Finite Group Databasebeta

← #28405357

Subgroups — #28405357

Order 14464.

1412 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20299434C211yesmaximal, characteristic, complemented
2C2×#16813670C211yesmaximal, characteristic, complemented
3#20299414C214yesmaximal, complemented
4#20299411C214yesmaximal, complemented
5#20299407C214yesmaximal, complemented
6#20299404C214yesmaximal, complemented
7#20299398C211yesmaximal, characteristic, complemented
8#20299397C214yesmaximal, complemented
9#20299396C211yesmaximal, characteristic, complemented
10C2×#16813653C211yesmaximal, characteristic, complemented
11#137×D113C214yesmaximal, complemented
12#20299364C211yesmaximal, characteristic, complemented
13C2×#137×C113C211yesmaximal, characteristic, complemented
14#16813670—22no—
15#16813670—22no—
16#16813670C2214yescomplemented
17#16813670C2214yescomplemented
18#16813670C2212yescomplemented
19#16813669—21no—
20#16813669C2212yescomplemented
21#16813668—21no—
22#16813668C2212yescomplemented
23#16813667—22no—
24#16813667C2212yescomplemented
25C2×#14268648—21no—
26C2×#14268648C2212yescomplemented
27C2×#14268647—22no—
28C2×#14268647—22no—
29C2×#14268647C2211yescharacteristic
30C2×#14268647C2211yescharacteristic
31C2×#14268647C2211yescharacteristic, complemented
32#41×D113—22no—
33#41×D113C2214yes—
34C2×Q8×D113—21no—
35C2×Q8×D113C2212yescomplemented
36#16813655C2212yescomplemented
37#16813655C2212yescomplemented
38#16813654C2218yescomplemented
39#16813654C2218yescomplemented
40#16813653C2214yescomplemented
41#16813653C2214yescomplemented
42#16813653C2212yescomplemented
43#16813652C2212yescomplemented
44#16813651C2218yescomplemented
45#16813650C2212yescomplemented
46#16813650C2212yescomplemented
47#16813649C2218yescomplemented
48#16813648C2212yescomplemented
49#16813647C2212yescomplemented
50#16813645C2218yescomplemented
51#16813644C2214yescomplemented
52#16813643C2218yescomplemented
53#16813642C2214yescomplemented
54C2×#14268637C2212yescomplemented
55Q16×D113C2218yescomplemented
56C2×#14268634C2212yescomplemented
57SD16×D113C2218yescomplemented
58#16813630C2214yescomplemented
59#16813628C2214yescomplemented
60C2×#14268631C2211yescharacteristic, complemented
61M16×D113C2214yescomplemented
62C2×#41×C113C2211yescharacteristic
63C22×Q8×C113C2211yescharacteristic, complemented
64#137×C113C2214yescomplemented
65#137×C113C2214yescomplemented
66C2×Q16×C113C2212yescomplemented
67C2×SD16×C113C2212yescomplemented
68C2×M16×C113C2211yescharacteristic, complemented
69#14268648—22no—
70#14268648—24no—
71#14268648—24no—
72#14268648—24no—
73#14268648C2318yes—
74#14268648C2314yes—
75#14268647—21no—
76#14268647—21no—
77#14268647—21no—
78#14268647—21no—
79#14268647—24no—
80#14268647—28no—
81#14268647—28no—
82#14268647—22no—
83#14268647—24no—
84#14268647—22no—
85#14268647D412yes—
86#14268647D412yes—
87#14268647C2312yes—
88#14268647C2312yes—
89#14268647C2312yes—
90#14268647C2312yes—
91#14268647C2312yes—
92#14268647C2312yes—
93#14268646—24no—
94#14268646—24no—
95#14268646—22no—
96#14268646C2314yes—
97C2×Dic226—22no—
98C2×Dic226—22no—
99C2×Dic226D411yescharacteristic
100C2×Dic226C2311yescharacteristic
101C2×Dic226C2311yescharacteristic
102Q8×D113—24no—
103Q8×D113—24no—
104Q8×D113—24no—
105Q8×D113—22no—
106Q8×D113C2318yes—
107Q8×D113C2314yes—
108C2×D452—22no—
109C2×D452—22no—
110C2×D452D411yescharacteristic
111C2×D452C2311yescharacteristic
112C2×D452C2311yescharacteristic
113C2×C22⋊D113—21no—
114C2×C22⋊D113—21no—
115C2×C22⋊D113—24no—
116C2×C22⋊D113D412yes—
117D4×D113—24no—
118D4×D113—24no—
119D4×D113—22no—
120D4×D113C2314yes—
121C2×C4×D113—22no—
122C2×C4×D113—21no—
123C2×C4×D113—21no—
124C2×C4×D113—22no—
125C2×C4×D113—24no—
126C2×C4×D113C2312yes—
127Dic452C2314yes—
128#14268637C2314yes—
129#14268637C2318yes—
130#14268636C2314yes—
131#14268636C2314yes—
132#14268635C2314yes—
133#14268634C2318yes—
134#14268634C2314yes—
135D904C2314yes—
136#14268632C2314yes—
137#14268631C2312yes—
138#14268631C2312yes—
139#14268630C2318yes—
140C2×#11744751C2312yes—
141C8×D113C2318yes—
142#41×C113—21no—
143#41×C113—21no—
144#41×C113C2312yes—
145#41×C113C2312yes—
146C2×Q8×C113—21no—
147C2×Q8×C113—21no—
148C2×Q8×C113—21no—
149C2×Q8×C113C2312yes—
150C2×Q8×C113C2312yes—
151C2×Q8×C113C2312yes—
152C2×Q8×C113C2311yescharacteristic
153C2×D4×C113—21no—
154C2×D4×C113C2311yescharacteristic
155C22×C4×C113—21no—
156C22×C4×C113—21no—
157C22×C4×C113C2311yescharacteristic
158Q16×C113C2318yes—
159SD16×C113C2318yes—
160M16×C113C2312yes—
161M16×C113C2312yes—
162C2×C8×C113C2312yes—
163C23×C113—21no—
164C23×C113C2×D411yescharacteristic
165C22×D113—21no—
166C22×D113—21no—
167C22×D113—24no—
168C22×D113C2×D412yes—
169Q8×C113—22no—
170Q8×C113—22no—
171Q8×C113—22no—
172Q8×C113—21no—
173Q8×C113C2414yes—
174Q8×C113C2412yes—
175D4×C113—22no—
176D4×C113—22no—
177D4×C113—21no—
178D4×C113C2412yes—
179C2×C4×C113—21no—
180C2×C4×C113—22no—
181C2×C4×C113—22no—
182C2×C4×C113—22no—
183C2×C4×C113—22no—
184C2×C4×C113—22no—
185C2×C4×C113C2×D411yescharacteristic
186C2×C4×C113C2×D411yescharacteristic
187C2×C4×C113C2×D411yescharacteristic
188C2×C4×C113C2411yescharacteristic
189C2×C4×C113C2411yescharacteristic
190C2×C4×C113C2411yescharacteristic
191C22⋊D113—22no—
192C22⋊D113—22no—
193C22⋊D113—22no—
194C22⋊D113—22no—
195C22⋊D113—24no—
196C22⋊D113—24no—
197C22⋊D113C2×D414yes—
198C22⋊D113C2×D414yes—
199C2×Dic113—21no—
200C2×Dic113—21no—
201C2×Dic113—24no—
202C2×Dic113C2×D412yes—
203D452—24no—
204D452—28no—
205D452—21no—
206D452—21no—
207D452C2×D412yes—
208D452C2×D412yes—
209D452C2412yes—
210D452C2412yes—
211C4×D113—24no—
212C4×D113—28no—
213C4×D113—22no—
214C4×D113—22no—
215C4×D113—22no—
216C4×D113—22no—
217C4×D113—24no—
218C4×D113—28no—
219C4×D113C2×D414yes—
220C4×D113C2414yes—
221Dic226—21no—
222Dic226—21no—
223Dic226—28no—
224Dic226—24no—
225Dic226C2×D412yes—
226Dic226C2×D412yes—
227Dic226C2412yes—
228Dic226C2412yes—
229C8×C113C2414yes—
230#11744751C2414yes—
231C22×C113—22no—
232C22×C113—41no—
233C22×C113—81no—
234C22×C113C22×D411yescharacteristic
235C22×C113C22×D411yescharacteristic
236C22×C113C22×D411yescharacteristic
237C2×D113—22no—
238C2×D113—22no—
239C2×D113—42no—
240C2×D113—42no—
241C2×D113—84no—
242C2×D113—81no—
243C2×D113C22×D414yes—
244C4×C113—22no—
245C4×C113—24no—
246C4×C113C22×D411yescharacteristic
247C4×C113C22×D411yescharacteristic
248C4×C113C22×D411yescharacteristic
249C4×C113C2511yescharacteristic
250Dic113—22no—
251Dic113—22no—
252Dic113C22×D414yes—
253C2×C113—21no—
254C2×C113—21no—
255C2×C113—21no—
256C2×C113—42no—
257C2×C113C23×D411yescharacteristic
258D113—24no—
259D113—42no—
260D113—42no—
261#3586—1131nomaximal, Sylow
262C113#358611yescharacteristic, complemented, Sylow
263#585—1131no—
264C2×2−1+4—1131no—
265#578—1134no—
266#578—1134no—
267#577—1134no—
268#577—1134no—
269#575—1131no—
270#575—1134no—
271#575—1131no—
272C2×#137—1131no—
273C2×#137—1134no—
274C2×#137D11311yescharacteristic, complemented
275#568—1131no—
2762−1+4—1132no—
2772−1+4—1134no—
2782−1+4—1134no—
2792−1+4—1132no—
2802−1+4—2262no—
2812−1+4—2262no—
2822−1+4—2261no—
283H(5,𝔽2)—1132no—
284H(5,𝔽2)—1132no—
285H(5,𝔽2)—2262no—
286H(5,𝔽2)—2261no—
287C2×#41—1131no—
288C2×#41—1134no—
289C2×#41—1132no—
290C2×#41—1131no—
291C2×#41—1131no—
292C2×#41—2262no—
293C2×#41—2262no—
294C2×#41—2261no—
295C2×#41—2262no—
296C2×#41C2×D11311yescharacteristic
297C22×Q8—1132no—
298C22×Q8—2261no—
299C22×Q8C2×D11311yescharacteristic, complemented
300#137—1132no—
301#137—1134no—
302#137—1134no—
303#137—1132no—
304#137—1138no—
305#137—1138no—
306#137—1138no—
307#137—1132no—
308#137—1132no—
309#137C2×D11314yescomplemented
310#137C2×D11314yescomplemented
311AGL(1,ℤ/8)—1132no—
312AGL(1,ℤ/8)—1132no—
313AGL(1,ℤ/8)—1138no—
314AGL(1,ℤ/8)—1132no—
315AGL(1,ℤ/8)—1132no—
316#135—1134no—
317#135—1138no—
318#135—1138no—
319#135—1134no—
320C2×Q16—1132no—
321C2×Q16—1138no—
322C2×Q16C2×D11312yescomplemented
323C2×SD16—1132no—
324C2×SD16—1138no—
325C2×SD16C2×D11312yescomplemented
326#131—1134no—
327#131—1134no—
328C2×M16—1131no—
329C2×M16—1134no—
330C2×M16C2×D11311yescharacteristic, complemented
331#41—21no—
332#41—21no—
333#41—1132no—
334#41—1132no—
335#41—1132no—
336#41—1132no—
337#41—1138no—
338#41—1134no—
339#41—1134no—
340#41—1132no—
341#41—1132no—
342#41—1132no—
343#41—1132no—
344#41—2264no—
345#41—2264no—
346#41—2262no—
347#41—2268no—
348#41—2268no—
349#41—2262no—
350#41—2264no—
351#41—2264no—
352#41—2264no—
353#41—2262no—
354#41—2262no—
355#41—2264no—
356#41—2264no—
357#41—2261no—
358#41—2261no—
359#41—2261no—
360#41—2261no—
361#41C22×D11312yes—
362#41C22×D11312yes—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—1131no—
367C2×Q8—1138no—
368C2×Q8—1134no—
369C2×Q8—1131no—
370C2×Q8—1131no—
371C2×Q8—2262no—
372C2×Q8—2262no—
373C2×Q8—2264no—
374C2×Q8—2264no—
375C2×Q8—2264no—
376C2×Q8—2262no—
377C2×Q8C22×D11312yes—
378C2×Q8C22×D11312yes—
379C2×Q8C22×D11311yescharacteristic
380C2×Q8C22×D11312yes—
381C2×D4—21no—
382C2×D4—1132no—
383C2×D4—1134no—
384C2×D4—1131no—
385C2×D4—1131no—
386C2×D4—1131no—
387C2×D4—2262no—
388C2×D4—2262no—
389C2×D4—2264no—
390C2×D4—2262no—
391C2×D4—2264no—
392C2×D4—2264no—
393C2×D4—2261no—
394C2×D4—2261no—
395C2×D4C22×D11311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—1132no—
399C22×C4—2262no—
400C22×C4—2264no—
401C22×C4—2262no—
402C22×C4—2261no—
403C22×C4—2261no—
404C22×C4C22×D11311yescharacteristic
405Q16—1134no—
406Q16—1138no—
407Q16—1134no—
408Q16C22×D11318yes—
409SD16—1138no—
410SD16—1134no—
411SD16—1134no—
412SD16—1134no—
413SD16—1134no—
414SD16C22×D11318yes—
415D8—1134no—
416D8—1134no—
417M16—1132no—
418M16—1132no—
419M16—1138no—
420M16C22×D11312yes—
421M16C22×D11312yes—
422C2×C8—1132no—
423C2×C8—1138no—
424C2×C8C22×D11312yes—
425C23—21no—
426C23—1132no—
427C23—2264no—
428C23—2261no—
429C23—2261no—
430C23D4×D11311yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—21no—
435Q8—1132no—
436Q8—1132no—
437Q8—1132no—
438Q8—1132no—
439Q8—2268no—
440Q8—2264no—
441Q8—2261no—
442Q8—2261no—
443Q8C23×D11314yes—
444Q8C23×D11312yes—
445D4—22no—
446D4—22no—
447D4—21no—
448D4—1134no—
449D4—1134no—
450D4—1132no—
451D4—1132no—
452D4—1132no—
453D4—1132no—
454D4—2264no—
455D4—2268no—
456D4—2264no—
457D4—2264no—
458D4—2262no—
459D4—2262no—
460D4—2262no—
461D4—2262no—
462D4—2261no—
463D4—2261no—
464D4C23×D11312yes—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—1132no—
472C2×C4—1134no—
473C2×C4—1134no—
474C2×C4—2264no—
475C2×C4—2268no—
476C2×C4—2264no—
477C2×C4—2268no—
478C2×C4—2264no—
479C2×C4—2261no—
480C2×C4—2261no—
481C2×C4—2262no—
482C2×C4—2262no—
483C2×C4—2262no—
484C2×C4—2262no—
485C2×C4D4×D11311yescharacteristic
486C2×C4D4×D11311yescharacteristic
487C2×C4D4×D11311yescharacteristic
488C2×C4C23×D11311yescharacteristic
489C2×C4C23×D11311yescharacteristic
490C2×C4C23×D11311yescharacteristic
491C8—1134no—
492C8C23×D11314yes—
493C22—22no—
494C22—41no—
495C22—81no—
496C22—1134no—
497C22—9041no—
498C22—9044no—
499C22—4522no—
500C22—4522no—
501C22—2262no—
502C22—2262no—
503C22C2×D4×D11311yescharacteristic
504C22C2×D4×D11311yescharacteristic
505C22C2×D4×D11311yescharacteristic
506C4—22no—
507C4—24no—
508C4—1134no—
509C4—2262no—
510C4—2262no—
511C4C2×D4×D11311yescharacteristic
512C4C2×D4×D11311yescharacteristic
513C4C2×D4×D11311yescharacteristic
514C4C24×D11311yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—4522no—
520C2—4522no—
521C2—2264no—
522C2C22×D4×D11311yescharacteristic
subgroupG/H
centerC2C22×D4×D113
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC2×#137×C113C2
socleC2×C113C23×D4
hypercenterC2×#137D113
hypocenterC113#3586
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. #20299434C2semidirect
2. C2×#16813670C2semidirect
3. #20299414C2semidirect
4. #20299411C2semidirect
5. #20299407C2semidirect
6. #20299404C2semidirect
7. #20299398C2semidirect
8. #20299397C2semidirect
9. #20299396C2semidirect
10. C2×#16813653C2semidirect
11. #137×D113C2semidirect
12. #20299364C2semidirect
13. C2×#137×C113C2semidirect
16. #16813670C22semidirect
17. #16813670C22semidirect
18. #16813670C22semidirect
20. #16813669C22semidirect
22. #16813668C22semidirect
24. #16813667C22semidirect
26. C2×#14268648C22semidirect
31. C2×#14268647C22semidirect
35. C2×Q8×D113C22semidirect
36. #16813655C22semidirect
37. #16813655C22semidirect
38. #16813654C22semidirect
39. #16813654C22semidirect
40. #16813653C22semidirect
41. #16813653C22semidirect
42. #16813653C22semidirect
43. #16813652C22semidirect
44. #16813651C22semidirect
45. #16813650C22semidirect
46. #16813650C22semidirect
47. #16813649C22semidirect
48. #16813648C22semidirect
49. #16813647C22semidirect
50. #16813645C22semidirect
51. #16813644C22semidirect
52. #16813643C22semidirect
53. #16813642C22semidirect
54. C2×#14268637C22semidirect
55. Q16×D113C22semidirect
56. C2×#14268634C22semidirect
57. SD16×D113C22semidirect
58. #16813630C22semidirect
59. #16813628C22semidirect
60. C2×#14268631C22semidirect
61. M16×D113C22semidirect
63. C22×Q8×C113C22semidirect
64. #137×C113C22semidirect
65. #137×C113C22semidirect
66. C2×Q16×C113C22semidirect
67. C2×SD16×C113C22semidirect
68. C2×M16×C113C22semidirect
262. C113#3586semidirect
274. C2×#137D113semidirect
299. C22×Q8C2×D113semidirect
309. #137C2×D113semidirect
310. #137C2×D113semidirect
322. C2×Q16C2×D113semidirect
325. C2×SD16C2×D113semidirect
330. C2×M16C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
C2×#168136702
#202994142
#202994112
#202994072
#202994042
#202993982
#202993972
#202993962
C2×#168136532
#137×D1132
#202993642
C2×#137×C1132
#3586113