Finite Group Databasebeta

← #30547461

Subgroups — #30547461

Order 15616.

2578 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460605C211yesmaximal, characteristic, complemented
2C2×#16840577C211yesmaximal, characteristic, complemented
3#21460531C211yesmaximal, characteristic, complemented
4#21455558C216yesmaximal, complemented
5#21455553C216yesmaximal, complemented
6#21455536C216yesmaximal, complemented
7C2×#16840035C216yesmaximal, complemented
8#21454428C211yesmaximal, characteristic, complemented
9#21454302C211yesmaximal, characteristic, complemented
10#21454301C211yesmaximal, characteristic, complemented
11C61×#3586C211yesmaximal, characteristic, complemented
12C2×#14519824C2211yescharacteristic
13#16840577—24no—
14#16840577—24no—
15#16840577—24no—
16#16840573—24no—
17#16840560C2216yes—
18C2×#14519808C2213yescomplemented
19C2×#14519807C2216yes—
20C2×#14519807C2213yescomplemented
21C2×#14519797C2213yescomplemented
22C22×#11758202C2213yescomplemented
23#16840529—24no—
24#16840529C2216yes—
25#16840529C2211yescharacteristic
26#16840529C2211yescharacteristic
27C2×#14519784—24no—
28C2×#14519784C2216yes—
29C2×#14519784C2211yescharacteristic
30#16840035C22112yescomplemented
31#16840035C22124yescomplemented
32#16840030C2214yescomplemented
33#16840027C22112yescomplemented
34#16840023C22112yescomplemented
35#16839810C2216yescomplemented
36C2×#14519712C2216yescomplemented
37#16839746C2216yescomplemented
38#16839744C2216yescomplemented
39C2×#14519696C2216yescomplemented
40C2×#14519695C2216yescomplemented
41C61×#585C2211yescharacteristic
42C2×2−1+4×C61C2211yescharacteristic
43C61×#578—24no—
44C61×#577—24no—
45C61×#575C2216yes—
46C2×#137×C61C2216yes—
47C61×#568C2211yescharacteristic
48#14519824—24no—
49#14519824—24no—
50C2×#11758222C2313yes—
51C2×#11758224—23no—
52C22×Dic122C2313yes—
53C2×Q8×D61—21no—
54#14519808—26no—
55#14519808—212no—
56#14519807—212no—
57#14519807—212no—
58#14519807C23112yes—
59#14519803—26no—
60#14519802C2314yes—
61#14519797—212no—
62#14519797—26no—
63#14519797—26no—
64C2×#11758202—26no—
65C2×#11758202—26no—
66C2×#11758202—26no—
67C2×#11758202C2313yes—
68C2×#11758202C2316yes—
69C2×#11758200—26no—
70C2×#11758200C2313yes—
71C2×#11758201C2313yes—
72C2×#11758199C2311yescharacteristic
73#14519784—24no—
74#14519784—24no—
75#14519784—212no—
76#14519784—26no—
77#14519784—26no—
78#14519784D414yes—
79#14519784D414yes—
80#14519784C2314yes—
81#14519784C23112yes—
82C2×#11758194—212no—
83C2×#11758194—26no—
84C2×#11758194C2316yes—
85C2×#11758194C2316yes—
86C2×#11758194C2313yes—
87C22×#92418—26no—
88C22×#92418C2313yes—
89#14519712C23112yes—
90#14519712C2314yes—
91#14519696C23112yes—
92#14519695C23112yes—
93#14519694C2314yes—
94#14519693C2314yes—
952−1+4×C61—21no—
962−1+4×C61—24no—
972−1+4×C61—24no—
982−1+4×C61D414yes—
99H(5,𝔽2)×C61—23no—
100H(5,𝔽2)×C61D414yes—
101C2×#41×C61—24no—
102C2×#41×C61—23no—
103C2×#41×C61C2316yes—
104C2×#41×C61C2313yes—
105C2×#41×C61C2311yescharacteristic
106C22×Q8×C61—21no—
107C22×Q8×C61C2313yes—
108#137×C61—212no—
109#137×C61—26no—
110#137×C61C23112yes—
111AGL(1,ℤ/8)×C61—26no—
112AGL(1,ℤ/8)×C61C2314yes—
113#135×C61—212no—
114C2×Q16×C61—26no—
115C2×SD16×C61—26no—
116#131×C61—24no—
117C2×M16×C61C2316yes—
118#41×C61—24no—
119#41×C61—212no—
120#41×C61—26no—
121#41×C61—26no—
122#41×C61—26no—
123#41×C61—26no—
124#41×C61—412no—
125#41×C61C2×D4112yes—
126#41×C61C2×D414yes—
127#41×C61C2×D414yes—
128#41×C61C2414yes—
129C2×Q8×C61—24no—
130C2×Q8×C61—23no—
131C2×Q8×C61—26no—
132C2×Q8×C61—26no—
133C2×Q8×C61—44no—
134C2×Q8×C61C2×D416yes—
135C2×Q8×C61#4111yescharacteristic
136C2×Q8×C61C2413yes—
137C2×D4×C61—26no—
138C2×D4×C61—212no—
139C2×D4×C61—23no—
140C2×D4×C61C2×D416yes—
141C2×D4×C61#4113yes—
142C2×D4×C61C2411yescharacteristic
143C22×C4×C61—26no—
144C22×C4×C61—23no—
145C22×C4×C61—23no—
146C22×C4×C61C2413yes—
147C2×C22⋊D61—23no—
148C22×Dic61—23no—
149Q8×D61—44no—
150#11758224—412no—
151#11758222—26no—
152#11758222—26no—
153C2×D244C2411yescharacteristic
154C2×C4×D61—23no—
155C2×Dic122—26no—
156C2×Dic122—26no—
157C2×Dic122—23no—
158C2×Dic122C2413yes—
159Q16×C61—212no—
160SD16×C61—24no—
161SD16×C61—212no—
162D8×C61—24no—
163M16×C61—26no—
164M16×C61C2414yes—
165C2×C8×C61—26no—
166#11758202—212no—
167#11758202—212no—
168#11758202—23no—
169#11758202—23no—
170#11758201—24no—
171#11758201—26no—
172#11758200—212no—
173#11758200—26no—
174#11758199—24no—
175#11758199—21no—
176#11758199—21no—
177#11758194—26no—
178#11758194—26no—
179#11758194—212no—
180#11758194—24no—
181#11758194C2×D4112yes—
182#11758194C2414yes—
183#11758194C2414yes—
184C2×#92418—26no—
185C2×#92418—26no—
186C2×#92418—24no—
187C2×#92418—212no—
188C2×#92418C2×D416yes—
189C2×#92418C2×D416yes—
190C2×#92418C2411yescharacteristic
191C2×#92418C2413yes—
192C23×C61—26no—
193C23×C61—21no—
194C23×C61C2×#4113yes—
195C22×D61—21no—
196Q8×C61—26no—
197Q8×C61—23no—
198Q8×C61—23no—
199Q8×C61—46no—
200Q8×C61#12114yes—
201Q8×C61C22×D414yes—
202D4×C61—212no—
203D4×C61—24no—
204D4×C61—26no—
205D4×C61—21no—
206D4×C61—21no—
207D4×C61—46no—
208D4×C61#121112yes—
209D4×C61C22×D414yes—
210C2×C4×C61—26no—
211C2×C4×C61—212no—
212C2×C4×C61—24no—
213C2×C4×C61—23no—
214C2×C4×C61—46no—
215C2×C4×C61—46no—
216C2×C4×C61C22×D416yes—
217C2×C4×C61C22×D416yes—
218C2×C4×C61C2×#4113yes—
219C2×C4×C61C2511yescharacteristic
220C22⋊D61—46no—
221C2×Dic61—23no—
222C2×Dic61—46no—
223D244—21no—
224D244—21no—
225C4×D61—46no—
226Dic122—23no—
227Dic122—23no—
228Dic122—46no—
229C8×C61—24no—
230#92418—24no—
231#92418—24no—
232#92418C22×D411yescharacteristic
233#92418C22×D416yes—
234#92418C22×D411yescharacteristic
235#60472—611nomaximal, Sylow
236C22×C61—24no—
237C22×C61—43no—
238C22×C61—41no—
239C22×C61—86no—
240C22×C61—81no—
241C22×C61C2×#12116yes—
242C22×C61C22×#4111yescharacteristic
243C2×D61—41no—
244C2×D61—81no—
245C4×C61—24no—
246C4×C61—43no—
247C4×C61C2×#12116yes—
248C4×C61C23×D411yescharacteristic
249C4×C61C23×D411yescharacteristic
250Dic61—43no—
251#3586—611no—
252#3586D6111yescharacteristic, complemented
253C2×#578—611no—
254#3573—611no—
255#3069—616no—
256#3069—616no—
257#3067—616no—
258C2×#473—616no—
259#2916—611no—
260#2889—611no—
261#2889—611no—
262C2×C61—21no—
263C2×C61—26no—
264C2×C61—44no—
265C2×C61—81no—
266C2×C61C22×#12111yescharacteristic
267D61—81no—
268#585C2×D6111yescharacteristic
269C2×2−1+4—611no—
270C2×2−1+4C2×D6111yescharacteristic
271#578—24no—
272#578—1224no—
273#578—1224no—
274#578—1224no—
275#577—24no—
276#577—1224no—
277#575—616no—
278#575C2×D6116yes—
279C2×#137—613no—
280C2×#137—613no—
281C2×#137—616no—
282C2×#137C2×D6116yes—
283C2×#135—613no—
284C22×Q16—613no—
285#568—611no—
286#568—611no—
287#568—616no—
288#568—1224no—
289#568C2×D6111yescharacteristic
290C2×#131—611no—
291C2×#131—616no—
292C2×#131—1224no—
293#473—6112no—
294#473—6124no—
295#472—614no—
296#471—6112no—
297#471—6112no—
298#430—616no—
299C2×#108—616no—
300#413—616no—
301#413—616no—
302C2×#101—616no—
303C2×#101—616no—
304C61#6047211yescharacteristic, complemented, Sylow
3052−1+4—21no—
3062−1+4—24no—
3072−1+4—24no—
3082−1+4—1224no—
3092−1+4—1224no—
3102−1+4C22⋊D6114yes—
311H(5,𝔽2)—23no—
312H(5,𝔽2)C22⋊D6114yes—
313C2×#41—24no—
314C2×#41—23no—
315C2×#41—613no—
316C2×#41—1223no—
317C2×#41C22×D6116yes—
318C2×#41C22×D6113yes—
319C2×#41C22×D6111yescharacteristic
320C22×Q8—21no—
321C22×Q8—613no—
322C22×Q8—1221no—
323C22×Q8C22×D6113yes—
324#137—212no—
325#137—26no—
326#137—6112no—
327#137—12212no—
328#137—12212no—
329#137—12212no—
330#137—1226no—
331#137C22×D61112yes—
332AGL(1,ℤ/8)—26no—
333AGL(1,ℤ/8)—614no—
334AGL(1,ℤ/8)—1226no—
335AGL(1,ℤ/8)C22×D6114yes—
336#135—212no—
337#135—1226no—
338#135—1226no—
339#135—12212no—
340C2×Q16—26no—
341C2×Q16—613no—
342C2×Q16—616no—
343C2×Q16—1226no—
344C2×Q16—1226no—
345C2×Q16—1226no—
346C2×SD16—26no—
347C2×SD16—613no—
348C2×SD16—613no—
349C2×SD16—1226no—
350C2×D8—611no—
351#131—24no—
352#131—614no—
353#131—614no—
354#131—614no—
355#131—6112no—
356#131—1226no—
357#131—1226no—
358#131—12212no—
359#131—1224no—
360#131—1224no—
361C2×M16—613no—
362C2×M16—616no—
363C2×M16—616no—
364C2×M16—1226no—
365C2×M16—12212no—
366C2×M16C22×D6116yes—
367C22×C8—613no—
368C22×C8—1226no—
369#108—614no—
370#108—6112no—
371#101—6112no—
372#101—6112no—
373#100—614no—
374#100—614no—
375#41—24no—
376#41—212no—
377#41—26no—
378#41—26no—
379#41—26no—
380#41—26no—
381#41—412no—
382#41—1226no—
383#41—1226no—
384#41—24412no—
385#41D4×D6114yes—
386#41C2×C22⋊D61112yes—
387#41C2×C22⋊D6114yes—
388#41C23×D6114yes—
389C2×Q8—24no—
390C2×Q8—23no—
391C2×Q8—26no—
392C2×Q8—26no—
393C2×Q8—44no—
394C2×Q8—613no—
395C2×Q8—1223no—
396C2×Q8—1226no—
397C2×Q8—1226no—
398C2×Q8—2444no—
399C2×Q8#1175822411yescharacteristic
400C2×Q8C2×C22⋊D6116yes—
401C2×Q8C23×D6113yes—
402C2×D4—26no—
403C2×D4—212no—
404C2×D4—23no—
405C2×D4—611no—
406C2×D4—1223no—
407C2×D4#1175822413yes—
408C2×D4C2×C22⋊D6116yes—
409C2×D4C23×D6111yescharacteristic
410C22×C4—26no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—1223no—
414C22×C4—1223no—
415C22×C4C23×D6113yes—
416Q16—212no—
417Q16—1223no—
418Q16—1223no—
419Q16—12212no—
420Q16—12212no—
421SD16—24no—
422SD16—212no—
423SD16—1226no—
424SD16—1226no—
425SD16—1224no—
426SD16—12212no—
427D8—24no—
428D8—1221no—
429D8—1221no—
430D8—1224no—
431M16—26no—
432M16—6112no—
433M16—614no—
434M16—614no—
435M16—1224no—
436M16—12212no—
437M16—1226no—
438M16—1226no—
439M16C23×D6114yes—
440C2×C8—26no—
441C2×C8—616no—
442C2×C8—616no—
443C2×C8—611no—
444C2×C8—613no—
445C2×C8—1224no—
446C2×C8—12212no—
447C2×C8—1226no—
448C2×C8—1226no—
449C23—26no—
450C23—21no—
451C23—1221no—
452C23C2×#1175822413yes—
453Q8—26no—
454Q8—23no—
455Q8—23no—
456Q8—46no—
457Q8—1223no—
458Q8—1223no—
459Q8—2446no—
460Q8#1451974914yes—
461Q8C22×C22⋊D6114yes—
462D4—212no—
463D4—24no—
464D4—26no—
465D4—21no—
466D4—21no—
467D4—46no—
468D4—1221no—
469D4—1221no—
470D4—2446no—
471D4#14519749112yes—
472D4C22×C22⋊D6114yes—
473C2×C4—26no—
474C2×C4—212no—
475C2×C4—24no—
476C2×C4—23no—
477C2×C4—46no—
478C2×C4—46no—
479C2×C4—1223no—
480C2×C4—2446no—
481C2×C4—2446no—
482C2×C4C2×D4×D6116yes—
483C2×C4C22×C22⋊D6116yes—
484C2×C4C2×#1175822413yes—
485C2×C4C24×D6111yescharacteristic
486C8—24no—
487C8—611no—
488C8—611no—
489C8—616no—
490C8—1224no—
491C8—1224no—
492C22—24no—
493C22—43no—
494C22—41no—
495C22—86no—
496C22—81no—
497C22—4881no—
498C22—2441no—
499C22C2×#1451974916yes—
500C22C22×#1175822411yescharacteristic
501C4—24no—
502C4—43no—
503C4—2443no—
504C4C2×#1451974916yes—
505C4C22×D4×D6111yescharacteristic
506C4C23×C22⋊D6111yescharacteristic
507C2—21no—
508C2—26no—
509C2—44no—
510C2—81no—
511C2—4881no—
512C2C22×#1451974911yescharacteristic
subgroupG/H
centerC2C22×#14519749
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3586C2
socleC2×C61C22×#121
hypercenter#3586D61
hypocenterC61#60472
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×#41 ◁ #3586

Complements

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

NH ≅ G/Ntype
1. #21460605C2semidirect
2. C2×#16840577C2semidirect
3. #21460531C2semidirect
4. #21455558C2semidirect
5. #21455553C2semidirect
6. #21455536C2semidirect
7. C2×#16840035C2semidirect
8. #21454428C2semidirect
9. #21454302C2semidirect
10. #21454301C2semidirect
11. C61×#3586C2semidirect
18. C2×#14519808C22semidirect
20. C2×#14519807C22semidirect
21. C2×#14519797C22semidirect
22. C22×#11758202C22semidirect
30. #16840035C22semidirect
31. #16840035C22semidirect
32. #16840030C22semidirect
33. #16840027C22semidirect
34. #16840023C22semidirect
35. #16839810C22semidirect
36. C2×#14519712C22semidirect
37. #16839746C22semidirect
38. #16839744C22semidirect
39. C2×#14519696C22semidirect
40. C2×#14519695C22semidirect
252. #3586D61semidirect
304. C61#60472semidirect

Maximal subgroups

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

subgroupindex
#214606052
C2×#168405772
#214605312
#214555582
#214555532
#214555362
C2×#168400352
#214544282
#214543022
#214543012
C61×#35862
#6047261