Finite Group Databasebeta

← C2×#20265987 #27185891

Subgroups — C2×#20265987 #27185891

Order 12928.

1460 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692302C211yesmaximal, characteristic, complemented
2C2×#15692300C211yesmaximal, characteristic, complemented
3#20265987C2116yesmaximal, direct product factor
4C2×#15692288C211yesmaximal, characteristic, complemented
5C2×#15692284C212yesmaximal, complemented
6C2×#15692279C212yesmaximal, complemented
7C2×#15692275C211yesmaximal, characteristic, complemented
8C2×#15692274C212yesmaximal, complemented
9C22×#13108062C211yesmaximal, characteristic, complemented
10C2×SD16×D101C212yesmaximal, complemented
11C2×#15692263C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
13#15692302—24no—
14#15692302C2218yescomplemented
15#15692300—24no—
16#15692300C2218yescomplemented
17C2×#13108074—21no—
18C2×#13108074C2211yescharacteristic
19C2×#13108074C2211yescharacteristic
20C2×#13108073—21no—
21C2×#13108073—22no—
22C2×#13108073C2212yes—
23C2×#13108073C2212yescomplemented
24C22×Dic202—21no—
25C22×Dic202C2211yescharacteristic, complemented
26C2×Q8×D101C2212yescomplemented
27C22×C22⋊D101—22no—
28C2×D4×D101—21no—
29C2×D4×D101C2212yes—
30#15692288C2218yescomplemented
31#15692284C22116yescomplemented
32#15692279C22116yescomplemented
33#15692275C2218yescomplemented
34#15692274C22116yescomplemented
35C2×Dic404C2212yescomplemented
36C2×#13108064C2211yescharacteristic, complemented
37C2×#13108063C2212yescomplemented
38C2×#13108062C2212yescomplemented
39C2×#13108062C2212yescomplemented
40C2×#13108062C2211yescharacteristic, complemented
41C2×#13108062C2218yescomplemented
42C2×#13108061C2211yescharacteristic, complemented
43SD16×D101C22116yescomplemented
44C2×#13108059C2211yescharacteristic, complemented
45#15692263C2218yescomplemented
46C2×#13108057C2212yescomplemented
47C22×#11717579C2211yescharacteristic, complemented
48C2×C8×D101C2212yescomplemented
49C2×#41×C101C2211yescharacteristic, complemented
50C22×D4×C101C2211yescharacteristic
51AGL(1,ℤ/8)×C101C2218yescomplemented
52C2×SD16×C101C2212yescomplemented
53C2×D8×C101C2212yescomplemented
54C2×M16×C101C2211yescharacteristic, complemented
55C23×D101—22no—
56#13108074—22no—
57#13108074—28no—
58#13108074D414yes—
59#13108074C2314yes—
60#13108074C2314yes—
61#13108073—216no—
62#13108073—24no—
63#13108073—24no—
64#13108073—28no—
65#13108073C2318yes—
66#13108073C2318yescomplemented
67C2×Dic202—21no—
68C2×Dic202—22no—
69C2×Dic202—22no—
70C2×Dic202—28no—
71C2×Dic202D411yescharacteristic
72C2×Dic202C2312yes—
73C2×Dic202C2311yescharacteristic
74C2×Dic202C2314yescomplemented
75Q8×D101—24no—
76Q8×D101C2318yescomplemented
77C2×D404D411yescharacteristic
78C2×D404C2311yescharacteristic
79C2×C22⋊D101—21no—
80C2×C22⋊D101—22no—
81C2×C22⋊D101—22no—
82C2×C22⋊D101—216no—
83C2×C22⋊D101—22no—
84C2×C22⋊D101D412yes—
85D4×D101—28no—
86D4×D101—24no—
87D4×D101C2318yes—
88C22×Dic101—21no—
89C22×Dic101—22no—
90C22×Dic101—22no—
91C2×C4×D101—22no—
92C2×C4×D101—21no—
93C2×C4×D101C2312yes—
94Dic404C2318yescomplemented
95#13108064C2314yescomplemented
96#13108063C2318yescomplemented
97#13108062C2318yescomplemented
98#13108062C2318yescomplemented
99#13108062C2314yescomplemented
100#13108061C2314yescomplemented
101#13108059C2314yescomplemented
102#13108057C2318yescomplemented
103C2×#11717579C2311yescharacteristic
104C2×#11717579C2311yescharacteristic
105C2×#11717579C2314yescomplemented
106C8×D101C2318yescomplemented
107C24×C101—21no—
108#41×C101—22no—
109#41×C101C2314yescomplemented
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101—21no—
113C2×D4×C101—22no—
114C2×D4×C101C2312yes—
115C2×D4×C101C2311yescharacteristic
116C2×D4×C101C2314yes—
117C22×C4×C101—21no—
118C22×C4×C101C2311yescharacteristic
119SD16×C101C2318yescomplemented
120D8×C101C2318yescomplemented
121M16×C101C2314yescomplemented
122C2×C8×C101C2312yes—
123C23×C101—21no—
124C23×C101—22no—
125C23×C101—24no—
126C23×C101—82no—
127C23×C101C2×D411yescharacteristic
128C22×D101—21no—
129C22×D101—28no—
130C22×D101—84no—
131C22×D101C2×D412yes—
132Q8×C101—21no—
133Q8×C101C2412yes—
134D4×C101—24no—
135D4×C101—22no—
136D4×C101—24no—
137D4×C101—21no—
138D4×C101C2412yes—
139D4×C101C2414yes—
140C2×C4×C101—21no—
141C2×C4×C101—24no—
142C2×C4×C101C2×D411yescharacteristic
143C2×C4×C101C2×D412yes—
144C2×C4×C101C2411yescharacteristic
145C2×C4×C101C2412yes—
146C22⋊D101—24no—
147C22⋊D101—28no—
148C22⋊D101—28no—
149C22⋊D101—28no—
150C22⋊D101C2×D418yes—
151C2×Dic101—21no—
152C2×Dic101—22no—
153C2×Dic101—24no—
154C2×Dic101—28no—
155C2×Dic101—28no—
156C2×Dic101C2×D412yes—
157D404—21no—
158D404C2×D414yes—
159D404C2412yes—
160C4×D101—28no—
161C4×D101—24no—
162C4×D101C2×D414yes—
163C4×D101C2414yes—
164Dic202—22no—
165Dic202—24no—
166Dic202—21no—
167Dic202—28no—
168Dic202C2×D414yes—
169Dic202C2414yes—
170Dic202C2412yes—
171C8×C101C2414yes—
172#11717579C2412yes—
173#11717579C2412yes—
174C22×C101—22no—
175C22×C101—22no—
176C22×C101—24no—
177C22×C101—44no—
178C22×C101—42no—
179C22×C101—84no—
180C22×C101C22×D412yes—
181C22×C101C22×D411yescharacteristic
182C2×D101—24no—
183C2×D101—22no—
184C2×D101—42no—
185C2×D101—88no—
186C2×D101C22×D414yes—
187C4×C101—22no—
188C4×C101C22×D411yescharacteristic
189C4×C101C22×D412yes—
190C4×C101C2511yescharacteristic
191Dic101—24no—
192Dic101—22no—
193Dic101C22×D414yes—
194C2×C101—22no—
195C2×C101—44no—
196C2×C101—42no—
197C2×C101#57712yescomplemented
198C2×C101C23×D411yescharacteristic
199D101—24no—
200D101—42no—
201C2×#577—1011nomaximal, Sylow
202C101C2×#57711yescharacteristic, complemented, Sylow
203C2×2−1+4—1011no—
204C2×H(5,𝔽2)—1011no—
205#577—10116no—
206C2×#137—1012no—
207C2×#137—1011no—
208C2×AGL(1,ℤ/8)—1012no—
209C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
210C2×#135—1011no—
211C2×#135—1012no—
212C22×SD16—1011no—
213C22×SD16—1012no—
214C2×#131—1011no—
2152−1+4—1018no—
2162−1+4—2024no—
217H(5,𝔽2)—1018no—
218H(5,𝔽2)—2024no—
219C2×#41—1011no—
220C2×#41—1012no—
221C2×#41—1012no—
222C2×#41—1011no—
223C2×#41—2021no—
224C2×#41—2022no—
225C2×#41—2021no—
226C2×#41C2×D10111yescharacteristic, complemented
227C22×Q8—1012no—
228C22×Q8—1011no—
229C22×Q8—2021no—
230C22×D4—1012no—
231C22×D4—2021no—
232C22×D4—2022no—
233C22×D4C2×D10111yescharacteristic
234#137—10116no—
235#137—1018no—
236AGL(1,ℤ/8)—10116no—
237AGL(1,ℤ/8)C2×D10118yescomplemented
238#135—1018no—
239#135—10116no—
240C2×Q16—1011no—
241C2×Q16—1012no—
242C2×SD16—1011no—
243C2×SD16—1012no—
244C2×SD16—1012no—
245C2×SD16—1018no—
246C2×SD16—1011no—
247C2×SD16—10116no—
248C2×SD16—1012no—
249C2×SD16C2×D10112yescomplemented
250C2×D8—1011no—
251C2×D8C2×D10112yescomplemented
252#131—1018no—
253C2×M16—1012no—
254C2×M16C2×D10111yescharacteristic, complemented
255C22×C8—1011no—
256C22×C8—1012no—
257C24—21no—
258C24—2022no—
259#41—22no—
260#41—1014no—
261#41—1014no—
262#41—1018no—
263#41—1018no—
264#41—1014no—
265#41—2028no—
266#41—2024no—
267#41—2024no—
268#41—20216no—
269#41—2028no—
270#41—2022no—
271#41C22×D10114yescomplemented
272C2×Q8—1011no—
273C2×Q8—1018no—
274C2×Q8—1012no—
275C2×Q8—1011no—
276C2×Q8—1014no—
277C2×Q8—2022no—
278C2×Q8—2028no—
279C2×Q8—2024no—
280C2×Q8—2021no—
281C2×Q8—2022no—
282C2×Q8C22×D10111yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—1012no—
287C2×D4—1018no—
288C2×D4—1011no—
289C2×D4—1011no—
290C2×D4—2024no—
291C2×D4—2022no—
292C2×D4—2028no—
293C2×D4—20216no—
294C2×D4—2022no—
295C2×D4—2022no—
296C2×D4—2021no—
297C2×D4C22×D10111yescharacteristic
298C2×D4C22×D10112yes—
299C2×D4C22×D10114yes—
300C22×C4—21no—
301C22×C4—1012no—
302C22×C4—2022no—
303C22×C4—2022no—
304C22×C4—2022no—
305C22×C4—2021no—
306C22×C4—2021no—
307C22×C4C22×D10111yescharacteristic
308Q16—1014no—
309Q16—1018no—
310SD16—1014no—
311SD16—1018no—
312SD16—1018no—
313SD16—1014no—
314SD16—1018no—
315SD16C22×D10118yescomplemented
316D8—1014no—
317D8C22×D10118yescomplemented
318M16—1018no—
319M16C22×D10114yescomplemented
320C2×C8—1011no—
321C2×C8—1011no—
322C2×C8—1014no—
323C2×C8—1018no—
324C2×C8C22×D10112yes—
325C23—21no—
326C23—22no—
327C23—24no—
328C23—82no—
329C23—8084no—
330C23—1012no—
331C23—2028no—
332C23—2021no—
333C23D4×D10111yescharacteristic
334Q8—21no—
335Q8—1014no—
336Q8—1014no—
337Q8—1012no—
338Q8—2028no—
339Q8—2024no—
340Q8—2022no—
341Q8—2021no—
342Q8C23×D10112yes—
343D4—24no—
344D4—24no—
345D4—22no—
346D4—21no—
347D4—1018no—
348D4—1014no—
349D4—1012no—
350D4—2028no—
351D4—2028no—
352D4—2028no—
353D4—2024no—
354D4—2021no—
355D4C23×D10112yes—
356D4C23×D10114yes—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—1012no—
360C2×C4—1014no—
361C2×C4—1014no—
362C2×C4—2028no—
363C2×C4—2028no—
364C2×C4—2028no—
365C2×C4—2024no—
366C2×C4—2021no—
367C2×C4—2022no—
368C2×C4—2024no—
369C2×C4D4×D10112yes—
370C2×C4D4×D10111yescharacteristic
371C2×C4C23×D10112yes—
372C2×C4C23×D10111yescharacteristic
373C8—1012no—
374C8—1012no—
375C8C23×D10114yes—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—84no—
382C22—8088no—
383C22—1014no—
384C22—4042no—
385C22—2022no—
386C22—2024no—
387C22C2×D4×D10112yes—
388C22C2×D4×D10111yescharacteristic
389C4—22no—
390C4—1014no—
391C4—2022no—
392C4—2024no—
393C4C2×D4×D10112yes—
394C4C2×D4×D10111yescharacteristic
395C4C24×D10111yescharacteristic
396C2—22no—
397C2—44no—
398C2—42no—
399C2—4042no—
400C2—2024no—
401C2#2026598712yesdirect product factor
402C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101C2×#577
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692302C2semidirect
2. C2×#15692300C2semidirect
3. #20265987C2direct (N × H)
4. C2×#15692288C2semidirect
5. C2×#15692284C2semidirect
6. C2×#15692279C2semidirect
7. C2×#15692275C2semidirect
8. C2×#15692274C2semidirect
9. C22×#13108062C2semidirect
10. C2×SD16×D101C2semidirect
11. C2×#15692263C2semidirect
12. C2×AGL(1,ℤ/8)×C101C2semidirect
14. #15692302C22semidirect
16. #15692300C22semidirect
23. C2×#13108073C22semidirect
25. C22×Dic202C22semidirect
26. C2×Q8×D101C22semidirect
30. #15692288C22semidirect
31. #15692284C22semidirect
32. #15692279C22semidirect
33. #15692275C22semidirect
34. #15692274C22semidirect
35. C2×Dic404C22semidirect
36. C2×#13108064C22semidirect
37. C2×#13108063C22semidirect
38. C2×#13108062C22semidirect
39. C2×#13108062C22semidirect
40. C2×#13108062C22semidirect
41. C2×#13108062C22semidirect
42. C2×#13108061C22semidirect
43. SD16×D101C22semidirect
44. C2×#13108059C22semidirect
45. #15692263C22semidirect
46. C2×#13108057C22semidirect
47. C22×#11717579C22semidirect
48. C2×C8×D101C22semidirect
49. C2×#41×C101C22semidirect
51. AGL(1,ℤ/8)×C101C22semidirect
52. C2×SD16×C101C22semidirect
53. C2×D8×C101C22semidirect
54. C2×M16×C101C22semidirect
66. #13108073C23semidirect
74. C2×Dic202C23semidirect
76. Q8×D101C23semidirect
94. Dic404C23semidirect
95. #13108064C23semidirect
96. #13108063C23semidirect
97. #13108062C23semidirect
98. #13108062C23semidirect
99. #13108062C23semidirect
100. #13108061C23semidirect
101. #13108059C23semidirect
102. #13108057C23semidirect
105. C2×#11717579C23semidirect
106. C8×D101C23semidirect
109. #41×C101C23semidirect
119. SD16×C101C23semidirect
120. D8×C101C23semidirect
121. M16×C101C23semidirect
197. C2×C101#577semidirect
202. C101C2×#577semidirect
209. C2×AGL(1,ℤ/8)D101semidirect
226. C2×#41C2×D101semidirect
237. AGL(1,ℤ/8)C2×D101semidirect
249. C2×SD16C2×D101semidirect
251. C2×D8C2×D101semidirect
254. C2×M16C2×D101semidirect
271. #41C22×D101semidirect
315. SD16C22×D101semidirect
317. D8C22×D101semidirect
319. M16C22×D101semidirect
401. C2#20265987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923022
C2×#156923002
#202659872
C2×#156922882
C2×#156922842
C2×#156922792
C2×#156922752
C2×#156922742
C22×#131080622
C2×SD16×D1012
C2×#156922632
C2×AGL(1,ℤ/8)×C1012
C2×#577101