Finite Group Databasebeta

← #29684026

Subgroups — #29684026

Order 15616.

1562 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5C22×#14519693C211yesmaximal, characteristic, complemented
6#21454237C211yesmaximal, characteristic, complemented
7#21446727C214yesmaximal, complemented
8#21446720C214yesmaximal, complemented
9C2×C61×#430C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519805—22no—
13C2×#14519805—22no—
14C2×#14519805—24no—
15C2×#14519805C418yescomplemented
16C22×#11758190C2212yes—
17C22×D488C2212yes—
18C22×#11758194C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839772C2212yescomplemented
21#16839768—22no—
22#16839768C2214yes—
23#16839768C2212yescomplemented
24#16839764—22no—
25#16839759—22no—
26C2×#14519699—22no—
27#16839744—24no—
28C2×#14519695C2212yescomplemented
29C2×#14519693—24no—
30C2×#14519693C2214yes—
31C2×#14519693C2212yescomplemented
32#16839720—22no—
33#16839717C2214yescomplemented
34M16×Dic61—22no—
35C22×M16×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C61×#430—24no—
39C2×#108×C61C2214yescomplemented
40C24×D61—21no—
41C2×#11758222—22no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244—21no—
54C22×D244—22no—
55C22×D244—22no—
56C22×D244—22no—
57C22×D244—21no—
58C22×D244C2×C414yes—
59C22×D244C2312yes—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63#14519805—216no—
64#14519805—216no—
65C2×#11758190—24no—
66C2×#11758190—24no—
67C2×#11758190C2×C418yescomplemented
68C2×D488—24no—
69C2×D488—24no—
70C2×D488C2×C418yescomplemented
71C2×#11758194—21no—
72C2×#11758194—22no—
73C2×#11758194D414yes—
74C2×#11758194C2312yes—
75C22×#92418—21no—
76#14519736—21no—
77#14519736—21no—
78#14519736D412yes—
79#14519736C2312yes—
80C2×C4⋊Dic61D412yes—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83#14519702—24no—
84#14519699—28no—
85#14519699—24no—
86#14519695—24no—
87#14519693—28no—
88#14519693—24no—
89#14519687—24no—
90C8×Dic61—24no—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C2×M16×C61—22no—
98C2×M16×C61C2×C414yes—
99C2×M16×C61D412yes—
100C2×M16×C61D412yes—
101C2×M16×C61D414yes—
102C2×M16×C61C2312yes—
103C2×M16×C61C2312yes—
104C22×C8×C61—21no—
105C22×C8×C61—21no—
106C22×C8×C61C2312yes—
107#108×C61—28no—
108C24×C61#4111yescharacteristic
109C23×D61—24no—
110C23×D61—21no—
111C23×D61—22no—
112C23×D61—41no—
113C23×D61—41no—
114C23×D61—42no—
115C23×D61—84no—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—42no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132#11758222—24no—
133#11758222—24no—
134#11758222—24no—
135#11758222—24no—
136#11758222—42no—
137#11758222—42no—
138#11758222—42no—
139#11758222—42no—
140C2×D244—24no—
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—22no—
151C2×D244—21no—
152C2×D244—24no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244—42no—
156C2×D244—42no—
157C2×D244—41no—
158C2×D244—41no—
159C2×D244—42no—
160C2×D244—42no—
161C2×D244—42no—
162C2×D244C22×C414yes—
163C2×D244C22×C412yes—
164C2×C4×D61—24no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61—42no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174M16×C61—22no—
175M16×C61—22no—
176M16×C61—22no—
177M16×C61—22no—
178M16×C61—42no—
179M16×C61—42no—
180M16×C61C22⋊C418yes—
181M16×C61C22⋊C418yes—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—42no—
185C2×C8×C61—42no—
186C2×C8×C61C22×C414yes—
187C2×C8×C61C2×D414yes—
188C2×C8×C61C2×D414yes—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C2×D414yes—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—24no—
196#11758194—22no—
197#11758194C2×D414yes—
198C2×#92418—24no—
199C2×#92418—22no—
200D488—216no—
201#11758190—216no—
202C23×C61—42no—
203C23×C61C4×D412yes—
204C23×C61#12111yescharacteristic
205C23×C61#12111yescharacteristic
206C23×C61#12412yes—
207C23×C61C2×#4111yescharacteristic
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—41no—
211C22×D61—42no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—84no—
217C22×D61—84no—
218C22×D61—88no—
219C22×D61—82no—
220C22×D61—84no—
221C22×D61—88no—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C4×D412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12412yes—
232C2×C4×C61#12712yes—
233C2×C4×C61#12712yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—44no—
241C22⋊D61—44no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—82no—
247D244—24no—
248D244—22no—
249D244—24no—
250D244—22no—
251D244—42no—
252D244—41no—
253D244—42no—
254D244—41no—
255D244—44no—
256D244—44no—
257C4×D61—44no—
258C4×D61—44no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—42no—
264C8×C61—42no—
265C8×C61C2×C22⋊C418yes—
266#92418—24no—
267#21249—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61#39012yes—
274C22×C61#39012yes—
275C22×C61C2×C4×D411yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12411yescharacteristic
278C2×D61—42no—
279C2×D61—44no—
280C2×D61—82no—
281C2×D61—88no—
282C2×D61—84no—
283C2×D61—88no—
284C2×D61—82no—
285C2×D61—84no—
286C2×D61—82no—
287C2×D61—84no—
288C4×C61#39012yes—
289C4×C61#39012yes—
290C4×C61C22×C22⋊C411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12711yescharacteristic
294Dic61—42no—
295Dic61—44no—
296C22×AGL(1,ℤ/8)—611no—
297C2×#430D6111yescharacteristic, complemented
298#2896—611no—
299C2×#418—611no—
300C2×#413—611no—
301C22×#100—611no—
302#2873—611no—
303#1978—614no—
304#1977—614no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—22no—
309C2×C61C2×#39011yescharacteristic
310C2×C61#306812yes—
311D61—82no—
312D61—84no—
313C22×#41—611no—
314C23×D4—611no—
315C2×AGL(1,ℤ/8)—618no—
316C2×AGL(1,ℤ/8)—1222no—
317C2×AGL(1,ℤ/8)—1222no—
318C2×AGL(1,ℤ/8)—1224no—
319C22×SD16—612no—
320C22×D8—612no—
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323C22×M16C2×D6111yescharacteristic
324C22×M16C2×D6111yescharacteristic
325C2×#117—611no—
326#430—24no—
327C2×#108C2×D6114yescomplemented
328#419—612no—
329#418—614no—
330#418—612no—
331#418—1222no—
332#417—1222no—
333#416—1222no—
334C2×#102—1222no—
335#413—1224no—
336C2×#101—612no—
337C2×#100—614no—
338C2×#100—612no—
339C2×#100—1224no—
340#405—614no—
341#405—1222no—
342C4×M16—1222no—
343C61#2124911yescharacteristic, complemented, Sylow
344C25—1221no—
345C2×#41—614no—
346C2×#41—1222no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41—1221no—
351C2×#41—1222no—
352C2×#41—1222no—
353C22×Q8—611no—
354C22×D4—614no—
355C22×D4—611no—
356C22×D4—612no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4—1221no—
361C22×D4—1221no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—1222no—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368AGL(1,ℤ/8)—12216no—
369AGL(1,ℤ/8)—12216no—
370C2×SD16—618no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×D8—618no—
374C2×D8—1224no—
375C2×D8—1224no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—614no—
383C2×M16—612no—
384C2×M16—1221no—
385C2×M16—1222no—
386C2×M16C4×D6114yes—
387C2×M16D24412yes—
388C2×M16D24412yes—
389C2×M16C22⋊D6114yes—
390C2×M16C22×D6112yes—
391C2×M16C22×D6112yes—
392C22×C8—21no—
393C22×C8—21no—
394C22×C8—1221no—
395C22×C8C22×D6112yes—
396#117—612no—
397#117—612no—
398#117—1221no—
399#117—1221no—
400C2×C4⋊C4—612no—
401C2×C22⋊C4—1221no—
402C2×C42—1221no—
403#108—28no—
404#103—1224no—
405#102—1228no—
406#102—1224no—
407#101—1224no—
408#100—1228no—
409#100—1224no—
410#97—1224no—
411C4×C8—1224no—
412C24—1221no—
413C24—1222no—
414C24—1224no—
415C24—4884no—
416C24—2441no—
417C24—2441no—
418C24—2442no—
419C24#1175822611yescharacteristic
420#41—1224no—
421#41—1224no—
422#41—1224no—
423#41—1224no—
424#41—2442no—
425#41—2442no—
426#41—2442no—
427#41—2442no—
428C2×Q8—612no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×D4—612no—
435C2×D4—614no—
436C2×D4—1224no—
437C2×D4—1222no—
438C2×D4—1221no—
439C2×D4—1222no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1222no—
443C2×D4—1221no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1221no—
447C2×D4—1224no—
448C2×D4—1224no—
449C2×D4—1224no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2442no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2442no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1224no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4—2442no—
468C22×C4C2×C4×D6112yes—
469C22×C4C2×D24411yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6112yes—
474C22×C4#1175822612yes—
475C22×C4#1175822611yescharacteristic
476C22×C4C2×C22⋊D6112yes—
477C22×C4C23×D6111yescharacteristic
478SD16—12216no—
479D8—12216no—
480M16—22no—
481M16—22no—
482M16—22no—
483M16—22no—
484M16—42no—
485M16—42no—
486M16—614no—
487M16—1222no—
488M16—1224no—
489M16#1175819818yes—
490M16#1175819818yes—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—1222no—
496C2×C8—1224no—
497C2×C8C2×C4×D6114yes—
498C2×C8C2×D24414yes—
499C2×C8C2×C22⋊D6114yes—
500C4⋊C4—614no—
501C4⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C42—1222no—
505C42—1222no—
506C23—42no—
507C23—1222no—
508C23—1224no—
509C23—4882no—
510C23—4884no—
511C23—4884no—
512C23—4884no—
513C23—4888no—
514C23—4888no—
515C23—2441no—
516C23—2442no—
517C23—2441no—
518C23—2442no—
519C23—2444no—
520C23—2444no—
521C23#1451973912yes—
522C23#1451975211yescharacteristic
523C23#1451975211yescharacteristic
524C23#1451976612yes—
525C23C2×#1175822611yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2441no—
529Q8—2441no—
530D4—1222no—
531D4—1224no—
532D4—1224no—
533D4—1222no—
534D4—2444no—
535D4—2444no—
536D4—2442no—
537D4—2441no—
538D4—2442no—
539D4—2441no—
540D4—2444no—
541D4—2444no—
542C2×C4—1222no—
543C2×C4—1224no—
544C2×C4—4882no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4C2×#1175819812yes—
550C2×C4C2×#1175819812yes—
551C2×C4#1451973912yes—
552C2×C4#1451973912yes—
553C2×C4#1451973912yes—
554C2×C4#1451975212yes—
555C2×C4#1451975212yes—
556C2×C4#1451975211yescharacteristic
557C2×C4#1451975211yescharacteristic
558C2×C4#1451976612yes—
559C2×C4#1451977312yes—
560C2×C4#1451977312yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×D24411yescharacteristic
566C2×C4C2×#1175822611yescharacteristic
567C8—42no—
568C8—42no—
569C8—1224no—
570C8C2×#1175819818yes—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—4882no—
577C22—4884no—
578C22—4884no—
579C22—4888no—
580C22—4888no—
581C22—4882no—
582C22—4884no—
583C22—4882no—
584C22—2442no—
585C22—2444no—
586C22#1683966312yes—
587C22#1683966312yes—
588C22C2×#1451973911yescharacteristic
589C22C2×#1451975211yescharacteristic
590C22C2×#1451976611yescharacteristic
591C4—2442no—
592C4—2444no—
593C4#1683966312yes—
594C4#1683966312yes—
595C4C22×#1175819811yescharacteristic
596C4C2×#1451973911yescharacteristic
597C4C2×#1451975211yescharacteristic
598C4C2×#1451977311yescharacteristic
599C2—21no—
600C2—21no—
601C2—22no—
602C2—22no—
603C2—4884no—
604C2—4882no—
605C2C2×#1683966311yescharacteristic
606C2#2145554812yes—
subgroupG/H
centerC22C2×#14519752
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC22×C61C2×#121
prime radicalC22×#14519805C2
cosocleC22×C4×C61C24
hypercenterC2×#430D61
hypocenterC61#21249
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
2. #21454339C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839744C2semidirect
5. C22×#14519693C2semidirect
6. #21454237C2semidirect
7. #21446727C2semidirect
8. #21446720C2semidirect
9. C2×C61×#430C2semidirect
15. C2×#14519805C4semidirect
20. #16839772C22semidirect
23. #16839768C22semidirect
28. C2×#14519695C22semidirect
31. C2×#14519693C22semidirect
33. #16839717C22semidirect
39. C2×#108×C61C22semidirect
67. C2×#11758190C2×C4semidirect
70. C2×D488C2×C4semidirect
297. C2×#430D61semidirect
327. C2×#108C2×D61semidirect
343. C61#21249semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
#214543392
C2×#168397682
C2×#168397442
C22×#145196932
#214542372
#214467272
#214467202
C2×C61×#4302
#2124961