Finite Group Databasebeta

← C2×#19134906 #25097802

Subgroups — C2×#19134906 #25097802

Order 11008.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063305C211yesmaximal, characteristic
2C22×#13063283C211yesmaximal, characteristic
3C22×#13063250C211yesmaximal, characteristic, complemented
4C22×#13063248C211yesmaximal, characteristic, complemented
5C22×(C2×D43)⋊C8C211yesmaximal, characteristic
6#19134906C218yesmaximal, direct product factor
7C2×#14582661C211yesmaximal, characteristic, complemented
8C2×C43×#340C211yesmaximal, characteristic, complemented
9C23×Dic86C2211yescharacteristic
10C23×C4×D43C2211yescharacteristic
11C23×C43⋊C8C2211yescharacteristic
12C2×#13063305—24no—
13C2×#13063305—22no—
14C2×#13063305C418yescomplemented
15C2×#13063305C2214yes—
16C2×#13063283—22no—
17C2×#13063283—24no—
18C2×#13063283—24no—
19C2×#13063283C2214yes—
20C22×C4⋊Dic43—21no—
21C22×C4⋊Dic43C2211yescharacteristic
22C22×(C2×D43)⋊C4—21no—
23C22×(C2×D43)⋊C4—21no—
24C22×C4×Dic43—21no—
25C2×#13063250—24no—
26C2×#13063250C2214yescomplemented
27C2×#13063248—24no—
28C2×#13063248C2214yescomplemented
29C2×(C2×D43)⋊C8—24no—
30C2×(C2×D43)⋊C8C2214yes—
31#14582661C2214yescomplemented
32C23×C8×C43C2211yescharacteristic
33C22×C4⋊C4×C43C2211yescharacteristic
34C22×C4⋊C4×C43C2211yescharacteristic
35C43×#340C2214yescomplemented
36C24×D43D411yescharacteristic
37C22×Dic86—22no—
38C22×Dic86—22no—
39C22×Dic86—22no—
40C22×Dic86—22no—
41C22×Dic86—21no—
42C22×Dic86—22no—
43C22×Dic86C2×C414yes—
44C22×Dic86C2312yes—
45C23×Dic43—21no—
46C23×Dic43—21no—
47C23×Dic43D411yescharacteristic
48C22×C4×D43—22no—
49C22×C4×D43—22no—
50C22×C4×D43—22no—
51C22×C4×D43—22no—
52C22×C4×D43C2×C414yes—
53C22×C4×D43D412yes—
54C22×C4×D43D414yes—
55C22×C4×D43C2312yes—
56C22×C43⋊C8—22no—
57C22×C43⋊C8—22no—
58C22×C43⋊C8—22no—
59C22×C43⋊C8C2312yes—
60#13063305—216no—
61#13063305—28no—
62#13063305C2×C4116yescomplemented
63#13063283—216no—
64#13063283—28no—
65#13063283—28no—
66C2×C4⋊Dic43—22no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43—28no—
69C2×C4⋊Dic43—24no—
70C2×C4⋊Dic43D414yes—
71C2×C4⋊Dic43C2312yes—
72C2×(C2×D43)⋊C4—28no—
73C2×(C2×D43)⋊C4—28no—
74C2×(C2×D43)⋊C4—24no—
75C2×(C2×D43)⋊C4—24no—
76C2×C4×Dic43—28no—
77C2×C4×Dic43—24no—
78#13063250—28no—
79#13063248—28no—
80(C2×D43)⋊C8—28no—
81C23×C4×C43—21no—
82C23×C4×C43—21no—
83C23×C4×C43C2311yescharacteristic
84C22×C8×C43—22no—
85C22×C8×C43—22no—
86C22×C8×C43—22no—
87C22×C8×C43C2312yes—
88C2×C4⋊C4×C43—21no—
89C2×C4⋊C4×C43—22no—
90C2×C4⋊C4×C43—21no—
91C2×C4⋊C4×C43—22no—
92C2×C4⋊C4×C43C2×C414yes—
93C2×C4⋊C4×C43D414yes—
94C2×C4⋊C4×C43C2312yes—
95C2×C4⋊C4×C43C2312yes—
96C24×C43C2×D411yescharacteristic
97C23×D43—22no—
98C23×D43—22no—
99C23×D43—22no—
100C23×D43—22no—
101C23×D43—22no—
102C23×D43—22no—
103C23×D43C22⋊C414yes—
104C23×D43C2×D412yes—
105C22×C4×C43—24no—
106C22×C4×C43—24no—
107C22×C4×C43—22no—
108C22×C4×C43—22no—
109C22×C4×C43—42no—
110C22×C4×C43—42no—
111C22×C4×C43—42no—
112C22×C4×C43—42no—
113C22×C4×C43C22×C412yes—
114C22×C4×C43C2×D412yes—
115C22×C4×C43C2×D412yes—
116C22×C4×C43C2×D412yes—
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43#4112yes—
119C22×C4×C43#4112yes—
120C22×C4×C43C2411yescharacteristic
121C22×Dic43—24no—
122C22×Dic43—22no—
123C22×Dic43—24no—
124C22×Dic43—22no—
125C22×Dic43—22no—
126C22×Dic43—22no—
127C22×Dic43—42no—
128C22×Dic43—42no—
129C22×Dic43—42no—
130C22×Dic43—42no—
131C22×Dic43C22⋊C414yes—
132C22×Dic43C2×D412yes—
133C2×C4×D43—28no—
134C2×C4×D43—28no—
135C2×C4×D43—24no—
136C2×C4×D43—28no—
137C2×C4×D43—28no—
138C2×C4×D43—24no—
139C2×C4×D43—22no—
140C2×C4×D43—22no—
141C2×C4×D43—22no—
142C2×C4×D43—22no—
143C2×C4×D43C22⋊C418yes—
144C2×C4×D43C22×C414yes—
145C2×C4×D43C2×D414yes—
146C2×Dic86—28no—
147C2×Dic86—28no—
148C2×Dic86—22no—
149C2×Dic86—22no—
150C2×Dic86—22no—
151C2×Dic86—22no—
152C2×Dic86—24no—
153C2×Dic86—22no—
154C2×Dic86—42no—
155C2×Dic86—44no—
156C2×Dic86—44no—
157C2×Dic86—42no—
158C2×Dic86C22×C414yes—
159C2×C8×C43—22no—
160C2×C8×C43—22no—
161C2×C8×C43—22no—
162C2×C8×C43—44no—
163C4⋊C4×C43—24no—
164C4⋊C4×C43—24no—
165C4⋊C4×C43—22no—
166C4⋊C4×C43—22no—
167C4⋊C4×C43C22×C414yes—
168C4⋊C4×C43C2×D414yes—
169(C2×D43)⋊C4—216no—
170(C2×D43)⋊C4—216no—
171C4⋊Dic43—24no—
172C4⋊Dic43—22no—
173C4⋊Dic43—216no—
174C4⋊Dic43C2×D414yes—
175C4×Dic43—216no—
176C2×C43⋊C8—22no—
177C2×C43⋊C8—22no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—44no—
180C23×C43C2×C22⋊C412yes—
181C23×C43#12012yes—
182C23×C43#12312yes—
183C23×C43#13512yes—
184C23×C43#13512yes—
185C23×C43#13712yes—
186C23×C43#13712yes—
187C23×C43C22×D411yescharacteristic
188C22×D43—22no—
189C22×D43—22no—
190C22×D43—22no—
191C22×D43—22no—
192C22×D43—22no—
193C22×D43—22no—
194C22×D43—28no—
195C22×D43—28no—
196C22×D43—28no—
197C22×D43—44no—
198C22×D43—44no—
199C22×D43—44no—
200C22×D43—44no—
201C22×D43C2×C22⋊C414yes—
202C2×C4×C43—24no—
203C2×C4×C43—24no—
204C2×C4×C43—42no—
205C2×C4×C43—42no—
206C2×C4×C43—42no—
207C2×C4×C43—42no—
208C2×C4×C43—48no—
209C2×C4×C43—48no—
210C2×C4×C43C2×C22⋊C412yes—
211C2×C4×C43C4×D414yes—
212C2×C4×C43C4×D414yes—
213C2×C4×C43#12012yes—
214C2×C4×C43#12114yes—
215C2×C4×C43#12114yes—
216C2×C4×C43#12312yes—
217C2×C4×C43C23×C411yescharacteristic
218C2×C4×C43C22×D411yescharacteristic
219C2×C4×C43C22×D411yescharacteristic
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43C2×#4111yescharacteristic
222C2×C4×C43C2×#4111yescharacteristic
223C2×Dic43—22no—
224C2×Dic43—22no—
225C2×Dic43—28no—
226C2×Dic43—24no—
227C2×Dic43—24no—
228C2×Dic43—42no—
229C2×Dic43—42no—
230C2×Dic43—42no—
231C2×Dic43—42no—
232C2×Dic43—48no—
233C2×Dic43—48no—
234C2×Dic43C2×C22⋊C414yes—
235C4×D43—24no—
236C4×D43—24no—
237C4×D43—24no—
238C4×D43—24no—
239C4×D43—216no—
240Dic86—24no—
241Dic86—24no—
242Dic86—42no—
243Dic86—42no—
244Dic86—48no—
245C8×C43—42no—
246C43⋊C8—42no—
247C2×#1873—431nomaximal, Sylow
248C22×C43#38614yes—
249C22×C43#41614yes—
250C22×C43#41914yes—
251C22×C43#45214yes—
252C22×C43#45214yes—
253C22×C43#48514yes—
254C22×C43#48514yes—
255C22×C43C22×C22⋊C411yescharacteristic
256C22×C43C2×#12011yescharacteristic
257C22×C43C2×#12311yescharacteristic
258C22×C43C2×#13511yescharacteristic
259C22×C43C2×#13511yescharacteristic
260C22×C43C2×#13711yescharacteristic
261C22×C43C2×#13711yescharacteristic
262C2×D43—24no—
263C2×D43—24no—
264C2×D43—24no—
265C2×D43—42no—
266C2×D43—42no—
267C2×D43—416no—
268C2×D43—42no—
269C2×D43—42no—
270C4×C43—44no—
271C4×C43—44no—
272C4×C43#38614yes—
273C4×C43C2×C4×D411yescharacteristic
274C4×C43C2×C4×D411yescharacteristic
275C4×C43C2×#12111yescharacteristic
276C4×C43C2×#12111yescharacteristic
277Dic43—24no—
278Dic43—44no—
279Dic43—44no—
280C22×C22⋊Q8—431no—
281C22×#117—431no—
282C22×#103—431no—
283C22×#103—431no—
284C22×C22⋊C8—431no—
285#1873—438no—
286C2×#340—431no—
287C2×#340D4311yescharacteristic, complemented
288C2×C43#187318yescomplemented
289C2×C43C2×#38611yescharacteristic
290C2×C43C2×#41611yescharacteristic
291C2×C43C2×#41911yescharacteristic
292C2×C43C2×#45211yescharacteristic
293C2×C43C2×#45211yescharacteristic
294C2×C43C2×#48511yescharacteristic
295C2×C43C2×#48511yescharacteristic
296D43—44no—
297C23×Q8—431no—
298C24×C4—431no—
299C23×C8—431no—
300C23×C8C2×D4311yescharacteristic
301C2×C22⋊Q8—434no—
302C2×C22⋊Q8—438no—
303C2×C22⋊Q8—862no—
304C2×C22⋊Q8—864no—
305C2×#117—434no—
306C2×#117—864no—
307C2×#117—862no—
308C2×#117—864no—
309C22×C4⋊C4—431no—
310C22×C4⋊C4—861no—
311C22×C4⋊C4C2×D4311yescharacteristic
312C22×C4⋊C4C2×D4311yescharacteristic
313C22×C22⋊C4—861no—
314C22×C22⋊C4—861no—
315C22×C42—861no—
316C2×#103—434no—
317C2×#103—434no—
318C2×#103—864no—
319C2×#103—864no—
320C2×C22⋊C8—434no—
321C2×C22⋊C8—864no—
322#340—434no—
323#340C2×D4314yescomplemented
324C43C2×#187311yescharacteristic, complemented, Sylow
325C25—431no—
326C22×Q8—434no—
327C22×Q8—432no—
328C22×Q8—862no—
329C22×Q8—861no—
330C22×Q8—862no—
331C22×Q8—862no—
332C22×Q8—862no—
333C22×Q8—862no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—431no—
337C23×C4—434no—
338C23×C4—432no—
339C23×C4—434no—
340C23×C4—432no—
341C23×C4—862no—
342C23×C4—862no—
343C23×C4—862no—
344C23×C4—862no—
345C23×C4—861no—
346C23×C4—861no—
347C23×C4C22×D4311yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—432no—
352C22×C8—862no—
353C22×C8—862no—
354C22×C8—862no—
355C22×C8C22×D4312yes—
356C22⋊Q8—4316no—
357C22⋊Q8—8616no—
358C22⋊Q8—868no—
359#117—868no—
360#117—8616no—
361#117—868no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—434no—
367C2×C4⋊C4—432no—
368C2×C4⋊C4—868no—
369C2×C4⋊C4—864no—
370C2×C4⋊C4—862no—
371C2×C4⋊C4—861no—
372C2×C4⋊C4C4×D4314yes—
373C2×C4⋊C4D17214yes—
374C2×C4⋊C4C22×D4312yes—
375C2×C4⋊C4C22×D4312yes—
376C2×C22⋊C4—864no—
377C2×C22⋊C4—868no—
378C2×C22⋊C4—864no—
379C2×C22⋊C4—868no—
380C2×C42—864no—
381C2×C42—868no—
382#103—868no—
383#103—868no—
384C22⋊C8—868no—
385C24—434no—
386C24—432no—
387C24—862no—
388C24—862no—
389C24—862no—
390C24—862no—
391C24—862no—
392C24—862no—
393C24D4×D4311yescharacteristic
394C2×Q8—434no—
395C2×Q8—864no—
396C2×Q8—862no—
397C2×Q8—868no—
398C2×Q8—868no—
399C2×Q8—862no—
400C2×Q8—862no—
401C2×Q8—862no—
402C2×Q8—862no—
403C2×Q8—1722no—
404C2×Q8—1724no—
405C2×Q8—1724no—
406C2×Q8—1722no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—434no—
416C22×C4—432no—
417C22×C4—438no—
418C22×C4—434no—
419C22×C4—434no—
420C22×C4—862no—
421C22×C4—862no—
422C22×C4—868no—
423C22×C4—868no—
424C22×C4—864no—
425C22×C4—868no—
426C22×C4—868no—
427C22×C4—864no—
428C22×C4—862no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—864no—
433C22×C4—864no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—1722no—
437C22×C4—1722no—
438C22×C4—1722no—
439C22×C4—1722no—
440C22×C4C2×C4×D4312yes—
441C22×C4C2×D17212yes—
442C22×C4#1062189812yes—
443C22×C4D4×D4311yescharacteristic
444C22×C4D4×D4312yes—
445C22×C4D4×D4312yes—
446C22×C4#1062190212yes—
447C22×C4C23×D4311yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—862no—
453C2×C8—862no—
454C2×C8—862no—
455C2×C8—1724no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—434no—
461C4⋊C4—8616no—
462C4⋊C4—864no—
463C4⋊C4—862no—
464C4⋊C4C2×C4×D4314yes—
465C4⋊C4C2×D17214yes—
466C22⋊C4—8616no—
467C22⋊C4—8616no—
468C42—8616no—
469C23—434no—
470C23—862no—
471C23—868no—
472C23—862no—
473C23—868no—
474C23—862no—
475C23—862no—
476C23—862no—
477C23—862no—
478C23—868no—
479C23—1724no—
480C23—1724no—
481C23—1724no—
482C23—1724no—
483C23C22⋊C4×D4312yes—
484C23#1306329512yes—
485C23#1306331012yes—
486C23#1306334412yes—
487C23#1306334612yes—
488C23#1306335412yes—
489C23#1306335712yes—
490C23C2×D4×D4311yescharacteristic
491Q8—864no—
492Q8—864no—
493Q8—1722no—
494Q8—1722no—
495Q8—1728no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—434no—
505C2×C4—862no—
506C2×C4—862no—
507C2×C4—868no—
508C2×C4—864no—
509C2×C4—864no—
510C2×C4—864no—
511C2×C4—864no—
512C2×C4—8616no—
513C2×C4—864no—
514C2×C4—864no—
515C2×C4—1728no—
516C2×C4—1722no—
517C2×C4—1722no—
518C2×C4—1722no—
519C2×C4—1722no—
520C2×C4—1728no—
521C2×C4C22⋊C4×D4312yes—
522C2×C4#1306328714yes—
523C2×C4C4×D17214yes—
524C2×C4#1306329512yes—
525C2×C4#1306329814yes—
526C2×C4#1306330014yes—
527C2×C4#1306331012yes—
528C2×C4C22×C4×D4311yescharacteristic
529C2×C4C2×D4×D4311yescharacteristic
530C2×C4C2×D4×D4311yescharacteristic
531C2×C4C22×D17211yescharacteristic
532C2×C4C2×#1062189811yescharacteristic
533C2×C4C2×#1062190211yescharacteristic
534C8—42no—
535C8—1722no—
536C22—864no—
537C22—864no—
538C22—864no—
539C22—1722no—
540C22—1722no—
541C22—1722no—
542C22—1722no—
543C22—17216no—
544C22#1458277714yes—
545C22#1458289214yes—
546C22#1458290614yes—
547C22#1458306214yes—
548C22#1458306414yes—
549C22#1458324214yes—
550C22#1458324314yes—
551C22C2×C22⋊C4×D4311yescharacteristic
552C22C2×#1306329511yescharacteristic
553C22C2×#1306331011yescharacteristic
554C22C2×#1306334411yescharacteristic
555C22C2×#1306334611yescharacteristic
556C22C2×#1306335411yescharacteristic
557C22C2×#1306335711yescharacteristic
558C4—44no—
559C4—44no—
560C4—864no—
561C4—1724no—
562C4—1724no—
563C4#1458277714yes—
564C4C2×#1306328711yescharacteristic
565C4C2×C4×D17211yescharacteristic
566C4C2×#1306329811yescharacteristic
567C4C2×#1306330011yescharacteristic
568C2—1724no—
569C2#1913490618yesdirect product factor
570C2C2×#1458277711yescharacteristic
571C2C2×#1458289211yescharacteristic
572C2C2×#1458290611yescharacteristic
573C2C2×#1458306211yescharacteristic
574C2C2×#1458306411yescharacteristic
575C2C2×#1458324211yescharacteristic
576C2C2×#1458324311yescharacteristic
subgroupG/H
centerC24D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#340C2
socleC24×C43C2×D4
prime radicalC24×D43D4
cosocleC22×C4×C43C24
hypercenterC2×#340D43
hypocenterC43C2×#1873
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#13063250C2semidirect
4. C22×#13063248C2semidirect
6. #19134906C2direct (N × H)
7. C2×#14582661C2semidirect
8. C2×C43×#340C2semidirect
14. C2×#13063305C4semidirect
26. C2×#13063250C22semidirect
28. C2×#13063248C22semidirect
31. #14582661C22semidirect
35. C43×#340C22semidirect
62. #13063305C2×C4semidirect
287. C2×#340D43semidirect
288. C2×C43#1873semidirect
323. #340C2×D43semidirect
324. C43C2×#1873semidirect
569. C2#19134906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130633052
C22×#130632832
C22×#130632502
C22×#130632482
C22×(C2×D43)⋊C82
#191349062
C2×#145826612
C2×C43×#3402
C2×#187343