Finite Group Databasebeta

← #29712171

Subgroups — #29712171

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×#16839900C211yesmaximal, characteristic, complemented
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21449184C211yesmaximal, characteristic, complemented
4#21446957C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446153C212yesmaximal, complemented
8#21446126C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445905C211yesmaximal, characteristic, complemented
11C2×C61×#547C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200C2212yes—
14C23×#92418C2211yescharacteristic
15C2×#14519756C2212yes—
16#16839900—24no—
17C2×#14519700—24no—
18C2×#14519700C2212yes—
19C2×#14519700C2212yes—
20C2×#14519700C2212yes—
21C2×#14519698C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2212yes—
24#16839682—21no—
25#16839674—21no—
26#16839668C2212yes—
27#16839642—21no—
28#16839634—21no—
29#16839622C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C22×#41×C61—21no—
35C61×#547—24no—
36C61×#547—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#123×C61—21no—
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61—21no—
41C2×#121×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C22×C4⋊C4×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—22no—
48C22×Dic122—24no—
49C22×Dic122—22no—
50C22×Dic122—21no—
51C22×Dic122—42no—
52C22×Dic122C2312yes—
53C23×Dic61—21no—
54C2×#11758200—24no—
55C2×#11758200—24no—
56C2×#11758200—42no—
57C22×#92418—22no—
58C22×#92418—42no—
59C22×#92418C2312yes—
60C22×#92418C2312yes—
61#14519756—24no—
62#14519756—42no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—21no—
67#14519700—24no—
68#14519700—24no—
69#14519700—24no—
70#14519698—42no—
71C22⋊#92418—22no—
72C22⋊#92418—42no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76C2×#41×C61—28no—
77C2×#41×C61—44no—
78C2×#41×C61—44no—
79C22×Q8×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61D411yescharacteristic
83C22×D4×C61C2311yescharacteristic
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61—22no—
91#123×C61—28no—
92#123×C61—28no—
93C22⋊Q8×C61—28no—
94#121×C61—22no—
95#121×C61—22no—
96#121×C61—22no—
97#121×C61D414yes—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D414yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61D411yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C24×C61C2×D411yescharacteristic
122#41×C61—432no—
123C2×Q8×C61—24no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—24no—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61SD1614yescomplemented
138C2×D4×C61C2×D412yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—24no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—44no—
167C22×Dic61—42no—
168C2×Dic122—28no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—48no—
175C2×Dic122—44no—
176C2×Dic122—44no—
177C2×Dic122—42no—
178C2×Dic122#4114yescomplemented
179C4⋊C4×C61—24no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C2×D412yes—
187C4⋊C4×C61C2×D412yes—
188C4⋊C4×C61C2×D412yes—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—24no—
193C22⋊C4×C61—22no—
194C42×C61—21no—
195C42×C61—21no—
196C22⋊Dic61—44no—
197C22⋊Dic61—42no—
198#11758200—48no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201#11758196—44no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—44no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—21no—
212C23×C61—22no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—42no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61#12012yes—
221C23×C61C2×SD1612yes—
222C23×C61C2×SD1612yes—
223C23×C61AGL(1,ℤ/8)12yes—
224C23×C61C22×D411yescharacteristic
225Q8×C61—48no—
226D4×C61—28no—
227D4×C61—24no—
228D4×C61—48no—
229D4×C61—48no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—48no—
246C2×C4×C61—48no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—48no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—42no—
257C2×C4×C61#12012yes—
258C2×C4×C61C2×SD1612yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4112yes—
262C2×C4×C61H(5,𝔽2)12yes—
263C2×Dic61—24no—
264C2×Dic61—48no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270Dic122—24no—
271Dic122—24no—
272Dic122—48no—
273Dic122—42no—
274Dic122—42no—
275#92418—42no—
276#92418—42no—
277#23961—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—22no—
283C22×C61—24no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—82no—
291C22×C61—82no—
292C22×C61#45014yes—
293C22×C61C2×#12011yescharacteristic
294C22×C61#54612yes—
295C22×C61C22×SD1611yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57812yes—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61#52511yescharacteristic
306C4×C61#53511yescharacteristic
307C4×C61#54612yes—
308Dic61—44no—
309Dic61—82no—
310C2×#547D6111yescharacteristic
311C2×#448—611no—
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—42no—
323C2×C61—82no—
324C2×C61#240411yescharacteristic
325C2×C61C2×#45011yescharacteristic
326C2×C61#301111yescharacteristic
327C2×C61#329812yes—
328C2×C61#330812yes—
329C22×#41—21no—
330C23×Q8—611no—
331C22×SD16—612no—
332C23×C8—611no—
333#547—24no—
334#547—24no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×#123—21no—
337C2×#123—21no—
338C2×C22⋊Q8—21no—
339C2×C22⋊Q8—612no—
340C2×#121C2×D6111yescharacteristic
341C2×#121C2×D6111yescharacteristic
342C2×C4×D4C2×D6111yescharacteristic
343C2×C4×D4C2×D6111yescharacteristic
344C22×C4⋊C4C2×D6111yescharacteristic
345C22×C4⋊C4C2×D6111yescharacteristic
346#448—1224no—
347C2×#103—612no—
348C2×#103—612no—
349C2×#103—612no—
350C2×#103—1224no—
351C2×#102—612no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—612no—
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—28no—
366C2×#41—44no—
367C2×#41—44no—
368C22×Q8—21no—
369C22×Q8—612no—
370C22×Q8—1224no—
371C22×Q8—1221no—
372C22×Q8—1222no—
373C22×Q8—1222no—
374C22×Q8—2442no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4C22⋊D6111yescharacteristic
378C22×D4C22×D6111yescharacteristic
379C23×C4—21no—
380C23×C4—21no—
381C23×C4—21no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C2×SD16—1224no—
386C2×SD16—1224no—
387C2×SD16—2442no—
388C22×C8—612no—
389C22×C8—612no—
390C22×C8—1222no—
391C22×C8—2442no—
392C4⋊Q8—22no—
393C4⋊Q8—22no—
394#123—28no—
395#123—28no—
396C22⋊Q8—28no—
397C22⋊Q8—1224no—
398C22⋊Q8—2442no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121C22⋊D6114yes—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22⋊D6114yes—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4C22⋊D6111yescharacteristic
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428#103—1224no—
429#103—1224no—
430#103—1224no—
431#102—2442no—
432C22⋊C8—1222no—
433C22⋊C8—2442no—
434#95—1221no—
435#95—1222no—
436#95—1222no—
437C24—21no—
438C24C2×C22⋊D6111yescharacteristic
439C24C2×C22⋊D6111yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—614no—
445C2×Q8—1228no—
446C2×Q8—1222no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—2444no—
452C2×Q8—2448no—
453C2×Q8—2444no—
454C2×Q8—2442no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4#1175820014yescomplemented
467C2×D4C2×C22⋊D6112yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—2444no—
489C22×C4—2442no—
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497SD16—2448no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—2444no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—2444no—
515C4⋊C4C2×C22⋊D6112yes—
516C4⋊C4C2×C22⋊D6112yes—
517C4⋊C4C2×C22⋊D6112yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—2444no—
524C22⋊C4—2442no—
525C42—21no—
526C42—21no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23#1451974612yes—
539C23C2×#1175820012yes—
540C23C2×#1175820012yes—
541C23#1451980312yes—
542C23C22×C22⋊D6111yescharacteristic
543Q8—48no—
544Q8—1224no—
545Q8—1224no—
546Q8—2448no—
547Q8—2442no—
548Q8—2442no—
549D4—28no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—1224no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2448no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4#1451974612yes—
588C2×C4C2×#1175820012yes—
589C2×C4C22×C22⋊D6111yescharacteristic
590C2×C4C22×C22⋊D6111yescharacteristic
591C2×C4#41×D6112yes—
592C2×C4#1451982312yes—
593C8—2442no—
594C8—2442no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—22no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22#1683991314yes—
610C22C2×#1451974611yescharacteristic
611C22#1684040012yes—
612C22C22×#1175820011yescharacteristic
613C22C2×#1451980311yescharacteristic
614C22#1684057712yes—
615C4—22no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—4882no—
623C4—2444no—
624C4#1684026911yescharacteristic
625C4#1684032611yescharacteristic
626C4#1684040012yes—
627C2—24no—
628C2—42no—
629C2—82no—
630C2#2144913411yescharacteristic
631C2C2×#1683991311yescharacteristic
632C2#2145511411yescharacteristic
633C2#2145767812yes—
634C2#2145780412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#547C2
socleC23×C61C22×D4
hypercenterC2×#547D61
hypocenterC61#23961
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16839900C2semidirect
2. C22×#14519700C2semidirect
3. #21449184C2semidirect
4. #21446957C2semidirect
5. #21446822C2semidirect
6. #21446248C2semidirect
7. #21446153C2semidirect
8. #21446126C2semidirect
9. #21446112C2semidirect
10. #21445905C2semidirect
137. C2×D4×C61SD16semidirect
178. C2×Dic122#41semidirect
364. C61#23961semidirect
466. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
C22×#145197002
#214491842
#214469572
#214468222
#214462482
#214461532
#214461262
#214461122
#214459052
C2×C61×#5472
#2396161