Finite Group Databasebeta

← #29681963

Subgroups — #29681963

Order 15616.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic
2C2×#16840528C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×#16839744C211yesmaximal, characteristic, complemented
6#21446335C214yesmaximal, complemented
7#21446328C214yesmaximal, complemented
8#21445545C211yesmaximal, characteristic, complemented
9C2×C61×#343C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12#16840562—24no—
13#16840562C418yescomplemented
14C2×#14519810—21no—
15C2×#14519810C2212yes—
16C2×#14519805—21no—
17C2×#14519805C2212yes—
18C2×#14519799—22no—
19#16840528—24no—
20#16840528—24no—
21C2×#14519785C2214yes—
22C2×#14519783—22no—
23C22×#11758194C2211yescharacteristic
24C22×#11758194C2211yescharacteristic
25C2×M16×D61—22no—
26#16839761—22no—
27#16839761—22no—
28#16839761—22no—
29#16839761C2214yes—
30C2×#14519700—21no—
31C2×#14519700C2212yescomplemented
32C2×#14519697—21no—
33C2×#14519697C2212yescomplemented
34#16839744—24no—
35C2×#14519695C2212yescomplemented
36C2×#14519693C2212yescomplemented
37#16839526C2214yescomplemented
38C22×M16×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#343C2214yescomplemented
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—21no—
48C2×#11758222—21no—
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222C2×C414yes—
54C2×#11758222C2×C414yes—
55C2×#11758222D414yes—
56C2×#11758222D412yes—
57C2×#11758222C2312yes—
58C22×Dic122D411yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244D411yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66#14519810—24no—
67#14519810—28no—
68#14519810C2×C418yescomplemented
69#14519805—24no—
70#14519805—28no—
71#14519805C2×C418yescomplemented
72#14519799—216no—
73C2×Dic244—22no—
74C2×#11758190—22no—
75C2×#11758190—22no—
76C2×D488—22no—
77#14519785—28no—
78#14519785—28no—
79#14519783—216no—
80C2×#11758194—22no—
81C2×#11758194—22no—
82C2×#11758194—21no—
83C2×#11758194—22no—
84C2×#11758194—22no—
85C2×#11758194D412yes—
86C2×#11758194D412yes—
87C2×#11758194C2312yes—
88C2×#11758189—24no—
89M16×D61—216no—
90C22×#92418—21no—
91C22×#92418C2312yes—
92C2×C8×D61—24no—
93#14519700—24no—
94#14519700—24no—
95#14519697—24no—
96#14519697—24no—
97#14519695—24no—
98#14519693—24no—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61—21no—
104C2×M16×C61—22no—
105C2×M16×C61C2×C414yes—
106C2×M16×C61D414yes—
107C2×M16×C61C2312yes—
108C2×M16×C61C2312yes—
109C22×C8×C61—21no—
110C22×C8×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2312yes—
117C24×C61#4111yescharacteristic
118C23×D61—21no—
119C23×D61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—82no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4111yescharacteristic
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—42no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144#11758222—24no—
145#11758222—24no—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—22no—
150#11758222—22no—
151#11758222—24no—
152#11758222—24no—
153#11758222—24no—
154#11758222—24no—
155#11758222—42no—
156#11758222—42no—
157#11758222—42no—
158#11758222—42no—
159#11758222C22⋊C418yes—
160#11758222C22×C414yes—
161#11758222C2×D414yes—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C22⋊C414yes—
170C2×D244C22×C412yes—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—24no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—24no—
177C2×C4×D61—41no—
178C2×C4×D61—41no—
179C2×C4×D61—42no—
180C2×Dic122—22no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122C22⋊C414yes—
188C2×Dic122C22×C412yes—
189C2×Dic122C2×D412yes—
190M16×C61—24no—
191M16×C61—24no—
192M16×C61—22no—
193M16×C61—22no—
194M16×C61C22×C414yes—
195M16×C61C2×D414yes—
196C2×C8×C61—24no—
197C2×C8×C61—24no—
198C2×C8×C61—22no—
199C2×C8×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—41no—
203C4⋊C4×C61—41no—
204#11758194—24no—
205#11758194—24no—
206#11758194—22no—
207#11758194—22no—
208#11758194—42no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—42no—
213C2×#92418C2×D414yes—
214Dic244—28no—
215D488—28no—
216#11758190—28no—
217#11758190—28no—
218#11758189—216no—
219C8×D61—216no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C4×D412yes—
223C23×C61#12111yescharacteristic
224C23×C61#12111yescharacteristic
225C23×C61#12312yes—
226C23×C61C2×#4111yescharacteristic
227C22×D61—22no—
228C22×D61—24no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—81no—
233C22×D61—81no—
234C22×D61—81no—
235C22×D61—81no—
236C2×C4×C61—42no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61—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—
268D244—22no—
269D244—22no—
270D244—22no—
271D244—24no—
272D244—41no—
273D244—41no—
274D244C2×C22⋊C414yes—
275C4×D61—24no—
276C4×D61—24no—
277C4×D61—28no—
278C4×D61—44no—
279C4×D61—44no—
280Dic122—22no—
281Dic122—24no—
282Dic122—22no—
283Dic122—22no—
284Dic122—41no—
285Dic122—41no—
286Dic122C2×C22⋊C414yes—
287C8×C61—24no—
288C8×C61—24no—
289#92418—24no—
290#92418—42no—
291#21004—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
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—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—84no—
310C2×D61—82no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—82no—
314C4×C61—82no—
315C4×C61#38612yes—
316C4×C61#38612yes—
317C4×C61C22×C22⋊C411yescharacteristic
318C4×C61C2×C4×D411yescharacteristic
319C4×C61C2×#12011yescharacteristic
320C4×C61C2×#12111yescharacteristic
321Dic61—24no—
322Dic61—42no—
323C2×#575—611no—
324C2×#568—611no—
325C2×#417—611no—
326C2×#417—611no—
327C2×#413—611no—
328#1908—614no—
329#1907—614no—
330C2×#343D6111yescharacteristic, complemented
331#1738—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61—22no—
335C2×C61—22no—
336C2×C61C2×#38611yescharacteristic
337C2×C61#306811yescharacteristic
338C2×C61#306911yescharacteristic
339D61—44no—
340D61—82no—
341C22×#41—611no—
342C22×#41—611no—
343#575—618no—
344#575—1224no—
345C2×#137—612no—
346C2×#137—1221no—
347C2×AGL(1,ℤ/8)—612no—
348C2×AGL(1,ℤ/8)—1221no—
349C2×#135—1222no—
350#568—1224no—
351#568—1224no—
352C2×#131—614no—
353C2×#131—1222no—
354C22×M16—611no—
355C22×M16—611no—
356C22×M16—1222no—
357C22×M16C2×D6111yescharacteristic
358C22×M16C2×D6111yescharacteristic
359C22×C4⋊C4C2×D6111yescharacteristic
360#417—614no—
361#417—1222no—
362#417—1222no—
363#417—1222no—
364C2×#103—612no—
365C2×#103—1221no—
366C2×#102—612no—
367C2×#102—1221no—
368#413—1224no—
369C2×#101—612no—
370C2×#100—612no—
371#343C2×D6114yescomplemented
372#341—614no—
373C61#2100411yescharacteristic, complemented, Sylow
374C2×#41—614no—
375C2×#41—614no—
376C2×#41—614no—
377C2×#41—612no—
378C2×#41—612no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41—1222no—
384C2×#41—1221no—
385C2×#41—1221no—
386C2×#41—1222no—
387C2×#41—1221no—
388C2×#41—1221no—
389C2×#41—1222no—
390C22×Q8—611no—
391C22×Q8—611no—
392C22×D4—611no—
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4—1221no—
396C23×C4—21no—
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4C22×D6111yescharacteristic
400#137—618no—
401#137—1228no—
402#137—1224no—
403AGL(1,ℤ/8)—618no—
404AGL(1,ℤ/8)—1228no—
405AGL(1,ℤ/8)—1224no—
406#135—12216no—
407C2×Q16—1222no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C2×D8—1222no—
411#131—12216no—
412#131—1228no—
413#131—1228no—
414C2×M16—21no—
415C2×M16—22no—
416C2×M16—21no—
417C2×M16—22no—
418C2×M16—612no—
419C2×M16—612no—
420C2×M16—612no—
421C2×M16—1224no—
422C2×M16—12216no—
423C2×M16—1222no—
424C2×M16—1222no—
425C2×M16—1221no—
426C2×M16—1222no—
427C2×M16—1222no—
428C2×M16C4×D6114yes—
429C2×M16D24414yes—
430C2×M16C22×D6112yes—
431C2×M16C22×D6112yes—
432C22×C8—21no—
433C22×C8—21no—
434C22×C8—612no—
435C22×C8—1224no—
436C22×C8—1221no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4C22×D6112yes—
443#103—1224no—
444#103—1224no—
445#102—1224no—
446#102—1224no—
447#101—1224no—
448#100—1224no—
449C24—1221no—
450C24—1221no—
451C24#1175822211yescharacteristic
452#41—614no—
453#41—614no—
454#41—618no—
455#41—1222no—
456#41—1222no—
457#41—1224no—
458#41—1224no—
459#41—1224no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—1224no—
464#41—1224no—
465#41—1224no—
466#41—2442no—
467#41—2442no—
468#41—2442no—
469#41—2442no—
470C2×Q8—612no—
471C2×Q8—612no—
472C2×Q8—614no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1222no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×D4—612no—
481C2×D4—612no—
482C2×D4—614no—
483C2×D4—1222no—
484C2×D4—1224no—
485C2×D4—1222no—
486C2×D4—1224no—
487C2×D4—1224no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1222no—
495C2×D4—2441no—
496C2×D4—2441no—
497C2×D4—2442no—
498C22×C4—22no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—82no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1224no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×D24412yes—
514C22×C4#1175822211yescharacteristic
515C22×C4D4×D6112yes—
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4#1175822612yes—
521C22×C4C23×D6111yescharacteristic
522Q16—1228no—
523SD16—1228no—
524SD16—1228no—
525D8—1228no—
526M16—22no—
527M16—22no—
528M16—24no—
529M16—24no—
530M16—12216no—
531M16—1224no—
532M16—1224no—
533M16—1222no—
534M16—1222no—
535M16—2442no—
536M16C2×C4×D6114yes—
537M16C2×D24414yes—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—24no—
541C2×C8—24no—
542C2×C8—614no—
543C2×C8—12216no—
544C2×C8—1222no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—2442no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C23—41no—
553C23—41no—
554C23—1224no—
555C23—1222no—
556C23—1222no—
557C23—4881no—
558C23—4881no—
559C23—4881no—
560C23—4881no—
561C23—2441no—
562C23—2441no—
563C23C4×D24412yes—
564C23#1451975011yescharacteristic
565C23#1451975011yescharacteristic
566C23#1451976212yes—
567C23C2×#1175822211yescharacteristic
568Q8—614no—
569Q8—1222no—
570Q8—1224no—
571Q8—1222no—
572Q8—1222no—
573Q8—2441no—
574Q8—2441no—
575D4—614no—
576D4—1224no—
577D4—1224no—
578D4—1228no—
579D4—1222no—
580D4—1222no—
581D4—1222no—
582D4—1224no—
583D4—2444no—
584D4—2444no—
585D4—2441no—
586D4—2441no—
587C2×C4—42no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—1222no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1228no—
596C2×C4—1222no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4C22⋊C4×D6112yes—
602C2×C4C22⋊C4×D6112yes—
603C2×C4#1451973912yes—
604C2×C4#1451973912yes—
605C2×C4C4×D24412yes—
606C2×C4#1451974712yes—
607C2×C4#1451974712yes—
608C2×C4#1451975011yescharacteristic
609C2×C4#1451975011yescharacteristic
610C2×C4#1451975212yes—
611C2×C4#1451975212yes—
612C2×C4#1451976212yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C22×D24411yescharacteristic
618C2×C4C2×#1175822611yescharacteristic
619C8—24no—
620C8—24no—
621C8—1224no—
622C8—2442no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—1224no—
632C22—4882no—
633C22—4882no—
634C22—4884no—
635C22—4881no—
636C22—4881no—
637C22—2442no—
638C22—2442no—
639C22—2442no—
640C22#1683964012yes—
641C22#1683964012yes—
642C22C2×C4×D24411yescharacteristic
643C22C2×#1451975011yescharacteristic
644C22C2×#1451976211yescharacteristic
645C4—82no—
646C4—1224no—
647C4—2442no—
648C4#1683964012yes—
649C4#1683964012yes—
650C4C2×C22⋊C4×D6111yescharacteristic
651C4C2×#1451973911yescharacteristic
652C4C2×#1451974711yescharacteristic
653C4C2×#1451975211yescharacteristic
654C2—21no—
655C2—21no—
656C2—22no—
657C2—22no—
658C2—4882no—
659C2—2444no—
660C2C2×#1683964011yescharacteristic
661C2#2145554611yescharacteristic
662C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C2×#121
prime radicalC2×#16840562C2
cosocleC22×C4×C61C24
hypercenterC2×#343D61
hypocenterC61#21004
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16840528C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839761C2semidirect
5. C2×#16839744C2semidirect
6. #21446335C2semidirect
7. #21446328C2semidirect
8. #21445545C2semidirect
9. C2×C61×#343C2semidirect
13. #16840562C4semidirect
31. C2×#14519700C22semidirect
33. C2×#14519697C22semidirect
35. C2×#14519695C22semidirect
36. C2×#14519693C22semidirect
37. #16839526C22semidirect
41. C61×#343C22semidirect
68. #14519810C2×C4semidirect
71. #14519805C2×C4semidirect
330. C2×#343D61semidirect
371. #343C2×D61semidirect
373. C61#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
C2×#168405282
C2×#168397612
C2×#168397442
#214463352
#214463282
#214455452
C2×C61×#3432
#2100461