Finite Group Databasebeta

← C2×#20261545 #27156495

Subgroups — C2×#20261545 #27156495

Order 12416.

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×#15690857C211yesmaximal, characteristic, complemented
2C2×#15690855C211yesmaximal, characteristic, complemented
3#20261545C2116yesmaximal, direct product factor
4C2×#15690843C211yesmaximal, characteristic, complemented
5C2×#15690839C212yesmaximal, complemented
6C2×#15690834C212yesmaximal, complemented
7C2×#15690830C211yesmaximal, characteristic, complemented
8C2×#15690829C212yesmaximal, complemented
9C22×#13095499C211yesmaximal, characteristic, complemented
10C2×SD16×D97C212yesmaximal, complemented
11C2×#15690818C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
13#15690857—24no—
14#15690857C2218yescomplemented
15#15690855—24no—
16#15690855C2218yescomplemented
17C2×#13095511—21no—
18C2×#13095511C2211yescharacteristic
19C2×#13095511C2211yescharacteristic
20C2×#13095510—21no—
21C2×#13095510—22no—
22C2×#13095510C2212yes—
23C2×#13095510C2212yescomplemented
24C22×Dic194—21no—
25C22×Dic194C2211yescharacteristic, complemented
26C2×Q8×D97C2212yescomplemented
27C22×C22⋊D97—22no—
28C2×D4×D97—21no—
29C2×D4×D97C2212yes—
30#15690843C2218yescomplemented
31#15690839C22116yescomplemented
32#15690834C22116yescomplemented
33#15690830C2218yescomplemented
34#15690829C22116yescomplemented
35C2×Dic388C2212yescomplemented
36C2×#13095501C2211yescharacteristic, complemented
37C2×#13095500C2212yescomplemented
38C2×#13095499C2212yescomplemented
39C2×#13095499C2212yescomplemented
40C2×#13095499C2211yescharacteristic, complemented
41C2×#13095499C2218yescomplemented
42C2×#13095498C2211yescharacteristic, complemented
43SD16×D97C22116yescomplemented
44C2×#13095496C2211yescharacteristic, complemented
45#15690818C2218yescomplemented
46C2×#13095494C2212yescomplemented
47C22×#11715895C2211yescharacteristic, complemented
48C2×C8×D97C2212yescomplemented
49C2×#41×C97C2211yescharacteristic, complemented
50C22×D4×C97C2211yescharacteristic
51AGL(1,ℤ/8)×C97C2218yescomplemented
52C2×SD16×C97C2212yescomplemented
53C2×D8×C97C2212yescomplemented
54C2×M16×C97C2211yescharacteristic, complemented
55C23×D97—22no—
56#13095511—22no—
57#13095511—28no—
58#13095511D414yes—
59#13095511C2314yes—
60#13095511C2314yes—
61#13095510—216no—
62#13095510—24no—
63#13095510—24no—
64#13095510—28no—
65#13095510C2318yes—
66#13095510C2318yescomplemented
67C2×Dic194—21no—
68C2×Dic194—22no—
69C2×Dic194—22no—
70C2×Dic194—28no—
71C2×Dic194D411yescharacteristic
72C2×Dic194C2312yes—
73C2×Dic194C2311yescharacteristic
74C2×Dic194C2314yescomplemented
75Q8×D97—24no—
76Q8×D97C2318yescomplemented
77C2×D388D411yescharacteristic
78C2×D388C2311yescharacteristic
79C2×C22⋊D97—21no—
80C2×C22⋊D97—22no—
81C2×C22⋊D97—22no—
82C2×C22⋊D97—216no—
83C2×C22⋊D97—22no—
84C2×C22⋊D97D412yes—
85D4×D97—28no—
86D4×D97—24no—
87D4×D97C2318yes—
88C22×Dic97—21no—
89C22×Dic97—22no—
90C22×Dic97—22no—
91C2×C4×D97—22no—
92C2×C4×D97—21no—
93C2×C4×D97C2312yes—
94Dic388C2318yescomplemented
95#13095501C2314yescomplemented
96#13095500C2318yescomplemented
97#13095499C2318yescomplemented
98#13095499C2318yescomplemented
99#13095499C2314yescomplemented
100#13095498C2314yescomplemented
101#13095496C2314yescomplemented
102#13095494C2318yescomplemented
103C2×#11715895C2311yescharacteristic
104C2×#11715895C2311yescharacteristic
105C2×#11715895C2314yescomplemented
106C8×D97C2318yescomplemented
107C24×C97—21no—
108#41×C97—22no—
109#41×C97C2314yescomplemented
110C2×Q8×C97C2311yescharacteristic
111C2×D4×C97—21no—
112C2×D4×C97—21no—
113C2×D4×C97—22no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2312yes—
116C2×D4×C97C2314yes—
117C22×C4×C97—21no—
118C22×C4×C97C2311yescharacteristic
119SD16×C97C2318yescomplemented
120D8×C97C2318yescomplemented
121M16×C97C2314yescomplemented
122C2×C8×C97C2312yes—
123C23×C97—21no—
124C23×C97—22no—
125C23×C97—24no—
126C23×C97—82no—
127C23×C97C2×D411yescharacteristic
128C22×D97—21no—
129C22×D97—28no—
130C22×D97—84no—
131C22×D97C2×D412yes—
132Q8×C97—21no—
133Q8×C97C2412yes—
134D4×C97—24no—
135D4×C97—22no—
136D4×C97—24no—
137D4×C97—21no—
138D4×C97C2412yes—
139D4×C97C2414yes—
140C2×C4×C97—21no—
141C2×C4×C97—24no—
142C2×C4×C97C2×D411yescharacteristic
143C2×C4×C97C2×D412yes—
144C2×C4×C97C2411yescharacteristic
145C2×C4×C97C2412yes—
146C22⋊D97—24no—
147C22⋊D97—28no—
148C22⋊D97—28no—
149C22⋊D97—28no—
150C22⋊D97C2×D418yes—
151C2×Dic97—21no—
152C2×Dic97—22no—
153C2×Dic97—24no—
154C2×Dic97—28no—
155C2×Dic97—28no—
156C2×Dic97C2×D412yes—
157D388—21no—
158D388C2×D414yes—
159D388C2412yes—
160C4×D97—28no—
161C4×D97—24no—
162C4×D97C2×D414yes—
163C4×D97C2414yes—
164Dic194—22no—
165Dic194—24no—
166Dic194—21no—
167Dic194—28no—
168Dic194C2×D414yes—
169Dic194C2414yes—
170Dic194C2412yes—
171C8×C97C2414yes—
172#11715895C2412yes—
173#11715895C2412yes—
174C22×C97—22no—
175C22×C97—22no—
176C22×C97—24no—
177C22×C97—44no—
178C22×C97—42no—
179C22×C97—84no—
180C22×C97C22×D412yes—
181C22×C97C22×D411yescharacteristic
182C2×D97—24no—
183C2×D97—22no—
184C2×D97—42no—
185C2×D97—88no—
186C2×D97C22×D414yes—
187C4×C97—22no—
188C4×C97C22×D411yescharacteristic
189C4×C97C22×D412yes—
190C4×C97C2511yescharacteristic
191Dic97—24no—
192Dic97—22no—
193Dic97C22×D414yes—
194C2×C97—22no—
195C2×C97—44no—
196C2×C97—42no—
197C2×C97#57712yescomplemented
198C2×C97C23×D411yescharacteristic
199D97—24no—
200D97—42no—
201C2×#577—971nomaximal, Sylow
202C97C2×#57711yescharacteristic, complemented, Sylow
203C2×2−1+4—971no—
204C2×H(5,𝔽2)—971no—
205#577—9716no—
206C2×#137—972no—
207C2×#137—971no—
208C2×AGL(1,ℤ/8)—972no—
209C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
210C2×#135—971no—
211C2×#135—972no—
212C22×SD16—971no—
213C22×SD16—972no—
214C2×#131—971no—
2152−1+4—978no—
2162−1+4—1944no—
217H(5,𝔽2)—978no—
218H(5,𝔽2)—1944no—
219C2×#41—971no—
220C2×#41—972no—
221C2×#41—972no—
222C2×#41—971no—
223C2×#41—1941no—
224C2×#41—1942no—
225C2×#41—1941no—
226C2×#41C2×D9711yescharacteristic, complemented
227C22×Q8—972no—
228C22×Q8—971no—
229C22×Q8—1941no—
230C22×D4—972no—
231C22×D4—1941no—
232C22×D4—1942no—
233C22×D4C2×D9711yescharacteristic
234#137—9716no—
235#137—978no—
236AGL(1,ℤ/8)—9716no—
237AGL(1,ℤ/8)C2×D9718yescomplemented
238#135—978no—
239#135—9716no—
240C2×Q16—971no—
241C2×Q16—972no—
242C2×SD16—971no—
243C2×SD16—972no—
244C2×SD16—972no—
245C2×SD16—978no—
246C2×SD16—971no—
247C2×SD16—9716no—
248C2×SD16—972no—
249C2×SD16C2×D9712yescomplemented
250C2×D8—971no—
251C2×D8C2×D9712yescomplemented
252#131—978no—
253C2×M16—972no—
254C2×M16C2×D9711yescharacteristic, complemented
255C22×C8—971no—
256C22×C8—972no—
257C24—21no—
258C24—1942no—
259#41—22no—
260#41—974no—
261#41—974no—
262#41—978no—
263#41—978no—
264#41—974no—
265#41—1948no—
266#41—1944no—
267#41—1944no—
268#41—19416no—
269#41—1948no—
270#41—1942no—
271#41C22×D9714yescomplemented
272C2×Q8—971no—
273C2×Q8—978no—
274C2×Q8—972no—
275C2×Q8—971no—
276C2×Q8—974no—
277C2×Q8—1942no—
278C2×Q8—1948no—
279C2×Q8—1944no—
280C2×Q8—1941no—
281C2×Q8—1942no—
282C2×Q8C22×D9711yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—972no—
287C2×D4—978no—
288C2×D4—971no—
289C2×D4—971no—
290C2×D4—1944no—
291C2×D4—1942no—
292C2×D4—1948no—
293C2×D4—19416no—
294C2×D4—1942no—
295C2×D4—1942no—
296C2×D4—1941no—
297C2×D4C22×D9711yescharacteristic
298C2×D4C22×D9712yes—
299C2×D4C22×D9714yes—
300C22×C4—21no—
301C22×C4—972no—
302C22×C4—1942no—
303C22×C4—1942no—
304C22×C4—1942no—
305C22×C4—1941no—
306C22×C4—1941no—
307C22×C4C22×D9711yescharacteristic
308Q16—974no—
309Q16—978no—
310SD16—974no—
311SD16—978no—
312SD16—978no—
313SD16—974no—
314SD16—978no—
315SD16C22×D9718yescomplemented
316D8—974no—
317D8C22×D9718yescomplemented
318M16—978no—
319M16C22×D9714yescomplemented
320C2×C8—971no—
321C2×C8—971no—
322C2×C8—974no—
323C2×C8—978no—
324C2×C8C22×D9712yes—
325C23—21no—
326C23—22no—
327C23—24no—
328C23—82no—
329C23—7764no—
330C23—972no—
331C23—1948no—
332C23—1941no—
333C23D4×D9711yescharacteristic
334Q8—21no—
335Q8—974no—
336Q8—974no—
337Q8—972no—
338Q8—1948no—
339Q8—1944no—
340Q8—1942no—
341Q8—1941no—
342Q8C23×D9712yes—
343D4—24no—
344D4—24no—
345D4—22no—
346D4—21no—
347D4—978no—
348D4—974no—
349D4—972no—
350D4—1948no—
351D4—1948no—
352D4—1948no—
353D4—1944no—
354D4—1941no—
355D4C23×D9712yes—
356D4C23×D9714yes—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—972no—
360C2×C4—974no—
361C2×C4—974no—
362C2×C4—1948no—
363C2×C4—1948no—
364C2×C4—1948no—
365C2×C4—1944no—
366C2×C4—1941no—
367C2×C4—1942no—
368C2×C4—1944no—
369C2×C4D4×D9712yes—
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9712yes—
372C2×C4C23×D9711yescharacteristic
373C8—972no—
374C8—972no—
375C8C23×D9714yes—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—84no—
382C22—7768no—
383C22—974no—
384C22—3882no—
385C22—1942no—
386C22—1944no—
387C22C2×D4×D9712yes—
388C22C2×D4×D9711yescharacteristic
389C4—22no—
390C4—974no—
391C4—1942no—
392C4—1944no—
393C4C2×D4×D9712yes—
394C4C2×D4×D9711yescharacteristic
395C4C24×D9711yescharacteristic
396C2—22no—
397C2—44no—
398C2—42no—
399C2—3882no—
400C2—1944no—
401C2#2026154512yesdirect product factor
402C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97C2×#577
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
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×#15690857C2semidirect
2. C2×#15690855C2semidirect
3. #20261545C2direct (N × H)
4. C2×#15690843C2semidirect
5. C2×#15690839C2semidirect
6. C2×#15690834C2semidirect
7. C2×#15690830C2semidirect
8. C2×#15690829C2semidirect
9. C22×#13095499C2semidirect
10. C2×SD16×D97C2semidirect
11. C2×#15690818C2semidirect
12. C2×AGL(1,ℤ/8)×C97C2semidirect
14. #15690857C22semidirect
16. #15690855C22semidirect
23. C2×#13095510C22semidirect
25. C22×Dic194C22semidirect
26. C2×Q8×D97C22semidirect
30. #15690843C22semidirect
31. #15690839C22semidirect
32. #15690834C22semidirect
33. #15690830C22semidirect
34. #15690829C22semidirect
35. C2×Dic388C22semidirect
36. C2×#13095501C22semidirect
37. C2×#13095500C22semidirect
38. C2×#13095499C22semidirect
39. C2×#13095499C22semidirect
40. C2×#13095499C22semidirect
41. C2×#13095499C22semidirect
42. C2×#13095498C22semidirect
43. SD16×D97C22semidirect
44. C2×#13095496C22semidirect
45. #15690818C22semidirect
46. C2×#13095494C22semidirect
47. C22×#11715895C22semidirect
48. C2×C8×D97C22semidirect
49. C2×#41×C97C22semidirect
51. AGL(1,ℤ/8)×C97C22semidirect
52. C2×SD16×C97C22semidirect
53. C2×D8×C97C22semidirect
54. C2×M16×C97C22semidirect
66. #13095510C23semidirect
74. C2×Dic194C23semidirect
76. Q8×D97C23semidirect
94. Dic388C23semidirect
95. #13095501C23semidirect
96. #13095500C23semidirect
97. #13095499C23semidirect
98. #13095499C23semidirect
99. #13095499C23semidirect
100. #13095498C23semidirect
101. #13095496C23semidirect
102. #13095494C23semidirect
105. C2×#11715895C23semidirect
106. C8×D97C23semidirect
109. #41×C97C23semidirect
119. SD16×C97C23semidirect
120. D8×C97C23semidirect
121. M16×C97C23semidirect
197. C2×C97#577semidirect
202. C97C2×#577semidirect
209. C2×AGL(1,ℤ/8)D97semidirect
226. C2×#41C2×D97semidirect
237. AGL(1,ℤ/8)C2×D97semidirect
249. C2×SD16C2×D97semidirect
251. C2×D8C2×D97semidirect
254. C2×M16C2×D97semidirect
271. #41C22×D97semidirect
315. SD16C22×D97semidirect
317. D8C22×D97semidirect
319. M16C22×D97semidirect
401. C2#20261545direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908572
C2×#156908552
#202615452
C2×#156908432
C2×#156908392
C2×#156908342
C2×#156908302
C2×#156908292
C22×#130954992
C2×SD16×D972
C2×#156908182
C2×AGL(1,ℤ/8)×C972
C2×#57797