Finite Group Databasebeta

← C2×#20275064 #27385417

Subgroups — C2×#20275064 #27385417

Order 13568.

2470 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D424C211yesmaximal, characteristic
2C22×#13133179C211yesmaximal, characteristic, complemented
3C22×#13133137C211yesmaximal, characteristic, complemented
4C22×#13133137C211yesmaximal, characteristic, complemented
5C2×#16810524C211yesmaximal, characteristic, complemented
6#20275064C218yesmaximal, direct product factor
7C2×#16810377C211yesmaximal, characteristic, complemented
8C2×C53×#340C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C23×D212C2211yescharacteristic
11C23×C4×D53—21no—
12C22×D424—24no—
13C22×D424—24no—
14C22×D424C418yescomplemented
15C22×D424C2214yes—
16C23×#88322C2211yescharacteristic
17C2×#13133179—24no—
18C2×#13133179—22no—
19C2×#13133179—24no—
20C2×#13133179C2214yescomplemented
21C22×#11719921—21no—
22C22×#11719921—21no—
23C22×#11719921—21no—
24C22×C4×Dic53C2211yescharacteristic
25C2×#13133137—24no—
26C2×#13133137—24no—
27C2×#13133137C2214yescomplemented
28C2×#13133137C2214yescomplemented
29#16810524—22no—
30#16810524C2214yescomplemented
31#16810377C2214yescomplemented
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#340C2214yescomplemented
36C24×D53—21no—
37C24×D53—21no—
38C22×D212—21no—
39C22×D212—22no—
40C22×D212—22no—
41C22×D212—22no—
42C22×D212—22no—
43C22×D212—22no—
44C22×D212—22no—
45C22×D212—22no—
46C22×D212—21no—
47C22×D212—22no—
48C22×D212C2×C414yes—
49C22×D212C2×C414yescomplemented
50C22×D212D414yes—
51C22×D212D414yes—
52C22×D212C2312yes—
53C22×D212C2312yes—
54C23×Dic53D411yescharacteristic
55C23×Dic53D411yescharacteristic
56C22×C4×D53—28no—
57C22×C4×D53—24no—
58C22×C4×D53—44no—
59C22×C4×D53—44no—
60C2×D424—232no—
61C2×D424—28no—
62C2×D424—28no—
63C2×D424C2×C4116yescomplemented
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322C2312yes—
68#13133179—216no—
69#13133179—28no—
70#13133179—28no—
71C2×#11719921—28no—
72C2×#11719921—24no—
73C2×#11719921—24no—
74C2×#11719921—24no—
75C2×#11719921—44no—
76C2×#11719921—44no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—22no—
79C2×C4×Dic53D412yes—
80C2×C4×Dic53C2312yes—
81#13133137—28no—
82#13133137—28no—
83C23×C4×C53—21no—
84C23×C4×C53—21no—
85C23×C4×C53C2311yescharacteristic
86C22×C8×C53—22no—
87C22×C8×C53—22no—
88C22×C8×C53C2×C414yescomplemented
89C22×C8×C53C2312yes—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53D414yes—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2312yes—
98C24×C53C2×D411yescharacteristic
99C23×D53—24no—
100C23×D53—24no—
101C23×D53—24no—
102C23×D53—22no—
103C23×D53—24no—
104C23×D53—22no—
105C23×D53—42no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C23×D53—42no—
113C23×D53—42no—
114C23×D53—42no—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D412yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53D814yescomplemented
137C22×Dic53C2×D412yes—
138C22×Dic53C2×D412yes—
139C2×D212—28no—
140C2×D212—24no—
141C2×D212—24no—
142C2×D212—28no—
143C2×D212—24no—
144C2×D212—28no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—22no—
148C2×D212—28no—
149C2×D212—28no—
150C2×D212—22no—
151C2×D212—22no—
152C2×D212—22no—
153C2×D212—22no—
154C2×D212—24no—
155C2×D212—22no—
156C2×D212—42no—
157C2×D212—44no—
158C2×D212—42no—
159C2×D212—44no—
160C2×D212C22⋊C4116yescomplemented
161C2×D212C22×C414yes—
162C2×D212C22×C414yescomplemented
163C2×D212C2×D414yes—
164C2×D212C2×D414yes—
165C2×C4×D53—216no—
166C2×C4×D53—48no—
167C2×C4×D53—432no—
168C2×C4×D53—48no—
169C2×C8×C53—28no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53C22×C414yescomplemented
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53C2×D414yes—
179#11719921—216no—
180#11719921—48no—
181#11719921—48no—
182C4×Dic53—24no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—44no—
189D424—232no—
190C23×C53C4×D412yes—
191C23×C53#12012yes—
192C23×C53#12112yes—
193C23×C53C2×D812yes—
194C23×C53#13512yes—
195C23×C53AGL(1,ℤ/8)12yes—
196C23×C53AGL(1,ℤ/8)12yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—216no—
202C22×D53—24no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—48no—
208C22×D53—48no—
209C22×D53—48no—
210C22×D53—48no—
211C22×D53—48no—
212C22×D53—48no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—48no—
220C22×D53—48no—
221C22×D53—48no—
222C22×D53—84no—
223C22×D53—84no—
224C22×D53—84no—
225C22×D53—84no—
226C2×C4×C53—24no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—48no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—84no—
234C2×C4×C53C2×C22⋊C414yes—
235C2×C4×C53C4×D412yes—
236C2×C4×C53C4×D414yes—
237C2×C4×C53#12012yes—
238C2×C4×C53#12112yes—
239C2×C4×C53#12114yes—
240C2×C4×C53#12314yes—
241C2×C4×C53C23×C411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—28no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—44no—
254C2×Dic53C2×D814yescomplemented
255D212—24no—
256D212—24no—
257D212—24no—
258D212—24no—
259D212—24no—
260D212—216no—
261D212—42no—
262D212—42no—
263D212—48no—
264D212C2×C22⋊C418yescomplemented
265C4×D53—432no—
266C8×C53—24no—
267#88322—42no—
268C2×#1879—531nomaximal, Sylow
269C22×C53#38614yes—
270C22×C53C4×D814yes—
271C22×C53#44214yes—
272C22×C53#44714yes—
273C22×C53#44914yes—
274C22×C53#45914yes—
275C22×C53#46314yes—
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×#12011yescharacteristic
278C22×C53C2×#12111yescharacteristic
279C22×C53C22×D811yescharacteristic
280C22×C53C2×#13511yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C2×D53—28no—
284C2×D53—432no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—44no—
288C2×D53—44no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53—816no—
295C2×D53—82no—
296C2×D53—82no—
297C2×D53—82no—
298C2×D53—82no—
299C4×C53—44no—
300C4×C53—82no—
301C4×C53#38614yes—
302C4×C53C22×C22⋊C411yescharacteristic
303C4×C53C2×C4×D411yescharacteristic
304C4×C53C2×#12111yescharacteristic
305C4×C53C2×#12311yescharacteristic
306Dic53—24no—
307Dic53—42no—
308C23×D8—531no—
309C22×C4×D4—531no—
310C22×#102—531no—
311C22×#102—531no—
312C2×#390—531no—
313#1879—538no—
314C2×#340D5311yescharacteristic, complemented
315C2×#336—531no—
316C2×C53#187918yescomplemented
317C2×C53C2×#38611yescharacteristic
318C2×C53C2×C4×D811yescharacteristic
319C2×C53C2×#44211yescharacteristic
320C2×C53C2×#44711yescharacteristic
321C2×C53C2×#44911yescharacteristic
322C2×C53C2×#45911yescharacteristic
323C2×C53C2×#46311yescharacteristic
324D53—48no—
325D53—84no—
326C23×D4—531no—
327C23×D4—531no—
328C24×C4—1061no—
329C22×D8—538no—
330C22×D8—534no—
331C22×D8—1064no—
332C22×D8—1064no—
333C23×C8—531no—
334C23×C8C2×D5311yescharacteristic
335C2×C4×D4—534no—
336C2×C4×D4—1062no—
337C2×C4×D4—1064no—
338C2×C4×D4—1064no—
339C22×C4⋊C4C2×D5311yescharacteristic
340C22×C4⋊C4C2×D5311yescharacteristic
341C22×C22⋊C4—1061no—
342C22×C22⋊C4—1061no—
343C22×C22⋊C4—1061no—
344C22×C42—531no—
345C2×#102—534no—
346C2×#102—534no—
347C2×#102—1064no—
348C2×#102—1064no—
349#390—534no—
350#390—1062no—
351#340C2×D5314yescomplemented
352#336—534no—
353C53C2×#187911yescharacteristic, complemented, Sylow
354C25—1061no—
355C25—1061no—
356C22×D4—534no—
357C22×D4—534no—
358C22×D4—534no—
359C22×D4—534no—
360C22×D4—532no—
361C22×D4—532no—
362C22×D4—1062no—
363C22×D4—1062no—
364C22×D4—1062no—
365C22×D4—1062no—
366C22×D4—1061no—
367C22×D4—1062no—
368C22×D4—1062no—
369C22×D4—1061no—
370C22×D4—1062no—
371C22×D4—1062no—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—1064no—
377C23×C4—1068no—
378C23×C4—2124no—
379C23×C4—2124no—
380C23×C4C22×D5311yescharacteristic
381C2×D8—5316no—
382C2×D8—10632no—
383C2×D8—1068no—
384C2×D8—1068no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—532no—
388C22×C8—1062no—
389C22×C8—1062no—
390C22×C8—1062no—
391C22×C8C4×D5314yescomplemented
392C22×C8C22×D5312yes—
393C4×D4—1068no—
394C4×D4—10616no—
395C4×D4—1068no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4D21214yes—
402C2×C4⋊C4C22×D5312yes—
403C2×C4⋊C4C22×D5312yes—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1068no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—2124no—
409C2×C22⋊C4—2124no—
410C2×C42—532no—
411C2×C42—532no—
412C2×C42—1062no—
413C2×C42—1062no—
414#102—1068no—
415#102—1068no—
416C24—1064no—
417C24—1064no—
418C24—1064no—
419C24—1062no—
420C24—1064no—
421C24—1062no—
422C24—2122no—
423C24—2122no—
424C24—2122no—
425C24—2122no—
426C24—2122no—
427C24—2122no—
428C24—2122no—
429C24—2122no—
430C24—2122no—
431C24—2122no—
432C24D4×D5311yescharacteristic
433C2×D4—5316no—
434C2×D4—534no—
435C2×D4—534no—
436C2×D4—534no—
437C2×D4—534no—
438C2×D4—1068no—
439C2×D4—1064no—
440C2×D4—1064no—
441C2×D4—1068no—
442C2×D4—1068no—
443C2×D4—1064no—
444C2×D4—1064no—
445C2×D4—1068no—
446C2×D4—1068no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4—1062no—
450C2×D4—1062no—
451C2×D4—1062no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—2122no—
456C2×D4—2122no—
457C2×D4—2124no—
458C2×D4—2124no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—10616no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—2128no—
477C22×C4—21232no—
478C22×C4—2128no—
479C22×C4C2×C4×D5312yes—
480C22×C4C2×D21212yes—
481C22×C4#1171994512yes—
482C22×C4D4×D5312yes—
483C22×C4D4×D5312yes—
484C22×C4D4×D5311yescharacteristic
485C22×C4#1171994912yes—
486C22×C4C23×D5311yescharacteristic
487D8—10632no—
488C2×C8—28no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—1062no—
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—2124no—
495C2×C8C2×C4×D5314yescomplemented
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4C2×D21214yes—
502C22⋊C4—10616no—
503C22⋊C4—2128no—
504C22⋊C4—2128no—
505C42—1064no—
506C42—1062no—
507C42—1062no—
508C23—10616no—
509C23—1064no—
510C23—1064no—
511C23—1064no—
512C23—1064no—
513C23—4244no—
514C23—4244no—
515C23—4244no—
516C23—4244no—
517C23—2128no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2128no—
525C23—2128no—
526C23—2128no—
527C23—2128no—
528C23—2128no—
529C23—2128no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2128no—
535C23—2128no—
536C23#1313317912yes—
537C23#1313318712yes—
538C23#1313319012yes—
539C23D8×D5312yes—
540C23#1313324012yes—
541C23#1313324412yes—
542C23#1313324412yes—
543C23C2×D4×D5311yescharacteristic
544D4—538no—
545D4—10616no—
546D4—1064no—
547D4—1064no—
548D4—1064no—
549D4—1064no—
550D4—1064no—
551D4—2128no—
552D4—2122no—
553D4—2122no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—84no—
562C2×C4—534no—
563C2×C4—1068no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—21232no—
570C2×C4—2124no—
571C2×C4C22⋊C4×D5314yes—
572C2×C4#1313317912yes—
573C2×C4C4×D21214yes—
574C2×C4#1313318712yes—
575C2×C4#1313319012yes—
576C2×C4#1313319214yes—
577C2×C4#1313320214yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C2×D4×D5311yescharacteristic
580C2×C4C2×D4×D5311yescharacteristic
581C2×C4C22×D21211yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4C2×#1171994911yescharacteristic
584C8—24no—
585C8—2122no—
586C22—1068no—
587C22—4242no—
588C22—42416no—
589C22—4242no—
590C22—4242no—
591C22—4242no—
592C22—2124no—
593C22—2124no—
594C22—2124no—
595C22—2124no—
596C22—2124no—
597C22—2124no—
598C22—21232no—
599C22—2124no—
600C22—2124no—
601C22—2124no—
602C22#1681050114yes—
603C22#1681070514yes—
604C22#1681073414yes—
605C22#1681075814yes—
606C22#1681077114yes—
607C22#1681081714yes—
608C22#1681084114yes—
609C22C2×#1313317911yescharacteristic
610C22C2×#1313318711yescharacteristic
611C22C2×#1313319011yescharacteristic
612C22C2×D8×D5311yescharacteristic
613C22C2×#1313324011yescharacteristic
614C22C2×#1313324411yescharacteristic
615C22C2×#1313324411yescharacteristic
616C4—44no—
617C4—82no—
618C4—1064no—
619C4—2122no—
620C4#1681050114yes—
621C4C2×C22⋊C4×D5311yescharacteristic
622C4C2×C4×D21211yescharacteristic
623C4C2×#1313319211yescharacteristic
624C4C2×#1313320211yescharacteristic
625C2—4244no—
626C2—2128no—
627C2#2027506418yesdirect product factor
628C2C2×#1681050111yescharacteristic
629C2C2×#1681070511yescharacteristic
630C2C2×#1681073411yescharacteristic
631C2C2×#1681075811yescharacteristic
632C2C2×#1681077111yescharacteristic
633C2C2×#1681081711yescharacteristic
634C2C2×#1681084111yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC24×C53C2×D4
prime radicalC23×D424C2
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53C2×#1879
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×#13133179C2semidirect
3. C22×#13133137C2semidirect
4. C22×#13133137C2semidirect
5. C2×#16810524C2semidirect
6. #20275064C2direct (N × H)
7. C2×#16810377C2semidirect
8. C2×C53×#340C2semidirect
14. C22×D424C4semidirect
20. C2×#13133179C22semidirect
27. C2×#13133137C22semidirect
28. C2×#13133137C22semidirect
30. #16810524C22semidirect
31. #16810377C22semidirect
35. C53×#340C22semidirect
49. C22×D212C2×C4semidirect
63. C2×D424C2×C4semidirect
88. C22×C8×C53C2×C4semidirect
136. C22×Dic53D8semidirect
160. C2×D212C22⋊C4semidirect
162. C2×D212C22×C4semidirect
172. C2×C8×C53C22×C4semidirect
254. C2×Dic53C2×D8semidirect
264. D212C2×C22⋊C4semidirect
314. C2×#340D53semidirect
316. C2×C53#1879semidirect
351. #340C2×D53semidirect
353. C53C2×#1879semidirect
391. C22×C8C4×D53semidirect
495. C2×C8C2×C4×D53semidirect
627. C2#20275064direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4242
C22×#131331792
C22×#131331372
C2×#168105242
#202750642
C2×#168103772
C2×C53×#3402
C2×#187953