Finite Group Databasebeta

← #27155374

Subgroups — #27155374

Order 12416.

1560 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690855C211yesmaximal, characteristic, complemented
2C22×#13095511C211yesmaximal, characteristic, complemented
3#20261386C214yesmaximal, complemented
4#20261383C214yesmaximal, complemented
5#20261377C214yesmaximal, complemented
6#20261375C214yesmaximal, complemented
7#20261307C212yesmaximal, complemented
8#20261297C211yesmaximal, characteristic, complemented
9#20261296C212yesmaximal, complemented
10#20261294C211yesmaximal, characteristic, complemented
11#20261243C212yesmaximal, complemented
12C2×#15690793C211yesmaximal, characteristic, complemented
13#123×D97C212yesmaximal, complemented
14#20261208C211yesmaximal, characteristic, complemented
15C2×#123×C97C211yesmaximal, characteristic, complemented
16#15690855—28no—
17C2×#13095511—28no—
18C2×#13095511C2214yescomplemented
19C2×#13095511C2212yescomplemented
20C2×#13095511C2214yescomplemented
21C2×#13095510C2212yes—
22C2×#13095510C2212yescomplemented
23C22×Dic194C2211yescharacteristic, complemented
24C22×D388C2211yescharacteristic, complemented
25C22×C22⋊D97C2211yescharacteristic
26C22×C22⋊D97C2211yescharacteristic
27C22×C22⋊D97C2211yescharacteristic, complemented
28C22×C22⋊D97C2212yescomplemented
29C2×D4×D97C2212yes—
30C2×D4×D97C2212yescomplemented
31C22×C4×D97C2211yescharacteristic, complemented
32C22×C4×D97C2211yescharacteristic, complemented
33#15690809C2214yescomplemented
34#15690796C2214yescomplemented
35#15690795C2214yescomplemented
36#15690795C2214yescomplemented
37#15690794C2212yescomplemented
38#15690794C2212yescomplemented
39#15690793C2212yescomplemented
40#15690793C2214yescomplemented
41#15690793C2212yescomplemented
42#15690793C2212yescomplemented
43#15690792C2214yescomplemented
44#15690791C2212yescomplemented
45#15690788C2214yescomplemented
46#15690788C2214yescomplemented
47#15690787C2214yescomplemented
48#15690785C2214yescomplemented
49#15690784C2214yescomplemented
50#15690783C2214yescomplemented
51#15690783C2214yescomplemented
52#15690782C2214yescomplemented
53#15690780C2212yescomplemented
54#15690780C2214yescomplemented
55#15690780C2212yescomplemented
56#15690779C2212yescomplemented
57#15690778C2212yescomplemented
58#15690772C2214yescomplemented
59C4×C22⋊D97C2214yescomplemented
60#15690770C2214yescomplemented
61#15690770C2214yescomplemented
62#15690766C2212yescomplemented
63C2×#13095490C2212yescomplemented
64C4⋊C4×D97C2214yescomplemented
65C2×C22⋊Dic97C2211yescharacteristic, complemented
66C2×#13095488C2212yescomplemented
67C22⋊C4×D97C2214yescomplemented
68C22⋊C4×D97C2212yescomplemented
69C22×D4×C97C2211yescharacteristic
70C23×C4×C97C2211yescharacteristic, complemented
71#123×C97C2214yescomplemented
72#123×C97C2212yescomplemented
73#123×C97C2212yescomplemented
74C2×C4⋊C4×C97C2212yescomplemented
75C2×C22⋊C4×C97C2211yescharacteristic, complemented
76C2×C22⋊C4×C97C2212yescomplemented
77C23×D97C2312yes—
78C23×D97C2311yescharacteristic
79C23×D97C2311yescharacteristic
80C23×D97C2311yescharacteristic
81#13095511—216no—
82#13095511—216no—
83#13095511—28no—
84#13095510—28no—
85#13095510—28no—
86C2×Dic194—24no—
87C2×Dic194C2314yes—
88C2×Dic194C2312yes—
89C2×D388—24no—
90C2×D388C2314yes—
91C2×D388C2312yes—
92C2×D388C2314yes—
93C2×C22⋊D97—24no—
94C2×C22⋊D97—24no—
95C2×C22⋊D97—28no—
96C2×C22⋊D97D418yescomplemented
97C2×C22⋊D97C2312yes—
98C2×C22⋊D97C2312yes—
99C2×C22⋊D97C2312yes—
100C2×C22⋊D97C2312yes—
101C2×C22⋊D97C2312yes—
102C2×C22⋊D97C2312yes—
103C2×C22⋊D97C2314yes—
104C2×C22⋊D97C2312yes—
105C2×C22⋊D97C2312yes—
106D4×D97—28no—
107D4×D97—28no—
108C22×Dic97C2311yescharacteristic
109C22×Dic97C2312yes—
110C22×Dic97C2311yescharacteristic
111C22×Dic97C2312yes—
112C22×Dic97C2311yescharacteristic
113C2×C4×D97—24no—
114C2×C4×D97—24no—
115C2×C4×D97C2314yes—
116C2×C4×D97C2312yes—
117C2×C4×D97C2314yes—
118C2×C4×D97C2312yes—
119C2×C4×D97C2314yes—
120C4⋊Dic97C2314yes—
121#13095490C2314yes—
122#13095490C2314yes—
123#13095490C2314yes—
124#13095490C2314yes—
125C22⋊Dic97C2312yes—
126C22⋊Dic97C2314yes—
127C22⋊Dic97C2312yes—
128#13095488C2314yes—
129#13095488C2314yes—
130#13095488C2314yes—
131#13095488C2314yes—
132#13095488C2314yes—
133#13095488C2314yes—
134C4×Dic97C2314yes—
135C24×C97C2311yescharacteristic
136C24×C97C2311yescharacteristic
137C2×D4×C97—24no—
138C2×D4×C97C2312yes—
139C2×D4×C97C2312yes—
140C22×C4×C97—24no—
141C22×C4×C97C2312yes—
142C22×C4×C97C2311yescharacteristic
143C22×C4×C97C2311yescharacteristic
144C22×C4×C97C2312yes—
145C22×C4×C97C2311yescharacteristic
146C22×C4×C97C2312yes—
147C4⋊C4×C97C2314yes—
148C4⋊C4×C97C2314yes—
149C22⋊C4×C97C2312yes—
150C22⋊C4×C97C2314yes—
151C22⋊C4×C97C2312yes—
152C22⋊C4×C97C2314yes—
153C23×C97—22no—
154C23×C97—42no—
155C23×C97—42no—
156C23×C97C2×D414yes—
157C23×C97C2412yes—
158C23×C97C2411yescharacteristic
159C23×C97C2411yescharacteristic
160C23×C97C2411yescharacteristic
161C22×D97—22no—
162C22×D97—22no—
163C22×D97—24no—
164C22×D97—42no—
165C22×D97—42no—
166C22×D97—44no—
167C22×D97—42no—
168C22×D97C2×D414yes—
169C22×D97C2412yes—
170C22×D97C2412yes—
171C22×D97C2412yes—
172D4×C97—24no—
173D4×C97—24no—
174C2×C4×C97—22no—
175C2×C4×C97—22no—
176C2×C4×C97—24no—
177C2×C4×C97—24no—
178C2×C4×C97—22no—
179C2×C4×C97—42no—
180C2×C4×C97C2414yes—
181C2×C4×C97C2412yes—
182C2×C4×C97C2412yes—
183C2×C4×C97C2412yes—
184C22⋊D97—24no—
185C22⋊D97—24no—
186C22⋊D97—24no—
187C22⋊D97—24no—
188C22⋊D97—24no—
189C22⋊D97—24no—
190C22⋊D97—44no—
191C22⋊D97C2×D418yes—
192C22⋊D97C2×D418yes—
193C2×Dic97—22no—
194C2×Dic97—22no—
195C2×Dic97—24no—
196C2×Dic97—42no—
197C2×Dic97C2×D414yes—
198C2×Dic97C2414yes—
199C2×Dic97C2412yes—
200C2×Dic97C2412yes—
201C2×Dic97C2412yes—
202D388—24no—
203D388—28no—
204D388—44no—
205C4×D97—28no—
206C4×D97—24no—
207C4×D97—28no—
208C4×D97—24no—
209C4×D97—44no—
210Dic194—28no—
211Dic194—24no—
212C22×C97—22no—
213C22×C97—22no—
214C22×C97—22no—
215C22×C97—22no—
216C22×C97—24no—
217C22×C97—44no—
218C22×C97—82no—
219C22×C97C22×D412yes—
220C22×C97C22×D412yes—
221C22×C97C22×D412yes—
222C22×C97C2511yescharacteristic
223C2×D97—24no—
224C2×D97—22no—
225C2×D97—22no—
226C2×D97—24no—
227C2×D97—24no—
228C2×D97—44no—
229C2×D97—44no—
230C2×D97—84no—
231C2×D97—82no—
232C2×D97C22×D414yes—
233C4×C97—22no—
234C4×C97—22no—
235C4×C97—22no—
236C4×C97—24no—
237Dic97—22no—
238Dic97—22no—
239Dic97—24no—
240Dic97C22×D414yes—
241C2×C97—22no—
242C2×C97—22no—
243C2×C97—22no—
244C2×C97—42no—
245C2×C97C23×D411yescharacteristic
246C2×C97#58512yes—
247D97—24no—
248D97—42no—
249D97—42no—
250#3485—971nomaximal, Sylow
251C97#348511yescharacteristic, complemented, Sylow
252C2×H(5,𝔽2)—971no—
253C22×#41—971no—
254#547—974no—
255#547—974no—
256#546—974no—
257#546—974no—
258#536—972no—
259#534—971no—
260#534—972no—
261#534—971no—
262#525—972no—
263C2×#123—971no—
264C2×#123—972no—
265C2×#123D9711yescharacteristic, complemented
266#518—971no—
267H(5,𝔽2)—1948no—
268C2×#41—972no—
269C2×#41—972no—
270C2×#41—974no—
271C2×#41—974no—
272C2×#41—972no—
273C2×#41—1948no—
274C22×Q8—971no—
275C22×D4—971no—
276C22×D4—971no—
277C22×D4—971no—
278C22×D4—972no—
279C22×D4—972no—
280C22×D4—972no—
281C22×D4—971no—
282C22×D4C2×D9711yescharacteristic
283C23×C4—971no—
284C23×C4—971no—
285C23×C4C2×D9711yescharacteristic, complemented
286C4⋊Q8—974no—
287#124—974no—
288#123—972no—
289#123—972no—
290#123—972no—
291#123—974no—
292#123—972no—
293#123—974no—
294#123—974no—
295#123—972no—
296#123—974no—
297#123—972no—
298#123C2×D9712yescomplemented
299#123C2×D9712yescomplemented
300#123C2×D9714yescomplemented
301C22⋊Q8—974no—
302C22⋊Q8—974no—
303C22⋊Q8—974no—
304#121—974no—
305#121—974no—
306#121—974no—
307#121—974no—
308#121—974no—
309#120—972no—
310#120—972no—
311#120—972no—
312#120—974no—
313#120—972no—
314C4×D4—974no—
315C4×D4—974no—
316C4×D4—974no—
317C4×D4—974no—
318#117—972no—
319C2×C4⋊C4—972no—
320C2×C4⋊C4—974no—
321C2×C4⋊C4C2×D9712yescomplemented
322C2×C22⋊C4—971no—
323C2×C22⋊C4—972no—
324C2×C22⋊C4—972no—
325C2×C22⋊C4—974no—
326C2×C22⋊C4C2×D9711yescharacteristic, complemented
327C2×C22⋊C4C2×D9712yescomplemented
328C24—971no—
329C24—972no—
330C24—971no—
331C24—971no—
332C24C22×D9711yescharacteristic
333C24C22×D9711yescharacteristic
334#41—19416no—
335#41—19416no—
336#41—1948no—
337#41—1948no—
338#41—1948no—
339C2×Q8—974no—
340C2×Q8—972no—
341C2×Q8—1944no—
342C2×D4—24no—
343C2×D4—978no—
344C2×D4—972no—
345C2×D4—972no—
346C2×D4—974no—
347C2×D4—972no—
348C2×D4—972no—
349C2×D4—972no—
350C2×D4—972no—
351C2×D4—972no—
352C2×D4—972no—
353C2×D4—974no—
354C2×D4—974no—
355C2×D4—972no—
356C2×D4—1944no—
357C2×D4—1948no—
358C2×D4—1948no—
359C2×D4—1948no—
360C2×D4—1944no—
361C2×D4—1944no—
362C2×D4C22×D9712yes—
363C2×D4C22×D9712yes—
364C22×C4—24no—
365C22×C4—971no—
366C22×C4—972no—
367C22×C4—971no—
368C22×C4—972no—
369C22×C4—971no—
370C22×C4—974no—
371C22×C4—974no—
372C22×C4—972no—
373C22×C4—974no—
374C22×C4—972no—
375C22×C4—1944no—
376C22×C4—1944no—
377C22×C4C22×D9712yes—
378C22×C4C22×D9712yes—
379C22×C4C22×D9711yescharacteristic
380C22×C4C22×D9711yescharacteristic
381C22×C4C22×D9711yescharacteristic
382C22×C4C22×D9712yes—
383C4⋊C4—974no—
384C4⋊C4—974no—
385C4⋊C4—974no—
386C4⋊C4—974no—
387C4⋊C4—974no—
388C4⋊C4C22×D9714yes—
389C4⋊C4C22×D9714yes—
390C22⋊C4—972no—
391C22⋊C4—972no—
392C22⋊C4—974no—
393C22⋊C4—974no—
394C22⋊C4—974no—
395C22⋊C4—974no—
396C22⋊C4—974no—
397C22⋊C4—974no—
398C22⋊C4—974no—
399C22⋊C4C22×D9712yes—
400C22⋊C4C22×D9714yes—
401C22⋊C4C22×D9712yes—
402C22⋊C4C22×D9714yes—
403C42—974no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—974no—
408C23—972no—
409C23—972no—
410C23—972no—
411C23—3882no—
412C23—3884no—
413C23—3882no—
414C23—3882no—
415C23—1944no—
416C23—1942no—
417C23—1942no—
418C23D4×D9714yes—
419C23C23×D9712yes—
420C23C23×D9711yescharacteristic
421C23C23×D9711yescharacteristic
422C23C23×D9711yescharacteristic
423Q8—1948no—
424Q8—1944no—
425D4—24no—
426D4—24no—
427D4—978no—
428D4—978no—
429D4—3884no—
430D4—3884no—
431D4—1944no—
432D4—1948no—
433D4—1944no—
434D4—1944no—
435D4—1944no—
436D4—1944no—
437D4—1944no—
438D4—1944no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—974no—
446C2×C4—972no—
447C2×C4—974no—
448C2×C4—972no—
449C2×C4—972no—
450C2×C4—3884no—
451C2×C4—3882no—
452C2×C4—1948no—
453C2×C4—1944no—
454C2×C4—1948no—
455C2×C4—1944no—
456C2×C4—1944no—
457C2×C4—1942no—
458C2×C4—1942no—
459C2×C4C23×D9714yes—
460C2×C4C23×D9712yes—
461C2×C4C23×D9712yes—
462C2×C4C23×D9712yes—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—24no—
468C22—44no—
469C22—82no—
470C22—7762no—
471C22—7764no—
472C22—974no—
473C22—3884no—
474C22—3884no—
475C22—1942no—
476C22—1942no—
477C22—1944no—
478C22—1944no—
479C22—1944no—
480C22C2×D4×D9712yes—
481C22C2×D4×D9712yes—
482C22C2×D4×D9712yes—
483C22C24×D9711yescharacteristic
484C4—22no—
485C4—22no—
486C4—22no—
487C4—24no—
488C4—974no—
489C4—1942no—
490C4—1942no—
491C4—1944no—
492C2—22no—
493C2—22no—
494C2—22no—
495C2—42no—
496C2—3882no—
497C2—3882no—
498C2—1944no—
499C2C22×D4×D9711yescharacteristic
500C2#2026157112yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
FittingC2×#123×C97C2
hypercenterC2×#123D97
hypocenterC97#3485
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#15690855C2semidirect
2. C22×#13095511C2semidirect
3. #20261386C2semidirect
4. #20261383C2semidirect
5. #20261377C2semidirect
6. #20261375C2semidirect
7. #20261307C2semidirect
8. #20261297C2semidirect
9. #20261296C2semidirect
10. #20261294C2semidirect
11. #20261243C2semidirect
12. C2×#15690793C2semidirect
13. #123×D97C2semidirect
14. #20261208C2semidirect
15. C2×#123×C97C2semidirect
18. C2×#13095511C22semidirect
19. C2×#13095511C22semidirect
20. C2×#13095511C22semidirect
22. C2×#13095510C22semidirect
23. C22×Dic194C22semidirect
24. C22×D388C22semidirect
27. C22×C22⋊D97C22semidirect
28. C22×C22⋊D97C22semidirect
30. C2×D4×D97C22semidirect
31. C22×C4×D97C22semidirect
32. C22×C4×D97C22semidirect
33. #15690809C22semidirect
34. #15690796C22semidirect
35. #15690795C22semidirect
36. #15690795C22semidirect
37. #15690794C22semidirect
38. #15690794C22semidirect
39. #15690793C22semidirect
40. #15690793C22semidirect
41. #15690793C22semidirect
42. #15690793C22semidirect
43. #15690792C22semidirect
44. #15690791C22semidirect
45. #15690788C22semidirect
46. #15690788C22semidirect
47. #15690787C22semidirect
48. #15690785C22semidirect
49. #15690784C22semidirect
50. #15690783C22semidirect
51. #15690783C22semidirect
52. #15690782C22semidirect
53. #15690780C22semidirect
54. #15690780C22semidirect
55. #15690780C22semidirect
56. #15690779C22semidirect
57. #15690778C22semidirect
58. #15690772C22semidirect
59. C4×C22⋊D97C22semidirect
60. #15690770C22semidirect
61. #15690770C22semidirect
62. #15690766C22semidirect
63. C2×#13095490C22semidirect
64. C4⋊C4×D97C22semidirect
65. C2×C22⋊Dic97C22semidirect
66. C2×#13095488C22semidirect
67. C22⋊C4×D97C22semidirect
68. C22⋊C4×D97C22semidirect
70. C23×C4×C97C22semidirect
71. #123×C97C22semidirect
72. #123×C97C22semidirect
73. #123×C97C22semidirect
74. C2×C4⋊C4×C97C22semidirect
75. C2×C22⋊C4×C97C22semidirect
76. C2×C22⋊C4×C97C22semidirect
96. C2×C22⋊D97D4semidirect
251. C97#3485semidirect
265. C2×#123D97semidirect
285. C23×C4C2×D97semidirect
298. #123C2×D97semidirect
299. #123C2×D97semidirect
300. #123C2×D97semidirect
321. C2×C4⋊C4C2×D97semidirect
326. C2×C22⋊C4C2×D97semidirect
327. C2×C22⋊C4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908552
C22×#130955112
#202613862
#202613832
#202613772
#202613752
#202613072
#202612972
#202612962
#202612942
#202612432
C2×#156907932
#123×D972
#202612082
C2×#123×C972
#348597