Finite Group Databasebeta

← #29711281

Subgroups — #29711281

Order 15616.

1806 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic, complemented
2C2×#16839924C212yesmaximal, complemented
3C2×#16839900C211yesmaximal, characteristic, complemented
4#21449184C211yesmaximal, characteristic, complemented
5#21446907C212yesmaximal, complemented
6#21446822C212yesmaximal, complemented
7#21446258C211yesmaximal, characteristic, complemented
8#21446160C212yesmaximal, complemented
9#21446117C211yesmaximal, characteristic, complemented
10#21445759C211yesmaximal, characteristic, complemented
11C2×C61×#550C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519807—28no—
14C2×#14519807—24no—
15C22×#11758202C2212yes—
16C22×#11758200C2212yes—
17C22×#11758200C2212yes—
18C22×#11758194C2211yescharacteristic
19C2×#14519756C2212yes—
20#16839924—28no—
21#16839900—24no—
22C2×#14519701C2212yes—
23C2×#14519700C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2212yes—
28#16839682—21no—
29#16839674—21no—
30#16839668C2212yes—
31#16839642—21no—
32#16839634—21no—
33#16839622C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36C22×#41×C61C2211yescharacteristic
37C22×#41×C61C2211yescharacteristic
38C61×#550—24no—
39C61×#550—24no—
40C2×#127×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C2×#121×C61—21no—
43C2×#121×C61—21no—
44C2×#121×C61C2211yescharacteristic
45C2×#121×C61C2211yescharacteristic
46C2×C4×Q8×C61C2211yescharacteristic
47C2×C4×D4×C61—21no—
48C2×C4×D4×C61C2211yescharacteristic
49C22×Dic122—22no—
50C22×Dic122—24no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—42no—
54C22×Dic122C2312yes—
55C23×Dic61—21no—
56#14519807—232no—
57C2×#11758202—24no—
58C2×#11758202—24no—
59C2×#11758202D418yescomplemented
60C2×#11758200—24no—
61C2×#11758200—24no—
62C2×#11758200—24no—
63C2×#11758200—24no—
64C2×#11758200—24no—
65C2×#11758200—42no—
66C2×#11758194—22no—
67C2×#11758194—24no—
68C22×#92418C2312yes—
69C22×#92418C2312yes—
70#14519756—24no—
71#14519756—42no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—21no—
76#14519701—24no—
77#14519700—24no—
78#14519698—24no—
79#14519698—42no—
80C22⋊#92418—22no—
81C22⋊#92418—24no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—24no—
89C2×#41×C61—24no—
90C2×#41×C61—42no—
91C2×#41×C61—42no—
92C2×#41×C61D414yes—
93C22×Q8×C61C2311yescharacteristic
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×D4×C61—21no—
99C22×D4×C61—21no—
100C22×D4×C61D411yescharacteristic
101C22×D4×C61C2311yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C23×C4×C61C2311yescharacteristic
107#127×C61—28no—
108#127×C61—22no—
109#127×C61—22no—
110#121×C61—28no—
111#121×C61—28no—
112#121×C61—22no—
113#121×C61—22no—
114#121×C61—22no—
115#121×C61D414yes—
116C4×Q8×C61—22no—
117C4×Q8×C61—22no—
118C4×D4×C61—28no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61—21no—
131C24×C61—21no—
132C24×C61C2×D411yescharacteristic
133C24×C61C2×D411yescharacteristic
134#41×C61—28no—
135#41×C61—44no—
136#41×C61—48no—
137#41×C61—48no—
138#41×C61—44no—
139#41×C61—44no—
140C2×Q8×C61—21no—
141C2×Q8×C61—22no—
142C2×Q8×C61C2×D414yes—
143C2×Q8×C61C2×D412yes—
144C2×D4×C61—24no—
145C2×D4×C61—24no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—24no—
151C2×D4×C61—22no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—41no—
159C2×D4×C61—44no—
160C2×D4×C61—44no—
161C2×D4×C61—42no—
162C2×D4×C61—42no—
163C2×D4×C61—82no—
164C2×D4×C61—82no—
165C2×D4×C61—82no—
166C2×D4×C61C2×D412yes—
167C2×D4×C61C2×D412yes—
168C22×C4×C61—21no—
169C22×C4×C61—22no—
170C22×C4×C61—22no—
171C22×C4×C61—24no—
172C22×C4×C61—21no—
173C22×C4×C61—24no—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—22no—
177C22×C4×C61—22no—
178C22×C4×C61—44no—
179C22×C4×C61—44no—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D412yes—
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61C2411yescharacteristic
187C22×Dic61—22no—
188C22×Dic61—22no—
189C22×Dic61—44no—
190C22×Dic61—42no—
191C2×Dic122—28no—
192C2×Dic122—22no—
193C2×Dic122—22no—
194C2×Dic122—22no—
195C2×Dic122—22no—
196C2×Dic122—22no—
197C2×Dic122—48no—
198C2×Dic122—44no—
199C2×Dic122—44no—
200C2×Dic122—42no—
201C2×Dic122C2×D414yes—
202C4⋊C4×C61—24no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61C2×D412yes—
206C22⋊C4×C61—24no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—24no—
209C22⋊C4×C61—22no—
210C42×C61—24no—
211C42×C61—21no—
212C42×C61—21no—
213C22⋊Dic61—44no—
214C22⋊Dic61—42no—
215#11758202—216no—
216#11758200—216no—
217#11758200—48no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220#11758196—44no—
221#11758194—28no—
222C2×#92418—22no—
223C2×#92418—22no—
224C2×#92418—22no—
225C2×#92418—22no—
226C2×#92418—22no—
227C2×#92418C2×D414yes—
228C23×C61—22no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61—21no—
233C23×C61—22no—
234C23×C61—22no—
235C23×C61—22no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—22no—
239C23×C61—42no—
240C23×C61—42no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—41no—
244C23×C61—41no—
245C23×C61—42no—
246C23×C61—42no—
247C23×C61#12012yes—
248C23×C61C22×D411yescharacteristic
249Q8×C61—22no—
250Q8×C61—24no—
251Q8×C61#12014yes—
252D4×C61—22no—
253D4×C61—22no—
254D4×C61—24no—
255D4×C61—44no—
256D4×C61—48no—
257D4×C61—48no—
258D4×C61—48no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—42no—
262D4×C61—84no—
263D4×C61—84no—
264D4×C61—84no—
265C2×C4×C61—21no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—24no—
269C2×C4×C61—21no—
270C2×C4×C61—22no—
271C2×C4×C61—22no—
272C2×C4×C61—24no—
273C2×C4×C61—21no—
274C2×C4×C61—22no—
275C2×C4×C61—21no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—48no—
279C2×C4×C61—48no—
280C2×C4×C61—42no—
281C2×C4×C61—41no—
282C2×C4×C61—41no—
283C2×C4×C61—42no—
284C2×C4×C61#12012yes—
285C2×C4×C61#13712yescomplemented
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D412yes—
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61H(5,𝔽2)12yes—
290C2×Dic61—24no—
291C2×Dic61—48no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297Dic122—24no—
298Dic122—24no—
299Dic122—48no—
300Dic122—42no—
301Dic122—42no—
302#92418—24no—
303#92418—42no—
304#23870—611nomaximal, Sylow
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—42no—
315C22×C61—42no—
316C22×C61—41no—
317C22×C61—42no—
318C22×C61—42no—
319C22×C61—42no—
320C22×C61—42no—
321C22×C61—82no—
322C22×C61—82no—
323C22×C61—82no—
324C22×C61—82no—
325C22×C61C2×#12011yescharacteristic
326C22×C61D4212yes—
327C22×C61C2×#13711yescharacteristic
328C22×C61#57511yescharacteristic
329C22×C61#57712yes—
330C4×C61—22no—
331C4×C61—24no—
332C4×C61—22no—
333C4×C61—41no—
334C4×C61—41no—
335C4×C61—42no—
336C4×C61C2×#12011yescharacteristic
337C4×C61#53411yescharacteristic
338C4×C61D4212yes—
339Dic61—44no—
340Dic61—82no—
341C22×#137—611no—
342C2×#550D6111yescharacteristic
343C2×#452—612no—
344C2×#448—611no—
345#2409—611no—
346#2014—612no—
347#2000—612no—
348#1895—611no—
349#1878—612no—
350#1870—611no—
351#1788—611no—
352C2×C61—42no—
353C2×C61—81no—
354C2×C61—82no—
355C2×C61—82no—
356C2×C61#240311yescharacteristic
357C2×C61#300711yescharacteristic
358C2×C61#300911yescharacteristic
359C2×C61#327112yes—
360C2×C61#327812yes—
361C22×#41C2×D6111yescharacteristic
362C22×#41C2×D6111yescharacteristic
363C23×Q8—611no—
364C2×#137—1228no—
365C2×#137—1224no—
366C22×Q16—612no—
367C22×SD16—612no—
368C22×SD16—612no—
369C22×M16—611no—
370#550—24no—
371#550—24no—
372C2×#127—21no—
373C2×#127C2×D6111yescharacteristic
374C2×C22⋊Q8—612no—
375C2×#121—21no—
376C2×#121—21no—
377C2×#121C2×D6111yescharacteristic
378C2×#121C2×D6111yescharacteristic
379C2×C4×Q8C2×D6111yescharacteristic
380C2×C4×D4—21no—
381C2×C4×D4C2×D6111yescharacteristic
382#452—1228no—
383#448—1224no—
384C2×#103—612no—
385C2×#103—612no—
386C2×#102—612no—
387C2×#102—612no—
388C2×C22⋊C8—611no—
389C2×C22⋊C8—612no—
390#395—1221no—
391#393—1221no—
392#391—612no—
393#386—1221no—
394#385—1221no—
395#380—611no—
396#340—612no—
397#336—612no—
398C61#2387011yescharacteristic, complemented, Sylow
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—24no—
403C2×#41—24no—
404C2×#41—42no—
405C2×#41—42no—
406C2×#41C22⋊D6114yes—
407C22×Q8—612no—
408C22×Q8—1224no—
409C22×Q8—1221no—
410C22×Q8—1222no—
411C22×Q8—1222no—
412C22×Q8—2442no—
413C22×Q8C22×D6111yescharacteristic
414C22×D4—21no—
415C22×D4—21no—
416C22×D4—21no—
417C22×D4—21no—
418C22×D4—21no—
419C22×D4—21no—
420C22×D4C22⋊D6111yescharacteristic
421C22×D4C22×D6111yescharacteristic
422C22×D4C22×D6111yescharacteristic
423C23×C4—21no—
424C23×C4—21no—
425C23×C4—1221no—
426C23×C4C22×D6111yescharacteristic
427C23×C4C22×D6111yescharacteristic
428#137—12232no—
429C2×Q16—618no—
430C2×Q16—1224no—
431C2×Q16—1224no—
432C2×SD16—1224no—
433C2×SD16—1224no—
434C2×SD16—1224no—
435C2×SD16—1224no—
436C2×SD16—1224no—
437C2×SD16—2442no—
438C2×M16—1224no—
439C2×M16—1222no—
440C22×C8—612no—
441C22×C8—612no—
442#127—28no—
443#127—22no—
444#127—22no—
445C22⋊Q8—1224no—
446C22⋊Q8—2442no—
447#121—28no—
448#121—28no—
449#121—22no—
450#121—22no—
451#121—22no—
452#121C22⋊D6114yes—
453C4×Q8—22no—
454C4×Q8—22no—
455C4×D4—28no—
456C4×D4—22no—
457C4×D4—22no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—612no—
460C2×C4⋊C4—1222no—
461C2×C4⋊C4C22⋊D6111yescharacteristic
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—21no—
464C2×C22⋊C4—21no—
465C2×C22⋊C4—1222no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4C22⋊D6111yescharacteristic
468C2×C22⋊C4C22⋊D6111yescharacteristic
469C2×C42—21no—
470C2×C42C22×D6111yescharacteristic
471#103—1224no—
472#103—1224no—
473#102—1224no—
474#102—2442no—
475C22⋊C8—1224no—
476C22⋊C8—1222no—
477#95—1221no—
478#95—1222no—
479#95—1222no—
480C24—21no—
481C24—21no—
482C24C2×C22⋊D6111yescharacteristic
483C24C2×C22⋊D6111yescharacteristic
484#41—28no—
485#41—44no—
486#41—48no—
487#41—48no—
488#41—44no—
489#41—44no—
490C2×Q8—21no—
491C2×Q8—22no—
492C2×Q8—614no—
493C2×Q8—1228no—
494C2×Q8—1222no—
495C2×Q8—1222no—
496C2×Q8—1222no—
497C2×Q8—1222no—
498C2×Q8—1222no—
499C2×Q8—2444no—
500C2×Q8—2448no—
501C2×Q8—2444no—
502C2×Q8—2442no—
503C2×Q8D4×D6114yes—
504C2×Q8C2×C22⋊D6112yes—
505C2×D4—24no—
506C2×D4—24no—
507C2×D4—22no—
508C2×D4—24no—
509C2×D4—24no—
510C2×D4—24no—
511C2×D4—24no—
512C2×D4—22no—
513C2×D4—22no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—41no—
520C2×D4—44no—
521C2×D4—44no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4—82no—
525C2×D4—82no—
526C2×D4—82no—
527C2×D4C2×C22⋊D6112yes—
528C2×D4C2×C22⋊D6112yes—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—24no—
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—21no—
536C22×C4—21no—
537C22×C4—22no—
538C22×C4—24no—
539C22×C4—44no—
540C22×C4—44no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—2444no—
544C22×C4—2442no—
545C22×C4C2×C22⋊D6111yescharacteristic
546C22×C4C2×C22⋊D6111yescharacteristic
547C22×C4C2×C22⋊D6112yes—
548C22×C4C2×C22⋊D6112yes—
549C22×C4C2×C22⋊D6111yescharacteristic
550C22×C4C2×C22⋊D6111yescharacteristic
551C22×C4C23×D6111yescharacteristic
552Q16—12216no—
553SD16—12216no—
554SD16—2448no—
555M16—1228no—
556C2×C8—614no—
557C2×C8—1222no—
558C2×C8—1222no—
559C2×C8—1222no—
560C2×C8—1222no—
561C2×C8—1222no—
562C4⋊C4—24no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—2444no—
568C4⋊C4C2×C22⋊D6112yes—
569C22⋊C4—24no—
570C22⋊C4—22no—
571C22⋊C4—24no—
572C22⋊C4—22no—
573C22⋊C4—2444no—
574C22⋊C4—2442no—
575C42—24no—
576C42—21no—
577C42—21no—
578C23—21no—
579C23—21no—
580C23—22no—
581C23—22no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23#1451974612yes—
598C23C22×C22⋊D6111yescharacteristic
599Q8—22no—
600Q8—24no—
601Q8—1224no—
602Q8—1224no—
603Q8—2448no—
604Q8—2442no—
605Q8—2442no—
606Q8#1451974514yes—
607D4—22no—
608D4—22no—
609D4—24no—
610D4—44no—
611D4—48no—
612D4—48no—
613D4—48no—
614D4—41no—
615D4—41no—
616D4—42no—
617D4—84no—
618D4—84no—
619D4—84no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—1224no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2448no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4#1451974612yes—
647C2×C4#1451980712yescomplemented
648C2×C4C2×D4×D6112yes—
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4C22×C22⋊D6111yescharacteristic
651C2×C4#1451982212yes—
652C8—1224no—
653C8—2442no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—42no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22C2×#1451974611yescharacteristic
675C22D4×C22⋊D6112yes—
676C22C2×#1451980711yescharacteristic
677C22#1684056011yescharacteristic
678C22#1684057312yes—
679C4—22no—
680C4—24no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—42no—
685C4—4882no—
686C4—2444no—
687C4C2×#1451974511yescharacteristic
688C4#1684032111yescharacteristic
689C4D4×C22⋊D6112yes—
690C2—42no—
691C2—81no—
692C2—82no—
693C2—82no—
694C2#2144912911yescharacteristic
695C2#2145507911yescharacteristic
696C2#2145509611yescharacteristic
697C2#2145740012yes—
698C2#2145748512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC23×C61C22×D4
hypercenterC2×#550D61
hypocenterC61#23870
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C22×#14519807C2semidirect
2. C2×#16839924C2semidirect
3. C2×#16839900C2semidirect
4. #21449184C2semidirect
5. #21446907C2semidirect
6. #21446822C2semidirect
7. #21446258C2semidirect
8. #21446160C2semidirect
9. #21446117C2semidirect
10. #21445759C2semidirect
59. C2×#11758202D4semidirect
285. C2×C4×C61#137semidirect
398. C61#23870semidirect
647. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C2×#168399242
C2×#168399002
#214491842
#214469072
#214468222
#214462582
#214461602
#214461172
#214457592
C2×C61×#5502
#2387061