Finite Group Databasebeta

← #29707667

Subgroups — #29707667

Order 15616.

1442 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×#16840340C211yesmaximal, characteristic
2C2×#16839780C211yesmaximal, characteristic, complemented
3#21454339C211yesmaximal, characteristic, complemented
4#21454338C211yesmaximal, characteristic, complemented
5#21448442C211yesmaximal, characteristic, complemented
6#21447047C214yesmaximal, complemented
7#21447042C214yesmaximal, complemented
8C2×#16839530C211yesmaximal, characteristic, complemented
9C2×C61×#421C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340—22no—
13#16840340—22no—
14#16840340—22no—
15#16840340C2214yes—
16C2×#14519756—21no—
17C2×#14519753—21no—
18C2×#14519751—21no—
19C2×#14519749—22no—
20C2×#14519748—21no—
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839780—24no—
24C2×#14519705C2214yescomplemented
25#16839772—21no—
26#16839771—21no—
27#16839768—21no—
28#16839767—21no—
29#16839764C2212yescomplemented
30#16839764C2212yescomplemented
31#16839763C2212yescomplemented
32#16839763C2212yescomplemented
33#16839759—22no—
34#16839758—22no—
35#16839661—22no—
36#16839637—22no—
37#16839632C2212yescomplemented
38#16839628C2212yescomplemented
39#16839530C2214yescomplemented
40C22×#41×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#421—24no—
44C2×#104×C61C2214yescomplemented
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222D412yes—
52C2×#11758222C2312yes—
53C22×Dic122D411yescharacteristic
54C22×D244D411yescharacteristic
55C22×C22⋊D61—21no—
56C22×C22⋊D61—22no—
57C23×Dic61D412yes—
58C22×C4×D61—21no—
59C2×#11758194—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2312yes—
63C22×#92418—21no—
64#14519756—24no—
65#14519756—24no—
66#14519753—24no—
67#14519753—24no—
68#14519751—24no—
69#14519751—24no—
70#14519749—28no—
71#14519749—28no—
72#14519748—24no—
73#14519748—24no—
74#14519736—21no—
75#14519736—21no—
76#14519736—21no—
77#14519736—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—24no—
83C2×C4⋊Dic61D412yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2312yes—
87C2×#11758196—22no—
88C2×C22⋊Dic61—22no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—22no—
91C2×C4×Dic61—21no—
92#14519705—28no—
93#14519702—22no—
94#14519702—22no—
95#14519701—22no—
96#14519701—22no—
97#14519699—22no—
98#14519699—22no—
99#14519698—22no—
100#14519698—22no—
101#14519684—24no—
102C2×#41×C61—21no—
103C2×#41×C61—21no—
104C2×#41×C61—22no—
105C2×#41×C61—22no—
106C2×#41×C61—22no—
107C2×#41×C61—22no—
108C2×#41×C61D412yes—
109C2×#41×C61C2312yes—
110C22×Q8×C61D411yescharacteristic
111C22×D4×C61—21no—
112C22×D4×C61D411yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61C2311yescharacteristic
115C2×M16×C61—21no—
116C2×M16×C61—22no—
117C2×M16×C61—22no—
118C2×M16×C61C2312yes—
119C22×C8×C61—21no—
120#117×C61—22no—
121#117×C61D414yescomplemented
122C2×C4⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61C2312yes—
125#104×C61—28no—
126C24×C61—21no—
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129#41×C61—24no—
130#41×C61—24no—
131#41×C61—22no—
132#41×C61—22no—
133#41×C61—22no—
134#41×C61—22no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—42no—
138#41×C61—42no—
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61C2×D412yes—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61#4112yes—
164C22×C4×C61#4112yes—
165C22×C4×C61#4112yes—
166C22×C4×C61C2411yescharacteristic
167C2×C22⋊D61—22no—
168C2×C22⋊D61—22no—
169C2×C22⋊D61—28no—
170C2×C22⋊D61—42no—
171C2×C22⋊D61—42no—
172C2×C22⋊D61—84no—
173C22×Dic61—21no—
174C22×Dic61—22no—
175C22×Dic61—22no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61—82no—
179C22×Dic61C2×D414yes—
180#11758222—24no—
181#11758222—24no—
182#11758222—22no—
183#11758222—22no—
184#11758222—22no—
185#11758222—22no—
186#11758222—42no—
187#11758222—42no—
188#11758222—42no—
189#11758222—42no—
190C2×D244—22no—
191C2×D244—22no—
192C2×D244—21no—
193C2×D244C2×D412yes—
194C2×C4×D61—22no—
195C2×C4×D61—22no—
196C2×C4×D61—42no—
197C2×C4×D61—42no—
198C2×Dic122—22no—
199C2×Dic122—22no—
200C2×Dic122—21no—
201C2×Dic122C2×D412yes—
202M16×C61—22no—
203M16×C61—22no—
204M16×C61—42no—
205C2×C8×C61—22no—
206C2×C8×C61—42no—
207C4⋊C4×C61—24no—
208C22⋊C4×C61—24no—
209C42×C61—22no—
210C42×C61C2×D414yescomplemented
211C22⋊Dic61—28no—
212C22⋊Dic61—41no—
213C22⋊Dic61—41no—
214#11758198—28no—
215C4⋊Dic61—22no—
216C4⋊Dic61—22no—
217C4⋊Dic61—22no—
218C4⋊Dic61—22no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—84no—
224#11758196—28no—
225C4×Dic61—41no—
226C4×Dic61—41no—
227#11758194—22no—
228#11758194—22no—
229#11758194—42no—
230C2×#92418—22no—
231C2×#92418—42no—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—42no—
236C23×C61—42no—
237C23×C61—42no—
238C23×C61#12011yescharacteristic
239C23×C61#12011yescharacteristic
240C23×C61#12112yes—
241C23×C61#12412yes—
242C23×C61C22×D411yescharacteristic
243C22×D61—22no—
244C22×D61—42no—
245C22×D61—42no—
246C22×D61—42no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—42no—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—42no—
253D4×C61—42no—
254D4×C61—44no—
255D4×C61—42no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—44no—
264C2×C4×C61#12011yescharacteristic
265C2×C4×C61#12011yescharacteristic
266C2×C4×C61#12112yes—
267C2×C4×C61#12112yes—
268C2×C4×C61#12112yes—
269C2×C4×C61#12112yes—
270C2×C4×C61#12112yes—
271C2×C4×C61#12312yes—
272C2×C4×C61#12312yes—
273C2×C4×C61#12312yes—
274C2×C4×C61#12312yes—
275C2×C4×C61#12412yes—
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C61C2×#4111yescharacteristic
281C2×C4×C61C2×#4111yescharacteristic
282C22⋊D61—42no—
283C22⋊D61—42no—
284C22⋊D61—44no—
285C22⋊D61—88no—
286C2×Dic61—22no—
287C2×Dic61—24no—
288C2×Dic61—44no—
289C2×Dic61—42no—
290C2×Dic61—44no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—82no—
296D244—22no—
297D244—22no—
298D244—42no—
299C4×D61—42no—
300C4×D61—42no—
301C4×D61—44no—
302Dic122—22no—
303Dic122—22no—
304Dic122—42no—
305C8×C61—42no—
306#92418—42no—
307#23499—611nomaximal, Sylow
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—42no—
315C22×C61—42no—
316C22×C61—42no—
317C22×C61—42no—
318C22×C61—42no—
319C22×C61—84no—
320C22×C61#39412yes—
321C22×C61#39412yes—
322C22×C61C2×#12011yescharacteristic
323C22×C61C2×#12111yescharacteristic
324C22×C61C2×#12411yescharacteristic
325C2×D61—42no—
326C2×D61—42no—
327C2×D61—42no—
328C2×D61—84no—
329C4×C61—44no—
330C4×C61—42no—
331C4×C61#39412yes—
332C4×C61#39412yes—
333C4×C61C2×#12111yescharacteristic
334C4×C61C2×#12111yescharacteristic
335C4×C61C2×#12311yescharacteristic
336C4×C61C2×#12311yescharacteristic
337Dic61—42no—
338Dic61—44no—
339Dic61—81no—
340Dic61—81no—
341C2×#537—611no—
342C2×#421—611no—
343C2×#421D6111yescharacteristic, complemented
344#2896—611no—
345#2896—611no—
346#2317—611no—
347#2040—614no—
348#2039—614no—
349C2×#343—611no—
350C2×C61—21no—
351C2×C61—22no—
352C2×C61—21no—
353C2×C61—22no—
354C2×C61—82no—
355C2×C61C2×#39411yescharacteristic
356C2×C61#301911yescharacteristic
357C2×C61#302011yescharacteristic
358D61—82no—
359C22×#41—611no—
360C22×#41C2×D6111yescharacteristic
361C22×M16—611no—
362C22×M16C2×D6111yescharacteristic
363#537—614no—
364#537—1222no—
365#537—1222no—
366#537—1222no—
367C2×C22⋊Q8—1221no—
368C2×C22⋊Q8—1221no—
369C2×#121—1221no—
370C2×#121—1222no—
371C2×#121—1221no—
372C2×#117—611no—
373C2×#117C2×D6111yescharacteristic
374C22×C4⋊C4—611no—
375#421—24no—
376#421—1224no—
377C2×#104—614no—
378C2×#104C2×D6114yescomplemented
379#419—1221no—
380#419—1221no—
381#418—1221no—
382#418—1221no—
383#417—612no—
384#417—612no—
385#417—612no—
386#417—612no—
387#416—1222no—
388#416—1222no—
389#389—1222no—
390#385—1222no—
391#384—612no—
392#382—612no—
393#343—614no—
394C61#2349911yescharacteristic, complemented, Sylow
395C2×#41—21no—
396C2×#41—21no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—612no—
402C2×#41—612no—
403C2×#41—1221no—
404C2×#41—1221no—
405C2×#41—1222no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41D24412yes—
410C2×#41C22×D6112yes—
411C22×Q8—611no—
412C22×Q8D24411yescharacteristic
413C22×D4—21no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4—1221no—
417C22×D4D24411yescharacteristic
418C23×C4—21no—
419C23×C4—612no—
420C23×C4—1221no—
421C23×C4C22×D6111yescharacteristic
422C2×M16—21no—
423C2×M16—22no—
424C2×M16—22no—
425C2×M16—612no—
426C2×M16—1221no—
427C2×M16—1222no—
428C2×M16—1222no—
429C2×M16C22×D6112yes—
430C22×C8—21no—
431C22×C8—1221no—
432C22⋊Q8—1224no—
433C22⋊Q8—1224no—
434C22⋊Q8—1224no—
435C22⋊Q8—1224no—
436#121—1224no—
437#121—1224no—
438#121—1228no—
439#121—1224no—
440#121—1228no—
441#121—1224no—
442#117—22no—
443#117—1221no—
444#117—1221no—
445#117—1221no—
446#117—1221no—
447#117C22⋊D6114yescomplemented
448C2×C4⋊C4—21no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1224no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4—1221no—
463C2×C42—1221no—
464C2×C42C22×D6112yes—
465#104—28no—
466#104—1228no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#103—1222no—
471#102—1222no—
472#102—1222no—
473#102—1222no—
474#102—1222no—
475#95—1224no—
476C24—21no—
477C24—1221no—
478C24D4×D6111yescharacteristic
479#41—22no—
480#41—22no—
481#41—22no—
482#41—22no—
483#41—24no—
484#41—24no—
485#41—42no—
486#41—42no—
487#41—42no—
488#41—42no—
489#41—1222no—
490#41—1222no—
491#41—1222no—
492#41—1222no—
493#41—1224no—
494#41—1224no—
495#41—2442no—
496#41—2442no—
497#41—2442no—
498#41—2442no—
499C2×Q8—21no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—612no—
503C2×Q8—1221no—
504C2×Q8—1222no—
505C2×Q8—1222no—
506C2×Q8C2×D24412yes—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—21no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—42no—
513C2×D4—42no—
514C2×D4—612no—
515C2×D4—1228no—
516C2×D4—1222no—
517C2×D4—1222no—
518C2×D4—1221no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—4884no—
522C2×D4—2442no—
523C2×D4—2442no—
524C2×D4C2×D24412yes—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—42no—
530C22×C4—42no—
531C22×C4—614no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—1221no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1221no—
538C22×C4—1221no—
539C22×C4—4882no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4C2×D24411yescharacteristic
543C22×C4C2×D24411yescharacteristic
544C22×C4#1175822212yes—
545C22×C4D4×D6111yescharacteristic
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822412yes—
549C22×C4#1175822412yes—
550C22×C4C2×C22⋊D6112yes—
551C22×C4C23×D6111yescharacteristic
552M16—22no—
553M16—22no—
554M16—42no—
555M16—1222no—
556M16—1222no—
557M16—2442no—
558C2×C8—22no—
559C2×C8—42no—
560C2×C8—1222no—
561C2×C8—2442no—
562C4⋊C4—24no—
563C4⋊C4—1228no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C4⋊C4—4884no—
573C22⋊C4—24no—
574C22⋊C4—1228no—
575C22⋊C4—1228no—
576C22⋊C4—2441no—
577C22⋊C4—2441no—
578C42—22no—
579C42—2441no—
580C42—2441no—
581C42C2×C22⋊D6114yescomplemented
582C23—22no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—1222no—
589C23—2442no—
590C23—2442no—
591C23—2442no—
592C23#1451974711yescharacteristic
593C23#1451974711yescharacteristic
594C23#1451974912yes—
595C23#1451976312yes—
596C23C2×D4×D6111yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—42no—
600Q8—1222no—
601Q8—1222no—
602Q8—2442no—
603D4—22no—
604D4—22no—
605D4—42no—
606D4—42no—
607D4—44no—
608D4—42no—
609D4—1222no—
610D4—1222no—
611D4—4888no—
612D4—2442no—
613D4—2442no—
614D4—2444no—
615D4—2442no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—1224no—
625C2×C4—1222no—
626C2×C4—4882no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2444no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4—2442no—
635C2×C4—2444no—
636C2×C4—2444no—
637C2×C4#1451974711yescharacteristic
638C2×C4#1451974711yescharacteristic
639C2×C4#1451974812yes—
640C2×C4#1451974812yes—
641C2×C4#1451974912yes—
642C2×C4#1451975112yes—
643C2×C4#1451975112yes—
644C2×C4#1451975912yes—
645C2×C4#1451975912yes—
646C2×C4#1451976112yes—
647C2×C4#1451976112yes—
648C2×C4#1451976312yes—
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C22×C22⋊D6111yescharacteristic
651C2×C4C22×D24411yescharacteristic
652C2×C4C2×#1175822411yescharacteristic
653C2×C4C2×#1175822411yescharacteristic
654C2×C4C2×#1175822211yescharacteristic
655C8—42no—
656C8—2442no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—84no—
669C22—4884no—
670C22—2442no—
671C22—2442no—
672C22—2442no—
673C22#1683967812yes—
674C22#1683967812yes—
675C22C2×#1451974711yescharacteristic
676C22C2×#1451974911yescharacteristic
677C22C2×#1451976311yescharacteristic
678C4—44no—
679C4—42no—
680C4—4881no—
681C4—4881no—
682C4—2442no—
683C4—2444no—
684C4#1683967812yes—
685C4#1683967812yes—
686C4C2×#1451974811yescharacteristic
687C4C2×#1451975111yescharacteristic
688C4C2×#1451975911yescharacteristic
689C4C2×#1451976111yescharacteristic
690C2—21no—
691C2—22no—
692C2—21no—
693C2—22no—
694C2—82no—
695C2—4882no—
696C2C2×#1683967811yescharacteristic
697C2#2145517811yescharacteristic
698C2#2145518511yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C2×#120
prime radicalC2×#16840340C2
hypercenterC2×#421D61
hypocenterC61#23499
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#16839780C2semidirect
3. #21454339C2semidirect
4. #21454338C2semidirect
5. #21448442C2semidirect
6. #21447047C2semidirect
7. #21447042C2semidirect
8. C2×#16839530C2semidirect
9. C2×C61×#421C2semidirect
24. C2×#14519705C22semidirect
29. #16839764C22semidirect
30. #16839764C22semidirect
31. #16839763C22semidirect
32. #16839763C22semidirect
37. #16839632C22semidirect
38. #16839628C22semidirect
39. #16839530C22semidirect
44. C2×#104×C61C22semidirect
121. #117×C61D4semidirect
210. C42×C61C2×D4semidirect
343. C2×#421D61semidirect
378. C2×#104C2×D61semidirect
394. C61#23499semidirect
447. #117C22⋊D61semidirect
581. C42C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403402
C2×#168397802
#214543392
#214543382
#214484422
#214470472
#214470422
C2×#168395302
C2×C61×#4212
#2349961