Finite Group Databasebeta

← #29712172

Subgroups — #29712172

Order 15616.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840412C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21449178C211yesmaximal, characteristic, complemented
4#21446958C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446155C212yesmaximal, complemented
8#21446123C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445906C211yesmaximal, characteristic, complemented
11C2×C61×#448C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C23×#92418C2211yescharacteristic
14#16840412—24no—
15#16840412—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519758—21no—
19C2×#14519754—21no—
20C2×#14519753C2212yes—
21C2×#14519748C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C2×C4×C22⋊D61C2211yescharacteristic
24C22×C4⋊Dic61C2211yescharacteristic
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×#14519701—24no—
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31#16839681—21no—
32#16839673—21no—
33#16839669C2212yes—
34#16839641—21no—
35#16839635—21no—
36#16839622C2211yescharacteristic
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#448—24no—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C2×#11758224—28no—
48C2×#11758224—44no—
49C2×#11758224—44no—
50C22×Dic122—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61D411yescharacteristic
53C23×Dic61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418—22no—
58C22×#92418—42no—
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776—22no—
62#14519759—28no—
63#14519758—28no—
64#14519754—28no—
65#14519753—24no—
66#14519753—42no—
67#14519748—22no—
68#14519748—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71C4×C22⋊D61—22no—
72C4×C22⋊D61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61D414yes—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×#11758196D411yescharacteristic
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×#11758198—22no—
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519701—24no—
93#14519701—24no—
94#14519698—42no—
95#14519689—42no—
96#14519684—21no—
97#14519684—22no—
98#14519684—22no—
99C22×Q8×C61—21no—
100C22×Q8×C61—24no—
101C22×Q8×C61—22no—
102C22×Q8×C61—22no—
103C22×Q8×C61—42no—
104C22×Q8×C61C2312yes—
105C22×D4×C61—21no—
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C2×SD16×C61—24no—
110C2×SD16×C61—24no—
111C2×SD16×C61—42no—
112C22×C8×C61C2312yes—
113#121×C61—22no—
114#121×C61D414yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117#103×C61—24no—
118C22⋊C8×C61—22no—
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—22no—
123C2×Q8×C61—28no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—44no—
129C2×Q8×C61—48no—
130C2×Q8×C61—44no—
131C2×Q8×C61—42no—
132C2×Q8×C61#4114yescomplemented
133C2×D4×C61—24no—
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—42no—
138C2×D4×C61—42no—
139C2×D4×C61C2×D412yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61—44no—
145C22×C4×C61—42no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61SD1614yescomplemented
156C22×Dic61—24no—
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168#11758224—432no—
169C2×C4×D61—24no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×Dic122—24no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179SD16×C61—48no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D412yes—
185C22⋊C4×C61—24no—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188C22⋊Dic61—22no—
189C22⋊Dic61—42no—
190#11758198—22no—
191#11758198—44no—
192C4⋊Dic61—24no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C2×D412yes—
198C4⋊Dic61C2×D412yes—
199#11758196—24no—
200#11758196—24no—
201#11758196—22no—
202#11758196—44no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—44no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—42no—
216C23×C61#12012yes—
217C23×C61C2×SD1612yes—
218C23×C61AGL(1,ℤ/8)12yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—22no—
222C22×D61—21no—
223C22×D61—22no—
224C22×D61C2×SD1612yes—
225Q8×C61—24no—
226Q8×C61—24no—
227Q8×C61—48no—
228Q8×C61—42no—
229Q8×C61—42no—
230D4×C61—24no—
231D4×C61—48no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—48no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61#12012yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4112yes—
249C2×C4×C61H(5,𝔽2)12yes—
250C22⋊D61—28no—
251C22⋊D61—48no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—48no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—48no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C2×SD1612yes—
270C4×D61—48no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—42no—
274Dic122—48no—
275C8×C61—42no—
276#92418—42no—
277#23961—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61#45014yes—
288C22×C61C2×#12011yescharacteristic
289C22×C61#54612yes—
290C22×C61C22×SD1611yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61#57812yes—
293C2×D61—22no—
294C2×D61—24no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—42no—
298C4×C61—22no—
299C4×C61—44no—
300C4×C61—42no—
301C4×C61#52511yescharacteristic
302C4×C61#53511yescharacteristic
303C4×C61#54612yes—
304Dic61—22no—
305Dic61—41no—
306Dic61—42no—
307Dic61—42no—
308Dic61—41no—
309Dic61—82no—
310C2×#547—611no—
311C2×#448D6111yescharacteristic, complemented
312C22×#103—611no—
313#2409—611no—
314#2024—612no—
315#2000—612no—
316#1893—611no—
317#1877—612no—
318#1872—612no—
319#1869—611no—
320#1823—611no—
321C2×C61—24no—
322C2×C61—82no—
323C2×C61#240411yescharacteristic
324C2×C61C2×#45011yescharacteristic
325C2×C61#301111yescharacteristic
326C2×C61#329812yes—
327C2×C61#330812yes—
328D61—42no—
329C22×#41—1221no—
330C23×Q8C2×D6111yescharacteristic
331C22×SD16C2×D6112yes—
332C23×C8—611no—
333#547—1224no—
334#547—1224no—
335C2×C4⋊Q8—611no—
336C2×#123—1221no—
337C2×#123—1221no—
338C2×C22⋊Q8—612no—
339C2×C22⋊Q8—1221no—
340C2×#121—611no—
341C2×#121C2×D6111yescharacteristic
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C22×C4⋊C4—611no—
345C22×C4⋊C4—611no—
346#448—24no—
347C2×#103—612no—
348C2×#103—612no—
349C2×#103—1224no—
350C2×#103C2×D6112yes—
351C2×#102—612no—
352C2×C22⋊C8—612no—
353C2×C22⋊C8C2×D6111yescharacteristic
354#395—1221no—
355#393—1221no—
356#391—612no—
357#386—1221no—
358#385—1221no—
359#380—611no—
360#340—612no—
361#340—612no—
362#340—612no—
363#336—612no—
364C61#2396111yescharacteristic, complemented, Sylow
365C2×#41—1228no—
366C2×#41—2444no—
367C2×#41—2444no—
368C22×Q8—21no—
369C22×Q8—24no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—42no—
373C22×Q8—1221no—
374C22×Q8C22×D6112yes—
375C22×D4—21no—
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4C22×D6111yescharacteristic
379C23×C4—21no—
380C23×C4—611no—
381C23×C4—1221no—
382C23×C4—1221no—
383C23×C4—1221no—
384C23×C4C22×D6111yescharacteristic
385C2×SD16—24no—
386C2×SD16—24no—
387C2×SD16—42no—
388C22×C8—612no—
389C22×C8—1222no—
390C22×C8—2442no—
391C22×C8C22×D6112yes—
392C4⋊Q8—1222no—
393C4⋊Q8—1222no—
394#123—1228no—
395#123—1228no—
396C22⋊Q8—1228no—
397C22⋊Q8—1224no—
398C22⋊Q8—2442no—
399#121—22no—
400#121—1222no—
401#121—1222no—
402#121C22⋊D6114yes—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1221no—
427C2×C42—611no—
428#103—24no—
429#103—1224no—
430#103—1224no—
431#102—2442no—
432C22⋊C8—22no—
433C22⋊C8—2442no—
434#95—1222no—
435#95—1222no—
436#95—1221no—
437C24—21no—
438C24—611no—
439C24D4×D6111yescharacteristic
440#41—24432no—
441C2×Q8—22no—
442C2×Q8—28no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—1224no—
452C2×Q8—2442no—
453C2×Q8—2442no—
454C2×Q8#1175822414yescomplemented
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—614no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1224no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4C2×C22⋊D6112yes—
496C22×C4C23×D6111yescharacteristic
497SD16—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—2444no—
505C4⋊C4—21no—
506C4⋊C4—612no—
507C4⋊C4—612no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1224no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2444no—
517C4⋊C4C2×C22⋊D6112yes—
518C22⋊C4—24no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—2444no—
524C22⋊C4—2442no—
525C42—1221no—
526C42—1221no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—612no—
535C23—1221no—
536C23—1221no—
537C23—1222no—
538C23—1222no—
539C23#1451974512yes—
540C23SD16×D6112yes—
541C23#1451980412yes—
542C23C2×D4×D6111yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—48no—
546Q8—42no—
547Q8—42no—
548Q8—2448no—
549D4—24no—
550D4—48no—
551D4—41no—
552D4—41no—
553D4—1228no—
554D4—2448no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—612no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—2448no—
576C2×C4—2448no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2448no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4#1451974512yes—
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C22×C22⋊D6111yescharacteristic
591C2×C4C2×#1175822412yes—
592C2×C4#1451982212yes—
593C8—42no—
594C8—2442no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—82no—
604C22—1224no—
605C22—1222no—
606C22—2441no—
607C22—2442no—
608C22—2441no—
609C22#1683991114yes—
610C22C2×#1451974511yescharacteristic
611C22#1684039912yes—
612C22C2×SD16×D6111yescharacteristic
613C22C2×#1451980411yescharacteristic
614C22#1684057812yes—
615C4—22no—
616C4—42no—
617C4—44no—
618C4—1222no—
619C4—4882no—
620C4—2441no—
621C4—2442no—
622C4—2442no—
623C4—2441no—
624C4#1684026811yescharacteristic
625C4#1684032711yescharacteristic
626C4#1684040612yes—
627C2—24no—
628C2—82no—
629C2—2442no—
630C2#2144913511yescharacteristic
631C2C2×#1683991111yescharacteristic
632C2#2145511311yescharacteristic
633C2#2145767712yes—
634C2#2145781412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#448C2
socleC23×C61C22×D4
prime radicalC2×#16840412C2
hypercenterC2×#448D61
hypocenterC61#23961
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
3. #21449178C2semidirect
4. #21446958C2semidirect
5. #21446819C2semidirect
6. #21446248C2semidirect
7. #21446155C2semidirect
8. #21446123C2semidirect
9. #21446109C2semidirect
10. #21445906C2semidirect
11. C2×C61×#448C2semidirect
132. C2×Q8×C61#41semidirect
155. C2×C22⋊D61SD16semidirect
311. C2×#448D61semidirect
364. C61#23961semidirect
454. C2×Q8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404122
C22×#145197012
#214491782
#214469582
#214468192
#214462482
#214461552
#214461232
#214461092
#214459062
C2×C61×#4482
#2396161