Finite Group Databasebeta

← #29681952

Subgroups — #29681952

Order 15616.

1686 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic
2C22×#14519783C211yesmaximal, characteristic, complemented
3#21454339C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5C2×#16839744C211yesmaximal, characteristic, complemented
6#21446333C214yesmaximal, complemented
7#21446326C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C2×C61×#341C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12#16840562—24no—
13#16840562C418yescomplemented
14C2×#14519810—21no—
15C2×#14519810C2212yes—
16C2×#14519805—21no—
17C2×#14519805C2212yes—
18C2×#14519799—22no—
19C2×#14519783—24no—
20C2×#14519783—22no—
21C2×#14519783—24no—
22C2×#14519783C2214yes—
23C22×#11758194C2211yescharacteristic
24C22×#11758189—21no—
25C22×C8×D61—21no—
26C2×#14519736C2211yescharacteristic
27#16839772—21no—
28#16839768—21no—
29#16839764C2214yes—
30#16839759—21no—
31#16839759—21no—
32#16839759—22no—
33#16839759—22no—
34C2×#14519702C2212yescomplemented
35C2×#14519699C2212yescomplemented
36#16839744—24no—
37C2×#14519695C2212yescomplemented
38C2×#14519693C2212yescomplemented
39#16839530C2214yescomplemented
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C23×C8×C61C2211yescharacteristic
43C61×#341C2214yescomplemented
44C2×#11758222—22no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—21no—
52C2×#11758222—21no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222C2×C414yes—
56C2×#11758222C2×C414yes—
57C2×#11758222D414yes—
58C2×#11758222D412yes—
59C2×#11758222C2312yes—
60C22×Dic122D411yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244D411yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C22⋊D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68#14519810—24no—
69#14519810—28no—
70#14519810C2×C418yes—
71#14519805—24no—
72#14519805—28no—
73#14519805C2×C418yes—
74#14519799—216no—
75C2×Dic244—22no—
76C2×#11758190—22no—
77C2×#11758190—22no—
78C2×D488—22no—
79#14519783—216no—
80#14519783—28no—
81#14519783—28no—
82C2×#11758194—22no—
83C2×#11758194—21no—
84C2×#11758194D414yes—
85C2×#11758194C2312yes—
86C2×#11758189—28no—
87C2×#11758189—24no—
88C22×#92418—21no—
89C2×C8×D61—28no—
90C2×C8×D61—24no—
91#14519736—21no—
92#14519736—21no—
93#14519736—21no—
94#14519736—21no—
95C2×C4⋊Dic61C2312yes—
96C2×C22⋊Dic61—21no—
97C2×C4×Dic61—21no—
98#14519702—24no—
99#14519702—22no—
100#14519702—22no—
101#14519699—24no—
102#14519699—22no—
103#14519699—22no—
104#14519695—24no—
105#14519693—24no—
106C23×C4×C61C2311yescharacteristic
107C2×M16×C61—21no—
108C2×M16×C61—22no—
109C2×M16×C61—21no—
110C2×M16×C61—22no—
111C2×M16×C61—22no—
112C2×M16×C61C2×C414yes—
113C2×M16×C61C2312yes—
114C2×M16×C61C2312yes—
115C22×C8×C61—22no—
116C22×C8×C61—22no—
117C22×C8×C61—21no—
118C22×C8×C61—21no—
119C22×C8×C61D411yescharacteristic
120C22×C8×C61D411yescharacteristic
121C22×C8×C61D412yes—
122C22×C8×C61C2312yes—
123C24×C61#4111yescharacteristic
124C23×D61—21no—
125C23×D61—21no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4111yescharacteristic
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61—24no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—42no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—22no—
153#11758222—22no—
154#11758222—24no—
155#11758222—24no—
156#11758222—24no—
157#11758222—24no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162#11758222C22⋊C418yes—
163#11758222C22×C414yes—
164#11758222C2×D414yes—
165C2×D244—22no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244C22⋊C414yes—
173C2×D244C22×C412yes—
174C2×D244C2×D412yes—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—24no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61—42no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122C22⋊C414yes—
191C2×Dic122C22×C412yes—
192C2×Dic122C2×D412yes—
193M16×C61—24no—
194M16×C61—22no—
195M16×C61—22no—
196M16×C61—22no—
197M16×C61—42no—
198M16×C61C22×C414yes—
199C2×C8×C61—24no—
200C2×C8×C61—22no—
201C2×C8×C61—22no—
202C2×C8×C61—24no—
203C2×C8×C61—22no—
204C2×C8×C61—22no—
205C2×C8×C61—22no—
206C2×C8×C61—22no—
207C2×C8×C61—42no—
208C2×C8×C61C2×D412yes—
209C2×C8×C61C2×D412yes—
210C22⋊Dic61—41no—
211C22⋊Dic61—41no—
212C4⋊Dic61—22no—
213C4⋊Dic61—22no—
214C4×Dic61—41no—
215C4×Dic61—41no—
216#11758194—24no—
217#11758194—22no—
218#11758194C2×D414yes—
219C2×#92418—24no—
220C2×#92418—22no—
221Dic244—28no—
222D488—28no—
223#11758190—28no—
224#11758190—28no—
225#11758189—216no—
226C8×D61—216no—
227C23×C61—42no—
228C23×C61C4×D412yes—
229C23×C61#12111yescharacteristic
230C23×C61#12111yescharacteristic
231C23×C61#12312yes—
232C23×C61C2×#4111yescharacteristic
233C22×D61—22no—
234C22×D61—24no—
235C22×D61—22no—
236C22×D61—44no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—82no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12012yes—
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12111yescharacteristic
250C2×C4×C61#12111yescharacteristic
251C2×C4×C61#12312yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—24no—
260C22⋊D61—28no—
261C22⋊D61—44no—
262C22⋊D61—44no—
263C2×Dic61—22no—
264C2×Dic61—24no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—82no—
270D244—22no—
271D244—22no—
272D244—22no—
273D244—24no—
274D244—41no—
275D244—41no—
276D244C2×C22⋊C414yes—
277C4×D61—24no—
278C4×D61—24no—
279C4×D61—28no—
280C4×D61—44no—
281C4×D61—44no—
282Dic122—22no—
283Dic122—24no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122C2×C22⋊C414yes—
289C8×C61—24no—
290C8×C61—22no—
291C8×C61—22no—
292C8×C61—42no—
293#92418—24no—
294#21003—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61#38612yes—
301C22×C61#38612yes—
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C2×#12111yescharacteristic
304C22×C61C2×#12311yescharacteristic
305C2×D61—24no—
306C2×D61—48no—
307C2×D61—44no—
308C2×D61—42no—
309C2×D61—82no—
310C2×D61—82no—
311C2×D61—82no—
312C4×C61#38612yes—
313C4×C61#38612yes—
314C4×C61C22×C22⋊C411yescharacteristic
315C4×C61C2×C4×D411yescharacteristic
316C4×C61C2×#12011yescharacteristic
317C4×C61C2×#12111yescharacteristic
318Dic61—24no—
319Dic61—42no—
320Dic61—82no—
321C2×#575—611no—
322C22×#131—611no—
323#2896—611no—
324C2×#416—611no—
325C2×#413—611no—
326#1908—614no—
327#1907—614no—
328#1742—611no—
329C2×#341D6111yescharacteristic, complemented
330C2×C61—22no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—22no—
334C2×C61C2×#38611yescharacteristic
335C2×C61#306712yes—
336D61—44no—
337D61—82no—
338C22×#41—611no—
339C22×#41—611no—
340#575—618no—
341#575—1224no—
342C2×#137—612no—
343C2×#137—1221no—
344C2×AGL(1,ℤ/8)—612no—
345C2×AGL(1,ℤ/8)—1221no—
346C2×#135—1222no—
347C2×#131—614no—
348C2×#131—1224no—
349C2×#131—1224no—
350C2×#131—1222no—
351C22×M16—611no—
352C22×M16—1221no—
353C22×M16C2×D6111yescharacteristic
354C22×M16C2×D6111yescharacteristic
355C23×C8—1221no—
356C23×C8C2×D6111yescharacteristic
357C2×#117—611no—
358#419—1221no—
359#418—1221no—
360#417—614no—
361#416—1222no—
362#416—1221no—
363#416—1221no—
364#416—1222no—
365C2×#103—612no—
366C2×#102—612no—
367#413—1224no—
368C2×#101—612no—
369C2×#100—612no—
370#343—614no—
371#341C2×D6114yescomplemented
372C61#2100311yescharacteristic, complemented, Sylow
373C2×#41—614no—
374C2×#41—614no—
375C2×#41—614no—
376C2×#41—612no—
377C2×#41—612no—
378C2×#41—1221no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1222no—
382C2×#41—1221no—
383C2×#41—1221no—
384C2×#41—1221no—
385C2×#41—1222no—
386C2×#41—1221no—
387C2×#41—1221no—
388C2×#41—1222no—
389C22×Q8—611no—
390C22×Q8—611no—
391C22×D4—611no—
392C22×D4—611no—
393C22×D4—1221no—
394C22×D4—1221no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398#137—618no—
399#137—1228no—
400#137—1224no—
401AGL(1,ℤ/8)—618no—
402AGL(1,ℤ/8)—1228no—
403AGL(1,ℤ/8)—1224no—
404#135—12216no—
405C2×Q16—1222no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×D8—1222no—
409#131—1228no—
410#131—1228no—
411#131—12216no—
412C2×M16—21no—
413C2×M16—22no—
414C2×M16—21no—
415C2×M16—22no—
416C2×M16—22no—
417C2×M16—614no—
418C2×M16—612no—
419C2×M16—1228no—
420C2×M16—1221no—
421C2×M16—1222no—
422C2×M16—1224no—
423C2×M16C4×D6114yes—
424C2×M16C22×D6112yes—
425C2×M16C22×D6112yes—
426C22×C8—22no—
427C22×C8—22no—
428C22×C8—21no—
429C22×C8—21no—
430C22×C8—1228no—
431C22×C8—1221no—
432C22×C8—1224no—
433C22×C8D24411yescharacteristic
434C22×C8D24411yescharacteristic
435C22×C8D24412yes—
436C22×C8C22×D6112yes—
437#117—1221no—
438#117—1221no—
439#117—1221no—
440#117—1221no—
441C2×C4⋊C4—612no—
442C2×C22⋊C4—1221no—
443C2×C42—1221no—
444#103—1224no—
445#103—1222no—
446#103—1222no—
447#102—1224no—
448#102—1222no—
449#102—1222no—
450#101—1224no—
451#100—1224no—
452C24—1221no—
453C24—1221no—
454C24#1175822611yescharacteristic
455#41—614no—
456#41—614no—
457#41—618no—
458#41—1222no—
459#41—1222no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—1224no—
464#41—1224no—
465#41—1224no—
466#41—1224no—
467#41—1224no—
468#41—1224no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—614no—
476C2×Q8—1221no—
477C2×Q8—1222no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—612no—
484C2×D4—612no—
485C2×D4—614no—
486C2×D4—1222no—
487C2×D4—1224no—
488C2×D4—1222no—
489C2×D4—1224no—
490C2×D4—1224no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1222no—
497C2×D4—1221no—
498C2×D4—2441no—
499C2×D4—2441no—
500C2×D4—2442no—
501C22×C4—1221no—
502C22×C4—1222no—
503C22×C4—1224no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1221no—
507C22×C4—1224no—
508C22×C4—1222no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×D24411yescharacteristic
515C22×C4#1175822212yes—
516C22×C4D4×D6112yes—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6112yes—
520C22×C4#1175822611yescharacteristic
521C22×C4C23×D6111yescharacteristic
522Q16—1228no—
523SD16—1228no—
524SD16—1228no—
525D8—1228no—
526M16—22no—
527M16—24no—
528M16—22no—
529M16—22no—
530M16—42no—
531M16—614no—
532M16—12216no—
533M16—1222no—
534M16—1224no—
535M16C2×C4×D6114yes—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—24no—
543C2×C8—22no—
544C2×C8—42no—
545C2×C8—12216no—
546C2×C8—1222no—
547C2×C8—1224no—
548C2×C8C2×D24412yes—
549C2×C8C2×D24412yes—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C22⋊C4—2441no—
553C22⋊C4—2441no—
554C42—2441no—
555C42—2441no—
556C23—42no—
557C23—1222no—
558C23—1224no—
559C23—1222no—
560C23—4882no—
561C23—2444no—
562C23—2441no—
563C23—2441no—
564C23#1451973912yes—
565C23#1451975211yescharacteristic
566C23#1451975211yescharacteristic
567C23#1451976212yes—
568C23C2×#1175822611yescharacteristic
569Q8—614no—
570Q8—1222no—
571Q8—1224no—
572Q8—1222no—
573Q8—1222no—
574Q8—2441no—
575Q8—2441no—
576D4—614no—
577D4—1224no—
578D4—1224no—
579D4—1228no—
580D4—1222no—
581D4—1222no—
582D4—1222no—
583D4—1224no—
584D4—2444no—
585D4—2444no—
586D4—2441no—
587D4—2441no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1228no—
593C2×C4—1222no—
594C2×C4—4882no—
595C2×C4—2442no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4C22⋊C4×D6112yes—
601C2×C4C22⋊C4×D6112yes—
602C2×C4#1451973912yes—
603C2×C4C4×D24412yes—
604C2×C4C4×D24412yes—
605C2×C4#1451974712yes—
606C2×C4#1451974712yes—
607C2×C4#1451975012yes—
608C2×C4#1451975012yes—
609C2×C4#1451975211yescharacteristic
610C2×C4#1451975211yescharacteristic
611C2×C4#1451976212yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C22×D24411yescharacteristic
617C2×C4C2×#1175822211yescharacteristic
618C8—24no—
619C8—22no—
620C8—22no—
621C8—42no—
622C8—1224no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—1224no—
629C22—4882no—
630C22—4882no—
631C22—4882no—
632C22—2448no—
633C22—2442no—
634C22—2444no—
635C22#1683964012yes—
636C22#1683964012yes—
637C22C2×#1451973911yescharacteristic
638C22C2×#1451975211yescharacteristic
639C22C2×#1451976211yescharacteristic
640C4—1224no—
641C4—4882no—
642C4—2442no—
643C4#1683964012yes—
644C4#1683964012yes—
645C4C2×C22⋊C4×D6111yescharacteristic
646C4C2×C4×D24411yescharacteristic
647C4C2×#1451974711yescharacteristic
648C4C2×#1451975011yescharacteristic
649C2—21no—
650C2—21no—
651C2—22no—
652C2—22no—
653C2—4882no—
654C2—2444no—
655C2C2×#1683964011yescharacteristic
656C2#2145554012yes—
subgroupG/H
centerC2×C4#14519752
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#341C2
socleC22×C61C2×#121
prime radicalC2×#16840562C2
cosocleC22×C4×C61C24
hypercenterC2×#341D61
hypocenterC61#21003
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
2. C22×#14519783C2semidirect
3. #21454339C2semidirect
4. C2×#16839759C2semidirect
5. C2×#16839744C2semidirect
6. #21446333C2semidirect
7. #21446326C2semidirect
8. #21445557C2semidirect
9. C2×C61×#341C2semidirect
13. #16840562C4semidirect
34. C2×#14519702C22semidirect
35. C2×#14519699C22semidirect
37. C2×#14519695C22semidirect
38. C2×#14519693C22semidirect
39. #16839530C22semidirect
43. C61×#341C22semidirect
329. C2×#341D61semidirect
371. #341C2×D61semidirect
372. C61#21003semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
C22×#145197832
#214543392
C2×#168397592
C2×#168397442
#214463332
#214463262
#214455572
C2×C61×#3412
#2100361