Finite Group Databasebeta

← #30614905

Subgroups — #30614905

Order 15616.

3754 conjugacy classes of subgroups; only the 358 normal subgroups are shown.

normal subgroups only

This group has 3754 conjugacy classes of subgroups — too many to draw. Only its 358 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#410C213yesmaximal, complemented
2#21462294C213yesmaximal, complemented
3#21461663C218yesmaximal, complemented
4#585×D61C211yesmaximal, characteristic
5C2×#14519856C2213yescomplemented
6C22×C4⋊#6923C2211yescharacteristic, complemented
7#16840743—23no—
8#16840742—23no—
9#16840741—26no—
10D61⋊#99C22112yescomplemented
11(C2×#6923)⋊Q8—23no—
12#16840714—23no—
13#16840710C22112yescomplemented
14#16840704—26no—
15#16840599—23no—
16#16840598—26no—
17#16840598C413yescomplemented
18#16840597—23no—
19#16840597C414yescomplemented
202−1+4×D61—23no—
21H(5,𝔽2)×D61—23no—
22H(5,𝔽2)×D61C2214yes—
23C2×#41×D61—26no—
24C2×#41×D61C2213yes—
25C61×#585C411yescharacteristic, complemented
26C23×#6923C2312yes—
27#14519856—26no—
28#14519856—26no—
29C2×C4⋊#6923—23no—
30C2×C4⋊#6923—23no—
31C2×C4⋊#6923—23no—
32C2×C22⋊#6923—212no—
33C2×C22⋊#6923C2316yes—
34C2×C4×#6923—23no—
35#14519836—212no—
36#14519835—212no—
37(C2×#6923)⋊C4—212no—
38#14519825—212no—
39#14519825—23no—
40#14519824—212no—
41#14519824—26no—
42#14519824—212no—
43#14519823—212no—
44#14519823—26no—
45#14519823—212no—
46#14519822—212no—
47#14519822—23no—
48#14519822—212no—
49#14519822—212no—
50#14519822C2×C4112yes—
51C2×#11758226—26no—
52C2×#11758226D413yes—
53C2×#11758222—212no—
54C2×#11758222—26no—
55C2×#11758222C2×C413yes—
56C2×#11758222D416yes—
57C2×#11758224—26no—
58C2×#11758224—212no—
59C2×#11758224C2×C416yes—
60C2×#11758224D413yes—
61#41×D61—24no—
62#41×D61—212no—
63#41×D61—212no—
64#41×D61—212no—
65#41×D61—212no—
66#41×D61—212no—
67#41×D61—26no—
68#41×D61—212no—
69#41×D61—26no—
70#41×D61—26no—
71#41×D61—26no—
72C2×Q8×D61—26no—
73C2×Q8×D61D413yes—
74C2×D4×D61—26no—
75C2×D4×D61—212no—
76C2×D4×D61D413yes—
77C2×D4×D61C2316yes—
78C22×C4×D61—24no—
79C22×C4×D61D416yes—
80C22×C4×D61C2311yescharacteristic
812−1+4×C61—23no—
82H(5,𝔽2)×C61—23no—
83H(5,𝔽2)×C61C2×C414yes—
84C2×#41×C61—26no—
85C2×#41×C61C2×C413yes—
86C23×D61—24no—
87C23×D61C2×D416yes—
88C23×D61C2411yescharacteristic
89C22×#6923—26no—
90C22×#6923—26no—
91C22×#6923—82no—
92#41×C61—24no—
93#41×C61—212no—
94#41×C61—26no—
95#41×C61—212no—
96#41×C61—26no—
97C2×Q8×C61—26no—
98C2×Q8×C61C22⋊C413yescomplemented
99C2×D4×C61—26no—
100C2×D4×C61—212no—
101C2×D4×C61C22⋊C413yescomplemented
102C2×D4×C61C22×C416yes—
103C22×C4×C61—24no—
104C22×C4×C61C22⋊C416yescomplemented
105C22×C4×C61C22×C411yescharacteristic
106C2×C22⋊D61—212no—
107C2×C22⋊D61—212no—
108C2×C22⋊D61—212no—
109C2×C22⋊D61C22⋊C416yescomplemented
110C2×C22⋊D61C22×C416yes—
111C2×C22⋊D61C2×D416yes—
112C22×Dic61—24no—
113C22×Dic61C22⋊C416yescomplemented
114C22×Dic61C22×C411yescharacteristic
115#11758226—24no—
116#11758226—212no—
117#11758226—212no—
118#11758226—26no—
119#11758226—26no—
120Q8×D61—24no—
121Q8×D61—212no—
122Q8×D61—212no—
123Q8×D61—26no—
124Q8×D61—26no—
125#11758224—212no—
126#11758224—212no—
127#11758224—212no—
128#11758224—224no—
129#11758224—212no—
130#11758224—224no—
131#11758224—212no—
132#11758224—26no—
133#11758224—26no—
134D4×D61—212no—
135D4×D61—224no—
136D4×D61—212no—
137D4×D61—212no—
138D4×D61—224no—
139D4×D61—212no—
140D4×D61—212no—
141D4×D61—26no—
142D4×D61—26no—
143#11758222—212no—
144#11758222—224no—
145#11758222—212no—
146#11758222—224no—
147#11758222—26no—
148#11758222—212no—
149#11758222—26no—
150#11758222—212no—
151#11758222—212no—
152C2×D244—212no—
153C2×D244—26no—
154C2×D244C22⋊C413yescomplemented
155C2×D244C2×D416yes—
156C2×C4×D61—212no—
157C2×C4×D61—212no—
158C2×C4×D61—212no—
159C2×C4×D61—26no—
160C2×C4×D61—26no—
161C2×C4×D61—26no—
162C2×C4×D61—26no—
163C2×C4×D61C22⋊C416yescomplemented
164C2×C4×D61C22⋊C416yescomplemented
165C2×C4×D61C2×D416yes—
166C2×C4×D61C2×D413yes—
167C2×C4×D61C2×D413yes—
168C2×C4×D61C2×D413yes—
169C2×C4×D61C2×D413yes—
170C2×Dic122—26no—
171C2×Dic122—212no—
172C2×Dic122C22⋊C416yescomplemented
173C2×Dic122C22⋊C413yescomplemented
174C22⋊#6923—212no—
175C22⋊#6923—812no—
176C4⋊#6923—41no—
177C4⋊#6923—43no—
178C4⋊#6923—43no—
179C4⋊#6923—43no—
180C4×#6923—46no—
181C23×C61—24no—
182C23×C61C2×C22⋊C416yes—
183C23×C61C23×C411yescharacteristic
184C22×D61—212no—
185C22×D61—26no—
186C22×D61—26no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—46no—
190C22×D61—46no—
191C22×D61—84no—
192C22×D61—84no—
193C22×D61C2×C22⋊C416yes—
194C22×D61C22×D413yes—
195C22×D61C22×D413yes—
196C2×#6923—46no—
197C2×#6923—412no—
198C2×#6923—46no—
199C2×#6923—46no—
200C2×#6923—42no—
201Q8×C61—24no—
202Q8×C61—26no—
203D4×C61—212no—
204D4×C61—212no—
205D4×C61—26no—
206C2×C4×C61—212no—
207C2×C4×C61—26no—
208C2×C4×C61—26no—
209C2×C4×C61C2×C22⋊C416yes—
210C2×C4×C61C2×C22⋊C413yes—
211C2×C4×C61C2×C22⋊C413yes—
212C22⋊D61—224no—
213C22⋊D61—212no—
214C22⋊D61—212no—
215C22⋊D61—212no—
216C2×Dic61—212no—
217C2×Dic61—26no—
218C2×Dic61—26no—
219C2×Dic61C2×C22⋊C416yes—
220C2×Dic61C2×C22⋊C413yes—
221C2×Dic61C2×C22⋊C413yes—
222D244—212no—
223D244—212no—
224D244—26no—
225C4×D61—224no—
226C4×D61—26no—
227C4×D61—26no—
228C4×D61—26no—
229C4×D61—26no—
230C4×D61—26no—
231C4×D61—26no—
232C4×D61#12016yes—
233C4×D61#12011yescharacteristic
234C4×D61#12013yes—
235C4×D61#12013yes—
236C4×D61#12013yes—
237Dic122—212no—
238Dic122—26no—
239Dic122—212no—
240#20610—611nomaximal, Sylow
241C22×C61—26no—
242C22×C61—41no—
243C22×C61—46no—
244C22×C61—84no—
245C22×C61C22×C22⋊C413yes—
246C2×D61—23no—
247C2×D61—23no—
248C2×D61—26no—
249C2×D61—43no—
250C2×D61—46no—
251C2×D61—46no—
252C2×D61—46no—
253C2×D61—86no—
254C2×D61—812no—
255C2×D61C2×#12013yes—
256C2×D61C2×#12011yescharacteristic
257#6923—86no—
258#6923—82no—
259C4×C61—26no—
260C4×C61#37913yes—
261C4×C61#37911yescharacteristic
262Dic61—26no—
263Dic61#37913yes—
264Dic61#37911yescharacteristic
265C2×#585—611no—
266C2×#410—613no—
267#2311—613no—
268#1790—618no—
269C2×C61—23no—
270C2×C61—46no—
271C2×C61C2×#37911yescharacteristic
272D61—23no—
273D61—46no—
274D61#302711yescharacteristic
275D61#302811yescharacteristic
276#585—614no—
277#585—613no—
278#585—1226no—
279#585—1223no—
280#585—1223no—
281#585#692311yescharacteristic, complemented
282C2×2−1+4—1223no—
283C2×H(5,𝔽2)—614no—
284C2×H(5,𝔽2)—1223no—
285C22×#41—613no—
286C22×#41—1226no—
287C2×#117—613no—
288C22×C4⋊C4—611no—
289#410—1226no—
290#410—1223no—
291#410—1223no—
292C2×#99—6112no—
293#391—1223no—
294#390—1223no—
295#386—6112no—
296#380—1226no—
297C61#2061011yescharacteristic, complemented, Sylow
2982−1+4—23no—
2992−1+4—12212no—
3002−1+4—12212no—
3012−1+4—1226no—
3022−1+4—12212no—
3032−1+4—1223no—
304H(5,𝔽2)—23no—
305H(5,𝔽2)—6112no—
306H(5,𝔽2)—12212no—
307H(5,𝔽2)—12212no—
308H(5,𝔽2)—12212no—
309H(5,𝔽2)—12212no—
310H(5,𝔽2)—1226no—
311H(5,𝔽2)—12212no—
312H(5,𝔽2)—1223no—
313H(5,𝔽2)C2×#692314yes—
314C2×#41—26no—
315C2×#41—613no—
316C2×#41—613no—
317C2×#41—616no—
318C2×#41—616no—
319C2×#41—613no—
320C2×#41—12212no—
321C2×#41—12212no—
322C2×#41—1226no—
323C2×#41—1226no—
324C2×#41—1226no—
325C2×#41—1226no—
326C2×#41—12212no—
327C2×#41—12212no—
328C2×#41—12212no—
329C2×#41—1226no—
330C2×#41—1226no—
331C2×#41—1224no—
332C2×#41—12212no—
333C2×#41—12212no—
334C2×#41—12212no—
335C2×#41—1226no—
336C2×#41C2×#692313yes—
337C22×Q8—613no—
338C22×Q8—1226no—
339C22×D4—613no—
340C22×D4—616no—
341C22×D4—12212no—
342C22×D4—1226no—
343C23×C4—611no—
344C23×C4—616no—
345C23×C4—612no—
346C23×C4—1224no—
347#117—1226no—
348#117—1226no—
349C2×C4⋊C4—1223no—
350C2×C4⋊C4—1223no—
351C2×C4⋊C4—1223no—
352C2×C22⋊C4—616no—
353C2×C22⋊C4—12212no—
354C2×C42—1223no—
355#99—12212no—
356#99—12212no—
357#95—12212no—
358C24—611no—
359C24—616no—
360C24—1224no—
361#41—24no—
362#41—212no—
363#41—26no—
364#41—212no—
365#41—26no—
366#41—12212no—
367#41—12212no—
368#41—12212no—
369#41—12224no—
370#41—12224no—
371#41—12212no—
372#41—12212no—
373#41—1226no—
374#41—12212no—
375#41—1226no—
376#41—12224no—
377#41—1226no—
378#41—1226no—
379#41—12212no—
380#41—12212no—
381#41—12224no—
382#41—12212no—
383#41—1224no—
384#41—12212no—
385#41—12212no—
386#41—1226no—
387#41—1226no—
388#41—12212no—
389C2×Q8—26no—
390C2×Q8—616no—
391C2×Q8—613no—
392C2×Q8—12212no—
393C2×Q8—1226no—
394C2×Q8—1226no—
395C2×Q8—1226no—
396C2×Q8—1224no—
397C2×Q8—12212no—
398C2×Q8—12212no—
399C2×Q8C22⋊#692313yescomplemented
400C2×D4—26no—
401C2×D4—212no—
402C2×D4—616no—
403C2×D4—616no—
404C2×D4—616no—
405C2×D4—616no—
406C2×D4—613no—
407C2×D4—12212no—
408C2×D4—1226no—
409C2×D4—1226no—
410C2×D4—1226no—
411C2×D4—12212no—
412C2×D4—12212no—
413C2×D4—12224no—
414C2×D4—12212no—
415C2×D4—12212no—
416C2×D4—12224no—
417C2×D4—12212no—
418C2×D4—12212no—
419C2×D4—12212no—
420C2×D4—12212no—
421C2×D4C22⋊#692313yescomplemented
422C2×D4C22×#692316yes—
423C22×C4—24no—
424C22×C4—611no—
425C22×C4—613no—
426C22×C4—613no—
427C22×C4—613no—
428C22×C4—613no—
429C22×C4—616no—
430C22×C4—616no—
431C22×C4—616no—
432C22×C4—616no—
433C22×C4—1226no—
434C22×C4—1226no—
435C22×C4—1226no—
436C22×C4—1226no—
437C22×C4—12212no—
438C22×C4—12212no—
439C22×C4—12212no—
440C22×C4—1224no—
441C22×C4—1226no—
442C22×C4—1226no—
443C22×C4—4882no—
444C22×C4C22⋊#692316yescomplemented
445C22×C4C22×#692311yescharacteristic
446C4⋊C4—2441no—
447C4⋊C4—2443no—
448C4⋊C4—2443no—
449C4⋊C4—2443no—
450C22⋊C4—12212no—
451C22⋊C4—48812no—
452C42—2446no—
453C23—24no—
454C23—613no—
455C23—613no—
456C23—616no—
457C23—1226no—
458C23—1226no—
459C23—12212no—
460C23—4884no—
461C23—4884no—
462C23—2441no—
463C23—2441no—
464C23—2446no—
465C23—2446no—
466C23C2×C22⋊#692316yes—
467C23C23×#692311yescharacteristic
468Q8—24no—
469Q8—26no—
470Q8—12212no—
471Q8—12212no—
472Q8—1226no—
473D4—212no—
474D4—212no—
475D4—26no—
476D4—12212no—
477D4—12212no—
478D4—12212no—
479D4—12212no—
480D4—12224no—
481D4—12212no—
482D4—1226no—
483C2×C4—212no—
484C2×C4—26no—
485C2×C4—26no—
486C2×C4—613no—
487C2×C4—613no—
488C2×C4—616no—
489C2×C4—611no—
490C2×C4—613no—
491C2×C4—613no—
492C2×C4—613no—
493C2×C4—616no—
494C2×C4—1226no—
495C2×C4—1226no—
496C2×C4—1226no—
497C2×C4—1226no—
498C2×C4—12224no—
499C2×C4—1226no—
500C2×C4—1226no—
501C2×C4—12212no—
502C2×C4—1226no—
503C2×C4—1226no—
504C2×C4—2446no—
505C2×C4—2442no—
506C2×C4—2446no—
507C2×C4—24412no—
508C2×C4—2446no—
509C2×C4C2×C22⋊#692316yes—
510C2×C4C2×C22⋊#692313yes—
511C2×C4C2×C22⋊#692313yes—
512C22—26no—
513C22—41no—
514C22—46no—
515C22—84no—
516C22—611no—
517C22—613no—
518C22—1223no—
519C22—1223no—
520C22—1226no—
521C22—4886no—
522C22—48812no—
523C22—2443no—
524C22—2446no—
525C22—2446no—
526C22—2446no—
527C22C22×C22⋊#692313yes—
528C4—26no—
529C4—611no—
530C4—613no—
531C4—1226no—
532C4—4882no—
533C4—4886no—
534C4#1684070311yescharacteristic
535C4#1684070313yes—
536C2—23no—
537C2—46no—
538C2—611no—
539C2—611no—
540C2—1223no—
541C2—2446no—
542C2C2×#1684070311yescharacteristic
subgroupG/H
centerC2C2×#16840703
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#585C4
socleC2×C61C2×#379
prime radical#585×D61C2
cosocleC23×D61C24
hypercenter#585#6923
hypocenterC61#20610
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. D61⋊#410C2semidirect
2. #21462294C2semidirect
3. #21461663C2semidirect
5. C2×#14519856C22semidirect
6. C22×C4⋊#6923C22semidirect
10. D61⋊#99C22semidirect
13. #16840710C22semidirect
17. #16840598C4semidirect
19. #16840597C4semidirect
25. C61×#585C4semidirect
98. C2×Q8×C61C22⋊C4semidirect
101. C2×D4×C61C22⋊C4semidirect
104. C22×C4×C61C22⋊C4semidirect
109. C2×C22⋊D61C22⋊C4semidirect
113. C22×Dic61C22⋊C4semidirect
154. C2×D244C22⋊C4semidirect
163. C2×C4×D61C22⋊C4semidirect
164. C2×C4×D61C22⋊C4semidirect
172. C2×Dic122C22⋊C4semidirect
173. C2×Dic122C22⋊C4semidirect
281. #585#6923semidirect
297. C61#20610semidirect
399. C2×Q8C22⋊#6923semidirect
421. C2×D4C22⋊#6923semidirect
444. C22×C4C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4102
#214622942
#214616632
#585×D612
#2061061