Finite Group Databasebeta

← #29681200

Subgroups — #29681200

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840528C211yesmaximal, characteristic, complemented
2#21459152C211yesmaximal, characteristic
3C2×#16839780C211yesmaximal, characteristic, complemented
4C22×#14519703C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic, complemented
6#21446120C218yesmaximal, complemented
7C2×#16839521C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758222C2211yescharacteristic
11#16840528—24no—
12#16840528—24no—
13C2×#14519785—22no—
14C2×#14519783—22no—
15C22×#11758194C2211yescharacteristic
16C2×M16×D61C2214yescomplemented
17#16840343C414yescomplemented
18#16840338—21no—
19#16840338C412yescomplemented
20#16840331—21no—
21#16840331C412yescomplemented
22#16840324—22no—
23#16840312—22no—
24#16840274—22no—
25#16840274C2214yes—
26C22×C4×Dic61C2211yescharacteristic
27#16839780—24no—
28C2×#14519705C2214yes—
29C2×#14519703—24no—
30C2×#14519703C2214yes—
31#16839667—21no—
32#16839663—21no—
33#16839639—22no—
34C4×#11758198C2214yescomplemented
35#16839521C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#339C2214yescomplemented
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222C2×C412yes—
51C2×#11758222C2×C412yes—
52C2×#11758222C2×C412yes—
53C2×#11758222D412yes—
54C2×#11758222D412yes—
55C2×#11758222C2312yes—
56C22×Dic122C2×C411yescharacteristic
57C22×Dic122D411yescharacteristic
58C22×D244C2×C411yescharacteristic
59C22×D244D411yescharacteristic
60C22×C22⋊D61—21no—
61C22×C22⋊D61D412yes—
62C23×Dic61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2312yes—
65#14519785—216no—
66#14519783—216no—
67C2×#11758194—21no—
68C2×#11758194—22no—
69C2×#11758194D414yescomplemented
70C2×#11758194C2312yes—
71C2×#11758189—24no—
72M16×D61—28no—
73M16×D61—28no—
74C22×#92418—21no—
75C2×C8×D61—24no—
76#14519777—22no—
77#14519774—22no—
78#14519765—22no—
79#14519765—22no—
80#14519761—24no—
81#14519761—24no—
82#14519757—24no—
83#14519757C2×C418yescomplemented
84#14519752—24no—
85#14519752C2×C418yescomplemented
86#14519751—28no—
87#14519747—24no—
88#14519747—24no—
89C4×D244—28no—
90C2×#11758198—24no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61—22no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2312yes—
95#14519705—28no—
96#14519703—28no—
97#14519684—24no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—21no—
100C2×M16×C61—22no—
101C2×M16×C61D414yescomplemented
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108#117×C61C2×C412yes—
109#117×C61C2312yes—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2×C412yes—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61C2312yes—
116C24×C61C2×D411yescharacteristic
117C23×D61—21no—
118C23×D61C2×D412yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61C22⋊C414yes—
138C2×C22⋊D61C22⋊C414yes—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61—42no—
143C22×Dic61C2×D412yes—
144#11758222—24no—
145#11758222—24no—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—24no—
153#11758222—24no—
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—22no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—21no—
167C2×D244—28no—
168C2×D244C22⋊C412yes—
169C2×D244C22⋊C412yes—
170C2×D244C22×C412yes—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61C22×C414yes—
180C2×C4×D61C2×D414yes—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—22no—
185C2×Dic122—21no—
186C2×Dic122C22⋊C412yes—
187C2×Dic122C22⋊C412yes—
188C2×Dic122C22×C412yes—
189M16×C61—24no—
190M16×C61—22no—
191M16×C61C2×D414yes—
192C2×C8×C61—24no—
193C2×C8×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—42no—
196C4⋊C4×C61C22×C414yes—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—42no—
200C42×C61—22no—
201C42×C61—22no—
202C42×C61—22no—
203C42×C61—22no—
204#11758198—28no—
205#11758198—28no—
206#11758198—48no—
207C4⋊Dic61—28no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—42no—
211#11758194—24no—
212#11758194—22no—
213#11758194C2×D414yes—
214C2×#92418—24no—
215C2×#92418—22no—
216#11758189—216no—
217C8×D61—216no—
218C23×C61—42no—
219C23×C61C2×C22⋊C411yescharacteristic
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61#12112yes—
222C23×C61#12112yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—44no—
228C22×D61—42no—
229C22×D61—82no—
230C22×D61C2×C22⋊C414yes—
231C2×C4×C61—24no—
232C2×C4×C61—44no—
233C2×C4×C61—42no—
234C2×C4×C61—82no—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C4×D412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12012yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12312yes—
246C2×C4×C61#12312yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C22⋊D61—28no—
254C22⋊D61—28no—
255C22⋊D61—42no—
256C22⋊D61—42no—
257C22⋊D61—44no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—42no—
263C2×Dic61—48no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61C2×C22⋊C414yes—
267D244—22no—
268D244—22no—
269D244—24no—
270D244—24no—
271D244—42no—
272D244—88no—
273C4×D61—24no—
274C4×D61—24no—
275C4×D61—24no—
276C4×D61—24no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—44no—
280Dic122—24no—
281Dic122—24no—
282Dic122—22no—
283Dic122—22no—
284Dic122—42no—
285C8×C61—24no—
286#92418—24no—
287#20903—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61#38612yes—
294C22×C61#38612yes—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C2×#12111yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C2×D61—28no—
299C2×D61—24no—
300C2×D61—44no—
301C2×D61—42no—
302C2×D61—82no—
303C2×D61—82no—
304C2×D61—84no—
305C4×C61—44no—
306C4×C61—82no—
307C4×C61#38612yes—
308C4×C61#38612yes—
309C4×C61C2×C4×D411yescharacteristic
310C4×C61C2×C4×D411yescharacteristic
311C4×C61C2×#12011yescharacteristic
312C4×C61C2×#12311yescharacteristic
313Dic61—24no—
314Dic61—44no—
315Dic61—42no—
316C2×#568—611no—
317#3449—611no—
318C2×#421—611no—
319C22×#104—611no—
320#2297—611no—
321#1871—618no—
322#1733D6111yescharacteristic, complemented
323C2×#339—611no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—22no—
327C2×C61—22no—
328C2×C61C2×#38611yescharacteristic
329C2×C61#290112yes—
330D61—44no—
331D61—82no—
332C22×#41—611no—
333C22×#41—611no—
334#568—1224no—
335#568—1224no—
336C2×#131—1222no—
337C2×#131—1222no—
338C22×M16—611no—
339C22×M16—614no—
340C22×M16C2×D6111yescharacteristic
341#537—614no—
342#536—612no—
343#536—1221no—
344#535—612no—
345#535—1221no—
346#534—1222no—
347#532—1222no—
348#525—614no—
349#525—1222no—
350C2×#117C2×D6111yescharacteristic
351C2×#117C2×D6111yescharacteristic
352C22×C42—611no—
353#421—1224no—
354C2×#104—614no—
355C2×#104—614no—
356C2×#104—1224no—
357#391—1221no—
358#390—1221no—
359#386—1222no—
360C4×C22⋊C4—614no—
361#339—614no—
362#339C2×D6114yescomplemented
363C61#2090311yescharacteristic, complemented, Sylow
364C2×#41—612no—
365C2×#41—612no—
366C2×#41—612no—
367C2×#41—612no—
368C2×#41—612no—
369C2×#41—612no—
370C2×#41—1222no—
371C2×#41—1222no—
372C2×#41—1222no—
373C2×#41—1221no—
374C2×#41—1221no—
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C22×Q8—611no—
381C22×Q8—611no—
382C22×D4—612no—
383C22×D4—611no—
384C22×D4—611no—
385C22×D4—1221no—
386C23×C4—612no—
387C23×C4—1221no—
388C23×C4—1221no—
389C23×C4C22×D6111yescharacteristic
390#131—12216no—
391#131—12216no—
392C2×M16—21no—
393C2×M16—22no—
394C2×M16—614no—
395C2×M16—612no—
396C2×M16—1224no—
397C2×M16—1228no—
398C2×M16—1222no—
399C2×M16—1221no—
400C2×M16—1228no—
401C2×M16D24414yescomplemented
402C2×M16C22×D6112yes—
403C22×C8—21no—
404C22×C8—1224no—
405C22×C8—1221no—
406C4⋊Q8—1222no—
407#127—1222no—
408#124—1222no—
409#124—1222no—
410#123—1224no—
411#123—1224no—
412C22⋊Q8—618no—
413C22⋊Q8—1224no—
414#121—618no—
415#121—1224no—
416#121—1228no—
417#120—1224no—
418#120—1224no—
419C4×D4—1228no—
420#117—21no—
421#117—21no—
422#117—22no—
423#117—22no—
424#117C4×D6112yes—
425#117C22×D6112yes—
426C2×C4⋊C4—21no—
427C2×C4⋊C4C4×D6112yes—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1224no—
431C2×C42—21no—
432C2×C42—612no—
433C2×C42—1222no—
434C2×C42—1222no—
435C2×C42—1221no—
436C2×C42C22×D6112yes—
437#104—1228no—
438#104—1228no—
439#95—1224no—
440C24—612no—
441C24—1221no—
442C24D4×D6111yescharacteristic
443#41—1224no—
444#41—1224no—
445#41—1224no—
446#41—1224no—
447#41—1224no—
448#41—1224no—
449#41—1224no—
450#41—1224no—
451#41—1222no—
452#41—1222no—
453#41—1222no—
454#41—1222no—
455#41—1224no—
456#41—1224no—
457#41—2442no—
458#41—2442no—
459#41—2442no—
460#41—2442no—
461C2×Q8—612no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—1222no—
469C2×D4—614no—
470C2×D4—614no—
471C2×D4—612no—
472C2×D4—612no—
473C2×D4—612no—
474C2×D4—1228no—
475C2×D4—1224no—
476C2×D4—1224no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—2442no—
485C2×D4—2442no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—612no—
489C22×C4—614no—
490C22×C4—614no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—2444no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×D24412yes—
506C22×C4#1175822212yes—
507C22×C4D4×D6112yes—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4#1175822612yes—
512C22×C4C23×D6111yescharacteristic
513M16—22no—
514M16—24no—
515M16—614no—
516M16—12216no—
517M16—1222no—
518M16—1224no—
519M16C2×D24414yes—
520C2×C8—22no—
521C2×C8—24no—
522C2×C8—12216no—
523C2×C8—1222no—
524C2×C8—1224no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—1228no—
528C4⋊C4C2×C4×D6114yes—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—42no—
532C22⋊C4—1228no—
533C22⋊C4—1228no—
534C22⋊C4—2448no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—1222no—
540C42—1222no—
541C42—2442no—
542C23—42no—
543C23—614no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—4882no—
548C23—2444no—
549C23—2442no—
550C23C22⋊C4×D6111yescharacteristic
551C23C22⋊C4×D6111yescharacteristic
552C23#1451975012yes—
553C23#1451975212yes—
554C23C2×D4×D6111yescharacteristic
555Q8—1224no—
556Q8—1224no—
557Q8—1222no—
558Q8—1222no—
559Q8—2442no—
560D4—1228no—
561D4—1228no—
562D4—1222no—
563D4—1222no—
564D4—1224no—
565D4—1224no—
566D4—4888no—
567D4—2442no—
568D4—2442no—
569D4—2444no—
570D4—2442no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—614no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1222no—
584C2×C4—2442no—
585C2×C4—2448no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2444no—
591C2×C4C22⋊C4×D6111yescharacteristic
592C2×C4C22⋊C4×D6111yescharacteristic
593C2×C4#1451973912yes—
594C2×C4#1451973912yes—
595C2×C4C4×D24412yes—
596C2×C4C4×D24412yes—
597C2×C4#1451974712yes—
598C2×C4#1451974712yes—
599C2×C4#1451975012yes—
600C2×C4#1451975212yes—
601C2×C4#1451976212yes—
602C2×C4#1451976212yes—
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C22×D24411yescharacteristic
607C2×C4C2×#1175822211yescharacteristic
608C2×C4C2×#1175822611yescharacteristic
609C8—24no—
610C8—1224no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—42no—
615C22—42no—
616C22—1228no—
617C22—1224no—
618C22—4884no—
619C22—4882no—
620C22—4882no—
621C22—2444no—
622C22—2442no—
623C22#1683964012yes—
624C22#1683964012yes—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22C2×#1451975011yescharacteristic
627C22C2×#1451975211yescharacteristic
628C4—44no—
629C4—82no—
630C4—1224no—
631C4—2444no—
632C4—2442no—
633C4#1683964012yes—
634C4#1683964012yes—
635C4C2×#1451973911yescharacteristic
636C4C2×C4×D24411yescharacteristic
637C4C2×#1451974711yescharacteristic
638C4C2×#1451976211yescharacteristic
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—4882no—
644C2—2444no—
645C2C2×#1683964011yescharacteristic
646C2#2145436512yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C22×C22⋊C4
prime radical#21459152C2
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20903
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. C2×#16840528C2semidirect
3. C2×#16839780C2semidirect
4. C22×#14519703C2semidirect
5. #21448307C2semidirect
6. #21446120C2semidirect
7. C2×#16839521C2semidirect
8. C61×#1733C2semidirect
16. C2×M16×D61C22semidirect
17. #16840343C4semidirect
19. #16840338C4semidirect
21. #16840331C4semidirect
34. C4×#11758198C22semidirect
35. #16839521C22semidirect
39. C61×#339C22semidirect
69. C2×#11758194D4semidirect
83. #14519757C2×C4semidirect
85. #14519752C2×C4semidirect
101. C2×M16×C61D4semidirect
322. #1733D61semidirect
362. #339C2×D61semidirect
363. C61#20903semidirect
401. C2×M16D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168405282
#214591522
C2×#168397802
C22×#145197032
#214483072
#214461202
C2×#168395212
C61×#17332
#2090361