Finite Group Databasebeta

← C2×#27161668 #38756054

Subgroups — C2×#27161668 #38756054

Order 25216.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161668C2116yesmaximal, direct product factor
2C2×#20263321C211yesmaximal, characteristic, complemented
3C2×#20263314C211yesmaximal, characteristic, complemented
4C2×#20263312C211yesmaximal, characteristic, complemented
5C2×#20263311C211yesmaximal, characteristic, complemented
6C2×#20263308C211yesmaximal, characteristic, complemented
7C2×#20263304C211yesmaximal, characteristic, complemented
8C2×#20263301C211yesmaximal, characteristic, complemented
9C2×#20263301C211yesmaximal, characteristic, complemented
10C2×#20263299C211yesmaximal, characteristic, complemented
11C2×#20263296C211yesmaximal, characteristic, complemented
12C2×#20263296C211yesmaximal, characteristic, complemented
13C2×C4×D788C211yesmaximal, characteristic, complemented
14C2×#20263288C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
16C2×#126×C197C211yesmaximal, characteristic, complemented
17C24×D197C2211yescharacteristic
18C22×Dic394C2211yescharacteristic, complemented
19C22×D788C2211yescharacteristic
20C22×D788C2211yescharacteristic
21C22×D788C2211yescharacteristic
22C22×D788C2211yescharacteristic
23C22×D788C2211yescharacteristic, complemented
24C22×C22⋊D197C2211yescharacteristic
25C22×C22⋊D197C2211yescharacteristic
26C22×C4×D197C2211yescharacteristic
27C22×C4×D197C2211yescharacteristic
28C22×C4×D197C2211yescharacteristic, complemented
29C22×C4×D197C2211yescharacteristic, complemented
30#20263321C2218yescomplemented
31#20263314C2218yescomplemented
32#20263312C2218yescomplemented
33#20263311C2218yescomplemented
34#20263308C2218yescomplemented
35#20263304C2218yescomplemented
36#20263301C2218yescomplemented
37#20263301C2218yescomplemented
38#20263299C2218yescomplemented
39#20263296C2218yescomplemented
40#20263296C2218yescomplemented
41C4×D788C2218yescomplemented
42#20263288C2218yescomplemented
43C2×C4⋊Dic197C2211yescharacteristic, complemented
44C2×#15691337C2211yescharacteristic, complemented
45C2×#15691337C2211yescharacteristic, complemented
46C2×C22⋊Dic197C2211yescharacteristic
47C2×#15691335C2211yescharacteristic
48C2×#15691335C2211yescharacteristic
49C2×#15691335C2211yescharacteristic
50C2×#15691335C2211yescharacteristic
51C2×#15691335C2211yescharacteristic
52C2×#15691335C2211yescharacteristic
53C2×#15691335C2211yescharacteristic, complemented
54C2×#15691335C2211yescharacteristic, complemented
55C2×#15691335C2211yescharacteristic, complemented
56C2×#15691335C2211yescharacteristic, complemented
57C22⋊C4×D197C2218yescomplemented
58C2×C4×Dic197C2211yescharacteristic, complemented
59#126×C197C2218yescomplemented
60C2×C4⋊C4×C197C2211yescharacteristic, complemented
61C2×C4⋊C4×C197C2211yescharacteristic, complemented
62C2×C4⋊C4×C197C2211yescharacteristic, complemented
63C2×C22⋊C4×C197C2211yescharacteristic
64C2×C22⋊C4×C197C2211yescharacteristic
65C2×C22⋊C4×C197C2211yescharacteristic
66C2×C42×C197C2211yescharacteristic, complemented
67C23×D197—24no—
68C23×D197—42no—
69C23×D197—42no—
70C23×D197C2311yescharacteristic
71C23×D197C2311yescharacteristic
72C23×D197C2311yescharacteristic
73C23×D197C2311yescharacteristic
74C23×D197C2314yes—
75C2×Dic394—42no—
76C2×Dic394C2314yescomplemented
77C2×D788—24no—
78C2×D788—24no—
79C2×D788—42no—
80C2×D788—42no—
81C2×D788—42no—
82C2×D788C2314yes—
83C2×D788C2314yes—
84C2×D788C2314yes—
85C2×D788C2314yes—
86C2×D788C2314yescomplemented
87C2×C22⋊D197—42no—
88C2×C22⋊D197—42no—
89C2×C22⋊D197C2314yes—
90C2×C22⋊D197C2314yes—
91C22×Dic197C2311yescharacteristic
92C22×Dic197C2311yescharacteristic
93C22×Dic197C2311yescharacteristic
94C22×Dic197C2311yescharacteristic
95C2×C4×D197—42no—
96C2×C4×D197—42no—
97C2×C4×D197—42no—
98C2×C4×D197—42no—
99C2×C4×D197C2314yes—
100C2×C4×D197C2314yes—
101C2×C4×D197C2314yescomplemented
102C2×C4×D197C2314yescomplemented
103C4⋊Dic197C2314yescomplemented
104#15691337C2314yescomplemented
105#15691337C2314yescomplemented
106C22⋊Dic197C2314yes—
107#15691335C2314yes—
108#15691335C2314yes—
109#15691335C2314yes—
110#15691335C2314yes—
111#15691335C2314yes—
112#15691335C2314yes—
113#15691335C2314yescomplemented
114#15691335C2314yescomplemented
115#15691335C2314yescomplemented
116#15691335C2314yescomplemented
117C4×Dic197C2314yescomplemented
118C24×C197C2311yescharacteristic
119C22×C4×C197C2311yescharacteristic
120C22×C4×C197C2311yescharacteristic
121C22×C4×C197C2311yescharacteristic
122C22×C4×C197C2311yescharacteristic
123C22×C4×C197C2311yescharacteristic
124C22×C4×C197C2311yescharacteristic
125C4⋊C4×C197C2314yescomplemented
126C4⋊C4×C197C2314yescomplemented
127C4⋊C4×C197C2314yescomplemented
128C22⋊C4×C197C2314yes—
129C22⋊C4×C197C2314yes—
130C22⋊C4×C197C2314yes—
131C42×C197C2314yescomplemented
132C22×D197—22no—
133C22×D197—22no—
134C22×D197—22no—
135C22×D197—22no—
136C22×D197—22no—
137C22×D197—22no—
138C22×D197—22no—
139C22×D197—22no—
140C22×D197—22no—
141C22×D197—22no—
142C22×D197—22no—
143C22×D197—28no—
144C22×D197—44no—
145C22×D197—44no—
146C22×D197—416no—
147C22×D197#4114yes—
148C22×D197C2412yes—
149C22×D197C2412yes—
150C22×D197C2412yes—
151C22×D197C2412yes—
152Dic394—44no—
153D788—28no—
154D788—28no—
155D788—44no—
156D788—44no—
157D788—44no—
158C22⋊D197—44no—
159C22⋊D197—44no—
160C2×Dic197—22no—
161C2×Dic197—22no—
162C2×Dic197—22no—
163C2×Dic197—22no—
164C2×Dic197C2412yes—
165C2×Dic197C2412yes—
166C2×Dic197C2412yes—
167C2×Dic197C2412yes—
168C4×D197—44no—
169C4×D197—44no—
170C4×D197—44no—
171C4×D197—44no—
172C23×C197—22no—
173C23×C197—22no—
174C23×C197—22no—
175C23×C197C2412yes—
176C23×C197C2411yescharacteristic
177C2×C4×C197—22no—
178C2×C4×C197—22no—
179C2×C4×C197—22no—
180C2×C4×C197—22no—
181C2×C4×C197—22no—
182C2×C4×C197—22no—
183C2×C4×C197C2412yes—
184C2×C4×C197C2412yes—
185C2×C4×C197C2412yes—
186C2×C4×C197C2412yes—
187C2×C4×C197C2412yes—
188C2×C4×C197C2412yes—
189C22×C197—22no—
190C22×C197—22no—
191C22×C197—22no—
192C22×C197—44no—
193C22×C197C2×#4112yes—
194C22×C197H(5,𝔽2)12yes—
195C22×C197H(5,𝔽2)12yes—
196C22×C197C2511yescharacteristic
197C2×D197—28no—
198C2×D197—22no—
199C2×D197—22no—
200C2×D197—22no—
201C2×D197—22no—
202C2×D197—22no—
203C2×D197—22no—
204C2×D197—22no—
205C2×D197—22no—
206C2×D197—22no—
207C2×D197—22no—
208C2×D197—22no—
209C2×D197—416no—
210C2×D197—44no—
211C2×D197—44no—
212C2×D197—44no—
213C2×D197C2×#4114yes—
214C4×C197—22no—
215C4×C197—22no—
216C4×C197—22no—
217C4×C197—22no—
218C4×C197—22no—
219C4×C197—22no—
220Dic197—22no—
221Dic197—22no—
222Dic197—22no—
223Dic197—22no—
224C2×C197—42no—
225C2×C197#53814yescomplemented
226C2×C197C22×#4111yescharacteristic
227C2×C197C2×H(5,𝔽2)11yescharacteristic
228C2×C197C2×H(5,𝔽2)11yescharacteristic
229D197—24no—
230D197—42no—
231D197—42no—
232D197—42no—
233C197C2×#53811yescharacteristic, complemented, Sylow
234C2×#538—1971nomaximal, Sylow
235#538—19716no—
236C2×#126—1971no—
237C2×#126D19711yescharacteristic, complemented
238C2×#124—1971no—
239C2×#124—1971no—
240C2×#123—1971no—
241C2×#123—1971no—
242C2×C22⋊Q8—1971no—
243C2×#121—1971no—
244C2×#121—1971no—
245C2×#121—1971no—
246C2×#120—1971no—
247C2×#120—1971no—
248C2×C4×D4—1971no—
249C2×C4×D4—1971no—
250C22×C22⋊C4—1971no—
251C25—1971no—
252C22×Q8—1971no—
253C22×D4—1971no—
254C22×D4—1971no—
255C22×D4—1971no—
256C22×D4—1971no—
257C22×D4—1971no—
258C22×D4—1971no—
259C22×D4—1971no—
260C23×C4—1971no—
261C23×C4—1971no—
262C23×C4—1971no—
263C23×C4—1971no—
264#126—1978no—
265#126C2×D19718yescomplemented
266#124—1978no—
267#124—1978no—
268#123—1978no—
269#123—1978no—
270C22⋊Q8—1978no—
271#121—1978no—
272#121—1978no—
273#121—1978no—
274#120—1978no—
275#120—1978no—
276C4×D4—1978no—
277C4×D4—1978no—
278C2×C4⋊C4—1971no—
279C2×C4⋊C4—1971no—
280C2×C4⋊C4—1971no—
281C2×C4⋊C4C2×D19711yescharacteristic, complemented
282C2×C4⋊C4C2×D19711yescharacteristic, complemented
283C2×C4⋊C4C2×D19711yescharacteristic, complemented
284C2×C22⋊C4—1971no—
285C2×C22⋊C4—1971no—
286C2×C22⋊C4—1971no—
287C2×C22⋊C4—1971no—
288C2×C22⋊C4—1971no—
289C2×C22⋊C4—1971no—
290C2×C22⋊C4—1971no—
291C2×C22⋊C4—1971no—
292C2×C22⋊C4—1971no—
293C2×C22⋊C4—1971no—
294C2×C22⋊C4—1971no—
295C2×C22⋊C4—1978no—
296C2×C22⋊C4C2×D19711yescharacteristic
297C2×C22⋊C4C2×D19711yescharacteristic
298C2×C22⋊C4C2×D19711yescharacteristic
299C2×C42—1971no—
300C2×C42C2×D19711yescharacteristic, complemented
301C24—7882no—
302C24—7882no—
303C24—3944no—
304C24—1974no—
305C24—1971no—
306C24—1971no—
307C24—1971no—
308C24—1971no—
309C24C22×D19711yescharacteristic
310C2×Q8—7882no—
311C2×Q8—1974no—
312C2×D4—7882no—
313C2×D4—7882no—
314C2×D4—7882no—
315C2×D4—7882no—
316C2×D4—7882no—
317C2×D4—3944no—
318C2×D4—3944no—
319C2×D4—1974no—
320C2×D4—1974no—
321C2×D4—1974no—
322C2×D4—1974no—
323C2×D4—1974no—
324C2×D4—1974no—
325C2×D4—1974no—
326C22×C4—7882no—
327C22×C4—7882no—
328C22×C4—7882no—
329C22×C4—7882no—
330C22×C4—1974no—
331C22×C4—1974no—
332C22×C4—1974no—
333C22×C4—1974no—
334C22×C4—1971no—
335C22×C4—1971no—
336C22×C4—1971no—
337C22×C4—1971no—
338C22×C4C22×D19711yescharacteristic
339C22×C4C22×D19711yescharacteristic
340C22×C4C22×D19711yescharacteristic
341C22×C4C22×D19711yescharacteristic
342C22×C4C22×D19711yescharacteristic
343C22×C4C22×D19711yescharacteristic
344C4⋊C4—1974no—
345C4⋊C4—1974no—
346C4⋊C4—1974no—
347C4⋊C4C22×D19714yescomplemented
348C4⋊C4C22×D19714yescomplemented
349C4⋊C4C22×D19714yescomplemented
350C22⋊C4—1974no—
351C22⋊C4—1974no—
352C22⋊C4—1974no—
353C22⋊C4—1974no—
354C22⋊C4—1974no—
355C22⋊C4—1974no—
356C22⋊C4—1974no—
357C22⋊C4—1974no—
358C22⋊C4—1974no—
359C22⋊C4—1974no—
360C22⋊C4—1974no—
361C22⋊C4C22×D19714yes—
362C22⋊C4C22×D19714yes—
363C22⋊C4C22×D19714yes—
364C42—1974no—
365C42C22×D19714yescomplemented
366C23—22no—
367C23—22no—
368C23—22no—
369C23—7884no—
370C23—7884no—
371C23—78816no—
372C23—3948no—
373C23—3942no—
374C23—3942no—
375C23—3942no—
376C23—3942no—
377C23—3942no—
378C23—3942no—
379C23—3942no—
380C23—3942no—
381C23—3942no—
382C23—3942no—
383C23—3942no—
384C23—1972no—
385C23—1972no—
386C23—1972no—
387C23—1974no—
388C23—1972no—
389C23C23×D19712yes—
390C23C23×D19711yescharacteristic
391Q8—7884no—
392D4—7884no—
393D4—7884no—
394D4—7884no—
395D4—7884no—
396D4—7884no—
397D4—3948no—
398D4—3948no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—7884no—
406C2×C4—7884no—
407C2×C4—7884no—
408C2×C4—7884no—
409C2×C4—3942no—
410C2×C4—3942no—
411C2×C4—3942no—
412C2×C4—3942no—
413C2×C4—1972no—
414C2×C4—1972no—
415C2×C4—1972no—
416C2×C4—1972no—
417C2×C4C23×D19712yes—
418C2×C4C23×D19712yes—
419C2×C4C23×D19712yes—
420C2×C4C23×D19712yes—
421C2×C4C23×D19712yes—
422C2×C4C23×D19712yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—78816no—
428C22—7884no—
429C22—7884no—
430C22—7884no—
431C22—3942no—
432C22—3942no—
433C22—3942no—
434C22—3942no—
435C22—3942no—
436C22—3942no—
437C22—3942no—
438C22—3942no—
439C22—3942no—
440C22—3942no—
441C22—3942no—
442C22—3948no—
443C22—1974no—
444C22#41×D19712yes—
445C22#2026337212yes—
446C22#2026337212yes—
447C22C24×D19711yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—3942no—
455C4—3942no—
456C4—3942no—
457C4—3942no—
458C2—42no—
459C2—7882no—
460C2—7882no—
461C2—7882no—
462C2—3944no—
463C2#2716166814yesdirect product factor
464C2C2×#41×D19711yescharacteristic
465C2C2×#2026337211yescharacteristic
466C2C2×#2026337211yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#126×C197C2
socleC23×C197C24
hypercenterC2×#126D197
hypocenterC197C2×#538
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #27161668C2direct (N × H)
2. C2×#20263321C2semidirect
3. C2×#20263314C2semidirect
4. C2×#20263312C2semidirect
5. C2×#20263311C2semidirect
6. C2×#20263308C2semidirect
7. C2×#20263304C2semidirect
8. C2×#20263301C2semidirect
9. C2×#20263301C2semidirect
10. C2×#20263299C2semidirect
11. C2×#20263296C2semidirect
12. C2×#20263296C2semidirect
13. C2×C4×D788C2semidirect
14. C2×#20263288C2semidirect
15. C2×C22⋊C4×D197C2semidirect
16. C2×#126×C197C2semidirect
18. C22×Dic394C22semidirect
23. C22×D788C22semidirect
28. C22×C4×D197C22semidirect
29. C22×C4×D197C22semidirect
30. #20263321C22semidirect
31. #20263314C22semidirect
32. #20263312C22semidirect
33. #20263311C22semidirect
34. #20263308C22semidirect
35. #20263304C22semidirect
36. #20263301C22semidirect
37. #20263301C22semidirect
38. #20263299C22semidirect
39. #20263296C22semidirect
40. #20263296C22semidirect
41. C4×D788C22semidirect
42. #20263288C22semidirect
43. C2×C4⋊Dic197C22semidirect
44. C2×#15691337C22semidirect
45. C2×#15691337C22semidirect
53. C2×#15691335C22semidirect
54. C2×#15691335C22semidirect
55. C2×#15691335C22semidirect
56. C2×#15691335C22semidirect
57. C22⋊C4×D197C22semidirect
58. C2×C4×Dic197C22semidirect
59. #126×C197C22semidirect
60. C2×C4⋊C4×C197C22semidirect
61. C2×C4⋊C4×C197C22semidirect
62. C2×C4⋊C4×C197C22semidirect
66. C2×C42×C197C22semidirect
76. C2×Dic394C23semidirect
86. C2×D788C23semidirect
101. C2×C4×D197C23semidirect
102. C2×C4×D197C23semidirect
103. C4⋊Dic197C23semidirect
104. #15691337C23semidirect
105. #15691337C23semidirect
113. #15691335C23semidirect
114. #15691335C23semidirect
115. #15691335C23semidirect
116. #15691335C23semidirect
117. C4×Dic197C23semidirect
125. C4⋊C4×C197C23semidirect
126. C4⋊C4×C197C23semidirect
127. C4⋊C4×C197C23semidirect
131. C42×C197C23semidirect
225. C2×C197#538semidirect
233. C197C2×#538semidirect
237. C2×#126D197semidirect
265. #126C2×D197semidirect
281. C2×C4⋊C4C2×D197semidirect
282. C2×C4⋊C4C2×D197semidirect
283. C2×C4⋊C4C2×D197semidirect
300. C2×C42C2×D197semidirect
347. C4⋊C4C22×D197semidirect
348. C4⋊C4C22×D197semidirect
349. C4⋊C4C22×D197semidirect
365. C42C22×D197semidirect
463. C2#27161668direct (N × H)

Maximal subgroups

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

subgroupindex
#271616682
C2×#202633212
C2×#202633142
C2×#202633122
C2×#202633112
C2×#202633082
C2×#202633042
C2×#202633012
C2×#202632992
C2×#202632962
C2×C4×D7882
C2×#202632882
C2×C22⋊C4×D1972
C2×#126×C1972
C2×#538197