Finite Group Databasebeta

← C2×#20261541 #27156491

Subgroups — C2×#20261541 #27156491

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×#15690858C211yesmaximal, characteristic, complemented
2C2×#15690855C211yesmaximal, characteristic, complemented
3#20261541C2116yesmaximal, direct product factor
4C2×#15690841C211yesmaximal, characteristic, complemented
5C2×#15690839C212yesmaximal, complemented
6C2×#15690837C212yesmaximal, complemented
7C2×#15690836C211yesmaximal, characteristic, complemented
8C2×#15690832C211yesmaximal, characteristic, complemented
9C2×#15690831C212yesmaximal, complemented
10C2×SD16×D97C212yesmaximal, complemented
11C2×#15690816C211yesmaximal, characteristic, complemented
12C22×SD16×C97C211yesmaximal, characteristic, complemented
13#15690858—24no—
14#15690858C2218yescomplemented
15#15690855—24no—
16#15690855C2218yescomplemented
17C2×#13095512—21no—
18C2×#13095512C2212yescomplemented
19C2×#13095511—22no—
20C2×#13095511C2211yescharacteristic
21C2×#13095511C2211yescharacteristic
22C2×#13095511C2211yescharacteristic, complemented
23C2×#13095510—21no—
24C2×#13095510C2212yes—
25C2×Q8×D97—21no—
26C2×Q8×D97C2212yescomplemented
27C22×C22⋊D97—22no—
28C2×D4×D97—21no—
29C2×D4×D97C2212yes—
30#15690841C2218yescomplemented
31#15690839C22116yescomplemented
32#15690837C22116yescomplemented
33#15690836C2218yescomplemented
34#15690832C2218yescomplemented
35#15690831C22116yescomplemented
36C2×Dic388C2211yescharacteristic, complemented
37C2×#13095501C2212yescomplemented
38C2×#13095500C2211yescharacteristic, complemented
39C2×#13095500C2211yescharacteristic, complemented
40C2×#13095499C2212yescomplemented
41C2×#13095498C2212yescomplemented
42SD16×D97C22116yescomplemented
43C2×D776C2211yescharacteristic, complemented
44C2×#13095496C2212yescomplemented
45#15690816C2218yescomplemented
46C2×#13095495C2211yescharacteristic, complemented
47C2×#13095494C2212yescomplemented
48C2×C8×D97C2212yescomplemented
49C22×Q8×C97C2211yescharacteristic, complemented
50C22×D4×C97C2211yescharacteristic
51C2×SD16×C97C2212yescomplemented
52C2×SD16×C97C2212yescomplemented
53C2×SD16×C97C2218yescomplemented
54C22×C8×C97C2211yescharacteristic, complemented
55C23×D97—22no—
56#13095512—28no—
57#13095512—24no—
58#13095512C2318yescomplemented
59#13095511—22no—
60#13095511—22no—
61#13095511—216no—
62#13095511D414yes—
63#13095511C2314yes—
64#13095511C2314yes—
65#13095511C2314yescomplemented
66#13095510—24no—
67#13095510—28no—
68#13095510C2318yes—
69C2×Dic194—22no—
70C2×Dic194D411yescharacteristic
71C2×Dic194C2311yescharacteristic
72C2×Dic194C2311yescharacteristic
73Q8×D97—24no—
74Q8×D97—28no—
75Q8×D97C2318yescomplemented
76C2×D388—22no—
77C2×D388D411yescharacteristic
78C2×D388C2311yescharacteristic
79C2×D388C2311yescharacteristic
80C2×C22⋊D97—21no—
81C2×C22⋊D97—21no—
82C2×C22⋊D97—22no—
83C2×C22⋊D97—216no—
84C2×C22⋊D97—22no—
85C2×C22⋊D97D412yes—
86D4×D97—28no—
87D4×D97—24no—
88D4×D97C2318yes—
89C22×Dic97—22no—
90C2×C4×D97—22no—
91C2×C4×D97—21no—
92C2×C4×D97—21no—
93C2×C4×D97—22no—
94C2×C4×D97C2312yes—
95Dic388C2314yescomplemented
96#13095501C2318yescomplemented
97#13095500C2314yescomplemented
98#13095500C2314yescomplemented
99#13095499C2318yescomplemented
100#13095498C2318yescomplemented
101D776C2314yescomplemented
102#13095496C2318yescomplemented
103#13095495C2314yescomplemented
104#13095494C2318yescomplemented
105C2×#11715895C2312yes—
106C8×D97C2318yescomplemented
107C24×C97—21no—
108C2×Q8×C97—21no—
109C2×Q8×C97—22no—
110C2×Q8×C97C2312yes—
111C2×Q8×C97C2314yescomplemented
112C2×D4×C97—21no—
113C2×D4×C97—22no—
114C2×D4×C97C2312yes—
115C2×D4×C97C2314yes—
116C22×C4×C97—21no—
117C22×C4×C97C2311yescharacteristic
118SD16×C97C2318yescomplemented
119SD16×C97C2318yescomplemented
120C2×C8×C97C2311yescharacteristic
121C2×C8×C97C2311yescharacteristic
122C2×C8×C97C2314yescomplemented
123C23×C97—22no—
124C23×C97—24no—
125C23×C97—82no—
126C23×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—28no—
130C22×D97—84no—
131C22×D97C2×D412yes—
132Q8×C97—22no—
133Q8×C97—24no—
134Q8×C97C2414yes—
135D4×C97—22no—
136D4×C97—24no—
137D4×C97C2414yes—
138C2×C4×C97—22no—
139C2×C4×C97—24no—
140C2×C4×C97C2×D411yescharacteristic
141C2×C4×C97C2×D412yes—
142C2×C4×C97C2411yescharacteristic
143C2×C4×C97C2412yes—
144C22⋊D97—24no—
145C22⋊D97—24no—
146C22⋊D97—28no—
147C22⋊D97—28no—
148C22⋊D97C2×D418yes—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97—28no—
152C2×Dic97C2×D412yes—
153D388—28no—
154D388—21no—
155D388—21no—
156D388C2×D414yes—
157D388C2412yes—
158D388C2412yes—
159C4×D97—28no—
160C4×D97—24no—
161C4×D97—24no—
162C4×D97—28no—
163C4×D97C2×D414yes—
164C4×D97C2414yes—
165Dic194—21no—
166Dic194—21no—
167Dic194—28no—
168Dic194C2×D414yes—
169Dic194C2412yes—
170Dic194C2412yes—
171C8×C97C2412yes—
172C8×C97C2412yes—
173#11715895C2414yes—
174C22×C97—22no—
175C22×C97—24no—
176C22×C97—44no—
177C22×C97—84no—
178C22×C97C22×D412yes—
179C22×C97C22×D411yescharacteristic
180C2×D97—24no—
181C2×D97—22no—
182C2×D97—22no—
183C2×D97—42no—
184C2×D97—42no—
185C2×D97—88no—
186C2×D97C22×D414yes—
187C4×C97—24no—
188C4×C97C22×D411yescharacteristic
189C4×C97C22×D412yes—
190C4×C97C2511yescharacteristic
191Dic97—22no—
192Dic97—22no—
193Dic97C22×D414yes—
194C2×C97—22no—
195C2×C97—44no—
196C2×C97#57712yescomplemented
197C2×C97C23×D411yescharacteristic
198D97—24no—
199D97—42no—
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—971no—
207C2×#137—972no—
208C2×AGL(1,ℤ/8)—971no—
209C2×AGL(1,ℤ/8)—972no—
210C2×#135—972no—
211C2×#135—971no—
212C22×SD16—972no—
213C22×SD16D9711yescharacteristic, complemented
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—971no—
224C2×#41—1942no—
225C2×#41—1941no—
226C2×#41—1941no—
227C22×Q8—972no—
228C22×Q8—1941no—
229C22×Q8C2×D9711yescharacteristic, complemented
230C22×D4—972no—
231C22×D4—1941no—
232C22×D4—1942no—
233C22×D4C2×D9711yescharacteristic
234#137—978no—
235#137—9716no—
236AGL(1,ℤ/8)—978no—
237AGL(1,ℤ/8)—9716no—
238#135—9716no—
239#135—978no—
240C2×Q16—972no—
241C2×Q16—971no—
242C2×SD16—972no—
243C2×SD16—972no—
244C2×SD16—9716no—
245C2×SD16—971no—
246C2×SD16—971no—
247C2×SD16C2×D9712yescomplemented
248C2×SD16C2×D9712yescomplemented
249C2×SD16C2×D9718yescomplemented
250C2×D8—972no—
251C2×D8—971no—
252#131—978no—
253C2×M16—971no—
254C2×M16—972no—
255C22×C8—972no—
256C22×C8C2×D9711yescharacteristic, complemented
257C24—21no—
258C24—1942no—
259#41—974no—
260#41—974no—
261#41—978no—
262#41—978no—
263#41—974no—
264#41—974no—
265#41—19416no—
266#41—1944no—
267#41—1944no—
268#41—1948no—
269#41—1948no—
270#41—1942no—
271#41—1942no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—971no—
275C2×Q8—978no—
276C2×Q8—971no—
277C2×Q8—971no—
278C2×Q8—1942no—
279C2×Q8—1944no—
280C2×Q8—1948no—
281C2×Q8C22×D9712yes—
282C2×Q8C22×D9714yescomplemented
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—972no—
286C2×D4—978no—
287C2×D4—971no—
288C2×D4—971no—
289C2×D4—971no—
290C2×D4—1942no—
291C2×D4—1944no—
292C2×D4—1942no—
293C2×D4—1948no—
294C2×D4—19416no—
295C2×D4—1942no—
296C2×D4—1941no—
297C2×D4—1941no—
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—978no—
309Q16—974no—
310SD16—978no—
311SD16—978no—
312SD16—974no—
313SD16—974no—
314SD16C22×D9718yescomplemented
315SD16C22×D9718yescomplemented
316D8—978no—
317D8—974no—
318M16—974no—
319M16—978no—
320C2×C8—972no—
321C2×C8—978no—
322C2×C8C22×D9711yescharacteristic
323C2×C8C22×D9711yescharacteristic
324C2×C8C22×D9714yescomplemented
325C23—22no—
326C23—24no—
327C23—82no—
328C23—7764no—
329C23—972no—
330C23—1948no—
331C23—1941no—
332C23—1941no—
333C23D4×D9711yescharacteristic
334Q8—22no—
335Q8—24no—
336Q8—974no—
337Q8—972no—
338Q8—972no—
339Q8—1948no—
340Q8—1941no—
341Q8—1941no—
342Q8C23×D9714yes—
343D4—22no—
344D4—24no—
345D4—978no—
346D4—974no—
347D4—972no—
348D4—972no—
349D4—1948no—
350D4—1948no—
351D4—1948no—
352D4—1944no—
353D4—1944no—
354D4—1941no—
355D4—1941no—
356D4C23×D9714yes—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—972no—
360C2×C4—974no—
361C2×C4—974no—
362C2×C4—1948no—
363C2×C4—1948no—
364C2×C4—1948no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4—1944no—
368C2×C4—1944no—
369C2×C4D4×D9712yes—
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9712yes—
372C2×C4C23×D9711yescharacteristic
373C8—974no—
374C8C23×D9712yes—
375C8C23×D9712yes—
376C22—22no—
377C22—24no—
378C22—44no—
379C22—84no—
380C22—7768no—
381C22—974no—
382C22—3882no—
383C22—3882no—
384C22—1942no—
385C22—1942no—
386C22—1944no—
387C22C2×D4×D9712yes—
388C22C2×D4×D9711yescharacteristic
389C4—24no—
390C4—974no—
391C4—1942no—
392C4—1942no—
393C4C2×D4×D9711yescharacteristic
394C4C2×D4×D9712yes—
395C4C24×D9711yescharacteristic
396C2—22no—
397C2—44no—
398C2—3882no—
399C2—3882no—
400C2—1944no—
401C2#2026154112yesdirect product factor
402C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC22×SD16×C97C2
socleC22×C97C22×D4
hypercenterC22×SD16D97
hypocenterC97C2×#577
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690858C2semidirect
2. C2×#15690855C2semidirect
3. #20261541C2direct (N × H)
4. C2×#15690841C2semidirect
5. C2×#15690839C2semidirect
6. C2×#15690837C2semidirect
7. C2×#15690836C2semidirect
8. C2×#15690832C2semidirect
9. C2×#15690831C2semidirect
10. C2×SD16×D97C2semidirect
11. C2×#15690816C2semidirect
12. C22×SD16×C97C2semidirect
14. #15690858C22semidirect
16. #15690855C22semidirect
18. C2×#13095512C22semidirect
22. C2×#13095511C22semidirect
26. C2×Q8×D97C22semidirect
30. #15690841C22semidirect
31. #15690839C22semidirect
32. #15690837C22semidirect
33. #15690836C22semidirect
34. #15690832C22semidirect
35. #15690831C22semidirect
36. C2×Dic388C22semidirect
37. C2×#13095501C22semidirect
38. C2×#13095500C22semidirect
39. C2×#13095500C22semidirect
40. C2×#13095499C22semidirect
41. C2×#13095498C22semidirect
42. SD16×D97C22semidirect
43. C2×D776C22semidirect
44. C2×#13095496C22semidirect
45. #15690816C22semidirect
46. C2×#13095495C22semidirect
47. C2×#13095494C22semidirect
48. C2×C8×D97C22semidirect
49. C22×Q8×C97C22semidirect
51. C2×SD16×C97C22semidirect
52. C2×SD16×C97C22semidirect
53. C2×SD16×C97C22semidirect
54. C22×C8×C97C22semidirect
58. #13095512C23semidirect
65. #13095511C23semidirect
75. Q8×D97C23semidirect
95. Dic388C23semidirect
96. #13095501C23semidirect
97. #13095500C23semidirect
98. #13095500C23semidirect
99. #13095499C23semidirect
100. #13095498C23semidirect
101. D776C23semidirect
102. #13095496C23semidirect
103. #13095495C23semidirect
104. #13095494C23semidirect
106. C8×D97C23semidirect
111. C2×Q8×C97C23semidirect
118. SD16×C97C23semidirect
119. SD16×C97C23semidirect
122. C2×C8×C97C23semidirect
196. C2×C97#577semidirect
202. C97C2×#577semidirect
213. C22×SD16D97semidirect
229. C22×Q8C2×D97semidirect
247. C2×SD16C2×D97semidirect
248. C2×SD16C2×D97semidirect
249. C2×SD16C2×D97semidirect
256. C22×C8C2×D97semidirect
282. C2×Q8C22×D97semidirect
314. SD16C22×D97semidirect
315. SD16C22×D97semidirect
324. C2×C8C22×D97semidirect
401. C2#20261541direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908582
C2×#156908552
#202615412
C2×#156908412
C2×#156908392
C2×#156908372
C2×#156908362
C2×#156908322
C2×#156908312
C2×SD16×D972
C2×#156908162
C22×SD16×C972
C2×#57797