Finite Group Databasebeta

← #29717313

Subgroups — #29717313

Order 15616.

1348 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840027C212yesmaximal, complemented
2#21455071C211yesmaximal, characteristic
3C2×#16839901C211yesmaximal, characteristic, complemented
4#21454268C211yesmaximal, characteristic, complemented
5#21447115C212yesmaximal, complemented
6#21446930C212yesmaximal, complemented
7#21446336C212yesmaximal, complemented
8#21446282C211yesmaximal, characteristic, complemented
9#21446024C211yesmaximal, characteristic, complemented
10#21445743C211yesmaximal, characteristic, complemented
11C22×#137×C61C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×#14519802C2211yescharacteristic
14C22×#11758201C2212yescomplemented
15C2×#14519784C2212yes—
16C2×#14519751C2211yescharacteristic
17#16840027—28no—
18#16839907—21no—
19#16839907—21no—
20#16839901—21no—
21#16839901—21no—
22#16839901—22no—
23#16839901—21no—
24#16839896—21no—
25C2×#14519712C2212yescomplemented
26#16839763C2211yescharacteristic
27#16839763C2211yescharacteristic
28C2×#14519701C2212yescomplemented
29C2×#14519696C2212yescomplemented
30C2×#14519696C2212yescomplemented
31C2×#14519693C2212yescomplemented
32#16839729—22no—
33#16839729C2212yes—
34#16839726—21no—
35#16839726C2211yescharacteristic
36#16839726C2211yescharacteristic
37C2×C22⋊#92418—21no—
38C2×C22⋊#92418C2211yescharacteristic
39#16839530C2212yescomplemented
40#16839526C2212yescomplemented
41C22×#41×C61C2211yescharacteristic
42C23×Q8×C61C2211yescharacteristic
43C2×#137×C61—21no—
44C2×#137×C61—21no—
45C2×#137×C61—28no—
46C2×#137×C61C2212yes—
47C2×#137×C61C2212yes—
48C22×Q16×C61—21no—
49C22×SD16×C61—21no—
50C22×M16×C61C2211yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53#14519802—24no—
54#14519802—42no—
55C2×#11758201—24no—
56C2×#11758201—21no—
57C2×#11758201—21no—
58C2×#11758201—22no—
59C2×#11758201—22no—
60C2×#11758201—42no—
61C2×#11758199—21no—
62C2×#11758199—21no—
63#14519784—24no—
64#14519784—42no—
65C2×#11758194—22no—
66C2×#11758194C2312yes—
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C22×#92418—22no—
70C22×#92418C2312yes—
71#14519751—21no—
72#14519751—21no—
73#14519751—21no—
74#14519751—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758198—21no—
77#14519712—24no—
78#14519701—21no—
79#14519701—21no—
80#14519701—22no—
81#14519701—22no—
82#14519698—21no—
83#14519698—21no—
84#14519696—24no—
85#14519696—42no—
86#14519693—24no—
87C22⋊#92418—21no—
88C22⋊#92418—21no—
89C22⋊#92418—21no—
90C22⋊#92418—21no—
91C22⋊#92418—22no—
92C22⋊#92418—22no—
93C22⋊#92418D411yescharacteristic
94C22⋊#92418D411yescharacteristic
95C22⋊#92418D412yes—
96C2×#41×C61—21no—
97C2×#41×C61—21no—
98C2×#41×C61—24no—
99C2×#41×C61—42no—
100C2×#41×C61D411yescharacteristic
101C2×#41×C61D411yescharacteristic
102C2×#41×C61C2311yescharacteristic
103C2×#41×C61C2311yescharacteristic
104C22×Q8×C61—21no—
105C22×Q8×C61—24no—
106C22×Q8×C61—21no—
107C22×Q8×C61—21no—
108C22×Q8×C61—21no—
109C22×Q8×C61—21no—
110C22×Q8×C61—42no—
111C22×Q8×C61D411yescharacteristic
112C22×Q8×C61C2312yes—
113C22×D4×C61—21no—
114C22×D4×C61D411yescharacteristic
115C23×C4×C61—21no—
116C23×C4×C61—21no—
117C23×C4×C61C2311yescharacteristic
118#137×C61—28no—
119#137×C61—28no—
120#137×C61—24no—
121#137×C61—24no—
122#137×C61—24no—
123#137×C61D418yescomplemented
124C2×Q16×C61—28no—
125C2×Q16×C61—22no—
126C2×Q16×C61—22no—
127C2×SD16×C61—28no—
128C2×SD16×C61—22no—
129C2×SD16×C61—22no—
130C2×M16×C61—21no—
131C2×M16×C61—24no—
132C2×M16×C61C2312yes—
133C22×C8×C61—21no—
134C24×C61—21no—
135C24×C61#4111yescharacteristic
136C23×D61—21no—
137#41×C61—22no—
138#41×C61—22no—
139#41×C61—22no—
140#41×C61—22no—
141#41×C61—22no—
142#41×C61—44no—
143#41×C61—41no—
144#41×C61—41no—
145#41×C61—44no—
146#41×C61C2×D414yes—
147C2×Q8×C61—21no—
148C2×Q8×C61—22no—
149C2×Q8×C61—24no—
150C2×Q8×C61—24no—
151C2×Q8×C61—22no—
152C2×Q8×C61—22no—
153C2×Q8×C61—22no—
154C2×Q8×C61—22no—
155C2×Q8×C61—22no—
156C2×Q8×C61—44no—
157C2×Q8×C61—41no—
158C2×Q8×C61—44no—
159C2×Q8×C61—44no—
160C2×Q8×C61—42no—
161C2×Q8×C61—42no—
162C2×Q8×C61—41no—
163C2×Q8×C61—41no—
164C2×Q8×C61C2×D411yescharacteristic
165C2×Q8×C61C2×D414yes—
166C2×Q8×C61C2×D411yescharacteristic
167C2×D4×C61—21no—
168C2×D4×C61—22no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61—41no—
174C2×D4×C61—44no—
175C2×D4×C61C2×D411yescharacteristic
176C2×D4×C61C2×D411yescharacteristic
177C22×C4×C61—22no—
178C22×C4×C61—21no—
179C22×C4×C61—22no—
180C22×C4×C61—21no—
181C22×C4×C61—22no—
182C22×C4×C61—21no—
183C22×C4×C61—21no—
184C22×C4×C61—21no—
185C22×C4×C61—44no—
186C22×C4×C61—44no—
187C22×C4×C61—41no—
188C22×C4×C61—41no—
189C22×C4×C61C2×D411yescharacteristic
190C22×C4×C61C2×D411yescharacteristic
191C22×C4×C61C2×D411yescharacteristic
192C22×C4×C61C2×D411yescharacteristic
193C22×C4×C61#4111yescharacteristic
194C22×C4×C61C2411yescharacteristic
195C2×C22⋊D61—41no—
196C2×C22⋊D61—41no—
197C2×C22⋊D61—82no—
198C22×Dic61—21no—
199C2×D244—22no—
200C2×D244—21no—
201C2×D244—21no—
202C2×D244—22no—
203Q16×C61—28no—
204Q16×C61—28no—
205SD16×C61—28no—
206SD16×C61—28no—
207M16×C61—24no—
208M16×C61—22no—
209M16×C61C2×D414yes—
210C2×C8×C61—24no—
211C2×C8×C61—22no—
212#11758201—44no—
213#11758201—44no—
214#11758201—42no—
215#11758201—42no—
216#11758199—42no—
217#11758199—42no—
218#11758198—41no—
219#11758198—41no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222#11758194—22no—
223#11758194—22no—
224#11758194—22no—
225#11758194—22no—
226#11758194—44no—
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—44no—
233C2×#92418C2×D412yes—
234C2×#92418C2×D412yes—
235C23×C61—21no—
236C23×C61—22no—
237C23×C61—22no—
238C23×C61—21no—
239C23×C61—41no—
240C23×C61—42no—
241C23×C61—42no—
242C23×C61—81no—
243C23×C61#12111yescharacteristic
244C23×C61#12111yescharacteristic
245C23×C61C2×#4111yescharacteristic
246C22×D61—41no—
247C22×D61—42no—
248C22×D61—81no—
249C22×D61—81no—
250Q8×C61—21no—
251Q8×C61—24no—
252Q8×C61—21no—
253Q8×C61—21no—
254Q8×C61—24no—
255Q8×C61—42no—
256Q8×C61—44no—
257Q8×C61—44no—
258Q8×C61—42no—
259Q8×C61—42no—
260Q8×C61#12012yes—
261D4×C61—24no—
262D4×C61—21no—
263D4×C61—21no—
264D4×C61—21no—
265D4×C61—42no—
266D4×C61—42no—
267D4×C61—42no—
268D4×C61—42no—
269D4×C61#12012yes—
270C2×C4×C61—21no—
271C2×C4×C61—22no—
272C2×C4×C61—21no—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—21no—
276C2×C4×C61—24no—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—21no—
280C2×C4×C61—24no—
281C2×C4×C61—42no—
282C2×C4×C61—42no—
283C2×C4×C61—42no—
284C2×C4×C61—44no—
285C2×C4×C61—44no—
286C2×C4×C61—42no—
287C2×C4×C61—42no—
288C2×C4×C61—42no—
289C2×C4×C61#12111yescharacteristic
290C2×C4×C61#12111yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D412yes—
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61H(5,𝔽2)12yes—
295C22⋊D61—82no—
296C22⋊D61—82no—
297C2×Dic61—41no—
298C2×Dic61—81no—
299D244—41no—
300D244—41no—
301D244—42no—
302C8×C61—24no—
303#92418—22no—
304#92418—22no—
305#92418—41no—
306#92418—41no—
307#92418—42no—
308#24401—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—22no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—44no—
318C22×C61—41no—
319C22×C61—41no—
320C22×C61—41no—
321C22×C61—41no—
322C22×C61—41no—
323C22×C61—84no—
324C22×C61#47112yescomplemented
325C22×C61C2×#12111yescharacteristic
326C22×C61#54612yes—
327C2×D61—81no—
328C2×D61—81no—
329C2×D61—81no—
330C2×D61—162no—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—22no—
334C4×C61—41no—
335C4×C61—44no—
336C4×C61C2×#12011yescharacteristic
337C4×C61#53411yescharacteristic
338C4×C61D4212yes—
339Dic61—81no—
340C22×#137D6111yescharacteristic, complemented
341C2×#471—612no—
342#3006—611no—
343C2×#448—611no—
344#2880—611no—
345#2052—612no—
346#2019—612no—
347#1908—612no—
348#1900—611no—
349#1855—611no—
350#1784—611no—
351C2×C61—22no—
352C2×C61—22no—
353C2×C61—41no—
354C2×C61—42no—
355C2×C61—81no—
356C2×C61#241011yescharacteristic
357C2×C61C2×#47111yescharacteristic
358C2×C61#306911yescharacteristic
359D61—161no—
360C22×#41C2×D6111yescharacteristic
361C23×Q8C2×D6111yescharacteristic
362C2×#137—21no—
363C2×#137—21no—
364C2×#137—28no—
365C2×#137C2×D6112yes—
366C2×#137C2×D6112yes—
367C2×AGL(1,ℤ/8)—611no—
368C2×AGL(1,ℤ/8)—611no—
369C22×Q16—21no—
370C22×SD16—21no—
371C22×SD16—612no—
372C2×#131—612no—
373C22×M16C2×D6111yescharacteristic
374C2×#121—611no—
375#471—1228no—
376#449—1221no—
377#449—1221no—
378#448—1221no—
379#448—1221no—
380#448—1222no—
381#448—1221no—
382#447—1221no—
383C2×#108—612no—
384#417—611no—
385#417—611no—
386C2×#103—612no—
387C2×#101—612no—
388C2×#101—612no—
389C2×#100—612no—
390#408—612no—
391#408—1222no—
392#407—611no—
393#407—611no—
394#407—1221no—
395C2×C22⋊C8—611no—
396C2×C22⋊C8—1221no—
397#343—612no—
398#341—612no—
399C61#2440111yescharacteristic, complemented, Sylow
400C2×#41—21no—
401C2×#41—21no—
402C2×#41—24no—
403C2×#41—42no—
404C2×#41C22⋊D6111yescharacteristic
405C2×#41C22⋊D6111yescharacteristic
406C2×#41C22×D6111yescharacteristic
407C2×#41C22×D6111yescharacteristic
408C22×Q8—21no—
409C22×Q8—24no—
410C22×Q8—21no—
411C22×Q8—21no—
412C22×Q8—21no—
413C22×Q8—21no—
414C22×Q8—42no—
415C22×Q8C22⋊D6111yescharacteristic
416C22×Q8C22×D6112yes—
417C22×D4—21no—
418C22×D4—611no—
419C22×D4—1221no—
420C22×D4C22⋊D6111yescharacteristic
421C23×C4—21no—
422C23×C4—21no—
423C23×C4C22×D6111yescharacteristic
424#137—28no—
425#137—28no—
426#137—24no—
427#137—24no—
428#137—24no—
429#137C22⋊D6118yescomplemented
430AGL(1,ℤ/8)—1224no—
431AGL(1,ℤ/8)—2442no—
432C2×Q16—28no—
433C2×Q16—22no—
434C2×Q16—22no—
435C2×SD16—28no—
436C2×SD16—22no—
437C2×SD16—22no—
438C2×SD16—1222no—
439C2×SD16—1222no—
440C2×SD16—1224no—
441C2×SD16—1221no—
442C2×SD16—1221no—
443C2×SD16—2442no—
444C2×D8—1221no—
445C2×D8—1221no—
446#131—1224no—
447#131—2442no—
448C2×M16—21no—
449C2×M16—24no—
450C2×M16—612no—
451C2×M16—611no—
452C2×M16—611no—
453C2×M16—1222no—
454C2×M16C22×D6112yes—
455C22×C8—21no—
456C22×C8—612no—
457C22×C8—1222no—
458#121—1221no—
459#121—1221no—
460#121—1221no—
461#121—1221no—
462C2×C4⋊C4—611no—
463C2×C22⋊C4—1221no—
464#108—1224no—
465#103—1222no—
466#103—1222no—
467#103—1221no—
468#103—1221no—
469#102—1221no—
470#102—1221no—
471#101—1224no—
472#101—2442no—
473#100—1224no—
474C22⋊C8—611no—
475C22⋊C8—611no—
476C22⋊C8—612no—
477C22⋊C8—1222no—
478C22⋊C8—1222no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481C22⋊C8—1221no—
482C22⋊C8—1221no—
483C24—21no—
484C24—1221no—
485C24#1175822411yescharacteristic
486#41—22no—
487#41—22no—
488#41—22no—
489#41—22no—
490#41—22no—
491#41—44no—
492#41—41no—
493#41—41no—
494#41—44no—
495#41C2×C22⋊D6114yes—
496C2×Q8—22no—
497C2×Q8—21no—
498C2×Q8—24no—
499C2×Q8—24no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—22no—
504C2×Q8—22no—
505C2×Q8—44no—
506C2×Q8—41no—
507C2×Q8—44no—
508C2×Q8—44no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—41no—
512C2×Q8—41no—
513C2×Q8C2×C22⋊D6114yes—
514C2×Q8C2×C22⋊D6111yescharacteristic
515C2×Q8C2×C22⋊D6111yescharacteristic
516C2×D4—21no—
517C2×D4—22no—
518C2×D4—21no—
519C2×D4—21no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—41no—
523C2×D4—44no—
524C2×D4—1221no—
525C2×D4—1222no—
526C2×D4—1221no—
527C2×D4—1222no—
528C2×D4—4882no—
529C2×D4—2441no—
530C2×D4—2441no—
531C2×D4C2×C22⋊D6111yescharacteristic
532C2×D4C2×C22⋊D6111yescharacteristic
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—44no—
542C22×C4—44no—
543C22×C4—41no—
544C22×C4—41no—
545C22×C4—1221no—
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822411yescharacteristic
549C22×C4C2×C22⋊D6111yescharacteristic
550C22×C4C2×C22⋊D6111yescharacteristic
551C22×C4C23×D6111yescharacteristic
552Q16—28no—
553Q16—28no—
554SD16—28no—
555SD16—28no—
556SD16—2444no—
557SD16—2444no—
558SD16—2442no—
559SD16—2442no—
560D8—2442no—
561D8—2442no—
562M16—24no—
563M16—22no—
564M16—1222no—
565M16—1222no—
566M16—1222no—
567M16—1222no—
568M16—2444no—
569M16C2×C22⋊D6114yes—
570C2×C8—24no—
571C2×C8—22no—
572C2×C8—612no—
573C2×C8—612no—
574C2×C8—1222no—
575C2×C8—1222no—
576C2×C8—1222no—
577C2×C8—1221no—
578C2×C8—1221no—
579C2×C8—2444no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C22⋊C4—2441no—
583C22⋊C4—2441no—
584C23—21no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—81no—
592C23—4881no—
593C23—4881no—
594C23—2442no—
595C23—2441no—
596C23#1451974911yescharacteristic
597C23#1451974911yescharacteristic
598C23C2×#1175822411yescharacteristic
599Q8—21no—
600Q8—24no—
601Q8—21no—
602Q8—21no—
603Q8—24no—
604Q8—42no—
605Q8—44no—
606Q8—44no—
607Q8—42no—
608Q8—42no—
609Q8#1451974612yes—
610D4—24no—
611D4—21no—
612D4—21no—
613D4—21no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—4882no—
619D4—4882no—
620D4—2441no—
621D4—2441no—
622D4—2442no—
623D4#1451974612yes—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—24no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—4881no—
644C2×C4—2441no—
645C2×C4#1451974911yescharacteristic
646C2×C4#1451974911yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C22×C22⋊D6111yescharacteristic
649C2×C4C22×C22⋊D6112yes—
650C2×C4#1451982212yes—
651C8—24no—
652C8—1222no—
653C8—1222no—
654C8—2441no—
655C8—2441no—
656C8—2442no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—44no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—84no—
672C22—9762no—
673C22—4881no—
674C22—4881no—
675C22—4881no—
676C22#1684002712yescomplemented
677C22C2×#1451974911yescharacteristic
678C22#1684040612yes—
679C4—21no—
680C4—21no—
681C4—22no—
682C4—41no—
683C4—44no—
684C4—4881no—
685C4C2×#1451974611yescharacteristic
686C4#1684032111yescharacteristic
687C4D4×C22⋊D6112yes—
688C2—22no—
689C2—22no—
690C2—41no—
691C2—42no—
692C2—81no—
693C2—9761no—
694C2#2144919311yescharacteristic
695C2C2×#1684002711yescharacteristic
696C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC22×C61C2×#121
prime radical#21455071C2
hypercenterC22×#137D61
hypocenterC61#24401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16840027C2semidirect
3. C2×#16839901C2semidirect
4. #21454268C2semidirect
5. #21447115C2semidirect
6. #21446930C2semidirect
7. #21446336C2semidirect
8. #21446282C2semidirect
9. #21446024C2semidirect
10. #21445743C2semidirect
11. C22×#137×C61C2semidirect
14. C22×#11758201C22semidirect
25. C2×#14519712C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519696C22semidirect
30. C2×#14519696C22semidirect
31. C2×#14519693C22semidirect
39. #16839530C22semidirect
40. #16839526C22semidirect
123. #137×C61D4semidirect
324. C22×C61#471semidirect
340. C22×#137D61semidirect
399. C61#24401semidirect
429. #137C22⋊D61semidirect
676. C22#16840027semidirect

Maximal subgroups

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

subgroupindex
C2×#168400272
#214550712
C2×#168399012
#214542682
#214471152
#214469302
#214463362
#214462822
#214460242
#214457432
C22×#137×C612
#2440161