Finite Group Databasebeta

← C2×#21447031 #29707600

Subgroups — C2×#21447031 #29707600

Order 15616.

1714 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840331C211yesmaximal, characteristic
2C22×#14519703C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic, complemented
5C2×#16839663C211yesmaximal, characteristic, complemented
6#21447031C218yesmaximal, direct product factor
7C2×#16839521C211yesmaximal, characteristic, complemented
8C2×C61×#428C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840331—22no—
13#16840331—22no—
14#16840331—22no—
15#16840331C2214yes—
16C2×#14519774—21no—
17C2×#14519765—21no—
18C2×#14519752—21no—
19C2×#14519752—21no—
20C2×#14519747—22no—
21C22×#11758198—22no—
22C22×C4×Dic61C2211yescharacteristic
23C2×#14519703—24no—
24C2×#14519703C2214yescomplemented
25#16839761—22no—
26#16839761C2214yescomplemented
27C2×#14519700—21no—
28C2×#14519697—21no—
29C2×#14519693—24no—
30C2×#14519693C2214yescomplemented
31#16839663—22no—
32#16839663C2214yescomplemented
33#16839521C2214yescomplemented
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#428—22no—
38C61×#428C2214yescomplemented
39C2×#107×C61—21no—
40C2×#106×C61—21no—
41C24×D61D412yes—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D412yes—
49C2×#11758222C2312yes—
50C22×Dic122D411yescharacteristic
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—21no—
54C22×D244—21no—
55C22×D244—21no—
56C22×D244—22no—
57C22×D244—22no—
58C22×D244—22no—
59C22×D244—22no—
60C22×D244—22no—
61C22×D244D412yes—
62C22×D244D411yescharacteristic
63C22×D244C2312yes—
64C22×C22⋊D61—21no—
65C23×Dic61—21no—
66C22×C4×D61—21no—
67C2×#11758194—21no—
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194C2312yes—
71C22×#92418—21no—
72#14519774—24no—
73#14519774—24no—
74#14519765—24no—
75#14519765—24no—
76#14519752—24no—
77#14519752—24no—
78#14519752—24no—
79#14519752—24no—
80#14519747—28no—
81#14519747—28no—
82C2×#11758198—22no—
83C2×#11758198—22no—
84C2×#11758198—28no—
85C2×#11758198—48no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2312yes—
90#14519703—28no—
91#14519700—24no—
92#14519697—24no—
93#14519693—28no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96C2×M16×C61—21no—
97C2×M16×C61—22no—
98C2×M16×C61D414yescomplemented
99C2×M16×C61C2312yes—
100C22×C8×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61D412yes—
104#117×C61C2312yes—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61—21no—
115#107×C61—24no—
116#106×C61—24no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C23×D61—22no—
120C23×D61—24no—
121C23×D61—22no—
122C23×D61—24no—
123C23×D61—44no—
124C23×D61—44no—
125C23×D61C2×D414yes—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—44no—
130C22×C4×C61—41no—
131C22×C4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—22no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151#11758222—24no—
152#11758222—24no—
153#11758222—22no—
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—42no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—24no—
167C2×D244—24no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—28no—
175C2×D244—28no—
176C2×D244—42no—
177C2×D244—42no—
178C2×D244—42no—
179C2×D244—42no—
180C2×D244—42no—
181C2×D244—42no—
182C2×D244—42no—
183C2×D244—42no—
184C2×D244—84no—
185C2×D244—84no—
186C2×D244C2×D412yes—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—42no—
190C2×C4×D61—42no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122—21no—
194C2×Dic122C2×D412yes—
195M16×C61—24no—
196M16×C61—22no—
197M16×C61C2×D414yescomplemented
198C2×C8×C61—24no—
199C2×C8×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—41no—
205C4⋊C4×C61—41no—
206C4⋊C4×C61C2×D412yes—
207C4⋊C4×C61C2×D412yes—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C42×C61—22no—
211C42×C61—22no—
212#11758198—28no—
213#11758198—28no—
214#11758198—416no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—42no—
218#11758194—22no—
219#11758194—22no—
220#11758194—42no—
221C2×#92418—22no—
222C2×#92418—42no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61#12011yescharacteristic
226C23×C61#12011yescharacteristic
227C23×C61#12112yes—
228C23×C61#12312yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—24no—
231C22×D61—22no—
232C22×D61—24no—
233C22×D61—24no—
234C22×D61—44no—
235C22×D61—44no—
236C22×D61—416no—
237C22×D61—42no—
238C22×D61—44no—
239C22×D61—44no—
240C22×D61—42no—
241C22×D61—44no—
242C22×D61—42no—
243C22×D61—44no—
244C22×D61—82no—
245C22×D61—82no—
246C22×D61—88no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—44no—
252C2×C4×C61—42no—
253C2×C4×C61—82no—
254C2×C4×C61—82no—
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12312yes—
263C2×C4×C61#12312yes—
264C2×C4×C61#12312yes—
265C2×C4×C61#12412yes—
266C2×C4×C61#12412yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C22⋊D61—42no—
274C22⋊D61—42no—
275C22⋊D61—44no—
276C2×Dic61—24no—
277C2×Dic61—22no—
278C2×Dic61—42no—
279C2×Dic61—48no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282D244—22no—
283D244—22no—
284D244—42no—
285D244—42no—
286D244—42no—
287D244—42no—
288D244—44no—
289D244—44no—
290D244—42no—
291D244—88no—
292D244—88no—
293C4×D61—42no—
294C4×D61—42no—
295C4×D61—44no—
296Dic122—22no—
297Dic122—22no—
298Dic122—42no—
299C8×C61—24no—
300#92418—42no—
301C2×#2037—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—24no—
307C22×C61—24no—
308C22×C61#39412yes—
309C22×C61#39412yes—
310C22×C61#45312yes—
311C22×C61#45512yes—
312C22×C61C2×#12011yescharacteristic
313C22×C61C2×#12111yescharacteristic
314C22×C61C2×#12311yescharacteristic
315C2×D61—48no—
316C2×D61—42no—
317C2×D61—44no—
318C2×D61—42no—
319C2×D61—44no—
320C2×D61—42no—
321C2×D61—44no—
322C2×D61—88no—
323C2×D61—84no—
324C2×D61—88no—
325C2×D61—84no—
326C4×C61—42no—
327C4×C61—42no—
328C4×C61—82no—
329C4×C61#39412yes—
330C4×C61#39412yes—
331C4×C61C2×#12111yescharacteristic
332C4×C61C2×#12111yescharacteristic
333C4×C61C2×#12311yescharacteristic
334C4×C61C2×#12411yescharacteristic
335Dic61—44no—
336Dic61—42no—
337C2×#535—611no—
338C2×#428D6111yescharacteristic, complemented
339C22×#104—611no—
340C2×#417—611no—
341C22×#100—611no—
342C2×#390—611no—
343#2037—618no—
344C2×#339—611no—
345C2×C61—21no—
346C2×C61—21no—
347C2×C61—22no—
348C2×C61#203714yescomplemented
349C2×C61C2×#39411yescharacteristic
350C2×C61C2×#45311yescharacteristic
351C2×C61C2×#45511yescharacteristic
352D61—84no—
353D61—82no—
354C22×#41—611no—
355C23×D4—611no—
356C22×M16—611no—
357C22×M16C2×D6111yescharacteristic
358#535—614no—
359#535—1222no—
360#535—1222no—
361#535—1222no—
362C2×#127—1221no—
363C2×#124—1221no—
364C2×#121—1221no—
365C2×#121—1221no—
366C2×#120—1222no—
367C2×#117C2×D6111yescharacteristic
368C22×C4⋊C4C2×D6111yescharacteristic
369C22×C22⋊C4—1222no—
370C22×C42—611no—
371#428—22no—
372#428C2×D6114yescomplemented
373C2×#107—21no—
374C2×#106—21no—
375C2×#104—614no—
376C2×#104—1224no—
377#417—614no—
378#417—1222no—
379C2×#103—1221no—
380C2×#102—1221no—
381C2×#100—614no—
382C2×#100—1224no—
383#390—614no—
384#390—1222no—
385#339—614no—
386C61C2×#203711yescharacteristic, complemented, Sylow
387C25—612no—
388C2×#41—612no—
389C2×#41—612no—
390C2×#41—1221no—
391C2×#41—1221no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41—1222no—
395C2×#41—1222no—
396C22×Q8—611no—
397C22×D4—611no—
398C22×D4—612no—
399C22×D4—612no—
400C22×D4—1222no—
401C22×D4—1222no—
402C22×D4—1221no—
403C22×D4—1222no—
404C22×D4—1222no—
405C22×D4—1221no—
406C22×D4—1221no—
407C22×D4—1222no—
408C22×D4—1222no—
409C22×D4—1221no—
410C22×D4—1221no—
411C23×C4—21no—
412C23×C4—1221no—
413C23×C4—1221no—
414C23×C4C22×D6111yescharacteristic
415C2×M16—22no—
416C2×M16—21no—
417C2×M16—612no—
418C2×M16—1221no—
419C2×M16—1222no—
420C2×M16—1222no—
421C2×M16D24414yescomplemented
422C2×M16C22×D6112yes—
423C22×C8—21no—
424C22×C8—1221no—
425#127—1224no—
426#127—1224no—
427#124—1224no—
428#124—1224no—
429#121—1224no—
430#121—1224no—
431#121—1224no—
432#121—1224no—
433#120—1228no—
434#120—1228no—
435#117—21no—
436#117—21no—
437#117C22⋊D6112yes—
438#117C22×D6112yes—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4C22⋊D6111yescharacteristic
445C2×C4⋊C4C22⋊D6111yescharacteristic
446C2×C4⋊C4C22×D6112yes—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C22⋊C4—1228no—
451C2×C22⋊C4—2448no—
452C2×C42—21no—
453C2×C42—612no—
454C2×C42—1222no—
455C2×C42—1222no—
456C2×C42—1221no—
457#107—24no—
458#106—24no—
459#104—1228no—
460#103—1224no—
461#102—1224no—
462#100—1228no—
463C24—614no—
464C24—1222no—
465C24—1224no—
466C24—1222no—
467C24—1221no—
468C24—1224no—
469C24—2444no—
470C24—2444no—
471C24D4×D6111yescharacteristic
472#41—1222no—
473#41—1222no—
474#41—1222no—
475#41—1222no—
476#41—1224no—
477#41—1224no—
478#41—2442no—
479#41—2442no—
480#41—2442no—
481#41—2442no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1222no—
485C2×Q8—1222no—
486C2×D4—612no—
487C2×D4—1228no—
488C2×D4—1228no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—1222no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—1222no—
501C2×D4—1224no—
502C2×D4—1222no—
503C2×D4—1224no—
504C2×D4—4884no—
505C2×D4—4884no—
506C2×D4—2442no—
507C2×D4—2442no—
508C2×D4—2442no—
509C2×D4—2442no—
510C2×D4—2442no—
511C2×D4—2442no—
512C2×D4—2442no—
513C2×D4—2442no—
514C2×D4—2442no—
515C2×D4—2442no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—44no—
520C22×C4—41no—
521C22×C4—41no—
522C22×C4—1224no—
523C22×C4—1222no—
524C22×C4—1221no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4—2442no—
531C22×C4C2×D24412yes—
532C22×C4#1175822212yes—
533C22×C4D4×D6111yescharacteristic
534C22×C4D4×D6111yescharacteristic
535C22×C4D4×D6111yescharacteristic
536C22×C4#1175822612yes—
537C22×C4#1175822612yes—
538C22×C4C2×C22⋊D6111yescharacteristic
539C22×C4C2×C22⋊D6111yescharacteristic
540C22×C4C23×D6111yescharacteristic
541M16—22no—
542M16—24no—
543M16—1222no—
544M16—1222no—
545M16—2442no—
546M16C2×D24414yescomplemented
547C2×C8—22no—
548C2×C8—24no—
549C2×C8—1222no—
550C2×C8—2442no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4C2×C22⋊D6112yes—
558C4⋊C4C2×C22⋊D6112yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—1228no—
562C22⋊C4—1228no—
563C22⋊C4—24416no—
564C42—22no—
565C42—22no—
566C42—1222no—
567C42—1222no—
568C42—2442no—
569C23—22no—
570C23—22no—
571C23—1224no—
572C23—1224no—
573C23—1222no—
574C23—1224no—
575C23—4888no—
576C23—4882no—
577C23—4882no—
578C23—24416no—
579C23—2444no—
580C23—2444no—
581C23—2442no—
582C23—2444no—
583C23—2442no—
584C23—2444no—
585C23—2442no—
586C23—2444no—
587C23—2444no—
588C23#1451974511yescharacteristic
589C23#1451974511yescharacteristic
590C23#1451975212yes—
591C23#1451976212yes—
592C23C2×D4×D6111yescharacteristic
593Q8—1222no—
594Q8—1222no—
595Q8—2442no—
596D4—1222no—
597D4—1222no—
598D4—4888no—
599D4—4888no—
600D4—2442no—
601D4—2442no—
602D4—2444no—
603D4—2442no—
604D4—2442no—
605D4—2442no—
606D4—2442no—
607D4—2444no—
608D4—2444no—
609D4—2442no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—1224no—
619C2×C4—1222no—
620C2×C4—2442no—
621C2×C4—2448no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2444no—
627C2×C4#1451974511yescharacteristic
628C2×C4#1451974511yescharacteristic
629C2×C4#1451975112yes—
630C2×C4#1451975112yes—
631C2×C4#1451975212yes—
632C2×C4#1451975212yes—
633C2×C4#1451975212yes—
634C2×C4#1451976212yes—
635C2×C4#1451976212yes—
636C2×C4#1451976212yes—
637C2×C4#1451976612yes—
638C2×C4#1451976612yes—
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C22×C22⋊D6111yescharacteristic
641C2×C4C22×D24411yescharacteristic
642C2×C4C2×#1175822211yescharacteristic
643C2×C4C2×#1175822611yescharacteristic
644C2×C4C2×#1175822611yescharacteristic
645C8—24no—
646C8—2442no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—24no—
652C22—24no—
653C22—4884no—
654C22—4888no—
655C22—4888no—
656C22—4884no—
657C22—2448no—
658C22—2442no—
659C22—2444no—
660C22—2442no—
661C22—2444no—
662C22—2442no—
663C22—2444no—
664C22#1683967912yes—
665C22#1683967912yes—
666C22#1683992812yes—
667C22#1683993812yes—
668C22C2×#1451974511yescharacteristic
669C22C2×#1451975211yescharacteristic
670C22C2×#1451976211yescharacteristic
671C4—42no—
672C4—42no—
673C4—82no—
674C4—2444no—
675C4—2442no—
676C4#1683967912yes—
677C4#1683967912yes—
678C4C2×#1451975111yescharacteristic
679C4C2×#1451975211yescharacteristic
680C4C2×#1451976211yescharacteristic
681C4C2×#1451976611yescharacteristic
682C2—21no—
683C2—22no—
684C2—21no—
685C2—4884no—
686C2—4882no—
687C2#2144703114yesdirect product factor
688C2C2×#1683967911yescharacteristic
689C2C2×#1683992811yescharacteristic
690C2C2×#1683993811yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61#120
prime radicalC2×#16840331C2
hypercenterC2×#428D61
hypocenterC61C2×#2037
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C22×#14519703C2semidirect
3. C2×#16839761C2semidirect
4. C22×#14519693C2semidirect
5. C2×#16839663C2semidirect
6. #21447031C2direct (N × H)
7. C2×#16839521C2semidirect
8. C2×C61×#428C2semidirect
24. C2×#14519703C22semidirect
26. #16839761C22semidirect
30. C2×#14519693C22semidirect
32. #16839663C22semidirect
33. #16839521C22semidirect
38. C61×#428C22semidirect
98. C2×M16×C61D4semidirect
197. M16×C61C2×D4semidirect
338. C2×#428D61semidirect
348. C2×C61#2037semidirect
372. #428C2×D61semidirect
386. C61C2×#2037semidirect
421. C2×M16D244semidirect
546. M16C2×D244semidirect
687. C2#21447031direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403312
C22×#145197032
C2×#168397612
C22×#145196932
C2×#168396632
#214470312
C2×#168395212
C2×C61×#4282
C2×#203761