Finite Group Databasebeta

← #29717314

Subgroups — #29717314

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
1C22×#14519807C211yesmaximal, characteristic, complemented
2C2×#16840023C212yesmaximal, complemented
3C2×#16839900C211yesmaximal, characteristic, complemented
4#21454268C211yesmaximal, characteristic, complemented
5#21447116C212yesmaximal, complemented
6#21446933C212yesmaximal, complemented
7#21446332C212yesmaximal, complemented
8#21446283C211yesmaximal, characteristic, complemented
9#21446023C211yesmaximal, characteristic, complemented
10#21445742C211yesmaximal, characteristic, complemented
11C61×#3006C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519807—28no—
14C2×#14519807—21no—
15C2×#14519807—21no—
16C2×#14519807C2212yes—
17C2×#14519807C2212yes—
18C22×#11758202—21no—
19C22×#11758200—21no—
20C22×#11758200C2212yescomplemented
21C2×#14519784C2212yes—
22C22×#11758194C2211yescharacteristic
23#16840023—28no—
24#16839900—21no—
25#16839900—21no—
26#16839900—22no—
27C2×#14519712C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519695C2212yescomplemented
30C2×#14519695C2212yescomplemented
31C2×#14519694C2212yescomplemented
32#16839729—22no—
33#16839729C2212yes—
34#16839726—21no—
35#16839726C2211yescharacteristic
36C2×C22⋊#92418—21no—
37C2×C22⋊#92418C2211yescharacteristic
38#16839531C2212yescomplemented
39#16839526C2212yescomplemented
40C22×#41×C61C2211yescharacteristic
41C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
42C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C61×#449—21no—
45C61×#449—21no—
46C61×#448—21no—
47C61×#447—21no—
48C61×#417C2211yescharacteristic
49C61×#417C2211yescharacteristic
50C61×#407C2211yescharacteristic
51C22×Dic122—21no—
52C22×Dic122—21no—
53C22×Dic122—24no—
54C22×Dic122—21no—
55C22×Dic122—21no—
56C22×Dic122—21no—
57C22×Dic122—42no—
58C22×Dic122C2312yes—
59C23×Dic61—21no—
60#14519807—28no—
61#14519807—28no—
62#14519807—24no—
63#14519807—24no—
64#14519807—24no—
65#14519807D418yescomplemented
66C2×#11758202—28no—
67C2×#11758202—22no—
68C2×#11758202—22no—
69C2×#11758200—28no—
70C2×#11758200—22no—
71C2×#11758200—22no—
72C2×#11758200—24no—
73C2×#11758200—22no—
74C2×#11758200—22no—
75C2×#11758200—42no—
76#14519784—24no—
77#14519784—42no—
78C2×#11758194—21no—
79C2×#11758194—24no—
80C2×#11758194—22no—
81C2×#11758194C2312yes—
82C2×#11758194C2312yes—
83C22×#92418—21no—
84C22×#92418—22no—
85C22×#92418C2312yes—
86#14519712—24no—
87#14519700—22no—
88#14519700—22no—
89#14519695—24no—
90#14519695—42no—
91#14519694—24no—
92C22⋊#92418—21no—
93C22⋊#92418—21no—
94C22⋊#92418—22no—
95C22⋊#92418—22no—
96C22⋊#92418D411yescharacteristic
97C22⋊#92418D411yescharacteristic
98C22⋊#92418D412yes—
99C2×#41×C61—21no—
100C2×#41×C61—21no—
101C2×#41×C61—24no—
102C2×#41×C61—42no—
103C2×#41×C61D411yescharacteristic
104C2×#41×C61D411yescharacteristic
105C2×#41×C61C2311yescharacteristic
106C2×#41×C61C2311yescharacteristic
107C22×Q8×C61D411yescharacteristic
108C22×D4×C61—21no—
109C22×D4×C61—21no—
110C22×D4×C61D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114AGL(1,ℤ/8)×C61—24no—
115AGL(1,ℤ/8)×C61—42no—
116C2×SD16×C61—21no—
117C2×SD16×C61—21no—
118C2×D8×C61—21no—
119C2×D8×C61—21no—
120C2×M16×C61C2311yescharacteristic
121C2×M16×C61C2311yescharacteristic
122#121×C61—21no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128#103×C61—21no—
129#103×C61—21no—
130#102×C61—21no—
131#102×C61—21no—
132C22⋊C8×C61—21no—
133C22⋊C8×C61—21no—
134C24×C61—21no—
135C24×C61—21no—
136C24×C61#4111yescharacteristic
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—41no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—41no—
163C2×D4×C61—44no—
164C2×D4×C61—41no—
165C2×D4×C61—41no—
166C2×D4×C61—82no—
167C2×D4×C61C2×D411yescharacteristic
168C2×D4×C61C2×D411yescharacteristic
169C22×C4×C61—22no—
170C22×C4×C61—21no—
171C22×C4×C61—22no—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—44no—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61#4111yescharacteristic
183C22×C4×C61C2411yescharacteristic
184C22×Dic61—22no—
185C22×Dic61—44no—
186C22×Dic61—41no—
187C22×Dic61—41no—
188C2×Dic122—24no—
189C2×Dic122—24no—
190C2×Dic122—22no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122—22no—
194C2×Dic122—22no—
195C2×Dic122—44no—
196C2×Dic122—44no—
197C2×Dic122—44no—
198C2×Dic122—42no—
199C2×Dic122—42no—
200C2×Dic122—41no—
201C2×Dic122—41no—
202C2×Dic122C2×D414yes—
203SD16×C61—42no—
204SD16×C61—42no—
205D8×C61—42no—
206D8×C61—42no—
207M16×C61—22no—
208M16×C61—22no—
209C2×C8×C61—21no—
210C2×C8×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C22⋊C4×C61—41no—
214C22⋊C4×C61—41no—
215#11758202—28no—
216#11758202—28no—
217#11758200—28no—
218#11758200—28no—
219#11758200—44no—
220#11758200—44no—
221#11758194—24no—
222#11758194—22no—
223#11758194—22no—
224#11758194—22no—
225#11758194—44no—
226#11758194C2×D414yes—
227C2×#92418—24no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—22no—
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—41no—
243C23×C61—42no—
244C23×C61—81no—
245C23×C61—81no—
246C23×C61—81no—
247C23×C61#12111yescharacteristic
248C23×C61#12111yescharacteristic
249C23×C61C2×#4111yescharacteristic
250Q8×C61—21no—
251Q8×C61—21no—
252Q8×C61—21no—
253Q8×C61—42no—
254Q8×C61#12012yes—
255D4×C61—24no—
256D4×C61—21no—
257D4×C61—21no—
258D4×C61—21no—
259D4×C61—42no—
260D4×C61—42no—
261D4×C61—42no—
262D4×C61—42no—
263D4×C61—41no—
264D4×C61—41no—
265D4×C61—42no—
266D4×C61—82no—
267D4×C61—82no—
268D4×C61#12012yes—
269C2×C4×C61—21no—
270C2×C4×C61—22no—
271C2×C4×C61—21no—
272C2×C4×C61—21no—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—24no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—21no—
279C2×C4×C61—42no—
280C2×C4×C61—42no—
281C2×C4×C61—42no—
282C2×C4×C61—42no—
283C2×C4×C61—41no—
284C2×C4×C61—81no—
285C2×C4×C61#12111yescharacteristic
286C2×C4×C61#12111yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C22×D412yes—
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61H(5,𝔽2)12yes—
291C2×Dic61—24no—
292C2×Dic61—44no—
293C2×Dic61—44no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296Dic122—24no—
297Dic122—24no—
298Dic122—44no—
299Dic122—44no—
300Dic122—42no—
301Dic122—42no—
302C8×C61—41no—
303C8×C61—41no—
304#92418—24no—
305#92418—22no—
306#92418—22no—
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—81no—
325C22×C61—81no—
326C22×C61—81no—
327C22×C61—162no—
328C22×C61#47112yescomplemented
329C22×C61C2×#12111yescharacteristic
330C22×C61#54612yes—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—22no—
334C4×C61—41no—
335C4×C61—81no—
336C4×C61C2×#12011yescharacteristic
337C4×C61#53411yescharacteristic
338C4×C61D4212yes—
339Dic61—44no—
340C22×#137—611no—
341C2×#471—612no—
342#3006D6111yescharacteristic
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—161no—
357C2×C61#241011yescharacteristic
358C2×C61C2×#47111yescharacteristic
359C2×C61#306911yescharacteristic
360C22×#41C2×D6111yescharacteristic
361C23×Q8—611no—
362C2×#137—612no—
363C2×#137—612no—
364C2×#137—1228no—
365C2×#137—1221no—
366C2×#137—1221no—
367C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
368C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
369C22×Q16—1221no—
370C22×SD16—612no—
371C22×SD16—1221no—
372C2×#131—612no—
373C22×M16—611no—
374C2×#121C2×D6111yescharacteristic
375#471—1228no—
376#449—21no—
377#449—21no—
378#448—21no—
379#448—1221no—
380#448—1221no—
381#448—1222no—
382#447—21no—
383C2×#108—612no—
384#417C2×D6111yescharacteristic
385#417C2×D6111yescharacteristic
386C2×#103—612no—
387C2×#101—612no—
388C2×#101—612no—
389C2×#100—612no—
390#408—612no—
391#408—1222no—
392#407—611no—
393#407—1221no—
394#407C2×D6111yescharacteristic
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—612no—
409C22×Q8—1224no—
410C22×Q8—1221no—
411C22×Q8—1221no—
412C22×Q8—1221no—
413C22×Q8—1221no—
414C22×Q8—1221no—
415C22×Q8—2442no—
416C22×Q8C22⋊D6111yescharacteristic
417C22×D4—21no—
418C22×D4—21no—
419C22×D4C22⋊D6111yescharacteristic
420C22×D4C22×D6111yescharacteristic
421C23×C4—21no—
422C23×C4—1221no—
423C23×C4C22×D6111yescharacteristic
424#137—618no—
425#137—1224no—
426#137—1224no—
427#137—1228no—
428#137—1228no—
429#137—1224no—
430AGL(1,ℤ/8)—24no—
431AGL(1,ℤ/8)—42no—
432C2×Q16—1228no—
433C2×Q16—1222no—
434C2×Q16—1222no—
435C2×SD16—21no—
436C2×SD16—21no—
437C2×SD16—1222no—
438C2×SD16—1222no—
439C2×SD16—1228no—
440C2×SD16—1222no—
441C2×SD16—1222no—
442C2×SD16—1224no—
443C2×SD16—2442no—
444C2×D8—21no—
445C2×D8—21no—
446#131—1224no—
447#131—2442no—
448C2×M16—612no—
449C2×M16—612no—
450C2×M16—1222no—
451C2×M16—1224no—
452C2×M16—1221no—
453C2×M16C22×D6111yescharacteristic
454C2×M16C22×D6111yescharacteristic
455C22×C8—612no—
456C22×C8—1222no—
457C22×C8—1221no—
458#121—21no—
459#121—21no—
460#121—21no—
461#121—21no—
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—21no—
464#108—1224no—
465#103—21no—
466#103—21no—
467#103—1222no—
468#103—1222no—
469#102—21no—
470#102—21no—
471#101—1224no—
472#101—2442no—
473#100—1224no—
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8—611no—
477C22⋊C8—611no—
478C22⋊C8—612no—
479C22⋊C8—1222no—
480C22⋊C8—1222no—
481C22⋊C8—1221no—
482C22⋊C8—1221no—
483C24—21no—
484C24—21no—
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#41D4×D6114yes—
496C2×Q8—21no—
497C2×Q8—22no—
498C2×Q8—41no—
499C2×Q8—614no—
500C2×Q8—1224no—
501C2×Q8—1224no—
502C2×Q8—1222no—
503C2×Q8—1222no—
504C2×Q8—1222no—
505C2×Q8—1222no—
506C2×Q8—1222no—
507C2×Q8—2444no—
508C2×Q8—2444no—
509C2×Q8—2444no—
510C2×Q8—2442no—
511C2×Q8—2442no—
512C2×Q8—2441no—
513C2×Q8—2441no—
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—22no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—22no—
525C2×D4—21no—
526C2×D4—41no—
527C2×D4—44no—
528C2×D4—41no—
529C2×D4—41no—
530C2×D4—82no—
531C2×D4C2×C22⋊D6111yescharacteristic
532C2×D4C2×C22⋊D6111yescharacteristic
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—44no—
542C22×C4—1222no—
543C22×C4—2444no—
544C22×C4—2441no—
545C22×C4—2441no—
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822411yescharacteristic
549C22×C4C2×C22⋊D6111yescharacteristic
550C22×C4C2×C22⋊D6111yescharacteristic
551C22×C4C23×D6111yescharacteristic
552Q16—1228no—
553Q16—1228no—
554SD16—42no—
555SD16—42no—
556SD16—1228no—
557SD16—1228no—
558SD16—2444no—
559SD16—2444no—
560D8—42no—
561D8—42no—
562M16—22no—
563M16—22no—
564M16—614no—
565M16—1224no—
566M16—1222no—
567M16—1222no—
568M16—1222no—
569M16—2444no—
570C2×C8—21no—
571C2×C8—21no—
572C2×C8—612no—
573C2×C8—612no—
574C2×C8—1224no—
575C2×C8—1222no—
576C2×C8—1222no—
577C2×C8—1222no—
578C2×C8—1222no—
579C2×C8—2444no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C22⋊C4—41no—
583C22⋊C4—41no—
584C23—21no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—42no—
593C23—81no—
594C23—81no—
595C23—81no—
596C23#1451974911yescharacteristic
597C23#1451974911yescharacteristic
598C23C2×#1175822411yescharacteristic
599Q8—21no—
600Q8—21no—
601Q8—21no—
602Q8—42no—
603Q8—1224no—
604Q8—1224no—
605Q8—2444no—
606Q8—2444no—
607Q8—2442no—
608Q8—2442no—
609Q8#1451974512yes—
610D4—24no—
611D4—21no—
612D4—21no—
613D4—21no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—41no—
619D4—41no—
620D4—42no—
621D4—82no—
622D4—82no—
623D4#1451974512yes—
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—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—81no—
640C2×C4—1224no—
641C2×C4—2444no—
642C2×C4—2444no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4#1451974911yescharacteristic
646C2×C4#1451974911yescharacteristic
647C2×C4C2×D4×D6112yes—
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4#1451982212yes—
651C8—41no—
652C8—41no—
653C8—1224no—
654C8—1222no—
655C8—1222no—
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—81no—
673C22—81no—
674C22—81no—
675C22—162no—
676C22#1684002312yescomplemented
677C22C2×#1451974911yescharacteristic
678C22#1684039912yes—
679C4—21no—
680C4—21no—
681C4—22no—
682C4—41no—
683C4—81no—
684C4—2444no—
685C4C2×#1451974511yescharacteristic
686C4#1684032111yescharacteristic
687C4#1684039412yes—
688C2—22no—
689C2—22no—
690C2—41no—
691C2—42no—
692C2—81no—
693C2—161no—
694C2#2144919411yescharacteristic
695C2C2×#1684002311yescharacteristic
696C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3006C2
socleC22×C61C2×#121
hypercenter#3006D61
hypocenterC61#24401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. C22×#14519807C2semidirect
2. C2×#16840023C2semidirect
3. C2×#16839900C2semidirect
4. #21454268C2semidirect
5. #21447116C2semidirect
6. #21446933C2semidirect
7. #21446332C2semidirect
8. #21446283C2semidirect
9. #21446023C2semidirect
10. #21445742C2semidirect
20. C22×#11758200C22semidirect
27. C2×#14519712C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519695C22semidirect
30. C2×#14519695C22semidirect
31. C2×#14519694C22semidirect
38. #16839531C22semidirect
39. #16839526C22semidirect
65. #14519807D4semidirect
328. C22×C61#471semidirect
399. C61#24401semidirect
676. C22#16840023semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C2×#168400232
C2×#168399002
#214542682
#214471162
#214469332
#214463322
#214462832
#214460232
#214457422
C61×#30062
#2440161