Finite Group Databasebeta

← #30547447

Subgroups — #30547447

Order 15616.

2604 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460607C211yesmaximal, characteristic, complemented
2#21460600C211yesmaximal, characteristic, complemented
3C2×#16840574C211yesmaximal, characteristic, complemented
4#21460530C211yesmaximal, characteristic, complemented
5#21455557C216yesmaximal, complemented
6#21455548C216yesmaximal, complemented
7#21455540C216yesmaximal, complemented
8C2×#16840026C216yesmaximal, complemented
9#21454301C211yesmaximal, characteristic, complemented
10#21454297C211yesmaximal, characteristic, complemented
11C61×#2916C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13C2×#14519824C2211yescharacteristic
14C2×#14519823C2211yescharacteristic
15#16840579—24no—
16#16840574—24no—
17#16840574—24no—
18#16840574—24no—
19#16840574—24no—
20#16840567—24no—
21#16840562C2216yes—
22#16840562C2216yescomplemented
23C2×#14519810C2213yescomplemented
24C2×#14519810C2216yescomplemented
25C2×#14519805C2216yes—
26C2×#14519805C2213yescomplemented
27C2×#14519799C2213yescomplemented
28C22×#11758190C2213yescomplemented
29#16840529C2211yescharacteristic
30#16840528C2216yescomplemented
31#16840527—24no—
32C2×#14519783C2216yescomplemented
33#131×D61—24no—
34#16840038C22112yescomplemented
35#16840033C22112yescomplemented
36#16840026C22112yescomplemented
37#16840026C22124yescomplemented
38#16840026C2214yescomplemented
39#16839744C2216yescomplemented
40#16839744C2216yescomplemented
41C2×#14519695C2216yescomplemented
42C2×#14519693C2216yescomplemented
43C61×#568C2211yescharacteristic
44C61×#568C2211yescharacteristic
45C2×#131×C61C2211yescharacteristic
46C61×#430C2216yescomplemented
47C2×#108×C61C2216yescomplemented
48#14519825—21no—
49#14519824—24no—
50#14519824—24no—
51#14519824D414yes—
52#14519823—24no—
53#14519823—24no—
54#14519823D414yes—
55#14519822—23no—
56C2×#11758226—21no—
57C2×#11758222C2316yes—
58C2×#11758222C2313yes—
59C2×#11758222C2313yes—
60C2×#11758224—23no—
61#41×D61—24no—
62C22×Dic122C2313yes—
63C2×Q8×D61—21no—
64C22×D244C2313yes—
65C2×D4×D61—23no—
66#14519810—212no—
67#14519810—26no—
68#14519810—26no—
69#14519810—212no—
70#14519810C23112yes—
71#14519810C2314yes—
72#14519805—212no—
73#14519805—26no—
74#14519805—26no—
75#14519805—212no—
76#14519805C23112yes—
77#14519805C2314yes—
78#14519799—212no—
79#14519799—212no—
80#14519799—26no—
81#14519799—26no—
82C2×Dic244—26no—
83C2×Dic244C2313yes—
84C2×#11758190—26no—
85C2×#11758190—26no—
86C2×#11758190—26no—
87C2×#11758190—26no—
88C2×#11758190C2313yes—
89C2×#11758190C2316yes—
90C2×#11758190C2311yescharacteristic
91C2×D488—26no—
92C2×D488C2313yes—
93#14519785—212no—
94#14519784—24no—
95#14519783—26no—
96#14519783—26no—
97#14519783C2314yes—
98#14519783C23112yes—
99C2×#11758194C2316yes—
100C2×#11758189—26no—
101M16×D61—212no—
102C2×C8×D61—26no—
103#14519695C23112yes—
104#14519695C2314yes—
105#14519693C23112yes—
106#14519693C2314yes—
107C2×#41×C61C2311yescharacteristic
108#131×C61—24no—
109#131×C61—24no—
110#131×C61D414yes—
111#131×C61D414yes—
112C2×M16×C61C2316yes—
113C2×M16×C61C2316yes—
114C2×M16×C61C2313yes—
115C22×C8×C61C2313yes—
116#108×C61C2314yes—
117#108×C61C23112yes—
118C23×D61—23no—
119#41×C61C2×D414yes—
120#41×C61C2×D414yes—
121C2×Q8×C61#4111yescharacteristic
122C2×D4×C61#4113yes—
123C22×C4×C61C2413yes—
124C2×C22⋊D61—26no—
125C2×C22⋊D61—23no—
126C2×C22⋊D61—23no—
127C22×Dic61—23no—
128#11758226—24no—
129#11758226—44no—
130Q8×D61—24no—
131Q8×D61—44no—
132#11758224—212no—
133#11758224—412no—
134D4×D61—212no—
135D4×D61—412no—
136#11758222—26no—
137#11758222—26no—
138#11758222—26no—
139#11758222—26no—
140#11758222—26no—
141#11758222—26no—
142#11758222C2×D4112yes—
143#11758222C2414yes—
144C2×D244—26no—
145C2×D244—26no—
146C2×D244—23no—
147C2×D244C2×D416yes—
148C2×D244C2413yes—
149C2×D244C2411yescharacteristic
150C2×C4×D61—26no—
151C2×C4×D61—23no—
152C2×C4×D61—23no—
153C2×Dic122—26no—
154C2×Dic122—26no—
155C2×Dic122—23no—
156C2×Dic122C2×D416yes—
157C2×Dic122C2413yes—
158C2×Dic122C2411yescharacteristic
159M16×C61—26no—
160M16×C61—26no—
161M16×C61C2×D4112yes—
162M16×C61C2414yes—
163M16×C61C2414yes—
164C2×C8×C61—26no—
165C2×C8×C61—26no—
166C2×C8×C61C2×D416yes—
167C2×C8×C61C2×D416yes—
168C2×C8×C61C2411yescharacteristic
169C2×C8×C61C2413yes—
170#11758194—26no—
171#11758194C2414yes—
172C2×#92418—26no—
173Dic244—212no—
174Dic244—24no—
175Dic244—26no—
176D488—212no—
177D488—24no—
178D488—26no—
179#11758190—212no—
180#11758190—24no—
181#11758190—24no—
182#11758190—212no—
183#11758190—212no—
184#11758190—23no—
185#11758190—23no—
186#11758190—21no—
187#11758190—21no—
188#11758189—212no—
189#11758189—24no—
190C8×D61—24no—
191C8×D61—212no—
192C23×C61C2×#4113yes—
193C22×D61—26no—
194C22×D61—21no—
195C22×D61—23no—
196C22×D61—46no—
197C22×D61—163no—
198Q8×C61#12114yes—
199D4×C61#121112yes—
200C2×C4×C61C22×D416yes—
201C2×C4×C61C22×D416yes—
202C2×C4×C61C2×#4113yes—
203C2×C4×C61C2511yescharacteristic
204C22⋊D61—212no—
205C22⋊D61—24no—
206C22⋊D61—46no—
207C22⋊D61—46no—
208C2×Dic61—26no—
209C2×Dic61—23no—
210C2×Dic61—21no—
211C2×Dic61—46no—
212D244—26no—
213D244—23no—
214D244—23no—
215D244—21no—
216D244—21no—
217D244—46no—
218D244C22×D414yes—
219C4×D61—212no—
220C4×D61—24no—
221C4×D61—46no—
222C4×D61—46no—
223Dic122—26no—
224Dic122—23no—
225Dic122—23no—
226Dic122—21no—
227Dic122—21no—
228Dic122—46no—
229Dic122C22×D414yes—
230C8×C61—24no—
231C8×C61—24no—
232C8×C61C22×D411yescharacteristic
233C8×C61C22×D416yes—
234C8×C61C22×D411yescharacteristic
235#92418—24no—
236#60470—611nomaximal, Sylow
237C22×C61—43no—
238C22×C61C2×#12116yes—
239C22×C61C22×#4111yescharacteristic
240C2×D61—24no—
241C2×D61—41no—
242C2×D61—43no—
243C2×D61—86no—
244C2×D61—81no—
245C2×D61—83no—
246C2×D61—166no—
247C4×C61C2×#12116yes—
248C4×C61C23×D411yescharacteristic
249C4×C61C23×D411yescharacteristic
250Dic61—24no—
251Dic61—43no—
252Dic61—41no—
253#3586—611no—
254#3585—611no—
255C2×#577—611no—
256#3573—611no—
257#3069—616no—
258#3068—616no—
259#3067—616no—
260C2×#471—616no—
261#2916D6111yescharacteristic, complemented
262#2889—611no—
263#2888—611no—
264C2×C61—21no—
265C2×C61—26no—
266C2×C61C22×#12111yescharacteristic
267D61—44no—
268D61—81no—
269D61—83no—
270#585—611no—
271C2×2−1+4—611no—
272C2×H(5,𝔽2)—611no—
273#578—1224no—
274#577—1224no—
275#577—1224no—
276#577—1224no—
277#577—1224no—
278#576—1224no—
279#575—616no—
280#575—616no—
281C2×#137—613no—
282C2×#137—616no—
283C2×AGL(1,ℤ/8)—613no—
284C2×AGL(1,ℤ/8)—616no—
285C2×#135—613no—
286C22×SD16—613no—
287#568—611no—
288#568—616no—
289#568—1224no—
290#568C2×D6111yescharacteristic
291#568C2×D6111yescharacteristic
292C2×#131—616no—
293C2×#131—1224no—
294C2×#131C2×D6111yescharacteristic
295#473—6112no—
296#472—6112no—
297#471—6112no—
298#471—6124no—
299#471—614no—
300#430C2×D6116yescomplemented
301C2×#108C2×D6116yescomplemented
302#413—616no—
303#413—616no—
304C2×#101—616no—
305C2×#100—616no—
306C61#6047011yescharacteristic, complemented, Sylow
3072−1+4—614no—
3082−1+4—1221no—
3092−1+4—1224no—
3102−1+4—1224no—
311H(5,𝔽2)—614no—
312H(5,𝔽2)—1223no—
313H(5,𝔽2)—1224no—
314H(5,𝔽2)—1224no—
315C2×#41—616no—
316C2×#41—613no—
317C2×#41—613no—
318C2×#41—1224no—
319C2×#41—1221no—
320C2×#41—1223no—
321C2×#41C22×D6111yescharacteristic
322C22×Q8—613no—
323C22×Q8—1221no—
324C22×D4—613no—
325C22×D4—1223no—
326#137—6112no—
327#137—614no—
328#137—1226no—
329#137—1226no—
330#137—12212no—
331#137—12212no—
332AGL(1,ℤ/8)—6112no—
333AGL(1,ℤ/8)—614no—
334AGL(1,ℤ/8)—1226no—
335AGL(1,ℤ/8)—1226no—
336AGL(1,ℤ/8)—12212no—
337AGL(1,ℤ/8)—12212no—
338#135—12212no—
339#135—12212no—
340#135—1226no—
341#135—1226no—
342C2×Q16—613no—
343C2×Q16—1226no—
344C2×SD16—613no—
345C2×SD16—616no—
346C2×SD16—611no—
347C2×SD16—1226no—
348C2×SD16—1226no—
349C2×SD16—1226no—
350C2×SD16—1226no—
351C2×D8—613no—
352C2×D8—1226no—
353#131—24no—
354#131—24no—
355#131—614no—
356#131—6112no—
357#131—1224no—
358#131—12212no—
359#131—1226no—
360#131—1226no—
361#131D24414yes—
362#131D24414yes—
363C2×M16—616no—
364C2×M16—12212no—
365C2×M16—1226no—
366C2×M16C22×D6116yes—
367C2×M16C22×D6116yes—
368C2×M16C22×D6113yes—
369C22×C8—1226no—
370C22×C8C22×D6113yes—
371#108C22×D6114yes—
372#108C22×D61112yes—
373#101—6112no—
374#101—614no—
375#100—6112no—
376#100—614no—
377C24—1223no—
378#41—6112no—
379#41—614no—
380#41—1224no—
381#41—12212no—
382#41—1226no—
383#41—1226no—
384#41—1226no—
385#41—1226no—
386#41—1226no—
387#41—1226no—
388#41—2444no—
389#41—24412no—
390#41C2×D24414yes—
391#41D4×D6114yes—
392C2×Q8—616no—
393C2×Q8—613no—
394C2×Q8—611no—
395C2×Q8—1224no—
396C2×Q8—1223no—
397C2×Q8—1226no—
398C2×Q8—1226no—
399C2×Q8—2444no—
400C2×Q8#1175822611yescharacteristic
401C2×D4—616no—
402C2×D4—611no—
403C2×D4—613no—
404C2×D4—1226no—
405C2×D4—12212no—
406C2×D4—1223no—
407C2×D4—1223no—
408C2×D4—1226no—
409C2×D4—1226no—
410C2×D4—1223no—
411C2×D4—24412no—
412C2×D4#1175822613yes—
413C22×C4—1226no—
414C22×C4—1223no—
415C22×C4—1223no—
416C22×C4—1223no—
417C22×C4C23×D6113yes—
418Q16—12212no—
419Q16—1224no—
420Q16—1226no—
421SD16—12212no—
422SD16—1224no—
423SD16—12212no—
424SD16—1224no—
425SD16—1223no—
426SD16—1223no—
427SD16—12212no—
428SD16—1221no—
429SD16—1221no—
430D8—12212no—
431D8—1224no—
432D8—1226no—
433M16—26no—
434M16—26no—
435M16—614no—
436M16—1226no—
437M16—12212no—
438M16—1224no—
439M16C2×D244112yes—
440M16C23×D6114yes—
441M16C23×D6114yes—
442C2×C8—26no—
443C2×C8—26no—
444C2×C8—1226no—
445C2×C8—1224no—
446C2×C8—12212no—
447C2×C8C2×D24416yes—
448C2×C8C2×D24416yes—
449C2×C8C23×D6111yescharacteristic
450C2×C8C23×D6113yes—
451C23—1226no—
452C23—1221no—
453C23—1223no—
454C23—9763no—
455C23—2446no—
456C23C2×#1175822613yes—
457Q8—614no—
458Q8—1226no—
459Q8—1223no—
460Q8—1221no—
461Q8—1223no—
462Q8—1221no—
463Q8—2446no—
464Q8#1451975214yes—
465D4—614no—
466D4—12212no—
467D4—1224no—
468D4—1226no—
469D4—1223no—
470D4—1221no—
471D4—1221no—
472D4—1223no—
473D4—2446no—
474D4—2446no—
475D4—2446no—
476D4#14519752112yes—
477C2×C4—1226no—
478C2×C4—12212no—
479C2×C4—1224no—
480C2×C4—1221no—
481C2×C4—1223no—
482C2×C4—2446no—
483C2×C4—2446no—
484C2×C4—2446no—
485C2×C4C2×D4×D6116yes—
486C2×C4C22×D24416yes—
487C2×C4C2×#1175822613yes—
488C2×C4C24×D6111yescharacteristic
489C8—24no—
490C8—24no—
491C8—1224no—
492C8C22×D24411yescharacteristic
493C8C22×D24416yes—
494C8C22×D24411yescharacteristic
495C22—43no—
496C22—1224no—
497C22—9766no—
498C22—4881no—
499C22—4883no—
500C22—4886no—
501C22—2441no—
502C22—2443no—
503C22C2×#1451975216yes—
504C22C22×#1175822611yescharacteristic
505C4—1224no—
506C4—2441no—
507C4—2443no—
508C4C2×#1451975216yes—
509C4C22×D4×D6111yescharacteristic
510C4C23×D24411yescharacteristic
511C2—21no—
512C2—26no—
513C2—4883no—
514C2—4881no—
515C2—2444no—
516C2C22×#1451975211yescharacteristic
subgroupG/H
centerC2C22×#14519752
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2916C2
socleC2×C61C22×#121
hypercenter#2916D61
hypocenterC61#60470
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×#41 ◁ #2916

Complements

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

NH ≅ G/Ntype
1. #21460607C2semidirect
2. #21460600C2semidirect
3. C2×#16840574C2semidirect
4. #21460530C2semidirect
5. #21455557C2semidirect
6. #21455548C2semidirect
7. #21455540C2semidirect
8. C2×#16840026C2semidirect
9. #21454301C2semidirect
10. #21454297C2semidirect
11. C61×#2916C2semidirect
22. #16840562C22semidirect
23. C2×#14519810C22semidirect
24. C2×#14519810C22semidirect
26. C2×#14519805C22semidirect
27. C2×#14519799C22semidirect
28. C22×#11758190C22semidirect
30. #16840528C22semidirect
32. C2×#14519783C22semidirect
34. #16840038C22semidirect
35. #16840033C22semidirect
36. #16840026C22semidirect
37. #16840026C22semidirect
38. #16840026C22semidirect
39. #16839744C22semidirect
40. #16839744C22semidirect
41. C2×#14519695C22semidirect
42. C2×#14519693C22semidirect
46. C61×#430C22semidirect
47. C2×#108×C61C22semidirect
261. #2916D61semidirect
300. #430C2×D61semidirect
301. C2×#108C2×D61semidirect
306. C61#60470semidirect

Maximal subgroups

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

subgroupindex
#214606072
#214606002
C2×#168405742
#214605302
#214555572
#214555482
#214555402
C2×#168400262
#214543012
#214542972
C61×#29162
#6047061