Finite Group Databasebeta

← C2×#20261410 #27155232

Subgroups — C2×#20261410 #27155232

Order 12416.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261410C2116yesmaximal, direct product factor
2C2×#15690803C211yesmaximal, characteristic, complemented
3C2×#15690796C211yesmaximal, characteristic, complemented
4C2×#15690795C211yesmaximal, characteristic, complemented
5C2×#15690795C211yesmaximal, characteristic, complemented
6C2×#15690792C211yesmaximal, characteristic, complemented
7C2×#15690790C211yesmaximal, characteristic, complemented
8C2×#15690788C211yesmaximal, characteristic, complemented
9C2×#15690780C211yesmaximal, characteristic
10C2×C4×D388C211yesmaximal, characteristic, complemented
11C2×#15690772C211yesmaximal, characteristic, complemented
12C2×#15690768C211yesmaximal, characteristic, complemented
13C2×#15690767C211yesmaximal, characteristic, complemented
14C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×#126×C97C211yesmaximal, characteristic, complemented
17C24×D97C2211yescharacteristic
18C22×Dic194C2211yescharacteristic, complemented
19C22×D388C2211yescharacteristic
20C22×D388C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic
25C22×C4×D97C2211yescharacteristic
26C22×C4×D97C2211yescharacteristic, complemented
27#15690803C2218yescomplemented
28#15690796C2218yescomplemented
29#15690795C2218yescomplemented
30#15690795C2218yescomplemented
31#15690792C2218yescomplemented
32#15690790C2218yescomplemented
33#15690788C2218yescomplemented
34#15690780C2218yes—
35C4×D388C2218yescomplemented
36#15690772C2218yescomplemented
37#15690768C2218yescomplemented
38#15690767C2218yescomplemented
39C2×C4⋊Dic97C2211yescharacteristic, complemented
40C2×C4⋊Dic97C2211yescharacteristic, complemented
41C2×#13095490C2211yescharacteristic, complemented
42C2×#13095490C2211yescharacteristic, complemented
43C2×#13095490C2211yescharacteristic, complemented
44C2×C22⋊Dic97C2211yescharacteristic
45C2×C22⋊Dic97C2211yescharacteristic
46C2×#13095488C2211yescharacteristic
47C2×#13095488C2211yescharacteristic
48C2×#13095488C2211yescharacteristic
49C2×#13095488C2211yescharacteristic
50C2×#13095488C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic, complemented
53C2×#13095488C2211yescharacteristic, complemented
54C2×#13095488C2211yescharacteristic, complemented
55C22⋊C4×D97C2218yescomplemented
56C22⋊C4×D97C2218yescomplemented
57C2×C4×Dic97C2211yescharacteristic, complemented
58C2×C4×Dic97C2211yescharacteristic, complemented
59#126×C97C2218yescomplemented
60C2×C4⋊C4×C97C2211yescharacteristic, complemented
61C2×C4⋊C4×C97C2211yescharacteristic, complemented
62C2×C4⋊C4×C97C2211yescharacteristic, complemented
63C2×C22⋊C4×C97C2211yescharacteristic
64C2×C22⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C42×C97C2211yescharacteristic, complemented
67C23×D97—24no—
68C23×D97—24no—
69C23×D97—42no—
70C23×D97C2311yescharacteristic
71C23×D97C2314yes—
72C23×D97C2311yescharacteristic
73C23×D97C2311yescharacteristic
74C2×Dic194—42no—
75C2×Dic194C2314yescomplemented
76C2×D388—24no—
77C2×D388—24no—
78C2×D388C2314yes—
79C2×D388C2314yes—
80C2×C22⋊D97—42no—
81C2×C22⋊D97C2314yes—
82C22×Dic97C2311yescharacteristic
83C22×Dic97C2311yescharacteristic
84C22×Dic97C2311yescharacteristic
85C22×Dic97C2311yescharacteristic
86C22×Dic97C2311yescharacteristic
87C2×C4×D97—24no—
88C2×C4×D97—24no—
89C2×C4×D97—42no—
90C2×C4×D97—42no—
91C2×C4×D97—42no—
92C2×C4×D97C2314yes—
93C2×C4×D97C2314yes—
94C2×C4×D97C2314yes—
95C2×C4×D97C2314yes—
96C2×C4×D97C2314yescomplemented
97C4⋊Dic97C2314yescomplemented
98C4⋊Dic97C2314yescomplemented
99#13095490C2314yescomplemented
100#13095490C2314yescomplemented
101#13095490C2314yescomplemented
102C22⋊Dic97C2314yes—
103C22⋊Dic97C2314yes—
104#13095488C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108#13095488C2314yes—
109#13095488C2314yes—
110#13095488C2314yescomplemented
111#13095488C2314yescomplemented
112#13095488C2314yescomplemented
113C4×Dic97C2314yescomplemented
114C4×Dic97C2314yescomplemented
115C24×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2311yescharacteristic
121C22×C4×C97C2311yescharacteristic
122C4⋊C4×C97C2314yescomplemented
123C4⋊C4×C97C2314yescomplemented
124C4⋊C4×C97C2314yescomplemented
125C22⋊C4×C97C2314yes—
126C22⋊C4×C97C2314yes—
127C22⋊C4×C97C2314yes—
128C42×C97C2314yescomplemented
129C23×C97—22no—
130C23×C97—22no—
131C23×C97—22no—
132C23×C97C2412yes—
133C23×C97C2411yescharacteristic
134C22×D97—22no—
135C22×D97—22no—
136C22×D97—22no—
137C22×D97—22no—
138C22×D97—22no—
139C22×D97—22no—
140C22×D97—22no—
141C22×D97—28no—
142C22×D97—28no—
143C22×D97—44no—
144C22×D97—416no—
145C22×D97#4114yes—
146C22×D97#4114yes—
147C22×D97C2412yes—
148C22×D97C2412yes—
149C22×D97C2412yes—
150C2×C4×C97—22no—
151C2×C4×C97—22no—
152C2×C4×C97—22no—
153C2×C4×C97—22no—
154C2×C4×C97—22no—
155C2×C4×C97—22no—
156C2×C4×C97C2412yes—
157C2×C4×C97C2412yes—
158C2×C4×C97C2412yes—
159C2×C4×C97C2412yes—
160C2×C4×C97C2412yes—
161C2×C4×C97C2412yes—
162C22⋊D97—44no—
163C2×Dic97—22no—
164C2×Dic97—22no—
165C2×Dic97—22no—
166C2×Dic97—22no—
167C2×Dic97—22no—
168C2×Dic97C2412yes—
169C2×Dic97C2412yes—
170C2×Dic97C2412yes—
171C2×Dic97C2412yes—
172C2×Dic97C2412yes—
173D388—28no—
174D388—28no—
175C4×D97—28no—
176C4×D97—28no—
177C4×D97—44no—
178C4×D97—44no—
179C4×D97—44no—
180Dic194—44no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—44no—
185C22×C97C2×#4112yes—
186C22×C97C2×#4112yes—
187C22×C97H(5,𝔽2)12yes—
188C22×C97C2511yescharacteristic
189C2×D97—28no—
190C2×D97—28no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—416no—
199C2×D97—44no—
200C2×D97C2×#4114yes—
201C2×D97C2×#4114yes—
202C4×C97—22no—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—22no—
206C4×C97—22no—
207C4×C97—22no—
208Dic97—22no—
209Dic97—22no—
210Dic97—22no—
211Dic97—22no—
212Dic97—22no—
213C2×C97—42no—
214C2×C97#55114yescomplemented
215C2×C97C22×#4111yescharacteristic
216C2×C97C22×#4111yescharacteristic
217C2×C97C2×H(5,𝔽2)11yescharacteristic
218D97—24no—
219D97—24no—
220D97—42no—
221C2×#551—971nomaximal, Sylow
222C97C2×#55111yescharacteristic, complemented, Sylow
223#551—9716no—
224C2×#126—971no—
225C2×#126D9711yescharacteristic, complemented
226C2×#124—971no—
227C2×#123—971no—
228C2×#123—971no—
229C2×#123—971no—
230C2×C22⋊Q8—971no—
231C2×C22⋊Q8—971no—
232C2×#120—971no—
233C2×C4×D4—971no—
234C2×C4×D4—971no—
235C2×#117—971no—
236C2×#117—971no—
237C22×C22⋊C4—971no—
238C22×C22⋊C4—971no—
239C25—971no—
240C22×Q8—971no—
241C22×D4—971no—
242C22×D4—971no—
243C22×D4—971no—
244C23×C4—971no—
245C23×C4—971no—
246C23×C4—971no—
247C23×C4—971no—
248C23×C4—971no—
249#126—978no—
250#126C2×D9718yescomplemented
251#124—978no—
252#123—978no—
253#123—978no—
254#123—978no—
255C22⋊Q8—978no—
256C22⋊Q8—978no—
257#120—978no—
258C4×D4—978no—
259C4×D4—978no—
260#117—978no—
261#117—978no—
262C2×C4⋊C4—971no—
263C2×C4⋊C4—971no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—971no—
266C2×C4⋊C4—971no—
267C2×C4⋊C4C2×D9711yescharacteristic, complemented
268C2×C4⋊C4C2×D9711yescharacteristic, complemented
269C2×C4⋊C4C2×D9711yescharacteristic, complemented
270C2×C22⋊C4—971no—
271C2×C22⋊C4—971no—
272C2×C22⋊C4—971no—
273C2×C22⋊C4—971no—
274C2×C22⋊C4—971no—
275C2×C22⋊C4—971no—
276C2×C22⋊C4—971no—
277C2×C22⋊C4—971no—
278C2×C22⋊C4—971no—
279C2×C22⋊C4—971no—
280C2×C22⋊C4—971no—
281C2×C22⋊C4—978no—
282C2×C22⋊C4—978no—
283C2×C22⋊C4C2×D9711yescharacteristic
284C2×C22⋊C4C2×D9711yescharacteristic
285C2×C22⋊C4C2×D9711yescharacteristic
286C2×C42—971no—
287C2×C42—971no—
288C2×C42C2×D9711yescharacteristic, complemented
289C24—971no—
290C24—971no—
291C24—974no—
292C24—971no—
293C24—3882no—
294C24—1944no—
295C24—1944no—
296C24C22×D9711yescharacteristic
297C2×Q8—974no—
298C2×Q8—3882no—
299C2×D4—974no—
300C2×D4—974no—
301C2×D4—974no—
302C2×D4—3882no—
303C2×D4—1944no—
304C2×D4—1944no—
305C22×C4—971no—
306C22×C4—971no—
307C22×C4—971no—
308C22×C4—971no—
309C22×C4—971no—
310C22×C4—974no—
311C22×C4—974no—
312C22×C4—974no—
313C22×C4—974no—
314C22×C4—974no—
315C22×C4—3882no—
316C22×C4—3882no—
317C22×C4—3882no—
318C22×C4—1944no—
319C22×C4—1944no—
320C22×C4C22×D9711yescharacteristic
321C22×C4C22×D9711yescharacteristic
322C22×C4C22×D9711yescharacteristic
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9711yescharacteristic
326C4⋊C4—974no—
327C4⋊C4—974no—
328C4⋊C4—974no—
329C4⋊C4—974no—
330C4⋊C4—974no—
331C4⋊C4C22×D9714yescomplemented
332C4⋊C4C22×D9714yescomplemented
333C4⋊C4C22×D9714yescomplemented
334C22⋊C4—974no—
335C22⋊C4—974no—
336C22⋊C4—974no—
337C22⋊C4—974no—
338C22⋊C4—974no—
339C22⋊C4—974no—
340C22⋊C4—974no—
341C22⋊C4—974no—
342C22⋊C4—974no—
343C22⋊C4—974no—
344C22⋊C4—974no—
345C22⋊C4C22×D9714yes—
346C22⋊C4C22×D9714yes—
347C22⋊C4C22×D9714yes—
348C42—974no—
349C42—974no—
350C42C22×D9714yescomplemented
351C23—22no—
352C23—22no—
353C23—22no—
354C23—974no—
355C23—974no—
356C23—972no—
357C23—972no—
358C23—972no—
359C23—3884no—
360C23—38816no—
361C23—1948no—
362C23—1948no—
363C23—1942no—
364C23—1942no—
365C23—1942no—
366C23—1942no—
367C23—1942no—
368C23—1942no—
369C23—1942no—
370C23C23×D9712yes—
371C23C23×D9711yescharacteristic
372Q8—3884no—
373D4—3884no—
374D4—1948no—
375D4—1948no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—972no—
383C2×C4—972no—
384C2×C4—972no—
385C2×C4—972no—
386C2×C4—972no—
387C2×C4—3884no—
388C2×C4—3884no—
389C2×C4—3884no—
390C2×C4—1948no—
391C2×C4—1948no—
392C2×C4—1942no—
393C2×C4—1942no—
394C2×C4—1942no—
395C2×C4—1942no—
396C2×C4—1942no—
397C2×C4C23×D9712yes—
398C2×C4C23×D9712yes—
399C2×C4C23×D9712yes—
400C2×C4C23×D9712yes—
401C2×C4C23×D9712yes—
402C2×C4C23×D9712yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—44no—
407C22—974no—
408C22—974no—
409C22—38816no—
410C22—3884no—
411C22—1942no—
412C22—1942no—
413C22—1942no—
414C22—1948no—
415C22—1948no—
416C22—1942no—
417C22—1942no—
418C22—1942no—
419C22—1942no—
420C22#41×D9712yes—
421C22#41×D9712yes—
422C22#1569085612yes—
423C22C24×D9711yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—1942no—
431C4—1942no—
432C4—1942no—
433C4—1942no—
434C4—1942no—
435C2—42no—
436C2—3882no—
437C2—1944no—
438C2—1944no—
439C2#2026141014yesdirect product factor
440C2C2×#41×D9711yescharacteristic
441C2C2×#41×D9711yescharacteristic
442C2C2×#1569085611yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#126×C97C2
socleC23×C97C24
prime radicalC2×#15690780C2
hypercenterC2×#126D97
hypocenterC97C2×#551
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20261410C2direct (N × H)
2. C2×#15690803C2semidirect
3. C2×#15690796C2semidirect
4. C2×#15690795C2semidirect
5. C2×#15690795C2semidirect
6. C2×#15690792C2semidirect
7. C2×#15690790C2semidirect
8. C2×#15690788C2semidirect
10. C2×C4×D388C2semidirect
11. C2×#15690772C2semidirect
12. C2×#15690768C2semidirect
13. C2×#15690767C2semidirect
14. C2×C22⋊C4×D97C2semidirect
15. C2×C22⋊C4×D97C2semidirect
16. C2×#126×C97C2semidirect
18. C22×Dic194C22semidirect
26. C22×C4×D97C22semidirect
27. #15690803C22semidirect
28. #15690796C22semidirect
29. #15690795C22semidirect
30. #15690795C22semidirect
31. #15690792C22semidirect
32. #15690790C22semidirect
33. #15690788C22semidirect
35. C4×D388C22semidirect
36. #15690772C22semidirect
37. #15690768C22semidirect
38. #15690767C22semidirect
39. C2×C4⋊Dic97C22semidirect
40. C2×C4⋊Dic97C22semidirect
41. C2×#13095490C22semidirect
42. C2×#13095490C22semidirect
43. C2×#13095490C22semidirect
52. C2×#13095488C22semidirect
53. C2×#13095488C22semidirect
54. C2×#13095488C22semidirect
55. C22⋊C4×D97C22semidirect
56. C22⋊C4×D97C22semidirect
57. C2×C4×Dic97C22semidirect
58. C2×C4×Dic97C22semidirect
59. #126×C97C22semidirect
60. C2×C4⋊C4×C97C22semidirect
61. C2×C4⋊C4×C97C22semidirect
62. C2×C4⋊C4×C97C22semidirect
66. C2×C42×C97C22semidirect
75. C2×Dic194C23semidirect
96. C2×C4×D97C23semidirect
97. C4⋊Dic97C23semidirect
98. C4⋊Dic97C23semidirect
99. #13095490C23semidirect
100. #13095490C23semidirect
101. #13095490C23semidirect
110. #13095488C23semidirect
111. #13095488C23semidirect
112. #13095488C23semidirect
113. C4×Dic97C23semidirect
114. C4×Dic97C23semidirect
122. C4⋊C4×C97C23semidirect
123. C4⋊C4×C97C23semidirect
124. C4⋊C4×C97C23semidirect
128. C42×C97C23semidirect
214. C2×C97#551semidirect
222. C97C2×#551semidirect
225. C2×#126D97semidirect
250. #126C2×D97semidirect
267. C2×C4⋊C4C2×D97semidirect
268. C2×C4⋊C4C2×D97semidirect
269. C2×C4⋊C4C2×D97semidirect
288. C2×C42C2×D97semidirect
331. C4⋊C4C22×D97semidirect
332. C4⋊C4C22×D97semidirect
333. C4⋊C4C22×D97semidirect
350. C42C22×D97semidirect
439. C2#20261410direct (N × H)

Maximal subgroups

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

subgroupindex
#202614102
C2×#156908032
C2×#156907962
C2×#156907952
C2×#156907922
C2×#156907902
C2×#156907882
C2×#156907802
C2×C4×D3882
C2×#156907722
C2×#156907682
C2×#156907672
C2×C22⋊C4×D972
C2×#126×C972
C2×#55197