Finite Group Databasebeta

← C2×#27161661 #38756047

Subgroups — C2×#27161661 #38756047

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#27161661C2116yesmaximal, direct product factor
2C2×#20263321C211yesmaximal, characteristic, complemented
3C2×#20263319C211yesmaximal, characteristic, complemented
4C2×#20263313C211yesmaximal, characteristic, complemented
5C2×#20263312C211yesmaximal, characteristic, complemented
6C2×#20263309C211yesmaximal, characteristic, complemented
7C2×#20263307C211yesmaximal, characteristic, complemented
8C2×#20263304C211yesmaximal, characteristic, complemented
9C2×#20263298C211yesmaximal, characteristic, complemented
10C2×#20263297C211yesmaximal, characteristic, complemented
11C2×#20263294C211yesmaximal, characteristic, complemented
12C2×#20263294C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
14C2×#20263286C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
16C2×#121×C197C211yesmaximal, characteristic, complemented
17C24×D197C2211yescharacteristic
18C22×Dic394C2211yescharacteristic, complemented
19C22×C22⋊D197C2211yescharacteristic
20C22×C22⋊D197C2211yescharacteristic
21C22×C22⋊D197C2211yescharacteristic
22C22×C22⋊D197C2211yescharacteristic
23C23×Dic197C2211yescharacteristic, complemented
24C22×C4×D197C2211yescharacteristic
25C22×C4×D197C2211yescharacteristic
26#20263321C2218yescomplemented
27#20263319C2218yescomplemented
28#20263313C2218yescomplemented
29#20263312C2218yescomplemented
30#20263309C2218yescomplemented
31#20263307C2218yescomplemented
32#20263304C2218yescomplemented
33#20263298C2218yescomplemented
34#20263297C2218yescomplemented
35#20263294C2218yescomplemented
36#20263294C2218yescomplemented
37C4×C22⋊D197C2218yescomplemented
38#20263286C2218yescomplemented
39C2×C4⋊Dic197C2211yescharacteristic, complemented
40C2×#15691337C2211yescharacteristic, complemented
41C2×#15691337C2211yescharacteristic, complemented
42C2×#15691337C2211yescharacteristic, complemented
43C2×#15691337C2211yescharacteristic, complemented
44C2×C22⋊Dic197C2211yescharacteristic
45C2×C22⋊Dic197C2211yescharacteristic
46C2×C22⋊Dic197C2211yescharacteristic
47C2×C22⋊Dic197C2211yescharacteristic, complemented
48C2×C22⋊Dic197C2211yescharacteristic, complemented
49C2×C22⋊Dic197C2211yescharacteristic, complemented
50C2×#15691335C2211yescharacteristic
51C2×#15691335C2211yescharacteristic
52C2×#15691335C2211yescharacteristic
53C2×#15691335C2211yescharacteristic
54C2×#15691335C2211yescharacteristic
55C2×#15691335C2211yescharacteristic
56C22⋊C4×D197C2218yescomplemented
57C2×C4×Dic197C2211yescharacteristic, complemented
58C2×C4×Dic197C2211yescharacteristic, complemented
59C22×D4×C197C2211yescharacteristic
60C22×D4×C197C2211yescharacteristic
61C22×D4×C197C2211yescharacteristic, complemented
62C23×C4×C197C2211yescharacteristic, complemented
63#121×C197C2218yescomplemented
64C2×C4⋊C4×C197C2211yescharacteristic, complemented
65C2×C22⋊C4×C197C2211yescharacteristic
66C2×C22⋊C4×C197C2211yescharacteristic, complemented
67C23×D197—24no—
68C23×D197—42no—
69C23×D197—42no—
70C23×D197C2311yescharacteristic
71C23×D197C2311yescharacteristic
72C23×D197C2314yes—
73C2×Dic394—42no—
74C2×Dic394C2314yescomplemented
75C2×C22⋊D197—24no—
76C2×C22⋊D197—24no—
77C2×C22⋊D197—42no—
78C2×C22⋊D197—42no—
79C2×C22⋊D197C2314yes—
80C2×C22⋊D197C2314yes—
81C2×C22⋊D197C2314yes—
82C2×C22⋊D197C2314yes—
83C22×Dic197—42no—
84C22×Dic197C2311yescharacteristic
85C22×Dic197C2311yescharacteristic
86C22×Dic197C2311yescharacteristic
87C22×Dic197C2311yescharacteristic
88C22×Dic197C2311yescharacteristic
89C22×Dic197C2311yescharacteristic
90C22×Dic197C2314yescomplemented
91C2×C4×D197—42no—
92C2×C4×D197—42no—
93C2×C4×D197C2314yes—
94C2×C4×D197C2314yes—
95C4⋊Dic197C2314yescomplemented
96#15691337C2314yescomplemented
97#15691337C2314yescomplemented
98#15691337C2314yescomplemented
99#15691337C2314yescomplemented
100C22⋊Dic197C2314yes—
101C22⋊Dic197C2314yes—
102C22⋊Dic197C2314yes—
103C22⋊Dic197C2314yescomplemented
104C22⋊Dic197C2314yescomplemented
105C22⋊Dic197C2314yescomplemented
106#15691335C2314yes—
107#15691335C2314yes—
108#15691335C2314yes—
109#15691335C2314yes—
110#15691335C2314yes—
111#15691335C2314yes—
112C4×Dic197C2314yescomplemented
113C4×Dic197C2314yescomplemented
114C24×C197C2311yescharacteristic
115C24×C197C2311yescharacteristic
116C24×C197C2311yescharacteristic
117C2×D4×C197—42no—
118C2×D4×C197—42no—
119C2×D4×C197—42no—
120C2×D4×C197C2314yes—
121C2×D4×C197C2314yes—
122C2×D4×C197C2314yescomplemented
123C22×C4×C197—42no—
124C22×C4×C197C2311yescharacteristic
125C22×C4×C197C2311yescharacteristic
126C22×C4×C197C2311yescharacteristic
127C22×C4×C197C2311yescharacteristic
128C22×C4×C197C2314yescomplemented
129C4⋊C4×C197C2314yescomplemented
130C22⋊C4×C197C2314yes—
131C22⋊C4×C197C2314yescomplemented
132C22×D197—22no—
133C22×D197—22no—
134C22×D197—22no—
135C22×D197—22no—
136C22×D197—22no—
137C22×D197—28no—
138C22×D197—44no—
139C22×D197—44no—
140C22×D197—416no—
141C22×D197#4114yes—
142C22×D197C2412yes—
143C22×D197C2412yes—
144Dic394—44no—
145C22⋊D197—28no—
146C22⋊D197—28no—
147C22⋊D197—44no—
148C22⋊D197—44no—
149C2×Dic197—22no—
150C2×Dic197—22no—
151C2×Dic197—22no—
152C2×Dic197—22no—
153C2×Dic197—22no—
154C2×Dic197—22no—
155C2×Dic197—44no—
156C2×Dic197C2412yes—
157C2×Dic197C2412yes—
158C2×Dic197C2412yes—
159C2×Dic197C2412yes—
160C2×Dic197C2412yes—
161C2×Dic197C2412yes—
162C4×D197—44no—
163C4×D197—44no—
164C23×C197—22no—
165C23×C197—22no—
166C23×C197—22no—
167C23×C197—22no—
168C23×C197—22no—
169C23×C197—22no—
170C23×C197—22no—
171C23×C197—22no—
172C23×C197—22no—
173C23×C197C2412yes—
174C23×C197C2412yes—
175C23×C197C2412yes—
176C23×C197C2411yescharacteristic
177D4×C197—44no—
178D4×C197—44no—
179D4×C197—44no—
180C2×C4×C197—22no—
181C2×C4×C197—22no—
182C2×C4×C197—22no—
183C2×C4×C197—22no—
184C2×C4×C197—44no—
185C2×C4×C197C2412yes—
186C2×C4×C197C2412yes—
187C2×C4×C197C2412yes—
188C2×C4×C197C2412yes—
189C22×C197—22no—
190C22×C197—22no—
191C22×C197—22no—
192C22×C197—22no—
193C22×C197—22no—
194C22×C197—22no—
195C22×C197—22no—
196C22×C197—22no—
197C22×C197—22no—
198C22×C197—44no—
199C22×C197—44no—
200C22×C197—44no—
201C22×C197C2×#4112yes—
202C22×C197H(5,𝔽2)12yes—
203C22×C197H(5,𝔽2)12yes—
204C22×C197C2511yescharacteristic
205C2×D197—28no—
206C2×D197—22no—
207C2×D197—22no—
208C2×D197—22no—
209C2×D197—22no—
210C2×D197—22no—
211C2×D197—416no—
212C2×D197—44no—
213C2×D197C2×#4114yes—
214C4×C197—22no—
215C4×C197—22no—
216C4×C197—22no—
217C4×C197—22no—
218Dic197—22no—
219Dic197—22no—
220Dic197—22no—
221Dic197—22no—
222Dic197—22no—
223Dic197—22no—
224C2×C197—42no—
225C2×C197—42no—
226C2×C197—42no—
227C2×C197#53814yescomplemented
228C2×C197C22×#4111yescharacteristic
229C2×C197C2×H(5,𝔽2)11yescharacteristic
230C2×C197C2×H(5,𝔽2)11yescharacteristic
231D197—24no—
232D197—42no—
233C197C2×#53811yescharacteristic, complemented, Sylow
234C2×#538—1971nomaximal, Sylow
235#538—19716no—
236C2×#126—1971no—
237C2×#126—1971no—
238C2×#124—1971no—
239C2×#124—1971no—
240C2×#123—1971no—
241C2×#123—1971no—
242C2×C22⋊Q8—1971no—
243C2×#121—1971no—
244C2×#121—1971no—
245C2×#121D19711yescharacteristic, complemented
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×D4C2×D19711yescharacteristic
258C22×D4C2×D19711yescharacteristic
259C22×D4C2×D19711yescharacteristic, complemented
260C23×C4—1971no—
261C23×C4—1971no—
262C23×C4—1971no—
263C23×C4C2×D19711yescharacteristic, complemented
264#126—1978no—
265#126—1978no—
266#124—1978no—
267#124—1978no—
268#123—1978no—
269#123—1978no—
270C22⋊Q8—1978no—
271#121—1978no—
272#121—1978no—
273#121C2×D19718yescomplemented
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⋊C4—1971no—
282C2×C4⋊C4—1971no—
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—1971no—
296C2×C22⋊C4—1978no—
297C2×C22⋊C4C2×D19711yescharacteristic
298C2×C22⋊C4C2×D19711yescharacteristic, complemented
299C2×C42—1971no—
300C2×C42—1971no—
301C24—7882no—
302C24—7882no—
303C24—3944no—
304C24—1974no—
305C24—1971no—
306C24—1971no—
307C24C22×D19711yescharacteristic
308C24C22×D19711yescharacteristic
309C24C22×D19711yescharacteristic
310C2×Q8—7882no—
311C2×Q8—1974no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
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×D4C22×D19714yes—
324C2×D4C22×D19714yes—
325C2×D4C22×D19714yescomplemented
326C22×C4—42no—
327C22×C4—7882no—
328C22×C4—7882no—
329C22×C4—7882no—
330C22×C4—1974no—
331C22×C4—1974no—
332C22×C4—1974no—
333C22×C4—1971no—
334C22×C4—1971no—
335C22×C4—1971no—
336C22×C4—1971no—
337C22×C4—1971no—
338C22×C4—1971no—
339C22×C4C22×D19711yescharacteristic
340C22×C4C22×D19711yescharacteristic
341C22×C4C22×D19711yescharacteristic
342C22×C4C22×D19711yescharacteristic
343C22×C4C22×D19714yescomplemented
344C4⋊C4—1974no—
345C4⋊C4—1974no—
346C4⋊C4—1974no—
347C4⋊C4—1974no—
348C4⋊C4—1974no—
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⋊C4—1974no—
362C22⋊C4C22×D19714yes—
363C22⋊C4C22×D19714yescomplemented
364C42—1974no—
365C42—1974no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—7884no—
376C23—7884no—
377C23—78816no—
378C23—3948no—
379C23—3942no—
380C23—3942no—
381C23—3942no—
382C23—3942no—
383C23—3942no—
384C23—1972no—
385C23—1974no—
386C23—1972no—
387C23C23×D19712yes—
388C23C23×D19712yes—
389C23C23×D19712yes—
390C23C23×D19711yescharacteristic
391Q8—7884no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—7884no—
396D4—7884no—
397D4—3948no—
398D4—3948no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—7884no—
405C2×C4—7884no—
406C2×C4—7884no—
407C2×C4—3942no—
408C2×C4—3942no—
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×C4—1972no—
418C2×C4—1972no—
419C2×C4C23×D19712yes—
420C2×C4C23×D19712yes—
421C2×C4C23×D19712yes—
422C2×C4C23×D19712yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—78816no—
436C22—7884no—
437C22—3942no—
438C22—3942no—
439C22—3942no—
440C22—3942no—
441C22—3942no—
442C22—3948no—
443C22—1974no—
444C22#41×D19712yes—
445C22#2026337112yes—
446C22#2026337112yes—
447C22C24×D19711yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—3942no—
453C4—3942no—
454C4—3942no—
455C4—3942no—
456C4—3942no—
457C4—3942no—
458C2—42no—
459C2—42no—
460C2—42no—
461C2—7882no—
462C2—3944no—
463C2#2716166114yesdirect product factor
464C2C2×#41×D19711yescharacteristic
465C2C2×#2026337111yescharacteristic
466C2C2×#2026337111yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#121×C197C2
socleC23×C197C24
hypercenterC2×#121D197
hypocenterC197C2×#538
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #27161661C2direct (N × H)
2. C2×#20263321C2semidirect
3. C2×#20263319C2semidirect
4. C2×#20263313C2semidirect
5. C2×#20263312C2semidirect
6. C2×#20263309C2semidirect
7. C2×#20263307C2semidirect
8. C2×#20263304C2semidirect
9. C2×#20263298C2semidirect
10. C2×#20263297C2semidirect
11. C2×#20263294C2semidirect
12. C2×#20263294C2semidirect
13. C2×C4×C22⋊D197C2semidirect
14. C2×#20263286C2semidirect
15. C2×C22⋊C4×D197C2semidirect
16. C2×#121×C197C2semidirect
18. C22×Dic394C22semidirect
23. C23×Dic197C22semidirect
26. #20263321C22semidirect
27. #20263319C22semidirect
28. #20263313C22semidirect
29. #20263312C22semidirect
30. #20263309C22semidirect
31. #20263307C22semidirect
32. #20263304C22semidirect
33. #20263298C22semidirect
34. #20263297C22semidirect
35. #20263294C22semidirect
36. #20263294C22semidirect
37. C4×C22⋊D197C22semidirect
38. #20263286C22semidirect
39. C2×C4⋊Dic197C22semidirect
40. C2×#15691337C22semidirect
41. C2×#15691337C22semidirect
42. C2×#15691337C22semidirect
43. C2×#15691337C22semidirect
47. C2×C22⋊Dic197C22semidirect
48. C2×C22⋊Dic197C22semidirect
49. C2×C22⋊Dic197C22semidirect
56. C22⋊C4×D197C22semidirect
57. C2×C4×Dic197C22semidirect
58. C2×C4×Dic197C22semidirect
61. C22×D4×C197C22semidirect
62. C23×C4×C197C22semidirect
63. #121×C197C22semidirect
64. C2×C4⋊C4×C197C22semidirect
66. C2×C22⋊C4×C197C22semidirect
74. C2×Dic394C23semidirect
90. C22×Dic197C23semidirect
95. C4⋊Dic197C23semidirect
96. #15691337C23semidirect
97. #15691337C23semidirect
98. #15691337C23semidirect
99. #15691337C23semidirect
103. C22⋊Dic197C23semidirect
104. C22⋊Dic197C23semidirect
105. C22⋊Dic197C23semidirect
112. C4×Dic197C23semidirect
113. C4×Dic197C23semidirect
122. C2×D4×C197C23semidirect
128. C22×C4×C197C23semidirect
129. C4⋊C4×C197C23semidirect
131. C22⋊C4×C197C23semidirect
227. C2×C197#538semidirect
233. C197C2×#538semidirect
245. C2×#121D197semidirect
259. C22×D4C2×D197semidirect
263. C23×C4C2×D197semidirect
273. #121C2×D197semidirect
283. C2×C4⋊C4C2×D197semidirect
298. C2×C22⋊C4C2×D197semidirect
325. C2×D4C22×D197semidirect
343. C22×C4C22×D197semidirect
349. C4⋊C4C22×D197semidirect
363. C22⋊C4C22×D197semidirect
463. C2#27161661direct (N × H)

Maximal subgroups

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

subgroupindex
#271616612
C2×#202633212
C2×#202633192
C2×#202633132
C2×#202633122
C2×#202633092
C2×#202633072
C2×#202633042
C2×#202632982
C2×#202632972
C2×#202632942
C2×C4×C22⋊D1972
C2×#202632862
C2×C22⋊C4×D1972
C2×#121×C1972
C2×#538197