Finite Group Databasebeta

← #29828814

Subgroups — #29828814

Order 15616.

1576 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519742C211yesmaximal, characteristic, complemented
2C2×#16839961C212yesmaximal, complemented
3C2×#16839903C211yesmaximal, characteristic, complemented
4C2×#16839900C211yesmaximal, characteristic, complemented
5#21454162C211yesmaximal, characteristic
6#21446175C212yesmaximal, complemented
7#21446171C212yesmaximal, complemented
8#21446136C211yesmaximal, characteristic, complemented
9#21445823C212yesmaximal, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×#11758190C2212yes—
14C22×#11758200C2212yes—
15C2×#14519773C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17C2×#14519749—21no—
18C2×#14519749—21no—
19C2×#14519742—28no—
20C2×#14519742—21no—
21C2×#14519742—21no—
22C2×#14519742C2212yes—
23C2×#14519742C2212yes—
24C22×#11758196—21no—
25C22×C4×Dic61—21no—
26C22×C4×Dic61C2211yescharacteristic
27#16839961—28no—
28#16839903—24no—
29#16839900—24no—
30C2×#14519706C2212yes—
31C2×#14519702C2212yes—
32C2×#14519700C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839662—21no—
38#16839662—21no—
39#16839631C2211yescharacteristic
40#16839524C2212yes—
41#16839516C2212yes—
42C2×#121×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—22no—
49C22×Dic122—24no—
50C22×Dic122—24no—
51C22×Dic122D414yescomplemented
52C22×Dic122C2312yes—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—21no—
56C22×C22⋊D61—21no—
57C23×Dic61—21no—
58C23×Dic61D411yescharacteristic
59C23×Dic61D411yescharacteristic
60C2×#11758190—24no—
61C2×#11758190—24no—
62C2×#11758190—24no—
63C2×#11758200—24no—
64C2×#11758200—24no—
65C2×#11758200—24no—
66C22×#92418C2312yes—
67#14519773—24no—
68#14519751—22no—
69#14519751—22no—
70#14519749—44no—
71#14519749—44no—
72#14519742—28no—
73#14519742—28no—
74#14519742—24no—
75#14519742—24no—
76#14519742—24no—
77#14519742D418yescomplemented
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—28no—
80C2×#11758196—22no—
81C2×#11758196—22no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—28no—
90C2×C4×Dic61—22no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61—24no—
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519706—24no—
98#14519702—24no—
99#14519700—24no—
100#14519697—24no—
101C22⋊#92418—22no—
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61C2312yes—
107#121×C61—22no—
108#121×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—24no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114#102×C61—24no—
115C22⋊C8×C61—22no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—41no—
156C22×Dic61SD1614yescomplemented
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—22no—
165C2×Dic122—24no—
166C2×Dic122—216no—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—216no—
173C2×Dic122C2×D414yes—
174C2×Dic122C2×D414yes—
175C2×Dic122#4114yes—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D414yes—
184C22⋊C4×C61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187#11758200—48no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—28no—
194#11758196—28no—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—28no—
198C4×Dic61—22no—
199C4×Dic61—28no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C2×D414yes—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206#11758190—48no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61#12112yes—
219C23×C61C2×SD1612yes—
220C23×C61AGL(1,ℤ/8)12yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—84no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—48no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61#12112yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D412yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—84no—
249C22⋊D61—84no—
250C22⋊D61—84no—
251C2×Dic61—216no—
252C2×Dic61—21no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—28no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—44no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61#12012yes—
268C2×Dic61#12012yes—
269C2×Dic61C2×SD1612yes—
270C2×Dic61C2×SD1612yes—
271C2×Dic61#13712yes—
272D244—42no—
273D244—42no—
274Dic122—216no—
275Dic122—24no—
276Dic122—24no—
277Dic122—24no—
278Dic122#12118yescomplemented
279C8×C61—42no—
280#92418—42no—
281#32903—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61#46014yescomplemented
296C22×C61C2×#12011yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61C22×SD1611yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61C2×#13711yescharacteristic
302C22×C61#57511yescharacteristic
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—22no—
311C4×C61—44no—
312C4×C61—81no—
313C4×C61C2×#12112yes—
314C4×C61#52512yes—
315Dic61—28no—
316Dic61—22no—
317Dic61—22no—
318Dic61—41no—
319Dic61—81no—
320Dic61#44814yescomplemented
321C22×C4×Q8—611no—
322C2×#460—612no—
323C2×#448—611no—
324C2×#448—611no—
325#2856—611no—
326#1880—612no—
327#1880—612no—
328#1874—611no—
329#1874D6111yescharacteristic, complemented
330#1803—612no—
331#1797—611no—
332C2×C61—24no—
333C2×C61—161no—
334C2×C61#284711yescharacteristic
335C2×C61C2×#44811yescharacteristic
336C2×C61#300611yescharacteristic
337C2×C61C2×#46011yescharacteristic
338C2×C61#303711yescharacteristic
339C2×C61#310311yescharacteristic
340C2×C61#311211yescharacteristic
341D61—161no—
342C23×Q8—611no—
343C22×SD16—612no—
344C22×SD16—612no—
345C2×#127—611no—
346C2×#121—611no—
347C2×#121—1221no—
348C2×#121—1221no—
349C2×#121C2×D6111yescharacteristic
350C2×C4×Q8—612no—
351C2×C4×Q8—612no—
352C2×C4×Q8—1221no—
353C2×C4×Q8—1221no—
354C2×C4×Q8—1228no—
355C22×C4⋊C4—1221no—
356C22×C4⋊C4C2×D6111yescharacteristic
357C22×C42—611no—
358C22×C42—1221no—
359#460—1228no—
360#448—1224no—
361#448—1224no—
362C2×C4⋊C8—612no—
363C2×#103—612no—
364C2×#103—612no—
365C2×#102—612no—
366C2×#102C2×D6112yes—
367C2×C22⋊C8—611no—
368C2×C22⋊C8C2×D6111yescharacteristic
369#390—611no—
370#390—611no—
371#390—1221no—
372#390—1221no—
373#384—611no—
374#340—612no—
375#340C2×D6112yes—
376#336—612no—
377C61#3290311yescharacteristic, complemented, Sylow
378C22×Q8—614no—
379C22×Q8—612no—
380C22×Q8—1224no—
381C22×Q8—1224no—
382C22×Q8—1222no—
383C22×Q8—1221no—
384C22×D4—21no—
385C22×D4—611no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4—1221no—
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—1221no—
394C23×C4C22×D6111yescharacteristic
395C2×SD16—1224no—
396C2×SD16—1224no—
397C2×SD16—1224no—
398C2×SD16—1224no—
399C2×SD16—1224no—
400C2×SD16—1224no—
401C22×C8—612no—
402C22×C8C22×D6112yes—
403#127—1224no—
404#121—22no—
405#121—22no—
406#121—1222no—
407#121—1222no—
408#121—2444no—
409#121—2444no—
410C4×Q8—618no—
411C4×Q8—1224no—
412C4×Q8—1224no—
413C4×Q8—1228no—
414C4×Q8—1228no—
415C4×Q8—1224no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1228no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6112yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1224no—
437C2×C42—1222no—
438C2×C42—1222no—
439C2×C42—1228no—
440C4⋊C8—1224no—
441#103—1224no—
442#103—1224no—
443#102—24no—
444#102—1224no—
445C22⋊C8—22no—
446C22⋊C8—1222no—
447C24—21no—
448C24—1221no—
449C24D4×D6111yescharacteristic
450C2×Q8—614no—
451C2×Q8—614no—
452C2×Q8—614no—
453C2×Q8—12216no—
454C2×Q8—1224no—
455C2×Q8—1224no—
456C2×Q8—12216no—
457C2×Q8—1222no—
458C2×Q8—1222no—
459C2×Q8—1222no—
460C2×Q8—1224no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—1222no—
469C2×D4—1221no—
470C2×D4—1222no—
471C2×D4—1221no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4—2442no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—614no—
492C22×C4—611no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1222no—
499C22×C4—1224no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—2441no—
503C22×C4#1175822212yes—
504C22×C4D4×D6112yes—
505C22×C4D4×D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507SD16—2448no—
508SD16—2448no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—1228no—
520C4⋊C4—1228no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4D4×D6114yes—
526C22⋊C4—42no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—614no—
533C42—1222no—
534C42—1222no—
535C42—1228no—
536C42—1228no—
537C42—1222no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23#1451975012yes—
557C23SD16×D6112yes—
558C23#1451980112yes—
559C23C2×D4×D6111yescharacteristic
560Q8—618no—
561Q8—12216no—
562Q8—1224no—
563Q8—1224no—
564Q8—1224no—
565D4—42no—
566D4—42no—
567D4—84no—
568D4—4884no—
569D4—4884no—
570D4—4884no—
571D4—2442no—
572D4—2442no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—1228no—
590C2×C4—12216no—
591C2×C4—1221no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—2444no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#1451975012yes—
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6112yes—
608C2×C4#41×D6112yes—
609C2×C4C2×#1175822211yescharacteristic
610C8—42no—
611C8—2442no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22#1683996114yescomplemented
633C22#120×D6111yescharacteristic
634C22C2×#1451975011yescharacteristic
635C22#1684027011yescharacteristic
636C22C2×SD16×D6111yescharacteristic
637C22C2×#1451980111yescharacteristic
638C22#137×D6111yescharacteristic
639C22#1684055811yescharacteristic
640C4—22no—
641C4—44no—
642C4—81no—
643C4—614no—
644C4—1228no—
645C4—1222no—
646C4—1222no—
647C4—4881no—
648C4—2441no—
649C4#121×D6112yes—
650C4#1684027112yes—
651C2—24no—
652C2—161no—
653C2—9761no—
654C2#2145410011yescharacteristic
655C2#448×D6111yescharacteristic
656C2#2145506611yescharacteristic
657C2C2×#1683996111yescharacteristic
658C2#2145529011yescharacteristic
659C2#2145584711yescharacteristic
660C2#2145593811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radical#21454162C2
hypercenter#1874D61
hypocenterC61#32903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#14519742C2semidirect
2. C2×#16839961C2semidirect
3. C2×#16839903C2semidirect
4. C2×#16839900C2semidirect
6. #21446175C2semidirect
7. #21446171C2semidirect
8. #21446136C2semidirect
9. #21445823C2semidirect
10. #21445797C2semidirect
11. C61×#1874C2semidirect
51. C22×Dic122D4semidirect
77. #14519742D4semidirect
156. C22×Dic61SD16semidirect
278. Dic122#121semidirect
295. C22×C61#460semidirect
320. Dic61#448semidirect
329. #1874D61semidirect
377. C61#32903semidirect
632. C22#16839961semidirect

Maximal subgroups

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

subgroupindex
C22×#145197422
C2×#168399612
C2×#168399032
C2×#168399002
#214541622
#214461752
#214461712
#214461362
#214458232
#214457972
C61×#18742
#3290361