Finite Group Databasebeta

← C2×#20243679 #26217395

Subgroups — C2×#20243679 #26217395

Order 12032.

2438 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C8×D47C211yesmaximal, characteristic, complemented
2C22×C4×D188C211yesmaximal, characteristic
3C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
5C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
6#20243679C218yesmaximal, direct product factor
7C2×#15688878C211yesmaximal, characteristic, complemented
8C2×C47×#336C211yesmaximal, characteristic, complemented
9C23×D188C411yescharacteristic
10C23×C4×D47C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14C22×C8×D47—24no—
15C22×C8×D47—24no—
16C22×C8×D47—24no—
17C22×C8×D47C2214yescomplemented
18C2×C4×D188—24no—
19C2×C4×D188—24no—
20C2×C4×D188C414yes—
21C2×C4×D188C2214yes—
22C22×C4⋊Dic47C411yescharacteristic
23C22×(C2×D47)⋊C4C411yescharacteristic
24C22×(C2×D47)⋊C4C411yescharacteristic
25C2×(C2×D47)⋊C8—24no—
26C2×(C2×D47)⋊C8—24no—
27C2×(C2×D47)⋊C8—24no—
28C2×(C2×D47)⋊C8C2214yescomplemented
29C2×(C2×D47)⋊C8C2214yescomplemented
30C2×(C2×D47)⋊C8C2214yescomplemented
31#15688878C2214yescomplemented
32C23×C8×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C47×#336C2214yescomplemented
36C24×D47C2×C411yescharacteristic
37C24×D47C2×C411yescharacteristic
38C22×D188—24no—
39C22×D188—24no—
40C22×D188C818yescomplemented
41C22×D188C2×C414yes—
42C23×Dic47C2×C411yescharacteristic
43C23×Dic47C2×C411yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47C2×C412yes—
55C22×C4×D47C2×C412yes—
56C22×C4×D47D414yes—
57C22×C4×D47D414yes—
58C22×C4×D47C2312yes—
59C22×C4×D47C2312yes—
60C22×C47⋊C8—22no—
61C22×C47⋊C8—22no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8D414yescomplemented
66C22×C47⋊C8C2312yes—
67C22×C47⋊C8C2312yes—
68C2×C8×D47—232no—
69C2×C8×D47—28no—
70C2×C8×D47—28no—
71C2×C8×D47—28no—
72C4×D188—216no—
73C4×D188—28no—
74C4×D188—28no—
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47C2×C414yes—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4C2×C414yes—
80C2×(C2×D47)⋊C4C2×C414yes—
81(C2×D47)⋊C8—28no—
82(C2×D47)⋊C8—28no—
83(C2×D47)⋊C8—28no—
84C23×C4×C47C2×C411yescharacteristic
85C23×C4×C47C2×C411yescharacteristic
86C23×C4×C47C2311yescharacteristic
87C22×C8×C47—22no—
88C22×C8×C47—22no—
89C22×C8×C47—22no—
90C22×C8×C47—22no—
91C22×C8×C47—22no—
92C22×C8×C47D414yescomplemented
93C22×C8×C47C2312yes—
94C22×C8×C47C2312yes—
95C2×C42×C47—22no—
96C2×C42×C47—22no—
97C2×C42×C47C2×C412yes—
98C2×C42×C47C2312yes—
99C24×C47C22×C411yescharacteristic
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—22no—
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—22no—
110C23×D47C22⋊C414yes—
111C23×D47C2×C814yes—
112C23×D47C2×C814yescomplemented
113C23×D47M1614yes—
114C23×D47C22×C412yes—
115C23×D47C22×C412yes—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47C2×C814yescomplemented
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47#4112yes—
130C22×C4×C47#4112yes—
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47C22⋊C414yes—
138C22×Dic47C22×C412yes—
139C22×Dic47C22×C412yes—
140C2×D188—232no—
141C2×D188—28no—
142C2×D188—28no—
143C2×D188C2×C8116yescomplemented
144C2×C4×D47—24no—
145C2×C4×D47—24no—
146C2×C4×D47—28no—
147C2×C4×D47—28no—
148C2×C4×D47—28no—
149C2×C4×D47—28no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—24no—
155C2×C4×D47—24no—
156C2×C4×D47—24no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—22no—
163C2×C4×D47—44no—
164C2×C4×D47—44no—
165C2×C4×D47—44no—
166C2×C4×D47—44no—
167C2×C4×D47C22⋊C418yes—
168C2×C4×D47C2×D414yes—
169C2×C4×D47C2×D414yes—
170C2×C8×C47—28no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—44no—
177C2×C8×C47C2×D414yescomplemented
178C42×C47—24no—
179C42×C47—22no—
180C42×C47—22no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—28no—
183C4⋊Dic47—28no—
184C2×C47⋊C8—28no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—44no—
191C2×C47⋊C8C2×D414yescomplemented
192C8×D47—232no—
193C23×C47C2×C22⋊C412yes—
194C23×C47C4×D412yes—
195C23×C47C4×D412yes—
196C23×C47C22×C812yes—
197C23×C47C2×M1612yes—
198C23×C47#13112yes—
199C23×C47#13112yes—
200C23×C47C23×C411yescharacteristic
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—28no—
206C22×D47—28no—
207C22×D47—28no—
208C22×D47—28no—
209C22×D47—28no—
210C22×D47—28no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—28no—
218C22×D47—28no—
219C22×D47—28no—
220C22×D47—44no—
221C22×D47—44no—
222C22×D47—44no—
223C22×D47—44no—
224C22×D47C22⋊C8116yescomplemented
225C22×D47C2×C22⋊C414yes—
226C22×D47C22×C814yes—
227C22×D47C22×C814yescomplemented
228C22×D47C2×M1614yes—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—28no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—44no—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47#12014yes—
240C2×C4×C47#12114yes—
241C2×C4×C47#12114yes—
242C2×C4×C47#12314yes—
243C2×C4×C47C22×C814yescomplemented
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—28no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—44no—
257C2×Dic47C2×C22⋊C414yes—
258D188—232no—
259C4×D47—24no—
260C4×D47—24no—
261C4×D47—24no—
262C4×D47—24no—
263C4×D47—216no—
264C4×D47—42no—
265C4×D47—42no—
266C4×D47—42no—
267C4×D47—42no—
268C4×D47—48no—
269C8×C47—24no—
270C8×C47—42no—
271C47⋊C8—24no—
272C47⋊C8—42no—
273C2×#1869—471nomaximal, Sylow
274C22×C47#38614yes—
275C22×C47C2×C22⋊C814yes—
276C22×C47#40814yes—
277C22×C47C8×D414yes—
278C22×C47C8×D414yes—
279C22×C47#43514yes—
280C22×C47#43514yes—
281C22×C47C22×C22⋊C411yescharacteristic
282C22×C47C2×C4×D411yescharacteristic
283C22×C47C2×C4×D411yescharacteristic
284C22×C47C23×C811yescharacteristic
285C22×C47C22×M1611yescharacteristic
286C22×C47C2×#13111yescharacteristic
287C22×C47C2×#13111yescharacteristic
288C2×D47—24no—
289C2×D47—232no—
290C2×D47—24no—
291C2×D47—24no—
292C2×D47—24no—
293C2×D47—24no—
294C2×D47—24no—
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—24no—
298C2×D47—42no—
299C2×D47—416no—
300C2×D47—42no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47C2×C22⋊C818yescomplemented
304C4×C47—24no—
305C4×C47—42no—
306C4×C47#38614yes—
307C4×C47C2×#12011yescharacteristic
308C4×C47C2×#12111yescharacteristic
309C4×C47C2×#12111yescharacteristic
310C4×C47C2×#12311yescharacteristic
311Dic47—24no—
312Dic47—42no—
313C24×C8—471no—
314C22×C4×D4—471no—
315C22×C22⋊C8—471no—
316C22×C22⋊C8—471no—
317C22×C22⋊C8—471no—
318#1869—478no—
319C2×#336—471no—
320C2×#336D4711yescharacteristic, complemented
321C2×C47#186918yescomplemented
322C2×C47C2×#38611yescharacteristic
323C2×C47C22×C22⋊C811yescharacteristic
324C2×C47C2×#40811yescharacteristic
325C2×C47C2×C8×D411yescharacteristic
326C2×C47C2×C8×D411yescharacteristic
327C2×C47C2×#43511yescharacteristic
328C2×C47C2×#43511yescharacteristic
329D47—28no—
330D47—44no—
331C23×D4—471no—
332C24×C4—471no—
333C24×C4—471no—
334C23×C8—471no—
335C23×C8—471no—
336C23×C8—474no—
337C23×C8—944no—
338C23×C8—944no—
339C23×C8—944no—
340C23×C8C2×D4711yescharacteristic
341C23×C8C2×D4711yescharacteristic
342C2×C4×D4—474no—
343C2×C4×D4—474no—
344C2×C4×D4—944no—
345C2×C4×D4—944no—
346C22×C4⋊C4—471no—
347C22×C22⋊C4—471no—
348C22×C22⋊C4—471no—
349C22×C42C2×D4711yescharacteristic
350C2×C22⋊C8—474no—
351C2×C22⋊C8—474no—
352C2×C22⋊C8—474no—
353C2×C22⋊C8—944no—
354C2×C22⋊C8—944no—
355C2×C22⋊C8—944no—
356#336—474no—
357#336C2×D4714yescomplemented
358C47C2×#186911yescharacteristic, complemented, Sylow
359C25—471no—
360C25—471no—
361C22×D4—478no—
362C22×D4—474no—
363C22×D4—944no—
364C22×D4—944no—
365C23×C4—471no—
366C23×C4—472no—
367C23×C4—474no—
368C23×C4—474no—
369C23×C4—472no—
370C23×C4—471no—
371C23×C4—472no—
372C23×C4—472no—
373C23×C4—942no—
374C23×C4—942no—
375C23×C4—942no—
376C23×C4—942no—
377C23×C4—942no—
378C23×C4—942no—
379C23×C4—942no—
380C23×C4—942no—
381C23×C4—942no—
382C23×C4—942no—
383C23×C4C4×D4711yescharacteristic
384C23×C4C4×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—22no—
391C22×C8—474no—
392C22×C8—472no—
393C22×C8—472no—
394C22×C8—948no—
395C22×C8—948no—
396C22×C8—9432no—
397C22×C8—942no—
398C22×C8—942no—
399C22×C8—948no—
400C22×C8—942no—
401C22×C8—942no—
402C22×C8—942no—
403C22×C8D18814yescomplemented
404C22×C8C22×D4712yes—
405C22×C8C22×D4712yes—
406C4×D4—9416no—
407C4×D4—948no—
408C4×D4—948no—
409C2×C4⋊C4—474no—
410C2×C4⋊C4—944no—
411C2×C22⋊C4—474no—
412C2×C22⋊C4—474no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42C4×D4712yes—
418C2×C42C22×D4712yes—
419C22⋊C8—948no—
420C22⋊C8—948no—
421C22⋊C8—948no—
422C24—474no—
423C24—474no—
424C24—474no—
425C24—472no—
426C24—474no—
427C24—472no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24—942no—
435C24—942no—
436C24—942no—
437C24—942no—
438C24C2×C4×D4711yescharacteristic
439C2×D4—4716no—
440C2×D4—948no—
441C2×D4—9432no—
442C2×D4—948no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—474no—
449C22×C4—472no—
450C22×C4—478no—
451C22×C4—474no—
452C22×C4—474no—
453C22×C4—472no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—944no—
457C22×C4—944no—
458C22×C4—948no—
459C22×C4—948no—
460C22×C4—948no—
461C22×C4—948no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—1884no—
479C22×C4—1884no—
480C22×C4—1884no—
481C22×C4—1884no—
482C22×C4C8×D4714yescomplemented
483C22×C4C2×C4×D4712yes—
484C22×C4C2×C4×D4712yes—
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4C2×D18812yes—
487C22×C4#1062535012yes—
488C22×C4D4×D4712yes—
489C22×C4D4×D4712yes—
490C22×C4D4×D4712yes—
491C22×C4#1062535412yes—
492C22×C4C23×D4711yescharacteristic
493C2×C8—28no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—44no—
500C2×C8—474no—
501C2×C8—9432no—
502C2×C8—948no—
503C2×C8—942no—
504C2×C8—942no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8—942no—
508C2×C8—1884no—
509C2×C8C2×D18814yescomplemented
510C4⋊C4—948no—
511C22⋊C4—948no—
512C22⋊C4—948no—
513C42—24no—
514C42—22no—
515C42—22no—
516C23—474no—
517C23—4716no—
518C23—474no—
519C23—474no—
520C23—474no—
521C23—948no—
522C23—942no—
523C23—942no—
524C23—948no—
525C23—942no—
526C23—942no—
527C23—948no—
528C23—948no—
529C23—948no—
530C23—948no—
531C23—942no—
532C23—942no—
533C23—942no—
534C23—948no—
535C23—942no—
536C23—942no—
537C23—942no—
538C23—948no—
539C23—948no—
540C23—1884no—
541C23—1884no—
542C23—1884no—
543C23—1884no—
544C23C22⋊C4×D4712yes—
545C23#1309398612yes—
546C23C4×D18812yes—
547C23C2×C8×D4712yes—
548C23M16×D4712yes—
549C23#1309403012yes—
550C23#1309403212yes—
551C23C22×C4×D4711yescharacteristic
552D4—9432no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—474no—
561C2×C4—948no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—9416no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—1884no—
573C2×C4—1888no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4C22⋊C4×D4712yes—
579C2×C4#1309398612yes—
580C2×C4C4×D18812yes—
581C2×C4#1309399414yes—
582C2×C4#1309399714yes—
583C2×C4#1309399914yes—
584C2×C4#1309400914yes—
585C2×C4C2×C8×D4714yescomplemented
586C2×C4C2×D4×D4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C2×D4×D4711yescharacteristic
589C2×C4C22×D18811yescharacteristic
590C2×C4C2×#1062535011yescharacteristic
591C2×C4C2×#1062535411yescharacteristic
592C8—24no—
593C8—42no—
594C8—944no—
595C8—1882no—
596C22—478no—
597C22—944no—
598C22—944no—
599C22—944no—
600C22—944no—
601C22—944no—
602C22—944no—
603C22—9432no—
604C22—944no—
605C22—944no—
606C22—944no—
607C22—1882no—
608C22—1882no—
609C22—1882no—
610C22—1882no—
611C22—18816no—
612C22#1568900114yes—
613C22C22⋊C8×D4714yes—
614C22#1568908814yes—
615C22#1568918814yes—
616C22C8×D18814yes—
617C22#1568919314yes—
618C22#1568919614yes—
619C22C2×C22⋊C4×D4711yescharacteristic
620C22C2×#1309398611yescharacteristic
621C22C2×C4×D18811yescharacteristic
622C22C22×C8×D4711yescharacteristic
623C22C2×M16×D4711yescharacteristic
624C22C2×#1309403011yescharacteristic
625C22C2×#1309403211yescharacteristic
626C4—24no—
627C4—42no—
628C4—944no—
629C4—1882no—
630C4#1568900114yes—
631C4C2×#1309399411yescharacteristic
632C4C2×#1309399711yescharacteristic
633C4C2×#1309399911yescharacteristic
634C4C2×#1309400911yescharacteristic
635C2—948no—
636C2—1884no—
637C2#2024367918yesdirect product factor
638C2C2×#1568900111yescharacteristic
639C2C2×C22⋊C8×D4711yescharacteristic
640C2C2×#1568908811yescharacteristic
641C2C2×#1568918811yescharacteristic
642C2C2×C8×D18811yescharacteristic
643C2C2×#1568919311yescharacteristic
644C2C2×#1568919611yescharacteristic
subgroupG/H
centerC23×C4C22×D47
commutatorC22×C47C23×C8
FrattiniC22×C4C23×D47
FittingC2×C47×#336C2
socleC24×C47C22×C4
prime radicalC23×D188C4
cosocleC22×C4×C47C24
hypercenterC2×#336D47
hypocenterC47C2×#1869
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
1. C23×C8×D47C2semidirect
3. C22×(C2×D47)⋊C8C2semidirect
4. C22×(C2×D47)⋊C8C2semidirect
5. C22×(C2×D47)⋊C8C2semidirect
6. #20243679C2direct (N × H)
7. C2×#15688878C2semidirect
8. C2×C47×#336C2semidirect
17. C22×C8×D47C22semidirect
28. C2×(C2×D47)⋊C8C22semidirect
29. C2×(C2×D47)⋊C8C22semidirect
30. C2×(C2×D47)⋊C8C22semidirect
31. #15688878C22semidirect
35. C47×#336C22semidirect
40. C22×D188C8semidirect
65. C22×C47⋊C8D4semidirect
92. C22×C8×C47D4semidirect
112. C23×D47C2×C8semidirect
121. C22×C4×C47C2×C8semidirect
143. C2×D188C2×C8semidirect
177. C2×C8×C47C2×D4semidirect
191. C2×C47⋊C8C2×D4semidirect
224. C22×D47C22⋊C8semidirect
227. C22×D47C22×C8semidirect
243. C2×C4×C47C22×C8semidirect
303. C2×D47C2×C22⋊C8semidirect
320. C2×#336D47semidirect
321. C2×C47#1869semidirect
357. #336C2×D47semidirect
358. C47C2×#1869semidirect
403. C22×C8D188semidirect
482. C22×C4C8×D47semidirect
509. C2×C8C2×D188semidirect
585. C2×C4C2×C8×D47semidirect
637. C2#20243679direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C8×D472
C22×C4×D1882
C22×(C2×D47)⋊C82
#202436792
C2×#156888782
C2×C47×#3362
C2×#186947