Finite Group Databasebeta

← C2×#20261534 #27156482

Subgroups — C2×#20261534 #27156482

Order 12416.

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×#13095512C211yesmaximal, characteristic, complemented
2C2×#41×D97C211yesmaximal, characteristic, complemented
3#20261534C2116yesmaximal, direct product factor
4C2×#15690842C212yesmaximal, complemented
5C2×#15690841C211yesmaximal, characteristic, complemented
6C2×#15690838C211yesmaximal, characteristic, complemented
7C2×#15690837C212yesmaximal, complemented
8C2×#15690835C211yesmaximal, characteristic, complemented
9C2×#15690833C212yesmaximal, complemented
10C2×#15690831C212yesmaximal, complemented
11C2×M16×D97C211yesmaximal, characteristic, complemented
12C2×#137×C97C211yesmaximal, characteristic, complemented
13C2×#13095512—21no—
14C2×#13095512—21no—
15C2×#13095512—24no—
16C2×#13095512C2212yes—
17C2×#13095512C2212yescomplemented
18C2×#13095512C2211yescharacteristic, complemented
19C2×#13095512C2218yescomplemented
20C2×#13095511—21no—
21C2×#13095511C2211yescharacteristic
22C2×#13095510—21no—
23C2×#13095510C2211yescharacteristic, complemented
24#41×D97—24no—
25#41×D97C2218yescomplemented
26C2×Q8×D97C2211yescharacteristic, complemented
27C22×D388—21no—
28C22×D388C2211yescharacteristic, complemented
29C2×D4×D97—21no—
30C2×D4×D97C2211yescharacteristic, complemented
31C22×C4×D97—21no—
32C22×C4×D97—21no—
33C22×C4×D97C2211yescharacteristic
34#15690842C22116yescomplemented
35#15690841C2218yescomplemented
36#15690838C2218yescomplemented
37#15690837C22116yescomplemented
38#15690835C2218yescomplemented
39#15690833C22116yescomplemented
40#15690831C22116yescomplemented
41C2×#13095501C2212yescomplemented
42C2×#13095500C2212yescomplemented
43C2×#13095498C2212yescomplemented
44C2×#13095498C2212yescomplemented
45C2×D776C2212yescomplemented
46C2×#13095496C2212yescomplemented
47C2×#13095495C2211yescharacteristic, complemented
48C2×#13095494C2212yescomplemented
49M16×D97C2218yescomplemented
50C2×C8×D97C2212yescomplemented
51C2×#41×C97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic, complemented
53#137×C97C2218yescomplemented
54C2×Q16×C97C2212yescomplemented
55C2×SD16×C97C2212yescomplemented
56C2×M16×C97C2211yescharacteristic, complemented
57C23×D97—21no—
58C23×D97—21no—
59C23×D97D411yescharacteristic
60#13095512—22no—
61#13095512—24no—
62#13095512—28no—
63#13095512—28no—
64#13095512—24no—
65#13095512C2318yes—
66#13095512C2318yescomplemented
67#13095512C2314yescomplemented
68#13095511—22no—
69#13095511—28no—
70#13095511C2314yes—
71#13095510—28no—
72#13095510—22no—
73#13095510C2314yescomplemented
74C2×Dic194—21no—
75C2×Dic194C2311yescharacteristic
76Q8×D97—22no—
77Q8×D97C2314yescomplemented
78C2×D388—21no—
79C2×D388—22no—
80C2×D388—28no—
81C2×D388—22no—
82C2×D388—22no—
83C2×D388—21no—
84C2×D388C2312yes—
85C2×D388C2311yescharacteristic
86C2×D388C2314yescomplemented
87C2×C22⋊D97—21no—
88C2×C22⋊D97—21no—
89C2×C22⋊D97—21no—
90D4×D97—28no—
91D4×D97—22no—
92D4×D97C2314yescomplemented
93C22×Dic97—21no—
94C22×Dic97D411yescharacteristic
95C2×C4×D97—21no—
96C2×C4×D97—21no—
97C2×C4×D97—21no—
98C2×C4×D97—22no—
99C2×C4×D97—28no—
100C2×C4×D97—22no—
101C2×C4×D97—28no—
102C2×C4×D97D411yescharacteristic
103C2×C4×D97D414yes—
104C2×C4×D97D411yescharacteristic
105C2×C4×D97C2311yescharacteristic
106C2×C4×D97C2314yes—
107C2×C4×D97C2311yescharacteristic
108#13095501C2318yescomplemented
109#13095500C2318yescomplemented
110#13095498C2318yescomplemented
111#13095498C2318yescomplemented
112D776C2318yescomplemented
113#13095496C2318yescomplemented
114#13095495C2314yescomplemented
115#13095494C2318yescomplemented
116C2×#11715895C2312yes—
117C8×D97C2318yescomplemented
118#41×C97—22no—
119#41×C97C2314yes—
120C2×Q8×C97—21no—
121C2×Q8×C97—22no—
122C2×Q8×C97C2311yescharacteristic
123C2×Q8×C97C2312yes—
124C2×Q8×C97C2314yescomplemented
125C2×D4×C97—21no—
126C2×D4×C97C2311yescharacteristic
127C22×C4×C97—21no—
128C22×C4×C97—21no—
129C22×C4×C97C2311yescharacteristic
130Q16×C97C2318yescomplemented
131SD16×C97C2318yescomplemented
132M16×C97C2314yescomplemented
133C2×C8×C97C2312yes—
134C23×C97—21no—
135C23×C97C2×D411yescharacteristic
136C22×D97—21no—
137C22×D97—22no—
138C22×D97—24no—
139C22×D97—24no—
140C22×D97—42no—
141C22×D97—44no—
142C22×D97—44no—
143C22×D97C2×D411yescharacteristic
144C22×D97C2×D411yescharacteristic
145C22×D97C2×D414yes—
146Q8×C97—22no—
147Q8×C97—24no—
148Q8×C97—21no—
149Q8×C97C2414yes—
150Q8×C97C2412yes—
151D4×C97—24no—
152D4×C97—21no—
153D4×C97C2412yes—
154C2×C4×C97—21no—
155C2×C4×C97—22no—
156C2×C4×C97—24no—
157C2×C4×C97—24no—
158C2×C4×C97C2×D411yescharacteristic
159C2×C4×C97C2×D412yes—
160C2×C4×C97C2411yescharacteristic
161C2×C4×C97C2412yes—
162C22⋊D97—24no—
163C22⋊D97—24no—
164C22⋊D97—24no—
165C2×Dic97—21no—
166C2×Dic97—24no—
167C2×Dic97C2×D411yescharacteristic
168C2×Dic97C2×D411yescharacteristic
169C2×Dic97C2×D414yes—
170D388—24no—
171D388—28no—
172D388—28no—
173D388—21no—
174D388—22no—
175D388—24no—
176D388C2414yes—
177D388C2412yes—
178C4×D97—24no—
179C4×D97—24no—
180C4×D97—24no—
181C4×D97—28no—
182C4×D97—28no—
183C4×D97C2×D414yes—
184C4×D97C2×D412yes—
185C4×D97C2×D412yes—
186C4×D97C2×D414yes—
187C4×D97C2412yes—
188C4×D97C2412yes—
189Dic194—21no—
190Dic194—24no—
191Dic194C2412yes—
192C8×C97C2414yes—
193#11715895C2414yes—
194C22×C97—22no—
195C22×C97—22no—
196C22×C97—42no—
197C22×C97C22×D412yes—
198C22×C97C22×D411yescharacteristic
199C2×D97—22no—
200C2×D97—22no—
201C2×D97—24no—
202C2×D97—22no—
203C2×D97—44no—
204C2×D97—48no—
205C2×D97—48no—
206C2×D97—42no—
207C2×D97—44no—
208C2×D97C22×D412yes—
209C2×D97C22×D412yes—
210C4×C97—22no—
211C4×C97—24no—
212C4×C97C22×D411yescharacteristic
213C4×C97C22×D412yes—
214C4×C97C2511yescharacteristic
215Dic97—22no—
216Dic97C22×D412yes—
217Dic97C22×D412yes—
218C2×C97—22no—
219C2×C97—42no—
220C2×C97#57512yescomplemented
221C2×C97C23×D411yescharacteristic
222D97—22no—
223D97—22no—
224D97—44no—
225D97—42no—
226C2×#575—971nomaximal, Sylow
227C97C2×#57511yescharacteristic, complemented, Sylow
228C22×#41—971no—
229C22×#41—971no—
230#575—9716no—
231C2×#137—971no—
232C2×#137—972no—
233C2×#137D9711yescharacteristic, complemented
234C2×AGL(1,ℤ/8)—971no—
235C2×AGL(1,ℤ/8)—972no—
236C2×AGL(1,ℤ/8)—971no—
237C2×#135—972no—
238C2×#135—972no—
239C22×M16—971no—
240C2×#41—978no—
241C2×#41—971no—
242C2×#41—972no—
243C2×#41—972no—
244C2×#41—978no—
245C2×#41—971no—
246C2×#41—971no—
247C2×#41—1941no—
248C2×#41—1941no—
249C2×#41—1941no—
250C2×#41—1944no—
251C2×#41—1941no—
252C2×#41—1944no—
253C2×#41C2×D9711yescharacteristic
254C22×Q8—971no—
255C22×Q8C2×D9711yescharacteristic, complemented
256C22×D4—971no—
257C22×D4—971no—
258C22×D4—1941no—
259C22×D4—1941no—
260C23×C4—971no—
261C23×C4—1941no—
262C23×C4—1941no—
263#137—978no—
264#137—9716no—
265#137C2×D9718yescomplemented
266AGL(1,ℤ/8)—978no—
267AGL(1,ℤ/8)—9716no—
268AGL(1,ℤ/8)—978no—
269#135—9716no—
270#135—9716no—
271C2×Q16—972no—
272C2×Q16C2×D9712yescomplemented
273C2×SD16—972no—
274C2×SD16—972no—
275C2×SD16—972no—
276C2×SD16C2×D9712yescomplemented
277C2×D8—972no—
278C2×D8—972no—
279C2×M16—971no—
280C2×M16—978no—
281C2×M16—972no—
282C2×M16C2×D9711yescharacteristic, complemented
283C22×C8—972no—
284C24—971no—
285C24—1941no—
286C24—1941no—
287#41—22no—
288#41—978no—
289#41—974no—
290#41—978no—
291#41—974no—
292#41—974no—
293#41—1948no—
294#41—1948no—
295#41—1942no—
296#41—1948no—
297#41—1944no—
298#41—1944no—
299#41—1948no—
300#41—1942no—
301#41—1942no—
302#41C22×D9714yes—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—974no—
306C2×Q8—971no—
307C2×Q8—1941no—
308C2×Q8—1942no—
309C2×Q8C22×D9711yescharacteristic
310C2×Q8C22×D9712yes—
311C2×Q8C22×D9714yescomplemented
312C2×D4—21no—
313C2×D4—974no—
314C2×D4—972no—
315C2×D4—971no—
316C2×D4—974no—
317C2×D4—1948no—
318C2×D4—1941no—
319C2×D4—1942no—
320C2×D4—1942no—
321C2×D4—1942no—
322C2×D4—1941no—
323C2×D4—1948no—
324C2×D4—1941no—
325C2×D4—1941no—
326C2×D4—1942no—
327C2×D4—1941no—
328C2×D4C22×D9711yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—974no—
334C22×C4—971no—
335C22×C4—971no—
336C22×C4—971no—
337C22×C4—974no—
338C22×C4—1948no—
339C22×C4—1941no—
340C22×C4—1942no—
341C22×C4—1942no—
342C22×C4—1948no—
343C22×C4—1941no—
344C22×C4—1941no—
345C22×C4—1941no—
346C22×C4C22×D9711yescharacteristic
347Q16—978no—
348Q16C22×D9718yescomplemented
349SD16—978no—
350SD16—978no—
351SD16—978no—
352SD16C22×D9718yescomplemented
353D8—978no—
354D8—978no—
355M16—974no—
356M16—978no—
357M16C22×D9714yescomplemented
358C2×C8—972no—
359C2×C8—978no—
360C2×C8C22×D9712yes—
361C23—21no—
362C23—974no—
363C23—971no—
364C23—971no—
365C23—3884no—
366C23—3882no—
367C23—3884no—
368C23—1944no—
369C23—1944no—
370C23—1942no—
371C23—1941no—
372C23D4×D9711yescharacteristic
373Q8—22no—
374Q8—24no—
375Q8—21no—
376Q8—972no—
377Q8—1944no—
378Q8—1941no—
379Q8C23×D9712yes—
380Q8C23×D9714yes—
381D4—24no—
382D4—21no—
383D4—974no—
384D4—972no—
385D4—1944no—
386D4—1948no—
387D4—1948no—
388D4—1944no—
389D4—1944no—
390D4—1944no—
391D4—1944no—
392D4—1941no—
393D4—1942no—
394D4C23×D9712yes—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—974no—
400C2×C4—971no—
401C2×C4—971no—
402C2×C4—974no—
403C2×C4—972no—
404C2×C4—972no—
405C2×C4—974no—
406C2×C4—972no—
407C2×C4—972no—
408C2×C4—1944no—
409C2×C4—1948no—
410C2×C4—1948no—
411C2×C4—1944no—
412C2×C4—1944no—
413C2×C4—1941no—
414C2×C4—1944no—
415C2×C4D4×D9712yes—
416C2×C4D4×D9711yescharacteristic
417C2×C4C23×D9712yes—
418C2×C4C23×D9711yescharacteristic
419C8—974no—
420C8C23×D9714yes—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—972no—
425C22—972no—
426C22—3888no—
427C22—3884no—
428C22—3888no—
429C22—3884no—
430C22—3882no—
431C22—1944no—
432C22—1942no—
433C22—1942no—
434C22—1942no—
435C22C2×D4×D9712yes—
436C22C2×D4×D9711yescharacteristic
437C4—22no—
438C4—24no—
439C4—972no—
440C4—972no—
441C4—1942no—
442C4C2×D4×D9712yes—
443C4C2×D4×D9711yescharacteristic
444C4C24×D9711yescharacteristic
445C2—22no—
446C2—42no—
447C2—3884no—
448C2—3882no—
449C2—1942no—
450C2—1942no—
451C2#2026153412yesdirect product factor
452C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
hypercenterC2×#137D97
hypocenterC97C2×#575
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. C2×#41×D97C2semidirect
3. #20261534C2direct (N × H)
4. C2×#15690842C2semidirect
5. C2×#15690841C2semidirect
6. C2×#15690838C2semidirect
7. C2×#15690837C2semidirect
8. C2×#15690835C2semidirect
9. C2×#15690833C2semidirect
10. C2×#15690831C2semidirect
11. C2×M16×D97C2semidirect
12. C2×#137×C97C2semidirect
17. C2×#13095512C22semidirect
18. C2×#13095512C22semidirect
19. C2×#13095512C22semidirect
23. C2×#13095510C22semidirect
25. #41×D97C22semidirect
26. C2×Q8×D97C22semidirect
28. C22×D388C22semidirect
30. C2×D4×D97C22semidirect
34. #15690842C22semidirect
35. #15690841C22semidirect
36. #15690838C22semidirect
37. #15690837C22semidirect
38. #15690835C22semidirect
39. #15690833C22semidirect
40. #15690831C22semidirect
41. C2×#13095501C22semidirect
42. C2×#13095500C22semidirect
43. C2×#13095498C22semidirect
44. C2×#13095498C22semidirect
45. C2×D776C22semidirect
46. C2×#13095496C22semidirect
47. C2×#13095495C22semidirect
48. C2×#13095494C22semidirect
49. M16×D97C22semidirect
50. C2×C8×D97C22semidirect
52. C22×Q8×C97C22semidirect
53. #137×C97C22semidirect
54. C2×Q16×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×M16×C97C22semidirect
66. #13095512C23semidirect
67. #13095512C23semidirect
73. #13095510C23semidirect
77. Q8×D97C23semidirect
86. C2×D388C23semidirect
92. D4×D97C23semidirect
108. #13095501C23semidirect
109. #13095500C23semidirect
110. #13095498C23semidirect
111. #13095498C23semidirect
112. D776C23semidirect
113. #13095496C23semidirect
114. #13095495C23semidirect
115. #13095494C23semidirect
117. C8×D97C23semidirect
124. C2×Q8×C97C23semidirect
130. Q16×C97C23semidirect
131. SD16×C97C23semidirect
132. M16×C97C23semidirect
220. C2×C97#575semidirect
227. C97C2×#575semidirect
233. C2×#137D97semidirect
255. C22×Q8C2×D97semidirect
265. #137C2×D97semidirect
272. C2×Q16C2×D97semidirect
276. C2×SD16C2×D97semidirect
282. C2×M16C2×D97semidirect
311. C2×Q8C22×D97semidirect
348. Q16C22×D97semidirect
352. SD16C22×D97semidirect
357. M16C22×D97semidirect
451. C2#20261534direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
C2×#41×D972
#202615342
C2×#156908422
C2×#156908412
C2×#156908382
C2×#156908372
C2×#156908352
C2×#156908332
C2×#156908312
C2×M16×D972
C2×#137×C972
C2×#57597