no—
156C22×D97—22no—
157C22×D97—44no—
158C22×D97—44no—
159C22×D97—44no—
160C22×D97—416no—
161C22×D97C2412yes—
162C22×D97C2412yes—
163C22×D97C2412yes—
164D4×C97—44no—
165D4×C97—44no—
166D4×C97—44no—
167C2×C4×C97—22no—
168C2×C4×C97—22no—
169C2×C4×C97—22no—
170C2×C4×C97—22no—
171C2×C4×C97—44no—
172C2×C4×C97C2412yes—
173C2×C4×C97C2412yes—
174C2×C4×C97C2412yes—
175C2×C4×C97C2412yes—
176C22⋊D97—44no—
177C22⋊D97—44no—
178C22⋊D97—44no—
179C22⋊D97—44no—
180C22⋊D97—44no—
181C22⋊D97—44no—
182C22⋊D97—44no—
183C2×Dic97—22no—
184C2×Dic97—22no—
185C2×Dic97—22no—
186C2×Dic97—22no—
187C2×Dic97—22no—
188C2×Dic97—44no—
189C2×Dic97C2412yes—
190C2×Dic97C2412yes—
191C2×Dic97C2412yes—
192C2×Dic97C2412yes—
193C2×Dic97C2412yes—
194D388—44no—
195D388—44no—
196C4×D97—44no—
197C22×C97—22no—
198C22×C97—22no—
199C22×C97—22no—
200C22×C97—22no—
201C22×C97—22no—
202C22×C97—22no—
203C22×C97—22no—
204C22×C97—22no—
205C22×C97—22no—
206C22×C97—44no—
207C22×C97—44no—
208C22×C97—44no—
209C22×C97H(5,𝔽2)12yes—
210C22×C97H(5,𝔽2)12yes—
211C22×C97H(5,𝔽2)12yes—
212C22×C97C2511yescharacteristic
213C2×D97—22no—
214C2×D97—22no—
215C2×D97—22no—
216C2×D97—22no—
217C2×D97—22no—
218C2×D97—22no—
219C2×D97—22no—
220C2×D97—22no—
221C2×D97—22no—
222C2×D97—416no—
223C2×D97—44no—
224C2×D97—44no—
225C2×D97—44no—
226C4×C97—22no—
227C4×C97—22no—
228C4×C97—22no—
229C4×C97—22no—
230Dic97—22no—
231Dic97—22no—
232Dic97—22no—
233Dic97—22no—
234Dic97—22no—
235C2×C97—42no—
236C2×C97—42no—
237C2×C97—42no—
238C2×C97#56014yescomplemented
239C2×C97C2×H(5,𝔽2)11yescharacteristic
240C2×C97C2×H(5,𝔽2)11yescharacteristic
241C2×C97C2×H(5,𝔽2)11yescharacteristic
242D97—42no—
243D97—42no—
244D97—42no—
245C2×#560—971nomaximal, Sylow
246C97C2×#56011yescharacteristic, complemented, Sylow
247#560—9716no—
248C2×#127—971no—
249C2×#126—971no—
250C2×#126—971no—
251C2×#123—971no—
252C2×#123—971no—
253C2×#123—971no—
254C2×#121—971no—
255C2×#121—971no—
256C2×#121—971no—
257C2×#121—971no—
258C2×#121—971no—
259C2×#121D9711yescharacteristic, complemented
260C2×#120—971no—
261C2×#120—971no—
262C2×#120—971no—
263C25—971no—
264C22×D4—971no—
265C22×D4—971no—
266C22×D4—971no—
267C22×D4—971no—
268C22×D4—971no—
269C22×D4—971no—
270C22×D4—971no—
271C22×D4—971no—
272C22×D4—971no—
273C22×D4C2×D9711yescharacteristic
274C22×D4C2×D9711yescharacteristic
275C22×D4C2×D9711yescharacteristic, complemented
276C23×C4—971no—
277C23×C4—971no—
278C23×C4C2×D9711yescharacteristic, complemented
279#127—978no—
280#126—978no—
281#126—978no—
282#123—978no—
283#123—978no—
284#123—978no—
285#121—978no—
286#121—978no—
287#121—978no—
288#121—978no—
289#121—978no—
290#121C2×D9718yescomplemented
291#120—978no—
292#120—978no—
293#120—978no—
294C2×C4⋊C4—971no—
295C2×C4⋊C4—971no—
296C2×C4⋊C4—971no—
297C2×C4⋊C4—971no—
298C2×C4⋊C4—971no—
299C2×C4⋊C4C2×D9711yescharacteristic, complemented
300C2×C22⋊C4—971no—
301C2×C22⋊C4—971no—
302C2×C22⋊C4—971no—
303C2×C22⋊C4—971no—
304C2×C22⋊C4—971no—
305C2×C22⋊C4—971no—
306C2×C22⋊C4—971no—
307C2×C22⋊C4—971no—
308C2×C22⋊C4—971no—
309C2×C22⋊C4—971no—
310C2×C22⋊C4C2×D9711yescharacteristic
311C2×C22⋊C4C2×D9711yescharacteristic, complemented
312C2×C42—971no—
313C24—974no—
314C24—971no—
315C24—971no—
316C24—971no—
317C24—3882no—
318C24—3882no—
319C24—3882no—
320C24C22×D9711yescharacteristic
321C24C22×D9711yescharacteristic
322C24C22×D9711yescharacteristic
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—974no—
327C2×D4—974no—
328C2×D4—974no—
329C2×D4—974no—
330C2×D4—974no—
331C2×D4—974no—
332C2×D4—974no—
333C2×D4—974no—
334C2×D4—974no—
335C2×D4—3882no—
336C2×D4—3882no—
337C2×D4—3882no—
338C2×D4—3882no—
339C2×D4—3882no—
340C2×D4—3882no—
341C2×D4—3882no—
342C2×D4—3882no—
343C2×D4—3882no—
344C2×D4C22×D9714yes—
345C2×D4C22×D9714yes—
346C2×D4C22×D9714yescomplemented
347C22×C4—42no—
348C22×C4—974no—
349C22×C4—971no—
350C22×C4—971no—
351C22×C4—971no—
352C22×C4—971no—
353C22×C4—971no—
354C22×C4—974no—
355C22×C4—3882no—
356C22×C4—3882no—
357C22×C4C22×D9711yescharacteristic
358C22×C4C22×D9711yescharacteristic
359C22×C4C22×D9711yescharacteristic
360C22×C4C22×D9711yescharacteristic
361C22×C4C22×D9714yescomplemented
362C4⋊C4—974no—
363C4⋊C4—974no—
364C4⋊C4—974no—
365C4⋊C4—974no—
366C4⋊C4—974no—
367C4⋊C4C22×D9714yescomplemented
368C22⋊C4—974no—
369C22⋊C4—974no—
370C22⋊C4—974no—
371C22⋊C4—974no—
372C22⋊C4—974no—
373C22⋊C4—974no—
374C22⋊C4—974no—
375C22⋊C4—974no—
376C22⋊C4—974no—
377C22⋊C4—974no—
378C22⋊C4C22×D9714yes—
379C22⋊C4C22×D9714yescomplemented
380C42—974no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—972no—
391C23—972no—
392C23—972no—
393C23—3884no—
394C23—3884no—
395C23—3884no—
396C23—38816no—
397C23—1942no—
398C23—1942no—
399C23—1942no—
400C23—1942no—
401C23—1942no—
402C23—1942no—
403C23—1942no—
404C23—1942no—
405C23—1942no—
406C23C23×D9712yes—
407C23C23×D9712yes—
408C23C23×D9712yes—
409C23C23×D9711yescharacteristic
410D4—44no—
411D4—44no—
412D4—44no—
413D4—3884no—
414D4—3884no—
415D4—3884no—
416D4—3884no—
417D4—3884no—
418D4—3884no—
419D4—3884no—
420D4—3884no—
421D4—3884no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—44no—
427C2×C4—972no—
428C2×C4—972no—
429C2×C4—972no—
430C2×C4—972no—
431C2×C4—972no—
432C2×C4—3884no—
433C2×C4—3884no—
434C2×C4—1942no—
435C2×C4—1942no—
436C2×C4—1942no—
437C2×C4—1942no—
438C2×C4—1942no—
439C2×C4C23×D9712yes—
440C2×C4C23×D9712yes—
441C2×C4C23×D9712yes—
442C2×C4C23×D9712yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—38816no—
456C22—3884no—
457C22—3884no—
458C22—3884no—
459C22—1942no—
460C22—1942no—
461C22—1942no—
462C22—1942no—
463C22—1942no—
464C22—1942no—
465C22—1942no—
466C22—1942no—
467C22—1942no—
468C22#1569085512yes—
469C22#1569085512yes—
470C22#1569085612yes—
471C22C24×D9711yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—1942no—
477C4—1942no—
478C4—1942no—
479C4—1942no—
480C4—1942no—
481C2—42no—
482C2—42no—
483C2—42no—
484C2—3882no—
485C2—3882no—
486C2—3882no—
487C2#2026147314yesdirect product factor
488C2C2×#1569085511yescharacteristic
489C2C2×#1569085511yescharacteristic
490C2C2×#1569085611yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#121×C97C2
socleC23×C97C24
hypercenterC2×#121D97
hypocenterC97C2×#560
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20261473C2direct (N × H)
2. C2×#15690806C2semidirect
3. C2×#15690805C2semidirect
4. C2×#15690803C2semidirect
5. C2×#15690795C2semidirect
6. C2×#15690794C2semidirect
7. C2×#15690793C2semidirect
8. C2×#15690784C2semidirect
9. C2×#15690783C2semidirect
10. C2×#15690782C2semidirect
11. C2×#15690782C2semidirect
12. C2×#15690781C2semidirect
13. C2×#15690780C2semidirect
14. C2×#15690778C2semidirect
15. C2×#15690778C2semidirect
16. C2×#121×C97C2semidirect
25. C22×C22⋊D97C22semidirect
26. C22×C22⋊D97C22semidirect
27. C23×Dic97C22semidirect
28. C22×C4×D97C22semidirect
29. #15690806C22semidirect
30. #15690805C22semidirect
31. #15690803C22semidirect
32. #15690795C22semidirect
33. #15690794C22semidirect
34. #15690793C22semidirect
35. #15690784C22semidirect
36. #15690783C22semidirect
37. #15690782C22semidirect
38. #15690782C22semidirect
39. #15690781C22semidirect
40. #15690780C22semidirect
41. #15690778C22semidirect
42. #15690778C22semidirect
43. C2×C4⋊Dic97C22semidirect
44. C2×C4⋊Dic97C22semidirect
45. C2×#13095490C22semidirect
46. C2×#13095490C22semidirect
47. C2×#13095490C22semidirect
50. C2×C22⋊Dic97C22semidirect
57. C2×#13095488C22semidirect
58. C2×C4×Dic97C22semidirect
61. C22×D4×C97C22semidirect
62. C23×C4×C97C22semidirect
63. #121×C97C22semidirect
64. C2×C4⋊C4×C97C22semidirect
66. C2×C22⋊C4×C97C22semidirect
90. C2×C22⋊D97C23semidirect
91. C2×C22⋊D97C23semidirect
98. C22×Dic97C23semidirect
100. C2×C4×D97C23semidirect
101. C4⋊Dic97C23semidirect
102. C4⋊Dic97C23semidirect
103. #13095490C23semidirect
104. #13095490C23semidirect
105. #13095490C23semidirect
108. C22⋊Dic97C23semidirect
115. #13095488C23semidirect
116. C4×Dic97C23semidirect
125. C2×D4×C97C23semidirect
131. C22×C4×C97C23semidirect
132. C4⋊C4×C97C23semidirect
134. C22⋊C4×C97C23semidirect
238. C2×C97#560semidirect
246. C97C2×#560semidirect
259. C2×#121D97semidirect
275. C22×D4C2×D97semidirect
278. C23×C4C2×D97semidirect
290. #121C2×D97semidirect
299. C2×C4⋊C4C2×D97semidirect
311. C2×C22⋊C4C2×D97semidirect
346. C2×D4C22×D97semidirect
361. C22×C4C22×D97semidirect
367. C4⋊C4C22×D97semidirect
379. C22⋊C4C22×D97semidirect
487. C2#20261473direct (N × H)

Maximal subgroups

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

subgroupindex
#202614732
C2×#156908062
C2×#156908052
C2×#156908032
C2×#156907952
C2×#156907942
C2×#156907932
C2×#156907842
C2×#156907832
C2×#156907822
C2×#156907812
C2×#156907802
C2×#156907782
C2×#121×C972
C2×#56097