Finite Group Databasebeta

← C2×#20261544 #27156494

Subgroups — C2×#20261544 #27156494

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×#15690856C211yesmaximal, characteristic, complemented
3#20261544C2116yesmaximal, direct product factor
4C2×#15690842C212yesmaximal, complemented
5C2×#15690838C211yesmaximal, characteristic, complemented
6C2×#15690837C212yesmaximal, complemented
7C2×#15690833C212yesmaximal, complemented
8C2×#15690830C211yesmaximal, characteristic, complemented
9C22×#13095498C211yesmaximal, characteristic, complemented
10C2×SD16×D97C212yesmaximal, complemented
11C2×#15690818C211yesmaximal, characteristic, complemented
12C2×#137×C97C211yesmaximal, characteristic, complemented
13#15690858—24no—
14#15690858C2218yescomplemented
15#15690856—24no—
16#15690856C2218yescomplemented
17C2×#13095512—21no—
18C2×#13095512C2212yes—
19C2×#13095512C2212yescomplemented
20C2×#13095511—21no—
21C2×#13095511—22no—
22C2×#13095511C2211yescharacteristic
23C2×#13095511C2211yescharacteristic, complemented
24C2×Q8×D97—21no—
25C2×Q8×D97C2212yescomplemented
26C22×D388—21no—
27C22×D388C2211yescharacteristic
28C2×D4×D97—22no—
29C2×D4×D97C2212yes—
30#15690842C22116yescomplemented
31#15690838C2218yescomplemented
32#15690837C22116yescomplemented
33#15690833C22116yescomplemented
34#15690830C2218yescomplemented
35C2×#13095501C2211yescharacteristic, complemented
36C2×#13095500C2212yescomplemented
37C2×#13095499C2211yescharacteristic, complemented
38C2×#13095498C2212yescomplemented
39C2×#13095498C2212yescomplemented
40C2×#13095498C2211yescharacteristic, complemented
41C2×#13095498C2218yescomplemented
42SD16×D97C22116yescomplemented
43C2×D776C2212yescomplemented
44C2×#13095496C2211yescharacteristic, complemented
45#15690818C2218yescomplemented
46C2×#13095494C2212yescomplemented
47C22×#11715895C2211yescharacteristic, complemented
48C2×C8×D97C2212yescomplemented
49C2×#41×C97C2211yescharacteristic
50C22×Q8×C97C2211yescharacteristic, complemented
51#137×C97C2218yescomplemented
52C2×Q16×C97C2212yescomplemented
53C2×SD16×C97C2212yescomplemented
54C2×M16×C97C2211yescharacteristic, complemented
55C23×D97—21no—
56C23×D97—22no—
57#13095512—28no—
58#13095512—24no—
59#13095512—24no—
60#13095512C2318yes—
61#13095512C2318yescomplemented
62#13095511—22no—
63#13095511—216no—
64#13095511—28no—
65#13095511D414yes—
66#13095511C2314yes—
67#13095511C2314yescomplemented
68C2×Dic194—22no—
69C2×Dic194D411yescharacteristic
70C2×Dic194C2311yescharacteristic
71Q8×D97—24no—
72Q8×D97—28no—
73Q8×D97C2318yescomplemented
74C2×D388—22no—
75C2×D388—22no—
76C2×D388—21no—
77C2×D388—22no—
78C2×D388—28no—
79C2×D388D411yescharacteristic
80C2×D388C2311yescharacteristic
81C2×D388C2312yes—
82C2×D388C2314yes—
83C2×C22⋊D97—21no—
84C2×C22⋊D97—22no—
85C2×C22⋊D97D412yes—
86D4×D97—216no—
87D4×D97—24no—
88D4×D97C2318yes—
89C2×C4×D97—22no—
90C2×C4×D97—21no—
91C2×C4×D97—22no—
92C2×C4×D97—22no—
93C2×C4×D97C2312yes—
94#13095501C2314yescomplemented
95#13095500C2318yescomplemented
96#13095499C2314yescomplemented
97#13095498C2318yescomplemented
98#13095498C2318yescomplemented
99#13095498C2314yescomplemented
100D776C2318yescomplemented
101#13095496C2314yescomplemented
102#13095494C2318yescomplemented
103C2×#11715895C2311yescharacteristic
104C2×#11715895C2311yescharacteristic
105C2×#11715895C2314yescomplemented
106C8×D97C2318yescomplemented
107#41×C97—22no—
108#41×C97C2314yes—
109C2×Q8×C97—21no—
110C2×Q8×C97—22no—
111C2×Q8×C97C2312yes—
112C2×Q8×C97C2311yescharacteristic
113C2×Q8×C97C2314yescomplemented
114C2×D4×C97—21no—
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117C22×C4×C97—21no—
118C22×C4×C97C2311yescharacteristic
119Q16×C97C2318yescomplemented
120SD16×C97C2318yescomplemented
121M16×C97C2314yescomplemented
122C2×C8×C97C2312yes—
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—22no—
127C22×D97—24no—
128C22×D97—28no—
129C22×D97—84no—
130C22×D97—82no—
131C22×D97C2×D412yes—
132Q8×C97—22no—
133Q8×C97—24no—
134Q8×C97—21no—
135Q8×C97C2414yes—
136Q8×C97C2412yes—
137D4×C97—24no—
138D4×C97—21no—
139D4×C97C2412yes—
140C2×C4×C97—21no—
141C2×C4×C97—22no—
142C2×C4×C97—24no—
143C2×C4×C97—24no—
144C2×C4×C97C2×D411yescharacteristic
145C2×C4×C97C2×D412yes—
146C2×C4×C97C2411yescharacteristic
147C2×C4×C97C2412yes—
148C22⋊D97—24no—
149C22⋊D97—28no—
150C22⋊D97C2×D418yes—
151C2×Dic97—21no—
152C2×Dic97C2×D412yes—
153D388—28no—
154D388—28no—
155D388—22no—
156D388—24no—
157D388—21no—
158D388C2×D414yes—
159D388C2412yes—
160D388C2414yes—
161C4×D97—28no—
162C4×D97—24no—
163C4×D97—28no—
164C4×D97—28no—
165C4×D97C2×D414yes—
166C4×D97C2414yes—
167Dic194—21no—
168Dic194—28no—
169Dic194C2×D414yes—
170Dic194C2412yes—
171C8×C97C2414yes—
172#11715895C2412yes—
173#11715895C2412yes—
174C22×C97—22no—
175C22×C97—22no—
176C22×C97—42no—
177C22×C97C22×D412yes—
178C22×C97C22×D411yescharacteristic
179C2×D97—24no—
180C2×D97—22no—
181C2×D97—24no—
182C2×D97—42no—
183C2×D97—44no—
184C2×D97—88no—
185C2×D97—84no—
186C2×D97C22×D414yes—
187C4×C97—24no—
188C4×C97—22no—
189C4×C97C22×D411yescharacteristic
190C4×C97C22×D412yes—
191C4×C97C2511yescharacteristic
192Dic97—22no—
193Dic97C22×D414yes—
194C2×C97—22no—
195C2×C97—42no—
196C2×C97#57712yescomplemented
197C2×C97C23×D411yescharacteristic
198D97—24no—
199D97—42no—
200D97—44no—
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×#137D9711yescharacteristic, complemented
208C2×AGL(1,ℤ/8)—972no—
209C2×AGL(1,ℤ/8)—971no—
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—1942no—
224C2×#41—1941no—
225C2×#41—1941no—
226C2×#41C2×D9711yescharacteristic
227C22×Q8—972no—
228C22×Q8—1941no—
229C22×Q8C2×D9711yescharacteristic, complemented
230C22×D4—972no—
231C22×D4—971no—
232C22×D4—1941no—
233C22×D4—1942no—
234#137—9716no—
235#137C2×D9718yescomplemented
236AGL(1,ℤ/8)—9716no—
237AGL(1,ℤ/8)—978no—
238#135—978no—
239#135—9716no—
240C2×Q16—971no—
241C2×Q16C2×D9712yescomplemented
242C2×SD16—972no—
243C2×SD16—972no—
244C2×SD16—971no—
245C2×SD16—978no—
246C2×SD16—971no—
247C2×SD16—9716no—
248C2×SD16—972no—
249C2×SD16C2×D9712yescomplemented
250C2×D8—971no—
251C2×D8—972no—
252#131—978no—
253C2×M16—972no—
254C2×M16C2×D9711yescharacteristic, complemented
255C22×C8—971no—
256C22×C8—972no—
257C24—1942no—
258C24—1941no—
259#41—22no—
260#41—974no—
261#41—974no—
262#41—978no—
263#41—978no—
264#41—974no—
265#41—19416no—
266#41—1948no—
267#41—1944no—
268#41—1944no—
269#41—1948no—
270#41—1942no—
271#41C22×D9714yes—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—971no—
275C2×Q8—978no—
276C2×Q8—971no—
277C2×Q8—1942no—
278C2×Q8—1944no—
279C2×Q8—1948no—
280C2×Q8C22×D9712yes—
281C2×Q8C22×D9711yescharacteristic
282C2×Q8C22×D9714yescomplemented
283C2×D4—21no—
284C2×D4—972no—
285C2×D4—978no—
286C2×D4—971no—
287C2×D4—971no—
288C2×D4—972no—
289C2×D4—974no—
290C2×D4—1948no—
291C2×D4—1942no—
292C2×D4—1942no—
293C2×D4—1944no—
294C2×D4—1942no—
295C2×D4—19416no—
296C2×D4—1941no—
297C2×D4—1941no—
298C2×D4—1942no—
299C2×D4C22×D9711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—972no—
303C22×C4—1942no—
304C22×C4—1942no—
305C22×C4—1942no—
306C22×C4—1941no—
307C22×C4C22×D9711yescharacteristic
308Q16—974no—
309Q16C22×D9718yescomplemented
310SD16—978no—
311SD16—978no—
312SD16—974no—
313SD16—974no—
314SD16—978no—
315SD16C22×D9718yescomplemented
316D8—974no—
317D8—978no—
318M16—978no—
319M16C22×D9714yescomplemented
320C2×C8—971no—
321C2×C8—971no—
322C2×C8—974no—
323C2×C8—978no—
324C2×C8C22×D9712yes—
325C23—21no—
326C23—7762no—
327C23—7764no—
328C23—972no—
329C23—1948no—
330C23—1944no—
331C23—1941no—
332C23—1942no—
333C23D4×D9711yescharacteristic
334Q8—22no—
335Q8—24no—
336Q8—21no—
337Q8—974no—
338Q8—972no—
339Q8—1948no—
340Q8—1941no—
341Q8C23×D9714yes—
342Q8C23×D9712yes—
343D4—24no—
344D4—21no—
345D4—978no—
346D4—974no—
347D4—972no—
348D4—974no—
349D4—1948no—
350D4—1948no—
351D4—1948no—
352D4—1944no—
353D4—1944no—
354D4—1942no—
355D4—1941no—
356D4C23×D9712yes—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—972no—
362C2×C4—974no—
363C2×C4—974no—
364C2×C4—1948no—
365C2×C4—1948no—
366C2×C4—1948no—
367C2×C4—1941no—
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—42no—
379C22—7764no—
380C22—7768no—
381C22—974no—
382C22—3884no—
383C22—3882no—
384C22—1942no—
385C22—1944no—
386C22—1944no—
387C22C2×D4×D9712yes—
388C22C2×D4×D9711yescharacteristic
389C4—22no—
390C4—24no—
391C4—974no—
392C4—1942no—
393C4C2×D4×D9712yes—
394C4C2×D4×D9711yescharacteristic
395C4C24×D9711yescharacteristic
396C2—22no—
397C2—42no—
398C2—3884no—
399C2—3882no—
400C2—1944no—
401C2#2026154412yesdirect product factor
402C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
hypercenterC2×#137D97
hypocenterC97C2×#577
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. C2×#15690858C2semidirect
2. C2×#15690856C2semidirect
3. #20261544C2direct (N × H)
4. C2×#15690842C2semidirect
5. C2×#15690838C2semidirect
6. C2×#15690837C2semidirect
7. C2×#15690833C2semidirect
8. C2×#15690830C2semidirect
9. C22×#13095498C2semidirect
10. C2×SD16×D97C2semidirect
11. C2×#15690818C2semidirect
12. C2×#137×C97C2semidirect
14. #15690858C22semidirect
16. #15690856C22semidirect
19. C2×#13095512C22semidirect
23. C2×#13095511C22semidirect
25. C2×Q8×D97C22semidirect
30. #15690842C22semidirect
31. #15690838C22semidirect
32. #15690837C22semidirect
33. #15690833C22semidirect
34. #15690830C22semidirect
35. C2×#13095501C22semidirect
36. C2×#13095500C22semidirect
37. C2×#13095499C22semidirect
38. C2×#13095498C22semidirect
39. C2×#13095498C22semidirect
40. C2×#13095498C22semidirect
41. C2×#13095498C22semidirect
42. SD16×D97C22semidirect
43. C2×D776C22semidirect
44. C2×#13095496C22semidirect
45. #15690818C22semidirect
46. C2×#13095494C22semidirect
47. C22×#11715895C22semidirect
48. C2×C8×D97C22semidirect
50. C22×Q8×C97C22semidirect
51. #137×C97C22semidirect
52. C2×Q16×C97C22semidirect
53. C2×SD16×C97C22semidirect
54. C2×M16×C97C22semidirect
61. #13095512C23semidirect
67. #13095511C23semidirect
73. Q8×D97C23semidirect
94. #13095501C23semidirect
95. #13095500C23semidirect
96. #13095499C23semidirect
97. #13095498C23semidirect
98. #13095498C23semidirect
99. #13095498C23semidirect
100. D776C23semidirect
101. #13095496C23semidirect
102. #13095494C23semidirect
105. C2×#11715895C23semidirect
106. C8×D97C23semidirect
113. C2×Q8×C97C23semidirect
119. Q16×C97C23semidirect
120. SD16×C97C23semidirect
121. M16×C97C23semidirect
196. C2×C97#577semidirect
202. C97C2×#577semidirect
207. C2×#137D97semidirect
229. C22×Q8C2×D97semidirect
235. #137C2×D97semidirect
241. C2×Q16C2×D97semidirect
249. C2×SD16C2×D97semidirect
254. C2×M16C2×D97semidirect
282. C2×Q8C22×D97semidirect
309. Q16C22×D97semidirect
315. SD16C22×D97semidirect
319. M16C22×D97semidirect
401. C2#20261544direct (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×#156908562
#202615442
C2×#156908422
C2×#156908382
C2×#156908372
C2×#156908332
C2×#156908302
C22×#130954982
C2×SD16×D972
C2×#156908182
C2×#137×C972
C2×#57797