Finite Group Databasebeta

← C2×#20287648 #28264606

Subgroups — C2×#20287648 #28264606

Order 13568.

1778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic106C211yesmaximal, characteristic
2C2×#16811349C211yesmaximal, characteristic
3#20287648C2116yesmaximal, direct product factor
4C2×#16811032C211yesmaximal, characteristic
5C2×#16811030C211yesmaximal, characteristic
6C2×#16811020C211yesmaximal, characteristic
7C2×#16810932C211yesmaximal, characteristic
8C2×#16810930C211yesmaximal, characteristic
9C2×#16810916C211yesmaximal, characteristic
10C2×#16810908C211yesmaximal, characteristic
11C2×#16810906C211yesmaximal, characteristic
12C2×#16810906C211yesmaximal, characteristic
13C2×#16810751C211yesmaximal, characteristic
14C2×#16810730C211yesmaximal, characteristic
15C2×Q16×Dic53C211yesmaximal, characteristic
16C2×C53×#441C211yesmaximal, characteristic
17Q8×Dic106—24no—
18Q8×Dic106C2218yes—
19#16811349—24no—
20#16811349C2218yes—
21C2×#13133217—21no—
22C2×#13133217C2211yescharacteristic
23C2×#13133216—21no—
24C2×#13133216—21no—
25C2×#13133216—21no—
26C2×#13133216C2211yescharacteristic
27C2×#13133216C2211yescharacteristic
28C2×#13133216C2211yescharacteristic
29C2×#13133209—21no—
30C2×#13133209C2211yescharacteristic
31C2×#13133208—21no—
32C2×#13133208C2211yescharacteristic
33C2×Q8×Dic53C2211yescharacteristic
34C2×Q8×Dic53C2211yescharacteristic
35C2×Q8×Dic53C2211yescharacteristic
36C2×Q8×Dic53C2211yescharacteristic
37C2×C4×Dic106—21no—
38C2×C4×Dic106—21no—
39C2×C4×Dic106C2211yescharacteristic
40#16811032C2218yes—
41#16811030C2218yes—
42#16811020C2218yes—
43#16810932C2218yes—
44#16810930C2218yes—
45#16810916C2218yes—
46#16810908C2218yes—
47#16810906C2218yes—
48#16810906C2218yes—
49#16810751C2218yes—
50#16810730C2218yes—
51Q16×Dic53C2218yes—
52C2×#13133151C2211yescharacteristic
53C2×#13133151C2211yescharacteristic
54C2×#13133150C2211yescharacteristic
55C2×#13133149C2211yescharacteristic
56C2×#13133149C2211yescharacteristic
57C2×#13133148C2211yescharacteristic
58C2×#13133148C2211yescharacteristic
59C2×#13133148C2211yescharacteristic
60C2×C4⋊#88322C2211yescharacteristic
61C2×#13133146C2211yescharacteristic
62C2×#13133146C2211yescharacteristic
63C2×#13133141C2211yescharacteristic
64C2×#13133141C2211yescharacteristic
65C2×#13133141C2211yescharacteristic
66C2×#13133141C2211yescharacteristic
67C2×#13133127C2211yescharacteristic
68C2×C8×Dic53C2211yescharacteristic
69C22×Q16×C53C2211yescharacteristic
70C2×C4×Q8×C53C2211yescharacteristic
71C2×C4×Q8×C53C2211yescharacteristic
72C53×#441C2218yes—
73C2×#106×C53C2211yescharacteristic
74C2×#103×C53C2211yescharacteristic
75C2×#103×C53C2211yescharacteristic
76C2×#97×C53C2211yescharacteristic
77C22×Dic106—21no—
78C22×Dic106—21no—
79C22×Dic106—21no—
80C22×Dic106—21no—
81C22×Dic106D411yescharacteristic
82C22×#88322C2311yescharacteristic
83C22×#88322C2311yescharacteristic
84#13133217—22no—
85#13133217—28no—
86#13133217C2314yes—
87#13133216—28no—
88#13133216—28no—
89#13133216—22no—
90#13133216—22no—
91#13133216—22no—
92#13133216—28no—
93#13133216C2314yes—
94#13133216C2314yes—
95#13133216C2314yes—
96#13133209—22no—
97#13133209—28no—
98#13133209C2314yes—
99#13133208—28no—
100#13133208—22no—
101#13133208C2314yes—
102Q8×Dic53—22no—
103Q8×Dic53—22no—
104Q8×Dic53—22no—
105Q8×Dic53—22no—
106Q8×Dic53C2314yes—
107Q8×Dic53C2314yes—
108Q8×Dic53C2314yes—
109Q8×Dic53C2314yes—
110C4×Dic106—28no—
111C4×Dic106—28no—
112C4×Dic106D414yes—
113C4×Dic106C2314yes—
114C2×C4⋊Dic53—21no—
115C2×C4⋊Dic53—21no—
116C2×C4⋊Dic53—21no—
117C2×C4⋊Dic53—21no—
118C2×C4⋊Dic53—21no—
119C2×C4⋊Dic53—21no—
120C2×C4⋊Dic53D411yescharacteristic
121C2×C4⋊Dic53C2311yescharacteristic
122C2×C4⋊Dic53C2311yescharacteristic
123C2×C4⋊Dic53C2311yescharacteristic
124C2×C4⋊Dic53C2311yescharacteristic
125C2×#11719919—21no—
126C2×#11719919—21no—
127C2×#11719919—21no—
128C2×#11719919—21no—
129C2×#11719919—21no—
130C2×#11719919—21no—
131C2×#11719919D411yescharacteristic
132C2×#11719919D411yescharacteristic
133C2×C4×Dic53—21no—
134C2×C4×Dic53—21no—
135C2×C4×Dic53—21no—
136C2×C4×Dic53—21no—
137C2×C4×Dic53C2311yescharacteristic
138C2×C4×Dic53C2311yescharacteristic
139#13133151C2314yes—
140#13133151C2314yes—
141#13133150C2314yes—
142#13133149C2314yes—
143#13133149C2314yes—
144#13133148C2314yes—
145#13133148C2314yes—
146#13133148C2314yes—
147C4⋊#88322C2314yes—
148#13133146C2314yes—
149#13133146C2314yes—
150#13133141C2314yes—
151#13133141C2314yes—
152#13133141C2314yes—
153#13133141C2314yes—
154#13133127C2314yes—
155C8×Dic53C2314yes—
156C22×Q8×C53C2311yescharacteristic
157C22×Q8×C53C2311yescharacteristic
158C2×Q16×C53Q818yescomplemented
159C2×Q16×C53C2314yes—
160C22×C8×C53C2311yescharacteristic
161C22×C8×C53C2311yescharacteristic
162C4×Q8×C53—22no—
163C4×Q8×C53—22no—
164C4×Q8×C53C2314yes—
165C4×Q8×C53C2314yes—
166C2×C4⋊C4×C53—21no—
167C2×C4⋊C4×C53—21no—
168C2×C4⋊C4×C53C2311yescharacteristic
169C2×C4⋊C4×C53C2311yescharacteristic
170C2×C42×C53—21no—
171C2×C42×C53—21no—
172C2×C42×C53C2311yescharacteristic
173#106×C53C2314yes—
174#103×C53C2314yes—
175#103×C53C2314yes—
176#97×C53C2314yes—
177C2×Q8×C53—21no—
178C2×Q8×C53—21no—
179C2×Q8×C53—22no—
180C2×Q8×C53—22no—
181C2×Q8×C53C2×Q814yes—
182C2×Q8×C53C2×Q814yes—
183C2×Q8×C53C2412yes—
184C2×Q8×C53C2412yes—
185C22×C4×C53—21no—
186C22×C4×C53—21no—
187C22×C4×C53—21no—
188C22×C4×C53—21no—
189C22×C4×C53C2×D411yescharacteristic
190C22×C4×C53C2×D411yescharacteristic
191C22×C4×C53C2411yescharacteristic
192C22×Dic53—21no—
193C22×Dic53—21no—
194C22×Dic53—21no—
195C22×Dic53—21no—
196C22×Dic53C2×D411yescharacteristic
197C22×Dic53C2×D411yescharacteristic
198C22×Dic53C2×D411yescharacteristic
199C22×Dic53C2×D411yescharacteristic
200C2×Dic106—24no—
201C2×Dic106—24no—
202C2×Dic106—24no—
203C2×Dic106—24no—
204C2×Dic106—24no—
205C2×Dic106—44no—
206C2×Dic106—82no—
207C2×Dic106—82no—
208C2×Dic106—82no—
209C2×Dic106C2×D414yes—
210Q16×C53C2×Q8116yescomplemented
211C2×C8×C53—22no—
212C2×C8×C53C2×Q814yes—
213C2×C8×C53C2412yes—
214C2×C8×C53C2412yes—
215C4⋊C4×C53—24no—
216C4⋊C4×C53—24no—
217C4⋊C4×C53—21no—
218C4⋊C4×C53—21no—
219C4⋊C4×C53C2412yes—
220C4⋊C4×C53C2412yes—
221C42×C53—24no—
222C42×C53—24no—
223C42×C53C2×D412yes—
224C42×C53C2412yes—
225C4⋊Dic53—24no—
226C4⋊Dic53—21no—
227C4⋊Dic53—21no—
228C4⋊Dic53—21no—
229C4⋊Dic53—21no—
230C4⋊Dic53—24no—
231C4⋊Dic53—24no—
232C4⋊Dic53—24no—
233C4⋊Dic53—24no—
234C4⋊Dic53—24no—
235C4⋊Dic53C2×D414yes—
236C4⋊Dic53C2412yes—
237C4⋊Dic53C2412yes—
238C4⋊Dic53C2412yes—
239C4⋊Dic53C2412yes—
240#11719919—24no—
241#11719919—24no—
242#11719919—24no—
243#11719919—24no—
244#11719919—24no—
245#11719919—24no—
246#11719919C2×D414yes—
247#11719919C2×D414yes—
248C4×Dic53—24no—
249C4×Dic53—24no—
250C4×Dic53—24no—
251C4×Dic53—24no—
252C4×Dic53C2×D412yes—
253C4×Dic53C2×D412yes—
254C4×Dic53C2412yes—
255C4×Dic53C2412yes—
256C2×#88322—22no—
257C2×#88322—22no—
258C2×#88322C2412yes—
259C2×#88322C2412yes—
260C23×C53C22×D411yescharacteristic
261Q8×C53—24no—
262Q8×C53—24no—
263Q8×C53—22no—
264Q8×C53—22no—
265Q8×C53C22×Q814yes—
266Q8×C53C22×Q814yes—
267C2×C4×C53—24no—
268C2×C4×C53—24no—
269C2×C4×C53—22no—
270C2×C4×C53—22no—
271C2×C4×C53—22no—
272C2×C4×C53—22no—
273C2×C4×C53—22no—
274C2×C4×C53—22no—
275C2×C4×C53—42no—
276C2×C4×C53—42no—
277C2×C4×C53C22×D412yes—
278C2×C4×C53C22×D412yes—
279C2×C4×C53C22×D411yescharacteristic
280C2×C4×C53C22×Q812yes—
281C2×C4×C532−1+412yes—
282C2×C4×C53C2511yescharacteristic
283C2×Dic53—22no—
284C2×Dic53—22no—
285C2×Dic53—22no—
286C2×Dic53—22no—
287C2×Dic53—22no—
288C2×Dic53—22no—
289C2×Dic53—22no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53—42no—
293C2×Dic53—42no—
294C2×Dic53#13714yes—
295C2×Dic53C22×D412yes—
296C2×Dic53C22×D412yes—
297C2×Dic53C22×D412yes—
298C2×Dic53C22×D412yes—
299Dic106—28no—
300Dic106—48no—
301Dic106—84no—
302Dic106—84no—
303Dic106—84no—
304C8×C53—22no—
305C8×C53C22×Q814yes—
306#88322—22no—
307#88322—22no—
308C2×#3387—531nomaximal, Sylow
309C22×C53D4×Q812yes—
310C22×C53C2×#13712yes—
311C22×C53#57712yes—
312C22×C53C23×D411yescharacteristic
313C4×C53—24no—
314C4×C53—24no—
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53D4×Q812yes—
320C4×C53C23×Q811yescharacteristic
321C4×C53C2×2−1+411yescharacteristic
322Dic53—22no—
323Dic53—22no—
324Dic53—22no—
325Dic53—42no—
326Dic53—42no—
327Dic53—42no—
328Dic53—42no—
329Dic53C2×#13714yes—
330C2×Q82—531no—
331C2×#557—531no—
332#3387—5316no—
333C2×#501—531no—
334C2×#501—531no—
335C2×#498—531no—
336C2×#479—531no—
337C2×#479—531no—
338C2×#477—531no—
339C2×#475—531no—
340C2×#475—531no—
341C2×#475—531no—
342C2×#446—531no—
343C2×#441—531no—
344C2×#441D5311yescharacteristic
345C2×C4×Q16—531no—
346C2×C53#338714yescomplemented
347C2×C53C2×D4×Q811yescharacteristic
348C2×C53C22×#13711yescharacteristic
349C2×C53C2×#57711yescharacteristic
350C22×Q16C2×D5311yescharacteristic
351Q82—538no—
352Q82—1064no—
353#557—538no—
354#557—1064no—
355C2×C4⋊Q8—531no—
356C2×C4⋊Q8—531no—
357C2×C4⋊Q8—531no—
358C2×C4⋊Q8—531no—
359C2×C4⋊Q8—1061no—
360C2×C4⋊Q8—1061no—
361C2×C4⋊Q8—1061no—
362C2×C4⋊Q8—1061no—
363C2×#125—531no—
364C2×#125—531no—
365C2×#125—1061no—
366C2×#125—1061no—
367C2×C4×Q8—531no—
368C2×C4×Q8—531no—
369C2×C4×Q8—531no—
370C2×C4×Q8—531no—
371C2×C4×Q8—531no—
372C2×C4×Q8—1061no—
373C2×C4×Q8—1061no—
374C2×C4×Q8C2×D5311yescharacteristic
375C2×C4×Q8C2×D5311yescharacteristic
376#501—538no—
377#501—538no—
378#498—538no—
379#479—538no—
380#479—538no—
381#477—538no—
382#475—538no—
383#475—538no—
384#475—538no—
385#446—538no—
386#441—538no—
387#441C2×D5318yes—
388C4×Q16—538no—
389C2×#107—531no—
390C2×#107—531no—
391C2×#107—531no—
392C2×#106—531no—
393C2×#106—531no—
394C2×#106—531no—
395C2×#106—531no—
396C2×#106—531no—
397C2×#106C2×D5311yescharacteristic
398C2×C4⋊C8—531no—
399C2×C4⋊C8—531no—
400C2×C4⋊C8—531no—
401C2×#103—531no—
402C2×#103—531no—
403C2×#103—531no—
404C2×#103—531no—
405C2×#103C2×D5311yescharacteristic
406C2×#103C2×D5311yescharacteristic
407C2×#97—531no—
408C2×#97C2×D5311yescharacteristic
409C2×C4×C8—531no—
410C53C2×#338711yescharacteristic, complemented, Sylow
411C22×Q8—531no—
412C22×Q8—1061no—
413C22×Q8—1061no—
414C22×Q8—1061no—
415C22×Q8—1061no—
416C22×Q8C22×D5311yescharacteristic
417C22×Q8C22×D5311yescharacteristic
418C2×Q16Dic10618yescomplemented
419C2×Q16C22×D5314yes—
420C22×C8—531no—
421C22×C8—531no—
422C22×C8C22×D5311yescharacteristic
423C22×C8C22×D5311yescharacteristic
424C4⋊Q8—534no—
425C4⋊Q8—534no—
426C4⋊Q8—534no—
427C4⋊Q8—534no—
428C4⋊Q8—1068no—
429C4⋊Q8—1062no—
430C4⋊Q8—1062no—
431C4⋊Q8—1062no—
432C4⋊Q8—1068no—
433C4⋊Q8—1068no—
434C4⋊Q8—1068no—
435C4⋊Q8—1062no—
436#125—534no—
437#125—534no—
438#125—1068no—
439#125—1062no—
440#125—1068no—
441#125—1062no—
442C4×Q8—22no—
443C4×Q8—22no—
444C4×Q8—534no—
445C4×Q8—534no—
446C4×Q8—534no—
447C4×Q8—534no—
448C4×Q8—534no—
449C4×Q8—534no—
450C4×Q8—1068no—
451C4×Q8—1068no—
452C4×Q8—1062no—
453C4×Q8—1062no—
454C4×Q8—1062no—
455C4×Q8—1062no—
456C4×Q8C22×D5314yes—
457C4×Q8C22×D5314yes—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—531no—
461C2×C4⋊C4—531no—
462C2×C4⋊C4—531no—
463C2×C4⋊C4—531no—
464C2×C4⋊C4—531no—
465C2×C4⋊C4—531no—
466C2×C4⋊C4—531no—
467C2×C4⋊C4—1061no—
468C2×C4⋊C4—1061no—
469C2×C4⋊C4—1061no—
470C2×C4⋊C4—1061no—
471C2×C4⋊C4—1061no—
472C2×C4⋊C4—1061no—
473C2×C4⋊C4—1061no—
474C2×C4⋊C4—1061no—
475C2×C4⋊C4—1061no—
476C2×C4⋊C4—1061no—
477C2×C4⋊C4—1061no—
478C2×C4⋊C4—1061no—
479C2×C4⋊C4C22×D5311yescharacteristic
480C2×C4⋊C4C22×D5311yescharacteristic
481C2×C42—21no—
482C2×C42—21no—
483C2×C42—531no—
484C2×C42—531no—
485C2×C42—1061no—
486C2×C42—1061no—
487C2×C42—1061no—
488C2×C42—1061no—
489C2×C42C22×D5311yescharacteristic
490#107—534no—
491#107—534no—
492#107—534no—
493#106—534no—
494#106—534no—
495#106—534no—
496#106—534no—
497#106—534no—
498#106C22×D5314yes—
499C4⋊C8—534no—
500C4⋊C8—534no—
501C4⋊C8—534no—
502#103—534no—
503#103—534no—
504#103—534no—
505#103—534no—
506#103C22×D5314yes—
507#103C22×D5314yes—
508#97—534no—
509#97C22×D5314yes—
510C4×C8—534no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—22no—
514C2×Q8—22no—
515C2×Q8—534no—
516C2×Q8—1064no—
517C2×Q8—1064no—
518C2×Q8—1064no—
519C2×Q8—1064no—
520C2×Q8—1064no—
521C2×Q8—4242no—
522C2×Q8—4242no—
523C2×Q8—4242no—
524C2×Q8—2124no—
525C2×Q8C2×Dic10614yes—
526C2×Q8C2×Dic10614yes—
527C2×Q8C23×D5312yes—
528C2×Q8C23×D5312yes—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—531no—
534C22×C4—531no—
535C22×C4—531no—
536C22×C4—531no—
537C22×C4—1061no—
538C22×C4—1061no—
539C22×C4—1061no—
540C22×C4—1061no—
541C22×C4D4×D5311yescharacteristic
542C22×C4D4×D5311yescharacteristic
543C22×C4C23×D5311yescharacteristic
544Q16C2×Dic106116yescomplemented
545C2×C8—22no—
546C2×C8—532no—
547C2×C8—532no—
548C2×C8—1062no—
549C2×C8—1062no—
550C2×C8C2×Dic10614yes—
551C2×C8C23×D5312yes—
552C2×C8C23×D5312yes—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—534no—
558C4⋊C4—534no—
559C4⋊C4—534no—
560C4⋊C4—532no—
561C4⋊C4—532no—
562C4⋊C4—532no—
563C4⋊C4—532no—
564C4⋊C4—1064no—
565C4⋊C4—1064no—
566C4⋊C4—1064no—
567C4⋊C4—1064no—
568C4⋊C4—1064no—
569C4⋊C4—1064no—
570C4⋊C4—1064no—
571C4⋊C4—1064no—
572C4⋊C4—1064no—
573C4⋊C4—1061no—
574C4⋊C4—1061no—
575C4⋊C4—1061no—
576C4⋊C4—1061no—
577C4⋊C4—1064no—
578C4⋊C4—1064no—
579C4⋊C4—1064no—
580C4⋊C4C23×D5312yes—
581C4⋊C4C23×D5312yes—
582C42—24no—
583C42—24no—
584C42—532no—
585C42—532no—
586C42—532no—
587C42—532no—
588C42—1064no—
589C42—1064no—
590C42—1064no—
591C42—1064no—
592C42D4×D5312yes—
593C42C23×D5312yes—
594C23C2×D4×D5311yescharacteristic
595Q8—24no—
596Q8—24no—
597Q8—22no—
598Q8—22no—
599Q8—1068no—
600Q8—4244no—
601Q8—4244no—
602Q8—4244no—
603Q8—2128no—
604Q8C22×Dic10614yes—
605Q8C22×Dic10614yes—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—534no—
617C2×C4—532no—
618C2×C4—532no—
619C2×C4—532no—
620C2×C4—532no—
621C2×C4—1062no—
622C2×C4—1062no—
623C2×C4—1062no—
624C2×C4—1062no—
625C2×C4—1062no—
626C2×C4—1062no—
627C2×C4—1062no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4C2×D4×D5312yes—
633C2×C4C2×D4×D5312yes—
634C2×C4C2×D4×D5311yescharacteristic
635C2×C4C22×Dic10612yes—
636C2×C4#1313326412yes—
637C2×C4C24×D5311yescharacteristic
638C8—22no—
639C8—1062no—
640C8—1062no—
641C8C22×Dic10614yes—
642C22D4×Dic10612yes—
643C22#137×D5312yes—
644C22#1681143212yes—
645C22C22×D4×D5311yescharacteristic
646C4—22no—
647C4—24no—
648C4—24no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—534no—
653C4—1062no—
654C4—1062no—
655C4—1062no—
656C4—2122no—
657C4—2122no—
658C4—2122no—
659C4—2122no—
660C4D4×Dic10612yes—
661C4C23×Dic10611yescharacteristic
662C4C2×#1313326411yescharacteristic
663C2#2028764814yesdirect product factor
664C2C2×D4×Dic10611yescharacteristic
665C2C2×#137×D5311yescharacteristic
666C2C2×#1681143211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#441C2
socle, prime radicalC23×C53C22×D4
hypercenterC2×#441D53
hypocenterC53C2×#3387
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
3. #20287648C2direct (N × H)
158. C2×Q16×C53Q8semidirect
210. Q16×C53C2×Q8semidirect
346. C2×C53#3387semidirect
410. C53C2×#3387semidirect
418. C2×Q16Dic106semidirect
544. Q16C2×Dic106semidirect
663. C2#20287648direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1062
C2×#168113492
#202876482
C2×#168110322
C2×#168110302
C2×#168110202
C2×#168109322
C2×#168109302
C2×#168109162
C2×#168109082
C2×#168109062
C2×#168107512
C2×#168107302
C2×Q16×Dic532
C2×C53×#4412
C2×#338753