Finite Group Databasebeta

← #29681289

Subgroups — #29681289

Order 15616.

1842 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
1C2×#16840343C211yesmaximal, characteristic
2C2×#16840242C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C2×#16839730C211yesmaximal, characteristic, complemented
6#21446140C212yesmaximal, complemented
7#21446135C212yesmaximal, complemented
8#21446130C212yesmaximal, complemented
9#21446125C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758222C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840343—24no—
16#16840343C418yescomplemented
17C2×#14519757—21no—
18C2×#14519757C2212yes—
19C2×#14519752—21no—
20C2×#14519752C2212yes—
21C2×#14519751—22no—
22#16840242—24no—
23#16840242—24no—
24C2×C4×Dic122—21no—
25C2×C4×D244—21no—
26C2×#14519737—22no—
27C2×#14519735C2212yes—
28C22×C4⋊Dic61C2211yescharacteristic
29C2×C4⋊C4×D61C2212yes—
30#16839764—24no—
31#16839756—24no—
32C2×#14519702C2212yes—
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519699C2212yes—
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38#16839730—24no—
39C2×#14519689C2212yes—
40C2×#14519689C2212yes—
41#16839524C2212yes—
42#16839524C2212yes—
43C22×M16×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C22×C4⋊C4×C61C2211yescharacteristic
46C61×#340C2212yes—
47C61×#340C2212yes—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222—22no—
56C2×#11758222—22no—
57C2×#11758222C2×C414yes—
58C2×#11758222C2×C414yescomplemented
59C2×#11758222D414yes—
60C22×Dic122D411yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244—22no—
63C22×D244D411yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C22⋊D61—21no—
66C22×C22⋊D61D412yes—
67C23×Dic61—21no—
68C22×C4×D61—21no—
69C22×C4×D61—22no—
70C22×C4×D61C2312yes—
71C22×#92418—22no—
72C22×#92418—22no—
73C22×#92418—22no—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519757—24no—
77#14519757—28no—
78#14519757C2×C418yescomplemented
79#14519752—28no—
80#14519752—24no—
81#14519752C2×C418yescomplemented
82#14519751—216no—
83C4×Dic122—28no—
84C4×D244—28no—
85#14519737—216no—
86#14519735—24no—
87#14519735—24no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—22no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61D414yescomplemented
92C2×C4⋊Dic61C2312yes—
93C2×#11758196—22no—
94C4⋊C4×D61—24no—
95C4⋊C4×D61—24no—
96C2×#11758198—22no—
97C2×#11758198—22no—
98C2×#11758198—22no—
99C2×C4×Dic61—22no—
100#14519702—24no—
101#14519700—22no—
102#14519700—22no—
103#14519699—24no—
104#14519697—22no—
105#14519697—22no—
106#14519689—24no—
107#14519689—24no—
108C23×C4×C61—21no—
109C23×C4×C61C2311yescharacteristic
110C2×M16×C61—22no—
111C2×M16×C61—22no—
112C2×M16×C61—22no—
113C22×C8×C61C2312yes—
114#117×C61—22no—
115#117×C61D414yescomplemented
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61C2×C414yescomplemented
119C2×C4⋊C4×C61C2312yes—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C42×C61—21no—
124C24×C61C2×D411yescharacteristic
125C23×D61—21no—
126C23×D61C2×D412yes—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—24no—
131C22×C4×C61—81no—
132C22×C4×C61—81no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61C22⋊C418yes—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—24no—
150C22×Dic61—81no—
151C22×Dic61—81no—
152C22×Dic61C2×D412yes—
153#11758222—28no—
154#11758222—28no—
155#11758222—28no—
156#11758222—28no—
157#11758222—28no—
158#11758222—28no—
159#11758222—44no—
160#11758222—44no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—28no—
169C2×D244—82no—
170C2×D244—82no—
171C2×D244C22⋊C414yes—
172C2×D244C22×C412yes—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—24no—
178C2×C4×D61—28no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×C4×D61—82no—
182C2×C4×D61—82no—
183C2×C4×D61C22×C414yes—
184C2×C4×D61C2×D414yes—
185C2×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122C22⋊C414yes—
193C2×Dic122C22×C412yes—
194M16×C61—44no—
195C2×C8×C61—22no—
196C2×C8×C61—22no—
197C2×C8×C61—22no—
198C4⋊C4×C61—24no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61C22×C414yes—
203C4⋊C4×C61C2×D412yes—
204C4⋊C4×C61C2×D412yes—
205C22⋊C4×C61—24no—
206C42×C61—24no—
207#11758198—28no—
208#11758198—28no—
209#11758198—28no—
210C4⋊Dic61—24no—
211C4⋊Dic61—22no—
212C4⋊Dic61—28no—
213C4⋊Dic61C2×D414yes—
214#11758196—28no—
215C4×Dic61—28no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—44no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61C2×C22⋊C412yes—
228C23×C61#12012yes—
229C23×C61#12312yes—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61C2×C22⋊C414yes—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—84no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C4×D414yes—
257C2×C4×C61C4×D414yes—
258C2×C4×C61#12012yes—
259C2×C4×C61#12114yes—
260C2×C4×C61#12114yes—
261C2×C4×C61#12312yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—216no—
269C22⋊D61—48no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—24no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—84no—
279C2×Dic61C2×C22⋊C414yes—
280D244—22no—
281D244—22no—
282D244—28no—
283D244—41no—
284D244—41no—
285D244—88no—
286C4×D61—24no—
287C4×D61—24no—
288C4×D61—24no—
289C4×D61—24no—
290C4×D61—48no—
291C4×D61—88no—
292Dic122—28no—
293Dic122—22no—
294Dic122—22no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—42no—
298#92418—41no—
299#92418—41no—
300#20915—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61#38614yes—
308C22×C61C22×C22⋊C411yescharacteristic
309C22×C61C2×#12011yescharacteristic
310C22×C61C2×#12311yescharacteristic
311C22×C61#57611yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C2×D61—24no—
316C2×D61—24no—
317C2×D61—24no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—81no—
326C2×D61—81no—
327C2×D61—81no—
328C2×D61—81no—
329C4×C61—44no—
330C4×C61—42no—
331C4×C61—42no—
332C4×C61#38614yes—
333C4×C61C2×C4×D411yescharacteristic
334C4×C61C2×C4×D411yescharacteristic
335C4×C61C2×#12111yescharacteristic
336C4×C61C2×#12111yescharacteristic
337Dic61—24no—
338Dic61—44no—
339Dic61—42no—
340C2×#537—611no—
341C2×#520—611no—
342C2×#417—611no—
343C2×#416—611no—
344C2×#408—611no—
345#1875—612no—
346#1874—612no—
347#1873—612no—
348#1872—612no—
349#1736D6111yescharacteristic, complemented
350C2×#340—611no—
351C2×C61—24no—
352C2×C61C2×#38611yescharacteristic
353C2×C61#289711yescharacteristic
354C2×C61#289811yescharacteristic
355C2×C61#301211yescharacteristic
356C2×C61#301311yescharacteristic
357C2×C61#309811yescharacteristic
358C2×C61#309911yescharacteristic
359D61—44no—
360D61—82no—
361C22×#41—611no—
362C22×#41—611no—
363C22×M16C2×D6111yescharacteristic
364C23×C8—611no—
365#537—618no—
366#537—1224no—
367C2×C22⋊Q8—612no—
368C2×C22⋊Q8—1221no—
369C2×#121—612no—
370C2×#121—1221no—
371C2×#121—1222no—
372#520—1224no—
373#520—1224no—
374C2×C4×Q8—1221no—
375C2×C4×D4—1221no—
376C2×C4×D4—1222no—
377C2×#117—612no—
378C2×#117C2×D6111yescharacteristic
379C22×C4⋊C4—611no—
380C22×C4⋊C4—612no—
381C22×C4⋊C4C2×D6111yescharacteristic
382#417—1224no—
383#416—1224no—
384C2×#103—612no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#102—612no—
388C2×#102—611no—
389C2×#102—611no—
390#408—1224no—
391C2×C22⋊C8—612no—
392C2×C22⋊C8—612no—
393#340—612no—
394#340—612no—
395#340C2×D6112yes—
396#340C2×D6112yes—
397C61#2091511yescharacteristic, complemented, Sylow
398C2×#41—614no—
399C2×#41—614no—
400C2×#41—614no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—1222no—
404C2×#41—1222no—
405C2×#41—1222no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—1222no—
410C22×Q8—611no—
411C22×Q8—611no—
412C22×D4—612no—
413C22×D4—611no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4—1221no—
417C23×C4—21no—
418C23×C4—612no—
419C23×C4—1222no—
420C23×C4—1221no—
421C23×C4—1221no—
422C23×C4C22×D6111yescharacteristic
423C2×M16—22no—
424C2×M16—22no—
425C2×M16—22no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8—1222no—
429C22×C8—1222no—
430C22×C8—1222no—
431C22×C8C22×D6112yes—
432C22⋊Q8—618no—
433C22⋊Q8—1224no—
434C22⋊Q8—1228no—
435#121—618no—
436#121—1224no—
437#121—1228no—
438#121—12216no—
439C4×Q8—1228no—
440C4×D4—1228no—
441C4×D4—12216no—
442#117—22no—
443#117—1224no—
444#117—1224no—
445#117D24414yescomplemented
446C2×C4⋊C4—21no—
447C2×C4⋊C4—22no—
448C2×C4⋊C4—614no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—1224no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1224no—
456C2×C4⋊C4C4×D6114yescomplemented
457C2×C4⋊C4C22×D6112yes—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C22⋊C4—21no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4—1222no—
463C2×C22⋊C4—1222no—
464C2×C42—21no—
465C2×C42—1222no—
466#103—1224no—
467#103—1222no—
468#103—1222no—
469#102—1224no—
470#102—1222no—
471#102—1222no—
472C22⋊C8—1224no—
473C22⋊C8—1224no—
474C24—612no—
475C24—1221no—
476C24D4×D6111yescharacteristic
477#41—1228no—
478#41—1228no—
479#41—1228no—
480#41—1228no—
481#41—1228no—
482#41—1228no—
483#41—2444no—
484#41—2444no—
485C2×Q8—614no—
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1222no—
489C2×Q8—1222no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×D4—618no—
495C2×D4—612no—
496C2×D4—614no—
497C2×D4—1228no—
498C2×D4—1224no—
499C2×D4—1224no—
500C2×D4—1224no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1222no—
507C2×D4—1222no—
508C2×D4—4882no—
509C2×D4—4882no—
510C2×D4—2442no—
511C2×D4—2442no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—24no—
516C22×C4—81no—
517C22×C4—81no—
518C22×C4—612no—
519C22×C4—614no—
520C22×C4—614no—
521C22×C4—1228no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1224no—
527C22×C4—1222no—
528C22×C4—1221no—
529C22×C4—1224no—
530C22×C4—4882no—
531C22×C4—4882no—
532C22×C4—4881no—
533C22×C4—4881no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4C2×C4×D6112yes—
537C22×C4C2×D24412yes—
538C22×C4#1175822212yes—
539C22×C4D4×D6112yes—
540C22×C4D4×D6112yes—
541C22×C4D4×D6111yescharacteristic
542C22×C4#1175822612yes—
543C22×C4C23×D6111yescharacteristic
544M16—44no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—2444no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—24no—
559C4⋊C4—614no—
560C4⋊C4—1228no—
561C4⋊C4—1228no—
562C4⋊C4—1222no—
563C4⋊C4—1224no—
564C4⋊C4C2×C4×D6114yes—
565C4⋊C4C2×D24412yes—
566C4⋊C4C2×D24412yes—
567C22⋊C4—24no—
568C22⋊C4—1228no—
569C22⋊C4—1228no—
570C22⋊C4—1228no—
571C42—24no—
572C42—1228no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—614no—
578C23—1222no—
579C23—1222no—
580C23—1222no—
581C23—1222no—
582C23—1222no—
583C23—1222no—
584C23—1222no—
585C23—2441no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23C22⋊C4×D6112yes—
592C23#1451974712yes—
593C23#1451976212yes—
594C23C2×D4×D6111yescharacteristic
595Q8—1228no—
596Q8—1222no—
597Q8—1222no—
598Q8—2441no—
599Q8—2441no—
600D4—12216no—
601D4—1222no—
602D4—1222no—
603D4—1228no—
604D4—4888no—
605D4—2448no—
606D4—2441no—
607D4—2441no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—84no—
618C2×C4—614no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1224no—
622C2×C4—1224no—
623C2×C4—1224no—
624C2×C4—1224no—
625C2×C4—1222no—
626C2×C4—1224no—
627C2×C4—4888no—
628C2×C4—4884no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2448no—
634C2×C4C22⋊C4×D6112yes—
635C2×C4#1451973914yes—
636C2×C4C4×D24414yes—
637C2×C4#1451974712yes—
638C2×C4#1451975014yes—
639C2×C4#1451975214yes—
640C2×C4#1451976212yes—
641C2×C4C22×C4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4C2×D4×D6111yescharacteristic
644C2×C4C22×D24411yescharacteristic
645C2×C4C2×#1175822211yescharacteristic
646C2×C4C2×#1175822611yescharacteristic
647C8—42no—
648C8—2441no—
649C8—2441no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—1224no—
657C22—1224no—
658C22—1224no—
659C22—4881no—
660C22—4881no—
661C22—4881no—
662C22—4881no—
663C22—2442no—
664C22—2442no—
665C22—2442no—
666C22—2442no—
667C22—2442no—
668C22—2442no—
669C22—2442no—
670C22#1683964014yes—
671C22C2×C22⋊C4×D6111yescharacteristic
672C22C2×#1451974711yescharacteristic
673C22C2×#1451976211yescharacteristic
674C22#1684056611yescharacteristic
675C22#1684057111yescharacteristic
676C22#1684057211yescharacteristic
677C22#1684057811yescharacteristic
678C4—44no—
679C4—42no—
680C4—42no—
681C4—1224no—
682C4—2444no—
683C4—2442no—
684C4#1683964014yes—
685C4C2×#1451973911yescharacteristic
686C4C2×C4×D24411yescharacteristic
687C4C2×#1451975011yescharacteristic
688C4C2×#1451975211yescharacteristic
689C2—24no—
690C2—4882no—
691C2—2444no—
692C2C2×#1683964011yescharacteristic
693C2#2145434411yescharacteristic
694C2#2145435111yescharacteristic
695C2#2145512511yescharacteristic
696C2#2145513611yescharacteristic
697C2#2145580111yescharacteristic
698C2#2145581211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC2×#16840343C2
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20915
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#16840242C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839756C2semidirect
5. C2×#16839730C2semidirect
6. #21446140C2semidirect
7. #21446135C2semidirect
8. #21446130C2semidirect
9. #21446125C2semidirect
10. C2×#16839524C2semidirect
11. C61×#1736C2semidirect
16. #16840343C4semidirect
58. C2×#11758222C2×C4semidirect
78. #14519757C2×C4semidirect
81. #14519752C2×C4semidirect
91. C2×C4⋊Dic61D4semidirect
115. #117×C61D4semidirect
118. C2×C4⋊C4×C61C2×C4semidirect
349. #1736D61semidirect
397. C61#20915semidirect
445. #117D244semidirect
456. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
C2×#168402422
C2×#168397642
C2×#168397562
C2×#168397302
#214461402
#214461352
#214461302
#214461252
C2×#168395242
C61×#17362
#2091561