Finite Group Databasebeta

← C2×#20265976 #27185878

Subgroups — C2×#20265976 #27185878

Order 12928.

1484 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
1C22×#13108075C211yesmaximal, characteristic, complemented
2C2×#41×D101C211yesmaximal, characteristic, complemented
3#20265976C2116yesmaximal, direct product factor
4C2×#15692287C212yesmaximal, complemented
5C2×#15692286C211yesmaximal, characteristic, complemented
6C2×#15692283C211yesmaximal, characteristic, complemented
7C2×#15692282C212yesmaximal, complemented
8C2×#15692280C211yesmaximal, characteristic, complemented
9C2×#15692278C212yesmaximal, complemented
10C2×#15692276C212yesmaximal, complemented
11C2×M16×D101C211yesmaximal, characteristic, complemented
12C2×#137×C101C211yesmaximal, characteristic, complemented
13C2×#13108075—21no—
14C2×#13108075—21no—
15C2×#13108075—24no—
16C2×#13108075C2212yes—
17C2×#13108075C2211yescharacteristic, complemented
18C2×#13108075C2212yescomplemented
19C2×#13108075C2218yescomplemented
20C2×#13108074—21no—
21C2×#13108074C2211yescharacteristic
22C2×#13108073—21no—
23C2×#13108073C2211yescharacteristic, complemented
24#41×D101—24no—
25#41×D101C2218yescomplemented
26C2×Q8×D101C2211yescharacteristic, complemented
27C22×D404—21no—
28C22×D404C2211yescharacteristic, complemented
29C2×D4×D101—21no—
30C2×D4×D101C2211yescharacteristic, complemented
31C22×C4×D101—21no—
32C22×C4×D101—21no—
33C22×C4×D101C2211yescharacteristic
34#15692287C22116yescomplemented
35#15692286C2218yescomplemented
36#15692283C2218yescomplemented
37#15692282C22116yescomplemented
38#15692280C2218yescomplemented
39#15692278C22116yescomplemented
40#15692276C22116yescomplemented
41C2×#13108064C2212yescomplemented
42C2×#13108063C2212yescomplemented
43C2×#13108061C2212yescomplemented
44C2×#13108061C2212yescomplemented
45C2×D808C2212yescomplemented
46C2×#13108059C2212yescomplemented
47C2×#13108058C2211yescharacteristic, complemented
48C2×#13108057C2212yescomplemented
49M16×D101C2218yescomplemented
50C2×C8×D101C2212yescomplemented
51C2×#41×C101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic, complemented
53#137×C101C2218yescomplemented
54C2×Q16×C101C2212yescomplemented
55C2×SD16×C101C2212yescomplemented
56C2×M16×C101C2211yescharacteristic, complemented
57C23×D101—21no—
58C23×D101—21no—
59C23×D101D411yescharacteristic
60#13108075—22no—
61#13108075—24no—
62#13108075—28no—
63#13108075—28no—
64#13108075—24no—
65#13108075C2318yes—
66#13108075C2314yescomplemented
67#13108075C2318yescomplemented
68#13108074—22no—
69#13108074—28no—
70#13108074C2314yes—
71#13108073—28no—
72#13108073—22no—
73#13108073C2314yescomplemented
74C2×Dic202—21no—
75C2×Dic202C2311yescharacteristic
76Q8×D101—22no—
77Q8×D101C2314yescomplemented
78C2×D404—21no—
79C2×D404—22no—
80C2×D404—28no—
81C2×D404—22no—
82C2×D404—22no—
83C2×D404—21no—
84C2×D404C2311yescharacteristic
85C2×D404C2312yes—
86C2×D404C2314yescomplemented
87C2×C22⋊D101—21no—
88C2×C22⋊D101—21no—
89C2×C22⋊D101—21no—
90D4×D101—28no—
91D4×D101—22no—
92D4×D101C2314yescomplemented
93C22×Dic101—21no—
94C22×Dic101D411yescharacteristic
95C2×C4×D101—21no—
96C2×C4×D101—21no—
97C2×C4×D101—21no—
98C2×C4×D101—22no—
99C2×C4×D101—28no—
100C2×C4×D101—22no—
101C2×C4×D101—28no—
102C2×C4×D101D411yescharacteristic
103C2×C4×D101D414yes—
104C2×C4×D101D411yescharacteristic
105C2×C4×D101C2311yescharacteristic
106C2×C4×D101C2314yes—
107C2×C4×D101C2311yescharacteristic
108#13108064C2318yescomplemented
109#13108063C2318yescomplemented
110#13108061C2318yescomplemented
111#13108061C2318yescomplemented
112D808C2318yescomplemented
113#13108059C2318yescomplemented
114#13108058C2314yescomplemented
115#13108057C2318yescomplemented
116C2×#11717579C2312yes—
117C8×D101C2318yescomplemented
118#41×C101—22no—
119#41×C101C2314yes—
120C2×Q8×C101—21no—
121C2×Q8×C101—22no—
122C2×Q8×C101C2311yescharacteristic
123C2×Q8×C101C2312yes—
124C2×Q8×C101C2314yescomplemented
125C2×D4×C101—21no—
126C2×D4×C101C2311yescharacteristic
127C22×C4×C101—21no—
128C22×C4×C101—21no—
129C22×C4×C101C2311yescharacteristic
130Q16×C101C2318yescomplemented
131SD16×C101C2318yescomplemented
132M16×C101C2314yescomplemented
133C2×C8×C101C2312yes—
134C23×C101—21no—
135C23×C101C2×D411yescharacteristic
136C22×D101—21no—
137C22×D101—22no—
138C22×D101—24no—
139C22×D101—24no—
140C22×D101—42no—
141C22×D101—44no—
142C22×D101—44no—
143C22×D101C2×D411yescharacteristic
144C22×D101C2×D411yescharacteristic
145C22×D101C2×D414yes—
146Q8×C101—22no—
147Q8×C101—24no—
148Q8×C101—21no—
149Q8×C101C2414yes—
150Q8×C101C2412yes—
151D4×C101—24no—
152D4×C101—21no—
153D4×C101C2412yes—
154C2×C4×C101—21no—
155C2×C4×C101—22no—
156C2×C4×C101—24no—
157C2×C4×C101—24no—
158C2×C4×C101C2×D411yescharacteristic
159C2×C4×C101C2×D412yes—
160C2×C4×C101C2411yescharacteristic
161C2×C4×C101C2412yes—
162C22⋊D101—24no—
163C22⋊D101—24no—
164C22⋊D101—24no—
165C2×Dic101—21no—
166C2×Dic101—24no—
167C2×Dic101C2×D411yescharacteristic
168C2×Dic101C2×D411yescharacteristic
169C2×Dic101C2×D414yes—
170D404—24no—
171D404—28no—
172D404—28no—
173D404—21no—
174D404—22no—
175D404—24no—
176D404C2414yes—
177D404C2412yes—
178C4×D101—24no—
179C4×D101—24no—
180C4×D101—24no—
181C4×D101—28no—
182C4×D101—28no—
183C4×D101C2×D414yes—
184C4×D101C2×D412yes—
185C4×D101C2×D412yes—
186C4×D101C2×D414yes—
187C4×D101C2412yes—
188C4×D101C2412yes—
189Dic202—21no—
190Dic202—24no—
191Dic202C2412yes—
192C8×C101C2414yes—
193#11717579C2414yes—
194C22×C101—22no—
195C22×C101—22no—
196C22×C101—42no—
197C22×C101C22×D412yes—
198C22×C101C22×D411yescharacteristic
199C2×D101—22no—
200C2×D101—22no—
201C2×D101—24no—
202C2×D101—22no—
203C2×D101—44no—
204C2×D101—48no—
205C2×D101—48no—
206C2×D101—42no—
207C2×D101—44no—
208C2×D101C22×D412yes—
209C2×D101C22×D412yes—
210C4×C101—22no—
211C4×C101—24no—
212C4×C101C22×D411yescharacteristic
213C4×C101C22×D412yes—
214C4×C101C2511yescharacteristic
215Dic101—22no—
216Dic101C22×D412yes—
217Dic101C22×D412yes—
218C2×C101—22no—
219C2×C101—42no—
220C2×C101#57512yescomplemented
221C2×C101C23×D411yescharacteristic
222D101—22no—
223D101—22no—
224D101—44no—
225D101—42no—
226C2×#575—1011nomaximal, Sylow
227C101C2×#57511yescharacteristic, complemented, Sylow
228C22×#41—1011no—
229C22×#41—1011no—
230#575—10116no—
231C2×#137—1011no—
232C2×#137—1012no—
233C2×#137D10111yescharacteristic, complemented
234C2×AGL(1,ℤ/8)—1011no—
235C2×AGL(1,ℤ/8)—1012no—
236C2×AGL(1,ℤ/8)—1011no—
237C2×#135—1012no—
238C2×#135—1012no—
239C22×M16—1011no—
240C2×#41—1018no—
241C2×#41—1012no—
242C2×#41—1011no—
243C2×#41—1012no—
244C2×#41—1018no—
245C2×#41—1011no—
246C2×#41—1011no—
247C2×#41—2021no—
248C2×#41—2021no—
249C2×#41—2021no—
250C2×#41—2024no—
251C2×#41—2021no—
252C2×#41—2024no—
253C2×#41C2×D10111yescharacteristic
254C22×Q8—1011no—
255C22×Q8C2×D10111yescharacteristic, complemented
256C22×D4—1011no—
257C22×D4—1011no—
258C22×D4—2021no—
259C22×D4—2021no—
260C23×C4—1011no—
261C23×C4—2021no—
262C23×C4—2021no—
263#137—1018no—
264#137—10116no—
265#137C2×D10118yescomplemented
266AGL(1,ℤ/8)—1018no—
267AGL(1,ℤ/8)—10116no—
268AGL(1,ℤ/8)—1018no—
269#135—10116no—
270#135—10116no—
271C2×Q16—1012no—
272C2×Q16C2×D10112yescomplemented
273C2×SD16—1012no—
274C2×SD16—1012no—
275C2×SD16—1012no—
276C2×SD16C2×D10112yescomplemented
277C2×D8—1012no—
278C2×D8—1012no—
279C2×M16—1011no—
280C2×M16—1018no—
281C2×M16—1012no—
282C2×M16C2×D10111yescharacteristic, complemented
283C22×C8—1012no—
284C24—1011no—
285C24—2021no—
286C24—2021no—
287#41—22no—
288#41—1018no—
289#41—1014no—
290#41—1018no—
291#41—1014no—
292#41—1014no—
293#41—2028no—
294#41—2028no—
295#41—2022no—
296#41—2028no—
297#41—2024no—
298#41—2024no—
299#41—2028no—
300#41—2022no—
301#41—2022no—
302#41C22×D10114yes—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—1014no—
306C2×Q8—1011no—
307C2×Q8—2021no—
308C2×Q8—2022no—
309C2×Q8C22×D10111yescharacteristic
310C2×Q8C22×D10112yes—
311C2×Q8C22×D10114yescomplemented
312C2×D4—21no—
313C2×D4—1014no—
314C2×D4—1011no—
315C2×D4—1012no—
316C2×D4—1014no—
317C2×D4—2028no—
318C2×D4—2021no—
319C2×D4—2022no—
320C2×D4—2022no—
321C2×D4—2022no—
322C2×D4—2021no—
323C2×D4—2028no—
324C2×D4—2021no—
325C2×D4—2021no—
326C2×D4—2022no—
327C2×D4—2021no—
328C2×D4C22×D10111yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—1011no—
332C22×C4—1011no—
333C22×C4—1014no—
334C22×C4—1011no—
335C22×C4—1011no—
336C22×C4—1011no—
337C22×C4—1014no—
338C22×C4—2028no—
339C22×C4—2021no—
340C22×C4—2022no—
341C22×C4—2022no—
342C22×C4—2028no—
343C22×C4—2021no—
344C22×C4—2021no—
345C22×C4—2021no—
346C22×C4C22×D10111yescharacteristic
347Q16—1018no—
348Q16C22×D10118yescomplemented
349SD16—1018no—
350SD16—1018no—
351SD16—1018no—
352SD16C22×D10118yescomplemented
353D8—1018no—
354D8—1018no—
355M16—1014no—
356M16—1018no—
357M16C22×D10114yescomplemented
358C2×C8—1012no—
359C2×C8—1018no—
360C2×C8C22×D10112yes—
361C23—21no—
362C23—1014no—
363C23—1011no—
364C23—1011no—
365C23—4044no—
366C23—4042no—
367C23—4044no—
368C23—2024no—
369C23—2024no—
370C23—2022no—
371C23—2021no—
372C23D4×D10111yescharacteristic
373Q8—22no—
374Q8—24no—
375Q8—21no—
376Q8—1012no—
377Q8—2024no—
378Q8—2021no—
379Q8C23×D10112yes—
380Q8C23×D10114yes—
381D4—24no—
382D4—21no—
383D4—1014no—
384D4—1012no—
385D4—2024no—
386D4—2028no—
387D4—2028no—
388D4—2024no—
389D4—2024no—
390D4—2024no—
391D4—2024no—
392D4—2021no—
393D4—2022no—
394D4C23×D10112yes—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—1014no—
400C2×C4—1011no—
401C2×C4—1011no—
402C2×C4—1014no—
403C2×C4—1012no—
404C2×C4—1012no—
405C2×C4—1014no—
406C2×C4—1012no—
407C2×C4—1012no—
408C2×C4—2024no—
409C2×C4—2028no—
410C2×C4—2028no—
411C2×C4—2024no—
412C2×C4—2024no—
413C2×C4—2021no—
414C2×C4—2024no—
415C2×C4D4×D10112yes—
416C2×C4D4×D10111yescharacteristic
417C2×C4C23×D10112yes—
418C2×C4C23×D10111yescharacteristic
419C8—1014no—
420C8C23×D10114yes—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—1012no—
425C22—1012no—
426C22—4048no—
427C22—4044no—
428C22—4048no—
429C22—4044no—
430C22—4042no—
431C22—2024no—
432C22—2022no—
433C22—2022no—
434C22—2022no—
435C22C2×D4×D10112yes—
436C22C2×D4×D10111yescharacteristic
437C4—22no—
438C4—24no—
439C4—1012no—
440C4—1012no—
441C4—2022no—
442C4C2×D4×D10112yes—
443C4C2×D4×D10111yescharacteristic
444C4C24×D10111yescharacteristic
445C2—22no—
446C2—42no—
447C2—4044no—
448C2—4042no—
449C2—2022no—
450C2—2022no—
451C2#2026597612yesdirect product factor
452C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
hypercenterC2×#137D101
hypocenterC101C2×#575
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C2×#41×D101C2semidirect
3. #20265976C2direct (N × H)
4. C2×#15692287C2semidirect
5. C2×#15692286C2semidirect
6. C2×#15692283C2semidirect
7. C2×#15692282C2semidirect
8. C2×#15692280C2semidirect
9. C2×#15692278C2semidirect
10. C2×#15692276C2semidirect
11. C2×M16×D101C2semidirect
12. C2×#137×C101C2semidirect
17. C2×#13108075C22semidirect
18. C2×#13108075C22semidirect
19. C2×#13108075C22semidirect
23. C2×#13108073C22semidirect
25. #41×D101C22semidirect
26. C2×Q8×D101C22semidirect
28. C22×D404C22semidirect
30. C2×D4×D101C22semidirect
34. #15692287C22semidirect
35. #15692286C22semidirect
36. #15692283C22semidirect
37. #15692282C22semidirect
38. #15692280C22semidirect
39. #15692278C22semidirect
40. #15692276C22semidirect
41. C2×#13108064C22semidirect
42. C2×#13108063C22semidirect
43. C2×#13108061C22semidirect
44. C2×#13108061C22semidirect
45. C2×D808C22semidirect
46. C2×#13108059C22semidirect
47. C2×#13108058C22semidirect
48. C2×#13108057C22semidirect
49. M16×D101C22semidirect
50. C2×C8×D101C22semidirect
52. C22×Q8×C101C22semidirect
53. #137×C101C22semidirect
54. C2×Q16×C101C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×M16×C101C22semidirect
66. #13108075C23semidirect
67. #13108075C23semidirect
73. #13108073C23semidirect
77. Q8×D101C23semidirect
86. C2×D404C23semidirect
92. D4×D101C23semidirect
108. #13108064C23semidirect
109. #13108063C23semidirect
110. #13108061C23semidirect
111. #13108061C23semidirect
112. D808C23semidirect
113. #13108059C23semidirect
114. #13108058C23semidirect
115. #13108057C23semidirect
117. C8×D101C23semidirect
124. C2×Q8×C101C23semidirect
130. Q16×C101C23semidirect
131. SD16×C101C23semidirect
132. M16×C101C23semidirect
220. C2×C101#575semidirect
227. C101C2×#575semidirect
233. C2×#137D101semidirect
255. C22×Q8C2×D101semidirect
265. #137C2×D101semidirect
272. C2×Q16C2×D101semidirect
276. C2×SD16C2×D101semidirect
282. C2×M16C2×D101semidirect
311. C2×Q8C22×D101semidirect
348. Q16C22×D101semidirect
352. SD16C22×D101semidirect
357. M16C22×D101semidirect
451. C2#20265976direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080752
C2×#41×D1012
#202659762
C2×#156922872
C2×#156922862
C2×#156922832
C2×#156922822
C2×#156922802
C2×#156922782
C2×#156922762
C2×M16×D1012
C2×#137×C1012
C2×#575101