Finite Group Databasebeta

← C2×#19138910 #25265770

Subgroups — C2×#19138910 #25265770

Order 11008.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D172C211yesmaximal, characteristic
2C22×C4⋊C4×D43C211yesmaximal, characteristic, complemented
3#19138910C2116yesmaximal, direct product factor
4C2×#14582841C212yesmaximal, complemented
5C2×#14582834C212yesmaximal, complemented
6C2×#14582797C211yesmaximal, characteristic, complemented
7C2×#14582777C212yesmaximal
8C2×#14582770C212yesmaximal, complemented
9C2×#14582756C211yesmaximal, characteristic, complemented
10C2×#14582756C211yesmaximal, characteristic, complemented
11C2×#14582756C211yesmaximal, characteristic, complemented
12C2×C43×#389C211yesmaximal, characteristic, complemented
13C23×D172C2211yescharacteristic
14C23×C4×D43C2211yescharacteristic
15C23×C4×D43C2211yescharacteristic
16C23×C4×D43C2212yes—
17C2×C4×D172—28no—
18C2×C4×D172C2218yes—
19C22×C4⋊Dic43C2211yescharacteristic
20C22×C4⋊Dic43C2211yescharacteristic
21C22×C4⋊Dic43C2211yescharacteristic
22C22×Dic43⋊C4C2212yes—
23C22×Dic43⋊C4C2212yes—
24C2×C4⋊C4×D43—28no—
25C2×C4⋊C4×D43C2218yescomplemented
26C22×(C2×D43)⋊C4C2211yescharacteristic
27C22×(C2×D43)⋊C4C2212yes—
28C22×(C2×D43)⋊C4C2211yescharacteristic
29C22×(C2×D43)⋊C4C2212yes—
30C22×(C2×D43)⋊C4C2212yes—
31#14582841C22116yescomplemented
32#14582834C22116yescomplemented
33#14582797C2218yescomplemented
34#14582777C22116yes—
35#14582770C22116yescomplemented
36#14582756C2218yescomplemented
37#14582756C2218yescomplemented
38#14582756C2218yescomplemented
39C2×C43⋊#95C2212yes—
40C2×C43⋊#95C2212yes—
41C2×C43⋊#95C2212yes—
42C2×C43⋊#95C2212yes—
43C22×C4⋊C4×C43C2211yescharacteristic
44C22×C4⋊C4×C43C2211yescharacteristic
45C22×C4⋊C4×C43C2212yes—
46C22×C42×C43C2211yescharacteristic
47C43×#389C2218yescomplemented
48C2×#95×C43C2212yes—
49C24×D43C2311yescharacteristic
50C24×D43C2311yescharacteristic
51C22×D172—24no—
52C22×D172—24no—
53C22×D172Q818yes—
54C22×D172C2314yes—
55C23×Dic43C2312yes—
56C23×Dic43C2312yes—
57C23×Dic43C2311yescharacteristic
58C23×Dic43C2311yescharacteristic
59C22×C4×D43—24no—
60C22×C4×D43—24no—
61C22×C4×D43—28no—
62C22×C4×D43—24no—
63C22×C4×D43—24no—
64C22×C4×D43—24no—
65C22×C4×D43—44no—
66C22×C4×D43—44no—
67C22×C4×D43D418yes—
68C22×C4×D43C2314yes—
69C22×C4×D43C2314yes—
70C22×C4×D43C2318yes—
71C4×D172—232no—
72C2×C4⋊Dic43—24no—
73C2×C4⋊Dic43—24no—
74C2×C4⋊Dic43—24no—
75C2×C4⋊Dic43C2314yes—
76C2×C4⋊Dic43C2314yes—
77C2×C4⋊Dic43C2314yes—
78C2×Dic43⋊C4—28no—
79C2×Dic43⋊C4—44no—
80C2×Dic43⋊C4C2318yes—
81C2×Dic43⋊C4C2318yes—
82C4⋊C4×D43—232no—
83C2×(C2×D43)⋊C4—24no—
84C2×(C2×D43)⋊C4—28no—
85C2×(C2×D43)⋊C4—24no—
86C2×(C2×D43)⋊C4—44no—
87C2×(C2×D43)⋊C4—44no—
88C2×(C2×D43)⋊C4C2314yes—
89C2×(C2×D43)⋊C4C2314yes—
90C2×(C2×D43)⋊C4C2318yes—
91C2×(C2×D43)⋊C4C2318yes—
92C2×(C2×D43)⋊C4C2318yes—
93C43⋊#95C2318yes—
94C43⋊#95C2318yes—
95C43⋊#95C2318yes—
96C43⋊#95C2318yes—
97C23×C4×C43C2311yescharacteristic
98C23×C4×C43C2312yes—
99C23×C4×C43C2312yes—
100C23×C4×C43C2311yescharacteristic
101C23×C4×C43C2311yescharacteristic
102C2×C4⋊C4×C43—24no—
103C2×C4⋊C4×C43—24no—
104C2×C4⋊C4×C43—44no—
105C2×C4⋊C4×C43C2314yes—
106C2×C4⋊C4×C43C2314yes—
107C2×C4⋊C4×C43C2318yes—
108C2×C42×C43—24no—
109C2×C42×C43C2314yes—
110#95×C43C2318yes—
111C24×C43C2411yescharacteristic
112C23×D43—22no—
113C23×D43—22no—
114C23×D43—22no—
115C23×D43—22no—
116C23×D43—22no—
117C23×D43—22no—
118C23×D43—22no—
119C23×D43—22no—
120C23×D43—22no—
121C23×D43—22no—
122C23×D43C2×D414yes—
123C23×D43C2×Q814yes—
124C23×D43C2×Q814yes—
125C23×D43#4114yes—
126C23×D43C2412yes—
127C23×D43C2412yes—
128C22×C4×C43—22no—
129C22×C4×C43—22no—
130C22×C4×C43—24no—
131C22×C4×C43—22no—
132C22×C4×C43—24no—
133C22×C4×C43—24no—
134C22×C4×C43—22no—
135C22×C4×C43—24no—
136C22×C4×C43—24no—
137C22×C4×C43—24no—
138C22×C4×C43—22no—
139C22×C4×C43C2×D414yes—
140C22×C4×C43C2×Q814yes—
141C22×C4×C43#4114yes—
142C22×C4×C43#4114yes—
143C22×C4×C43C2414yes—
144C22×C4×C43C2414yes—
145C22×C4×C43C2412yes—
146C22×C4×C43C2412yes—
147C22×C4×C43C2412yes—
148C22×Dic43—22no—
149C22×Dic43—24no—
150C22×Dic43—22no—
151C22×Dic43—22no—
152C22×Dic43—24no—
153C22×Dic43—24no—
154C22×Dic43—22no—
155C22×Dic43—22no—
156C22×Dic43—24no—
157C22×Dic43—24no—
158C22×Dic43—24no—
159C22×Dic43C2×D414yes—
160C22×Dic43C2412yes—
161C22×Dic43C2414yes—
162C22×Dic43C2414yes—
163C22×Dic43C2412yes—
164C2×D172—232no—
165C2×D172—28no—
166C2×D172—28no—
167C2×D172C2×Q8116yes—
168C2×C4×D43—232no—
169C2×C4×D43—28no—
170C2×C4×D43—28no—
171C2×C4×D43—28no—
172C2×C4×D43—28no—
173C2×C4×D43—28no—
174C2×C4×D43—216no—
175C2×C4×D43—48no—
176C2×C4×D43—48no—
177C2×C4×D43—432no—
178C2×C4×D43—416no—
179C2×C4×D43C2×D4116yes—
180C4⋊C4×C43—28no—
181C4⋊C4×C43—28no—
182C4⋊C4×C43—48no—
183C42×C43—28no—
184(C2×D43)⋊C4—28no—
185(C2×D43)⋊C4—28no—
186(C2×D43)⋊C4—216no—
187(C2×D43)⋊C4—48no—
188(C2×D43)⋊C4—48no—
189C4⋊Dic43—28no—
190C4⋊Dic43—28no—
191C4⋊Dic43—28no—
192Dic43⋊C4—216no—
193Dic43⋊C4—48no—
194C23×C43C22×D412yes—
195C23×C43C22×Q812yes—
196C23×C43C2×#4112yes—
197C23×C43C2×#4112yes—
198C23×C43C2×#4112yes—
199C23×C43H(5,𝔽2)12yes—
200C23×C43H(5,𝔽2)12yes—
201C23×C43C2511yescharacteristic
202C22×D43—28no—
203C22×D43—28no—
204C22×D43—28no—
205C22×D43—28no—
206C22×D43—28no—
207C22×D43—28no—
208C22×D43—22no—
209C22×D43—22no—
210C22×D43—22no—
211C22×D43—22no—
212C22×D43—28no—
213C22×D43—28no—
214C22×D43—28no—
215C22×D43—22no—
216C22×D43—22no—
217C22×D43—22no—
218C22×D43—22no—
219C22×D43—22no—
220C22×D43—22no—
221C22×D43—44no—
222C22×D43—44no—
223C22×D43—44no—
224C22×D43—44no—
225C22×D43C22⋊Q8116yes—
226C22×D43C22×D414yes—
227C22×D43C22×Q814yes—
228C22×D43C22×Q814yes—
229C22×D43C2×#4114yes—
230C2×C4×C43—28no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—24no—
234C2×C4×C43—22no—
235C2×C4×C43—24no—
236C2×C4×C43—24no—
237C2×C4×C43—22no—
238C2×C4×C43—24no—
239C2×C4×C43—24no—
240C2×C4×C43—22no—
241C2×C4×C43—24no—
242C2×C4×C43—48no—
243C2×C4×C43—44no—
244C2×C4×C43—48no—
245C2×C4×C43#123116yes—
246C2×C4×C43C22×D414yes—
247C2×C4×C43C22×Q814yes—
248C2×C4×C43C2×#4114yes—
249C2×C4×C43C2×#4114yes—
250C2×Dic43—28no—
251C2×Dic43—22no—
252C2×Dic43—22no—
253C2×Dic43—24no—
254C2×Dic43—22no—
255C2×Dic43—24no—
256C2×Dic43—24no—
257C2×Dic43—22no—
258C2×Dic43—22no—
259C2×Dic43—24no—
260C2×Dic43—24no—
261C2×Dic43—24no—
262C2×Dic43—44no—
263C2×Dic43—48no—
264C2×Dic43—48no—
265C2×Dic43C22×D414yes—
266D172—232no—
267C4×D43—232no—
268C4×D43—432no—
269C4×D43—416no—
270C2×#2487—431nomaximal, Sylow
271C22×C43C2×C22⋊Q814yes—
272C22×C43C2×#12314yes—
273C22×C43#53514yes—
274C22×C43#55314yes—
275C22×C43#55314yes—
276C22×C43#55414yes—
277C22×C43#55414yes—
278C22×C43C23×D411yescharacteristic
279C22×C43C23×Q811yescharacteristic
280C22×C43C22×#4111yescharacteristic
281C22×C43C22×#4111yescharacteristic
282C22×C43C22×#4111yescharacteristic
283C22×C43C2×H(5,𝔽2)11yescharacteristic
284C22×C43C2×H(5,𝔽2)11yescharacteristic
285C2×D43—24no—
286C2×D43—24no—
287C2×D43—24no—
288C2×D43—24no—
289C2×D43—24no—
290C2×D43—24no—
291C2×D43—232no—
292C2×D43—24no—
293C2×D43—24no—
294C2×D43—24no—
295C2×D43—42no—
296C2×D43—42no—
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—416no—
300C2×D43C2×C22⋊Q818yes—
301C4×C43—24no—
302C4×C43—44no—
303C4×C43—42no—
304C4×C43—44no—
305C4×C43C2×#12318yes—
306Dic43—24no—
307Dic43—42no—
308Dic43—44no—
309Dic43—44no—
310C22×C4×D4—431no—
311C23×C4⋊C4—431no—
312#2487—4316no—
313C2×#400—432no—
314C2×#399—432no—
315C2×#389—431no—
316C2×#389D4311yescharacteristic, complemented
317C2×#386—432no—
318C2×#385—432no—
319C2×#380—431no—
320C2×#380—431no—
321C2×#380—431no—
322C2×C43#248718yescomplemented
323C2×C43C22×C22⋊Q811yescharacteristic
324C2×C43C22×#12311yescharacteristic
325C2×C43C2×#53511yescharacteristic
326C2×C43C2×#55311yescharacteristic
327C2×C43C2×#55311yescharacteristic
328C2×C43C2×#55411yescharacteristic
329C2×C43C2×#55411yescharacteristic
330D43—28no—
331D43—44no—
332C23×D4—431no—
333C24×C4—431no—
334C24×C4—431no—
335C24×C4—432no—
336C2×C4×D4—438no—
337C2×C4×D4—868no—
338C22×C4⋊C4—432no—
339C22×C4⋊C4—431no—
340C22×C4⋊C4—431no—
341C22×C4⋊C4—431no—
342C22×C4⋊C4—432no—
343C22×C4⋊C4—438no—
344C22×C4⋊C4—868no—
345C22×C4⋊C4C2×D4311yescharacteristic
346C22×C4⋊C4C2×D4311yescharacteristic
347C22×C4⋊C4C2×D4312yes—
348C22×C22⋊C4—431no—
349C22×C22⋊C4—431no—
350C22×C22⋊C4—432no—
351C22×C22⋊C4—432no—
352C22×C22⋊C4—432no—
353C22×C42C2×D4311yescharacteristic
354#400—4316no—
355#399—4316no—
356#389—438no—
357#389C2×D4318yescomplemented
358#386—4316no—
359#385—4316no—
360#380—438no—
361#380—438no—
362#380—438no—
363C2×#95—432no—
364C2×#95—432no—
365C2×#95—432no—
366C2×#95—432no—
367C2×#95C2×D4312yes—
368C43C2×#248711yescharacteristic, complemented, Sylow
369C25—431no—
370C25—431no—
371C22×D4—438no—
372C22×D4—434no—
373C22×D4—864no—
374C22×D4—864no—
375C23×C4—431no—
376C23×C4—438no—
377C23×C4—434no—
378C23×C4—432no—
379C23×C4—432no—
380C23×C4—431no—
381C23×C4—434no—
382C23×C4—438no—
383C23×C4—868no—
384C23×C4—864no—
385C23×C4—864no—
386C23×C4—864no—
387C23×C4—864no—
388C23×C4—864no—
389C23×C4—1724no—
390C23×C4—1724no—
391C23×C4C22×D4311yescharacteristic
392C23×C4C22×D4312yes—
393C23×C4C22×D4312yes—
394C23×C4C22×D4311yescharacteristic
395C23×C4C22×D4311yescharacteristic
396C4×D4—8632no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—44no—
400C2×C4⋊C4—438no—
401C2×C4⋊C4—434no—
402C2×C4⋊C4—434no—
403C2×C4⋊C4—434no—
404C2×C4⋊C4—438no—
405C2×C4⋊C4—8632no—
406C2×C4⋊C4—868no—
407C2×C4⋊C4—864no—
408C2×C4⋊C4—864no—
409C2×C4⋊C4—864no—
410C2×C4⋊C4—1724no—
411C2×C4⋊C4C22×D4314yes—
412C2×C4⋊C4C22×D4314yes—
413C2×C4⋊C4C22×D4318yes—
414C2×C22⋊C4—434no—
415C2×C22⋊C4—434no—
416C2×C22⋊C4—438no—
417C2×C22⋊C4—438no—
418C2×C22⋊C4—438no—
419C2×C22⋊C4—864no—
420C2×C22⋊C4—864no—
421C2×C22⋊C4—868no—
422C2×C22⋊C4—1724no—
423C2×C22⋊C4—1724no—
424C2×C42—24no—
425C2×C42C22×D4314yes—
426#95—438no—
427#95—438no—
428#95—438no—
429#95—438no—
430#95C22×D4318yes—
431C24—434no—
432C24—434no—
433C24—434no—
434C24—432no—
435C24—434no—
436C24—432no—
437C24—862no—
438C24—862no—
439C24—862no—
440C24—862no—
441C24—862no—
442C24—862no—
443C24—862no—
444C24—862no—
445C24—862no—
446C24—862no—
447C24C23×D4311yescharacteristic
448C2×D4—4316no—
449C2×D4—868no—
450C2×D4—8632no—
451C2×D4—868no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—434no—
464C22×C4—432no—
465C22×C4—4316no—
466C22×C4—432no—
467C22×C4—434no—
468C22×C4—434no—
469C22×C4—8616no—
470C22×C4—862no—
471C22×C4—862no—
472C22×C4—8632no—
473C22×C4—868no—
474C22×C4—868no—
475C22×C4—862no—
476C22×C4—864no—
477C22×C4—864no—
478C22×C4—862no—
479C22×C4—864no—
480C22×C4—862no—
481C22×C4—864no—
482C22×C4—864no—
483C22×C4—864no—
484C22×C4—868no—
485C22×C4—868no—
486C22×C4—868no—
487C22×C4—1728no—
488C22×C4—1728no—
489C22×C4—17232no—
490C22×C4—17216no—
491C22×C4#1062189814yes—
492C22×C4D4×D4314yes—
493C22×C4Q8×D4314yes—
494C22×C4#1062190214yes—
495C22×C4C23×D4314yes—
496C22×C4C23×D4314yes—
497C22×C4C23×D4312yes—
498C22×C4C23×D4312yes—
499C22×C4C23×D4312yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—48no—
503C4⋊C4—8616no—
504C4⋊C4—868no—
505C4⋊C4—868no—
506C4⋊C4—868no—
507C4⋊C4—1728no—
508C22⋊C4—868no—
509C22⋊C4—868no—
510C22⋊C4—8616no—
511C22⋊C4—1728no—
512C22⋊C4—1728no—
513C42—28no—
514C23—4316no—
515C23—434no—
516C23—434no—
517C23—434no—
518C23—434no—
519C23—868no—
520C23—868no—
521C23—868no—
522C23—868no—
523C23—868no—
524C23—868no—
525C23—862no—
526C23—862no—
527C23—862no—
528C23—862no—
529C23—868no—
530C23—868no—
531C23—868no—
532C23—862no—
533C23—862no—
534C23—862no—
535C23—862no—
536C23—862no—
537C23—862no—
538C23—1724no—
539C23—1724no—
540C23—1724no—
541C23—1724no—
542C23C2×D4×D4312yes—
543C23C2×Q8×D4312yes—
544C23#41×D4312yes—
545C23C2×#1062189812yes—
546C23C2×#1062190212yes—
547C23#1306337012yes—
548C23#1306337112yes—
549C23C24×D4311yescharacteristic
550D4—8632no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—434no—
567C2×C4—868no—
568C2×C4—862no—
569C2×C4—862no—
570C2×C4—8632no—
571C2×C4—862no—
572C2×C4—864no—
573C2×C4—864no—
574C2×C4—862no—
575C2×C4—864no—
576C2×C4—862no—
577C2×C4—864no—
578C2×C4—864no—
579C2×C4—864no—
580C2×C4—17232no—
581C2×C4—1728no—
582C2×C4—1724no—
583C2×C4—1728no—
584C2×C4—17216no—
585C2×C4#13063310116yes—
586C2×C4C2×D4×D4314yes—
587C2×C4C2×Q8×D4314yes—
588C2×C4C2×#1062189814yes—
589C2×C4C2×#1062190214yes—
590C22—438no—
591C22—864no—
592C22—864no—
593C22—864no—
594C22—8632no—
595C22—864no—
596C22—864no—
597C22—864no—
598C22—864no—
599C22—864no—
600C22—864no—
601C22—1722no—
602C22—1722no—
603C22—1722no—
604C22—17216no—
605C22—1722no—
606C22C22⋊Q8×D4314yes—
607C22C2×#1306331014yes—
608C22#1458346514yes—
609C22#1458359914yes—
610C22#1458360214yes—
611C22#1458360914yes—
612C22#1458361014yes—
613C22C22×D4×D4311yescharacteristic
614C22C22×Q8×D4311yescharacteristic
615C22C2×#41×D4311yescharacteristic
616C22C22×#1062189811yescharacteristic
617C22C22×#1062190211yescharacteristic
618C22C2×#1306337011yescharacteristic
619C22C2×#1306337111yescharacteristic
620C4—24no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—864no—
625C4—1722no—
626C4—1724no—
627C4—1724no—
628C4C2×#1306331018yes—
629C2—868no—
630C2—1724no—
631C2#1913891018yesdirect product factor
632C2C2×C22⋊Q8×D4311yescharacteristic
633C2C22×#1306331011yescharacteristic
634C2C2×#1458346511yescharacteristic
635C2C2×#1458359911yescharacteristic
636C2C2×#1458360211yescharacteristic
637C2C2×#1458360911yescharacteristic
638C2C2×#1458361011yescharacteristic
subgroupG/H
centerC24C23×D43
commutator, cosocleC23×C43C25
FrattiniC23C24×D43
FittingC2×C43×#389C2
socleC24×C43C24
prime radicalC23×D172C22
hypercenterC2×#389D43
hypocenterC43C2×#2487
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D43C2semidirect
3. #19138910C2direct (N × H)
4. C2×#14582841C2semidirect
5. C2×#14582834C2semidirect
6. C2×#14582797C2semidirect
8. C2×#14582770C2semidirect
9. C2×#14582756C2semidirect
10. C2×#14582756C2semidirect
11. C2×#14582756C2semidirect
12. C2×C43×#389C2semidirect
25. C2×C4⋊C4×D43C22semidirect
31. #14582841C22semidirect
32. #14582834C22semidirect
33. #14582797C22semidirect
35. #14582770C22semidirect
36. #14582756C22semidirect
37. #14582756C22semidirect
38. #14582756C22semidirect
47. C43×#389C22semidirect
316. C2×#389D43semidirect
322. C2×C43#2487semidirect
357. #389C2×D43semidirect
368. C43C2×#2487semidirect
631. C2#19138910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D1722
C22×C4⋊C4×D432
#191389102
C2×#145828412
C2×#145828342
C2×#145827972
C2×#145827772
C2×#145827702
C2×#145827562
C2×C43×#3892
C2×#248743