Finite Group Databasebeta

← C2×#20247854 #26385334

Subgroups — C2×#20247854 #26385334

Order 12032.

4222 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094009C212yesmaximal, complemented
2C22×#13094006C212yesmaximal, complemented
3C22×#13093997C211yesmaximal, characteristic
4C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
5#20247854C2116yesmaximal, direct product factor
6C2×#15689060C211yesmaximal, characteristic, complemented
7C2×#15689059C211yesmaximal, characteristic, complemented
8C2×#15689058C211yesmaximal, characteristic, complemented
9C2×#15689023C212yesmaximal, complemented
10C2×#15688992C212yesmaximal, complemented
11C2×#15688970C211yesmaximal, characteristic, complemented
12C2×C47×#399C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C22⋊D47C2211yescharacteristic
15C23×C22⋊D47C2211yescharacteristic
16C23×C4×D47C2212yes—
17C23×C4×D47C2211yescharacteristic
18C2×#13094009—216no—
19C2×#13094009C22116yescomplemented
20C2×#13094006—216no—
21C2×#13094006C22116yescomplemented
22C2×#13093997—28no—
23C2×#13093997C2218yes—
24C22×C4⋊Dic47C2212yes—
25C22×C4⋊Dic47C2211yescharacteristic
26C22×Dic47⋊C4C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×Dic47⋊C4C2212yes—
29C22×Dic47⋊C4C2212yes—
30C2×C4⋊C4×D47—28no—
31C2×C4⋊C4×D47C2218yescomplemented
32C22×C22⋊Dic47C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34C22×(C2×D47)⋊C4C2212yes—
35C22×(C2×D47)⋊C4C2211yescharacteristic
36C22×(C2×D47)⋊C4C2212yes—
37C22×C4×Dic47C2212yes—
38#15689060C2218yescomplemented
39#15689059C2218yescomplemented
40#15689058C2218yescomplemented
41#15689023C22116yescomplemented
42#15688992C22116yescomplemented
43#15688970C2218yescomplemented
44C2×C47⋊#95C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C22×C22⋊C4×C47C2211yescharacteristic
49C47×#399C2218yescomplemented
50C2×#95×C47C2211yescharacteristic
51C24×D47C2311yescharacteristic
52C24×D47C2311yescharacteristic
53C22×D188—24no—
54C22×D188—28no—
55C22×D188C2314yes—
56C22×C22⋊D47—24no—
57C22×C22⋊D47—28no—
58C22×C22⋊D47—24no—
59C22×C22⋊D47—28no—
60C22×C22⋊D47C2314yes—
61C22×C22⋊D47C2314yes—
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2312yes—
65C23×Dic47C2311yescharacteristic
66C22×C4×D47—28no—
67C22×C4×D47—28no—
68C22×C4×D47—28no—
69C22×C4×D47—28no—
70C22×C4×D47D418yes—
71C22×C4×D47C2318yes—
72C22×C4×D47C2314yes—
73#13094009—264no—
74#13094006—264no—
75#13093997—232no—
76C2×C4⋊Dic47—24no—
77C2×C4⋊Dic47—28no—
78C2×C4⋊Dic47C2318yes—
79C2×C4⋊Dic47C2314yes—
80C2×Dic47⋊C4—28no—
81C2×Dic47⋊C4—24no—
82C2×Dic47⋊C4—28no—
83C2×Dic47⋊C4—28no—
84C2×Dic47⋊C4C2314yes—
85C2×Dic47⋊C4C2318yes—
86C2×Dic47⋊C4C2318yes—
87C2×Dic47⋊C4C2318yes—
88C4⋊C4×D47—232no—
89C2×C22⋊Dic47—28no—
90C2×C22⋊Dic47C2318yes—
91C2×(C2×D47)⋊C4—24no—
92C2×(C2×D47)⋊C4—28no—
93C2×(C2×D47)⋊C4—28no—
94C2×(C2×D47)⋊C4D4116yescomplemented
95C2×(C2×D47)⋊C4C2318yes—
96C2×(C2×D47)⋊C4C2318yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×(C2×D47)⋊C4C2318yes—
99C2×C4×Dic47—28no—
100C2×C4×Dic47C2318yes—
101C47⋊#95C2314yes—
102C25×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2312yes—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—28no—
109C2×C4⋊C4×C47C2318yes—
110C2×C4⋊C4×C47C2314yes—
111C2×C22⋊C4×C47—24no—
112C2×C22⋊C4×C47—28no—
113C2×C22⋊C4×C47C2318yes—
114C2×C22⋊C4×C47C2314yes—
115#95×C47C2314yes—
116C24×C47—24no—
117C24×C47—22no—
118C24×C47—24no—
119C24×C47—22no—
120C24×C47—22no—
121C24×C47C2412yes—
122C24×C47C2411yescharacteristic
123C23×D47—24no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—24no—
127C23×D47—22no—
128C23×D47—24no—
129C23×D47—22no—
130C23×D47—22no—
131C23×D47C2×D418yes—
132C23×D47C2×D414yes—
133C23×D47C2412yes—
134C23×D47C2412yes—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47C2×D418yes—
144C22×C4×C47C2×D414yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2412yes—
147C22×C4×C47C2412yes—
148C22×C4×C47C2414yes—
149C2×C22⋊D47—28no—
150C2×C22⋊D47—216no—
151C2×C22⋊D47—28no—
152C2×C22⋊D47—216no—
153C2×C22⋊D47—416no—
154C2×C22⋊D47—416no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47C2×D418yes—
163C22×Dic47C2×D414yes—
164C22×Dic47#4118yes—
165C22×Dic47C2414yes—
166C22×Dic47C2412yes—
167C22×Dic47C2414yes—
168C22×Dic47C2412yes—
169C2×D188—28no—
170C2×D188—216no—
171C2×D188—416no—
172C2×C4×D47—232no—
173C2×C4×D47—216no—
174C2×C4×D47—216no—
175C2×C4×D47—216no—
176C2×C4×D47—216no—
177C2×C4×D47—432no—
178C2×C4×D47C2×D4116yes—
179C4⋊C4×C47—28no—
180C4⋊C4×C47—216no—
181C22⋊C4×C47—28no—
182C22⋊C4×C47—216no—
183C22⋊Dic47—216no—
184(C2×D47)⋊C4—28no—
185(C2×D47)⋊C4—216no—
186(C2×D47)⋊C4—216no—
187(C2×D47)⋊C4C2×D4132yescomplemented
188C4⋊Dic47—28no—
189C4⋊Dic47—216no—
190Dic47⋊C4—216no—
191Dic47⋊C4—28no—
192Dic47⋊C4—216no—
193Dic47⋊C4—216no—
194C4×Dic47—216no—
195C23×C47—24no—
196C23×C47—22no—
197C23×C47—24no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—44no—
201C23×C47—44no—
202C23×C47—44no—
203C23×C47—48no—
204C23×C47—48no—
205C23×C47C22×D412yes—
206C23×C47C22×D414yes—
207C23×C47C2×#4114yes—
208C23×C472−1+412yes—
209C23×C472−1+412yes—
210C23×C47C2511yescharacteristic
211C22×D47—28no—
212C22×D47—28no—
213C22×D47—216no—
214C22×D47—216no—
215C22×D47—24no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—24no—
219C22×D47—22no—
220C22×D47—24no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—44no—
224C22×D47—44no—
225C22×D47—44no—
226C22×D47—48no—
227C22×D47—48no—
228C22×D47#127116yes—
229C22×D47C22×D418yes—
230C22×D47C22×D414yes—
231C2×C4×C47—28no—
232C2×C4×C47—216no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—48no—
242C2×C4×C47—44no—
243C2×C4×C47C22×D418yes—
244C2×C4×C47C22×D414yes—
245C22⋊D47—416no—
246C22⋊D47—416no—
247C2×Dic47—216no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—48no—
256C2×Dic47—44no—
257C2×Dic47#123116yescomplemented
258C2×Dic47C22×D418yes—
259C2×Dic47C22×D414yes—
260C2×Dic47C2×#4118yes—
261D188—416no—
262C4×D47—232no—
263C4×D47—432no—
264C2×#2501—471nomaximal, Sylow
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47—88no—
271C22×C47C2×#12314yes—
272C22×C47C2×#12714yes—
273C22×C47#53614yes—
274C22×C47#54718yes—
275C22×C47#54718yes—
276C22×C47C23×D411yescharacteristic
277C22×C47C23×D412yes—
278C22×C47C22×#4112yes—
279C22×C47C2×2−1+411yescharacteristic
280C22×C47C2×2−1+411yescharacteristic
281C2×D47—24no—
282C2×D47—24no—
283C2×D47—28no—
284C2×D47—28no—
285C2×D47—232no—
286C2×D47—42no—
287C2×D47—42no—
288C2×D47—42no—
289C2×D47—44no—
290C2×D47—44no—
291C2×D47—88no—
292C2×D47C2×#12718yes—
293C4×C47—24no—
294C4×C47—28no—
295C4×C47—44no—
296C4×C47—42no—
297Dic47—28no—
298Dic47—44no—
299Dic47—42no—
300Dic47C2×#12318yescomplemented
301C22×#123—472no—
302C22×#123—472no—
303C22×#121—471no—
304C23×C4⋊C4—471no—
305#2501—4716no—
306C2×#399—471no—
307C2×#399—471no—
308C2×#399—471no—
309C2×#399D4711yescharacteristic, complemented
310C2×#390—472no—
311C2×#384—472no—
312C2×C4×C22⋊C4—471no—
313C2×C47—82no—
314C2×C47#250118yescomplemented
315C2×C47C22×#12311yescharacteristic
316C2×C47C22×#12711yescharacteristic
317C2×C47C2×#53611yescharacteristic
318C2×C47C2×#54712yes—
319C2×C47C2×#54712yes—
320D47—28no—
321D47—82no—
322C23×D4—471no—
323C23×D4—471no—
324C23×D4—471no—
325C24×C4—471no—
326C24×C4—472no—
327C2×#123—4716no—
328C2×#123—4716no—
329C2×#123—9416no—
330C2×#123—9416no—
331C2×#121—478no—
332C2×#121—948no—
333C22×C4⋊C4—471no—
334C22×C4⋊C4—472no—
335C22×C4⋊C4—472no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4—472no—
338C22×C4⋊C4—471no—
339C22×C4⋊C4—478no—
340C22×C4⋊C4—948no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4712yes—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—472no—
347C22×C22⋊C4—472no—
348C22×C22⋊C4C2×D4711yescharacteristic
349C22×C22⋊C4C2×D4712yes—
350C22×C42—472no—
351#399—478no—
352#399—478no—
353#399—478no—
354#399C2×D4718yescomplemented
355#390—4716no—
356#384—4716no—
357C4×C22⋊C4—478no—
358C2×#95—471no—
359C2×#95C2×D4711yescharacteristic
360C47C2×#250111yescharacteristic, complemented, Sylow
361C25—471no—
362C25—471no—
363C25C22×D4711yescharacteristic
364C22×D4—474no—
365C22×D4—474no—
366C22×D4—474no—
367C22×D4—944no—
368C22×D4—948no—
369C22×D4—944no—
370C22×D4—948no—
371C22×D4—944no—
372C22×D4—948no—
373C23×C4—471no—
374C23×C4—472no—
375C23×C4—478no—
376C23×C4—474no—
377C23×C4—472no—
378C23×C4—471no—
379C23×C4—478no—
380C23×C4—948no—
381C23×C4—948no—
382C23×C4—948no—
383C23×C4—948no—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4712yes—
387C23×C4C22×D4711yescharacteristic
388#123—9464no—
389#123—9464no—
390#121—9432no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—478no—
394C2×C4⋊C4—474no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—478no—
397C2×C4⋊C4—478no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—9432no—
400C2×C4⋊C4—948no—
401C2×C4⋊C4—944no—
402C2×C4⋊C4—948no—
403C2×C4⋊C4—948no—
404C2×C4⋊C4—944no—
405C2×C4⋊C4—948no—
406C2×C4⋊C4C22×D4714yes—
407C2×C4⋊C4C22×D4718yes—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—28no—
410C2×C22⋊C4—4716no—
411C2×C22⋊C4—478no—
412C2×C22⋊C4—478no—
413C2×C22⋊C4—474no—
414C2×C22⋊C4—478no—
415C2×C22⋊C4—478no—
416C2×C22⋊C4—948no—
417C2×C22⋊C4—948no—
418C2×C22⋊C4—944no—
419C2×C22⋊C4—948no—
420C2×C22⋊C4C22×D4714yes—
421C2×C22⋊C4C22×D4718yes—
422C2×C42—478no—
423C2×C42—948no—
424#95—474no—
425#95C22×D4714yes—
426C24—24no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—22no—
431C24—478no—
432C24—474no—
433C24—472no—
434C24—472no—
435C24—944no—
436C24—942no—
437C24—942no—
438C24—944no—
439C24—942no—
440C24—944no—
441C24—942no—
442C24—942no—
443C24C23×D4712yes—
444C24C23×D4711yescharacteristic
445C2×D4—948no—
446C2×D4—9416no—
447C2×D4—948no—
448C2×D4—9416no—
449C2×D4—948no—
450C2×D4—9416no—
451C2×D4—18816no—
452C2×D4—18816no—
453C2×D4—18816no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—474no—
463C22×C4—478no—
464C22×C4—472no—
465C22×C4—478no—
466C22×C4—474no—
467C22×C4—4716no—
468C22×C4—474no—
469C22×C4—472no—
470C22×C4—9432no—
471C22×C4—9416no—
472C22×C4—9416no—
473C22×C4—9416no—
474C22×C4—9416no—
475C22×C4—944no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—942no—
480C22×C4—944no—
481C22×C4—944no—
482C22×C4—18832no—
483C22×C4D4×D4718yes—
484C22×C4D4×D4714yes—
485C22×C4C23×D4714yes—
486C22×C4C23×D4712yes—
487C22×C4C23×D4712yes—
488C22×C4C23×D4714yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—9416no—
492C4⋊C4—948no—
493C4⋊C4—9416no—
494C4⋊C4—9416no—
495C4⋊C4—948no—
496C4⋊C4—9416no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—4732no—
500C22⋊C4—9416no—
501C22⋊C4—9416no—
502C22⋊C4—948no—
503C22⋊C4—9416no—
504C42—9416no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—4716no—
516C23—478no—
517C23—474no—
518C23—9416no—
519C23—9416no—
520C23—948no—
521C23—944no—
522C23—948no—
523C23—942no—
524C23—942no—
525C23—944no—
526C23—942no—
527C23—944no—
528C23—942no—
529C23—942no—
530C23—1888no—
531C23—1888no—
532C23—1884no—
533C23—1884no—
534C23—1884no—
535C23C2×D4×D4714yes—
536C23C2×D4×D4712yes—
537C23#41×D4714yes—
538C23#1309407112yes—
539C23#1309407212yes—
540C23C24×D4711yescharacteristic
541D4—18816no—
542D4—18816no—
543D4—18816no—
544C2×C4—216no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—4716no—
557C2×C4—474no—
558C2×C4—478no—
559C2×C4—478no—
560C2×C4—9432no—
561C2×C4—9416no—
562C2×C4—944no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—942no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—18832no—
570C2×C4—1888no—
571C2×C4—1884no—
572C2×C4C2×D4×D4718yes—
573C2×C4C2×D4×D4714yes—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—88no—
580C22—478no—
581C22—9432no—
582C22—948no—
583C22—948no—
584C22—944no—
585C22—944no—
586C22—3768no—
587C22—1884no—
588C22—1884no—
589C22—1882no—
590C22—1882no—
591C22—1882no—
592C22#123×D4714yes—
593C22#127×D4714yes—
594C22#1568969614yes—
595C22#1568977118yes—
596C22#1568977518yes—
597C22C22×D4×D4712yes—
598C22C22×D4×D4711yescharacteristic
599C22C2×#41×D4712yes—
600C22C2×#1309407111yescharacteristic
601C22C2×#1309407211yescharacteristic
602C4—28no—
603C4—24no—
604C4—44no—
605C4—42no—
606C4—478no—
607C4—948no—
608C4—1884no—
609C4—1882no—
610C2—82no—
611C2—948no—
612C2—3762no—
613C2#2024785418yesdirect product factor
614C2C2×#123×D4711yescharacteristic
615C2C2×#127×D4711yescharacteristic
616C2C2×#1568969611yescharacteristic
617C2C2×#1568977112yes—
618C2C2×#1568977512yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#399C2
socleC24×C47C24
prime radicalC22×#13093997C2
hypercenterC2×#399D47
hypocenterC47C2×#2501
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#13094009C2semidirect
2. C22×#13094006C2semidirect
4. C22×C4⋊C4×D47C2semidirect
5. #20247854C2direct (N × H)
6. C2×#15689060C2semidirect
7. C2×#15689059C2semidirect
8. C2×#15689058C2semidirect
9. C2×#15689023C2semidirect
10. C2×#15688992C2semidirect
11. C2×#15688970C2semidirect
12. C2×C47×#399C2semidirect
19. C2×#13094009C22semidirect
21. C2×#13094006C22semidirect
31. C2×C4⋊C4×D47C22semidirect
38. #15689060C22semidirect
39. #15689059C22semidirect
40. #15689058C22semidirect
41. #15689023C22semidirect
42. #15688992C22semidirect
43. #15688970C22semidirect
49. C47×#399C22semidirect
94. C2×(C2×D47)⋊C4D4semidirect
187. (C2×D47)⋊C4C2×D4semidirect
257. C2×Dic47#123semidirect
300. Dic47C2×#123semidirect
309. C2×#399D47semidirect
314. C2×C47#2501semidirect
354. #399C2×D47semidirect
360. C47C2×#2501semidirect
613. C2#20247854direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940092
C22×#130940062
C22×#130939972
C22×C4⋊C4×D472
#202478542
C2×#156890602
C2×#156890592
C2×#156890582
C2×#156890232
C2×#156889922
C2×#156889702
C2×C47×#3992
C2×#250147