Finite Group Databasebeta

← C2×#21444765 #29652204

Subgroups — C2×#21444765 #29652204

Order 15616.

1716 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×#16839912C211yesmaximal, characteristic, complemented
2C2×#16839895C211yesmaximal, characteristic, complemented
3C2×#16839671C211yesmaximal, characteristic, complemented
4#21444765C218yesmaximal, direct product factor
5C2×#16839499C211yesmaximal, characteristic, complemented
6C2×#16839496C211yesmaximal, characteristic, complemented
7C2×#16839486C211yesmaximal, characteristic, complemented
8C2×C61×#538C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C23×C22⋊D61—21no—
11C22×#11758201—21no—
12C22×#11758199—21no—
13C22×#11758198—21no—
14#16839912—22no—
15#16839912C2214yescomplemented
16#16839895—22no—
17#16839895C2214yescomplemented
18C2×#14519697—21no—
19C2×#14519697—21no—
20C2×C22⋊#92418C2211yescharacteristic
21C2×C22⋊#92418C2211yescharacteristic
22#16839671—22no—
23#16839671C2214yescomplemented
24C2×#14519684C2211yescharacteristic
25#16839499C2214yescomplemented
26#16839496C2214yescomplemented
27#16839486C2214yescomplemented
28C61×#538—22no—
29C61×#538—22no—
30C61×#538—22no—
31C61×#538C2214yescomplemented
32C2×#126×C61—21no—
33C2×#124×C61—21no—
34C2×#123×C61—21no—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C2×#121×C61—21no—
37C2×#121×C61C2211yescharacteristic
38C2×#120×C61—21no—
39C2×C4×D4×C61—21no—
40C22×C22⋊C4×C61C2211yescharacteristic
41C24×D61—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—44no—
48C22×D244C2312yes—
49C22×C22⋊D61—24no—
50C22×C22⋊D61—44no—
51C22×C22⋊D61—44no—
52C22×C22⋊D61—44no—
53C23×Dic61—21no—
54C2×#11758201—24no—
55C2×#11758201—44no—
56C2×#11758199—24no—
57C2×#11758199—44no—
58C22×#92418—21no—
59C22×#92418—21no—
60C2×#11758198—24no—
61C2×#11758198—44no—
62#14519697—24no—
63#14519697—24no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66C22⋊#92418C2312yes—
67C22⋊#92418C2312yes—
68#14519684—21no—
69#14519684C2312yes—
70C25×C61D411yescharacteristic
71C22×Q8×C61D411yescharacteristic, complemented
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61D411yescharacteristic, complemented
76C23×C4×C61—21no—
77C23×C4×C61D411yescharacteristic
78C23×C4×C61C2311yescharacteristic
79#126×C61—24no—
80#126×C61—24no—
81#124×C61—24no—
82#124×C61—24no—
83#123×C61—24no—
84#123×C61—24no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61D412yescomplemented
88C22⋊Q8×C61C2312yes—
89#121×C61—24no—
90#121×C61—24no—
91#121×C61—21no—
92#121×C61—21no—
93#121×C61D412yescomplemented
94#121×C61C2312yes—
95#120×C61—24no—
96#120×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic, complemented
102C2×C4⋊C4×C61D411yescharacteristic, complemented
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D412yes—
113C2×C22⋊C4×C61C2312yes—
114C2×C42×C61—21no—
115C24×C61—21no—
116C24×C61—21no—
117C24×C61—22no—
118C24×C61—21no—
119C24×C61—22no—
120C24×C61—42no—
121C24×C61—42no—
122C24×C61C2×D411yescharacteristic
123C24×C61C2×D412yes—
124C23×D61—24no—
125C23×D61—22no—
126C23×D61—42no—
127C23×D61—42no—
128C23×D61—42no—
129C23×D61—41no—
130C23×D61—41no—
131C23×D61—42no—
132C23×D61—42no—
133C2×Q8×C61—21no—
134C2×Q8×C61—42no—
135C2×Q8×C61C2×D412yescomplemented
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61—24no—
140C2×D4×C61—21no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61—82no—
145C2×D4×C61—82no—
146C2×D4×C61C2×D412yescomplemented
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—21no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61—82no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2411yescharacteristic
163C2×C22⋊D61—48no—
164C2×C22⋊D61—48no—
165C2×C22⋊D61—48no—
166C2×C22⋊D61—432no—
167C22×Dic61—24no—
168C22×Dic61—22no—
169C22×Dic61—42no—
170C22×Dic61—42no—
171C2×D244—24no—
172C2×D244—24no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—44no—
176C2×D244—44no—
177C2×D244—44no—
178C2×D244—42no—
179C2×D244—42no—
180C2×D244—44no—
181C2×D244—41no—
182C2×D244—41no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61C2×D412yescomplemented
189C4⋊C4×C61C2×D412yescomplemented
190C22⋊C4×C61—24no—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—24no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C42×C61—24no—
202#11758201—48no—
203#11758199—48no—
204#11758198—48no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—42no—
208C2×#92418—42no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—48no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—81no—
226C23×C61—81no—
227C23×C61—84no—
228C23×C61#12011yescharacteristic
229C23×C61#12011yescharacteristic
230C23×C61#13512yes—
231C23×C61AGL(1,ℤ/8)12yes—
232C23×C61C22×D411yescharacteristic
233C22×D61—24no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—44no—
237C22×D61—48no—
238C22×D61—42no—
239C22×D61—48no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61—44no—
244C22×D61—84no—
245C22×D61—84no—
246C22×D61—82no—
247C22×D61—82no—
248C22×D61—82no—
249C22×D61—82no—
250Q8×C61—42no—
251Q8×C61—81no—
252D4×C61—42no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—44no—
256D4×C61—81no—
257D4×C61—84no—
258D4×C61—84no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—42no—
274C2×C4×C61—81no—
275C2×C4×C61—84no—
276C2×C4×C61#12011yescharacteristic
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C22⋊D61—432no—
281C2×Dic61—24no—
282C2×Dic61—48no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285D244—24no—
286D244—24no—
287D244—42no—
288D244—42no—
289D244—44no—
290D244—44no—
291#92418—42no—
292#92418—42no—
293C2×#1600—611nomaximal, Sylow
294C22×C61—24no—
295C22×C61—24no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—44no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—82no—
308C22×C61—84no—
309C22×C61—84no—
310C22×C61—84no—
311C22×C61#44912yes—
312C22×C61#44912yes—
313C22×C61#45312yes—
314C22×C61B(4,𝔽2)12yes—
315C22×C61C2×#12011yescharacteristic
316C22×C61C2×#13511yescharacteristic
317C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
318C2×D61—44no—
319C2×D61—44no—
320C2×D61—44no—
321C2×D61—82no—
322C2×D61—82no—
323C2×D61—88no—
324C2×D61—88no—
325C2×D61—82no—
326C2×D61—82no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—42no—
332C4×C61—42no—
333C4×C61—42no—
334C4×C61—42no—
335Dic61—44no—
336C2×#538D6111yescharacteristic, complemented
337C2×#450—611no—
338C2×#447—611no—
339C2×#392—611no—
340#1600—618no—
341C2×#328—611no—
342C2×#327—611no—
343C2×#323—611no—
344C2×C61—21no—
345C2×C61—21no—
346C2×C61—22no—
347C2×C61—82no—
348C2×C61—82no—
349C2×C61#160014yescomplemented
350C2×C61C2×#44911yescharacteristic
351C2×C61C2×#45311yescharacteristic
352C2×C61C2×B(4,𝔽2)11yescharacteristic
353D61—84no—
354C23×D4—611no—
355C23×D4—1221no—
356C22×SD16—1221no—
357C22×D8—1221no—
358#538—22no—
359#538—22no—
360#538—22no—
361#538C2×D6114yescomplemented
362C2×#126—21no—
363C2×#124—21no—
364C2×#123—21no—
365C2×C22⋊Q8C2×D6111yescharacteristic
366C2×#121—21no—
367C2×#121C2×D6111yescharacteristic
368C2×#120—21no—
369C2×C4×D4—21no—
370C22×C22⋊C4—1221no—
371C22×C22⋊C4C2×D6111yescharacteristic
372#450—614no—
373#450—1222no—
374#447—614no—
375#447—1222no—
376C2×#102—1221no—
377C2×#102—1221no—
378C2×C22⋊C8—611no—
379C2×C22⋊C8—611no—
380#392—614no—
381#392—1222no—
382C2×#95—611no—
383#328—614no—
384#327—614no—
385#323—614no—
386C61C2×#160011yescharacteristic, complemented, Sylow
387C25—1221no—
388C25C22⋊D6111yescharacteristic
389C22×Q8C22⋊D6111yescharacteristic, complemented
390C22×D4—21no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—612no—
394C22×D4—1224no—
395C22×D4—1221no—
396C22×D4—1221no—
397C22×D4—1221no—
398C22×D4—1221no—
399C22×D4—1221no—
400C22×D4—2444no—
401C22×D4—2444no—
402C22×D4—2444no—
403C22×D4—2444no—
404C22×D4C22⋊D6111yescharacteristic, complemented
405C23×C4—21no—
406C23×C4—1221no—
407C23×C4C22⋊D6111yescharacteristic
408C23×C4C22×D6111yescharacteristic
409C2×SD16—1224no—
410C2×SD16—2444no—
411C2×D8—1224no—
412C2×D8—2444no—
413C22×C8—1221no—
414C22×C8—1221no—
415#126—24no—
416#126—24no—
417#124—24no—
418#124—24no—
419#123—24no—
420#123—24no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8C22⋊D6112yescomplemented
424C22⋊Q8C22×D6112yes—
425#121—24no—
426#121—24no—
427#121—21no—
428#121—21no—
429#121C22⋊D6112yescomplemented
430#121C22×D6112yes—
431#120—24no—
432#120—24no—
433C4×D4—24no—
434C4×D4—24no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
438C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1224no—
449C2×C22⋊C4—2444no—
450C2×C22⋊C4C22⋊D6112yes—
451C2×C22⋊C4C22×D6112yes—
452C2×C42—21no—
453#102—1224no—
454#102—1224no—
455C22⋊C8—612no—
456C22⋊C8—612no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459#95—612no—
460#95—1221no—
461C24—21no—
462C24—21no—
463C24—22no—
464C24—21no—
465C24—22no—
466C24—42no—
467C24—42no—
468C24—1224no—
469C24—1222no—
470C24—2442no—
471C24—2442no—
472C24—2442no—
473C24—2441no—
474C24—2441no—
475C24—2442no—
476C24—2442no—
477C24C2×C22⋊D6111yescharacteristic
478C24C2×C22⋊D6112yes—
479C2×Q8—21no—
480C2×Q8—42no—
481C2×Q8C2×C22⋊D6112yescomplemented
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4—1224no—
493C2×D4—1224no—
494C2×D4—1222no—
495C2×D4—1222no—
496C2×D4—2442no—
497C2×D4—2442no—
498C2×D4—2444no—
499C2×D4—2444no—
500C2×D4—2444no—
501C2×D4—2444no—
502C2×D4—2448no—
503C2×D4—2448no—
504C2×D4—2448no—
505C2×D4—24432no—
506C2×D4—2441no—
507C2×D4—2441no—
508C2×D4C2×C22⋊D6112yescomplemented
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—24no—
514C22×C4—21no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—82no—
518C22×C4—1224no—
519C22×C4—1222no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6112yes—
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C2×C22⋊D6111yescharacteristic
528C22×C4C23×D6111yescharacteristic
529SD16—2448no—
530D8—2448no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—2442no—
534C2×C8—2442no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4C2×C22⋊D6112yescomplemented
541C4⋊C4C2×C22⋊D6112yescomplemented
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—2448no—
554C42—24no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—81no—
572C23—81no—
573C23—84no—
574C23—1224no—
575C23—4882no—
576C23—4884no—
577C23—4884no—
578C23—4882no—
579C23—4882no—
580C23—4882no—
581C23—2442no—
582C23—2442no—
583C23—2442no—
584C23—2444no—
585C23—2448no—
586C23—2448no—
587C23—2444no—
588C23—2442no—
589C23—2442no—
590C23—2442no—
591C23#1451974611yescharacteristic
592C23#1451974611yescharacteristic
593C23#1451979712yes—
594C23#1451980212yes—
595C23C22×C22⋊D6111yescharacteristic
596Q8—42no—
597Q8—81no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—44no—
602D4—81no—
603D4—84no—
604D4—84no—
605D4—1224no—
606D4—1224no—
607D4—2442no—
608D4—2442no—
609D4—2444no—
610D4—2444no—
611D4—24432no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—81no—
628C2×C4—84no—
629C2×C4—1224no—
630C2×C4—2442no—
631C2×C4—2448no—
632C2×C4—2442no—
633C2×C4#1451974611yescharacteristic
634C2×C4#1451974611yescharacteristic
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4C22×C22⋊D6111yescharacteristic
637C8—2442no—
638C8—2442no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—24no—
644C22—24no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—4882no—
657C22—4888no—
658C22—4882no—
659C22—4882no—
660C22—4888no—
661C22—4882no—
662C22—2444no—
663C22—2444no—
664C22—2444no—
665C22#1683990712yes—
666C22#1683990712yes—
667C22#1683993012yes—
668C22#1683994912yes—
669C22C2×#1451974611yescharacteristic
670C22C2×#1451979711yescharacteristic
671C22C2×#1451980211yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—2444no—
681C2—21no—
682C2—22no—
683C2—21no—
684C2—82no—
685C2—82no—
686C2—4884no—
687C2#2144476514yesdirect product factor
688C2C2×#1683990711yescharacteristic
689C2C2×#1683993011yescharacteristic
690C2C2×#1683994911yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#538C2
socleC23×C61#120
hypercenterC2×#538D61
hypocenterC61C2×#1600
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. C2×#16839912C2semidirect
2. C2×#16839895C2semidirect
3. C2×#16839671C2semidirect
4. #21444765C2direct (N × H)
5. C2×#16839499C2semidirect
6. C2×#16839496C2semidirect
7. C2×#16839486C2semidirect
8. C2×C61×#538C2semidirect
15. #16839912C22semidirect
17. #16839895C22semidirect
23. #16839671C22semidirect
25. #16839499C22semidirect
26. #16839496C22semidirect
27. #16839486C22semidirect
31. C61×#538C22semidirect
71. C22×Q8×C61D4semidirect
75. C22×D4×C61D4semidirect
87. C22⋊Q8×C61D4semidirect
93. #121×C61D4semidirect
101. C2×C4⋊C4×C61D4semidirect
102. C2×C4⋊C4×C61D4semidirect
135. C2×Q8×C61C2×D4semidirect
146. C2×D4×C61C2×D4semidirect
188. C4⋊C4×C61C2×D4semidirect
189. C4⋊C4×C61C2×D4semidirect
336. C2×#538D61semidirect
349. C2×C61#1600semidirect
361. #538C2×D61semidirect
386. C61C2×#1600semidirect
389. C22×Q8C22⋊D61semidirect
404. C22×D4C22⋊D61semidirect
423. C22⋊Q8C22⋊D61semidirect
429. #121C22⋊D61semidirect
437. C2×C4⋊C4C22⋊D61semidirect
438. C2×C4⋊C4C22⋊D61semidirect
481. C2×Q8C2×C22⋊D61semidirect
508. C2×D4C2×C22⋊D61semidirect
540. C4⋊C4C2×C22⋊D61semidirect
541. C4⋊C4C2×C22⋊D61semidirect
687. C2#21444765direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168399122
C2×#168398952
C2×#168396712
#214447652
C2×#168394992
C2×#168394962
C2×#168394862
C2×C61×#5382
C2×#160061