Finite Group Databasebeta

← C2×#19135632 #25123498

Subgroups — C2×#19135632 #25123498

Order 11008.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063356C211yesmaximal, characteristic
2C2×#14583475C211yesmaximal, characteristic
3C22×#13063323C211yesmaximal, characteristic, complemented
4C22×#13063251C211yesmaximal, characteristic
5C22×#13063243C211yesmaximal, characteristic, complemented
6#19135632C218yesmaximal, direct product factor
7C2×#14582658C211yesmaximal, characteristic, complemented
8C2×C43×#419C211yesmaximal, characteristic, complemented
9C22×#10621898C2211yescharacteristic
10C23×Dic86C2211yescharacteristic
11C2×#13063356—22no—
12C2×#13063356—24no—
13C2×#13063356—24no—
14C2×#13063356C2214yes—
15C22×#10621885—21no—
16C22×#10621884—21no—
17C22×#10621877C2211yescharacteristic
18#14583475—22no—
19#14583475—22no—
20#14583475—22no—
21#14583475C2214yes—
22C2×#13063325—21no—
23C2×#13063323—22no—
24C2×#13063323—24no—
25C2×#13063323—24no—
26C2×#13063323C2214yescomplemented
27C2×#13063313—21no—
28C2×#13063309—22no—
29C2×#13063305—22no—
30C22×Dic43⋊C4—22no—
31C22×C4×Dic43C2211yescharacteristic
32C2×#13063251—24no—
33C2×#13063251C2214yes—
34C2×#13063243—24no—
35C2×#13063243C2214yescomplemented
36#14582658C2214yescomplemented
37C23×Q8×C43C2211yescharacteristic
38C22×M16×C43C2211yescharacteristic
39C2×#117×C43C2211yescharacteristic
40C43×#419—22no—
41C43×#419C2214yescomplemented
42C2×#103×C43—22no—
43C2×#10621898—21no—
44C2×#10621898—21no—
45C2×#10621898—22no—
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898D412yes—
50C2×#10621898C2312yes—
51C22×Dic86—22no—
52C22×Dic86—21no—
53C22×Dic86—21no—
54C22×Dic86—22no—
55C22×Dic86—22no—
56C22×Dic86—21no—
57C22×Dic86—21no—
58C22×Dic86—22no—
59C22×Dic86D412yes—
60C22×Dic86D411yescharacteristic
61C22×Dic86C2312yes—
62C22×D172D411yescharacteristic
63C22×C22⋊D43—21no—
64C23×Dic43—21no—
65C23×Dic43D412yes—
66C22×C4×D43—21no—
67#13063356—216no—
68#13063356—28no—
69#13063356—28no—
70C2×#10621885—28no—
71C2×#10621885—24no—
72C2×#10621884—28no—
73C2×#10621884—24no—
74C2×#10621877—22no—
75C2×#10621877—21no—
76C2×#10621877D414yes—
77C2×#10621877C2312yes—
78C22×C43⋊C8—21no—
79#13063325—24no—
80#13063325—24no—
81#13063323—216no—
82#13063323—28no—
83#13063323—28no—
84#13063313—24no—
85#13063313—24no—
86#13063309—28no—
87#13063309—28no—
88#13063305—28no—
89#13063305—28no—
90C2×C4⋊Dic43—24no—
91C2×Dic43⋊C4—216no—
92C2×Dic43⋊C4—28no—
93C2×(C2×D43)⋊C4—24no—
94C2×C4×Dic43—22no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43D414yes—
97C2×C4×Dic43C2312yes—
98#13063251—28no—
99#13063243—28no—
100C22×Q8×C43—22no—
101C22×Q8×C43—21no—
102C22×Q8×C43—21no—
103C22×Q8×C43—21no—
104C22×Q8×C43—22no—
105C22×Q8×C43—21no—
106C22×Q8×C43—21no—
107C22×Q8×C43—22no—
108C22×Q8×C43D414yes—
109C22×Q8×C43C2312yes—
110C23×C4×C43—21no—
111C23×C4×C43C2311yescharacteristic
112C2×M16×C43—21no—
113C2×M16×C43—22no—
114C2×M16×C43—22no—
115C2×M16×C43C2312yes—
116C22×C8×C43—21no—
117#117×C43—21no—
118#117×C43—21no—
119#117×C43D412yes—
120#117×C43C2312yes—
121C2×C4⋊C4×C43D412yes—
122C2×C22⋊C4×C43—21no—
123C2×C42×C43—21no—
124#103×C43—28no—
125C24×C43C2×D411yescharacteristic
126C23×D43—21no—
127C2×Q8×C43—24no—
128C2×Q8×C43—22no—
129C2×Q8×C43—24no—
130C2×Q8×C43—24no—
131C2×Q8×C43—24no—
132C2×Q8×C43—24no—
133C2×Q8×C43—22no—
134C2×Q8×C43—22no—
135C2×Q8×C43—22no—
136C2×Q8×C43—22no—
137C2×Q8×C43—41no—
138C2×Q8×C43—41no—
139C2×Q8×C43—42no—
140C2×Q8×C43—42no—
141C2×Q8×C43—42no—
142C2×Q8×C43—42no—
143C2×Q8×C43—42no—
144C2×Q8×C43C2×D414yes—
145C22×C4×C43—22no—
146C22×C4×C43—24no—
147C22×C4×C43—22no—
148C22×C4×C43—41no—
149C22×C4×C43—41no—
150C22×C4×C43—42no—
151C22×C4×C43C2×D412yes—
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D412yes—
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D412yes—
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43#4112yes—
160C22×C4×C43C2411yescharacteristic
161C2×C22⋊D43—22no—
162C2×C22⋊D43—22no—
163C2×C22⋊D43—42no—
164C2×C22⋊D43—42no—
165C22×Dic43—28no—
166C22×Dic43—24no—
167C22×Dic43—22no—
168C22×Dic43—24no—
169C22×Dic43—21no—
170C22×Dic43—22no—
171C22×Dic43—24no—
172C22×Dic43—22no—
173C22×Dic43C2×D414yes—
174#10621898—24no—
175#10621898—24no—
176#10621898—22no—
177#10621898—22no—
178#10621898—22no—
179#10621898—22no—
180#10621898—42no—
181#10621898—42no—
182#10621898—42no—
183#10621898—42no—
184C2×D172—21no—
185C2×D172—22no—
186C2×D172—22no—
187C2×D172C2×D412yes—
188C2×C4×D43—22no—
189C2×C4×D43—22no—
190C2×C4×D43—42no—
191C2×C4×D43—42no—
192C2×Dic86—22no—
193C2×Dic86—22no—
194C2×Dic86—22no—
195C2×Dic86—22no—
196C2×Dic86—24no—
197C2×Dic86—21no—
198C2×Dic86—24no—
199C2×Dic86—22no—
200C2×Dic86—22no—
201C2×Dic86—22no—
202C2×Dic86—22no—
203C2×Dic86—22no—
204C2×Dic86—22no—
205C2×Dic86—42no—
206C2×Dic86—42no—
207C2×Dic86—42no—
208C2×Dic86—42no—
209C2×Dic86—42no—
210C2×Dic86—42no—
211C2×Dic86—42no—
212C2×Dic86—42no—
213C2×Dic86C2×D412yes—
214M16×C43—22no—
215M16×C43—22no—
216M16×C43—42no—
217C2×C8×C43—22no—
218C2×C8×C43—42no—
219C4⋊C4×C43—22no—
220C4⋊C4×C43C2×D414yes—
221C22⋊C4×C43—22no—
222C22⋊C4×C43—22no—
223C42×C43—22no—
224C42×C43—22no—
225#10621885—216no—
226#10621884—216no—
227(C2×D43)⋊C4—216no—
228C4⋊Dic43—216no—
229Dic43⋊C4—232no—
230C4×Dic43—24no—
231C4×Dic43—22no—
232C4×Dic43C2×D414yes—
233#10621877—24no—
234#10621877—22no—
235#10621877C2×D414yes—
236C2×C43⋊C8—24no—
237C2×C43⋊C8—22no—
238C23×C43—24no—
239C23×C43#12011yescharacteristic
240C23×C43#12011yescharacteristic
241C23×C43#12012yes—
242C23×C43#12112yes—
243C23×C43C22×D411yescharacteristic
244C22×D43—22no—
245C22×D43—42no—
246C22×D43—44no—
247Q8×C43—24no—
248Q8×C43—24no—
249Q8×C43—42no—
250Q8×C43—42no—
251Q8×C43—44no—
252Q8×C43—44no—
253C2×C4×C43—24no—
254C2×C4×C43—24no—
255C2×C4×C43—42no—
256C2×C4×C43—42no—
257C2×C4×C43—44no—
258C2×C4×C43—44no—
259C2×C4×C43—44no—
260C2×C4×C43#12011yescharacteristic
261C2×C4×C43#12011yescharacteristic
262C2×C4×C43#12012yes—
263C2×C4×C43#12012yes—
264C2×C4×C43#12012yes—
265C2×C4×C43#12112yes—
266C2×C4×C43#12112yes—
267C2×C4×C43#12112yes—
268C2×C4×C43#12112yes—
269C2×C4×C43#12112yes—
270C2×C4×C43#12712yes—
271C2×C4×C43#12712yes—
272C2×C4×C43C22×D411yescharacteristic
273C2×C4×C43C22×D411yescharacteristic
274C2×C4×C43C22×D411yescharacteristic
275C2×C4×C43C22×D411yescharacteristic
276C2×C4×C43C22×D411yescharacteristic
277C2×C4×C43C2×#4111yescharacteristic
278C22⋊D43—42no—
279C22⋊D43—42no—
280C22⋊D43—44no—
281C2×Dic43—28no—
282C2×Dic43—216no—
283C2×Dic43—24no—
284C2×Dic43—24no—
285C2×Dic43—22no—
286C2×Dic43—24no—
287C2×Dic43—24no—
288C2×Dic43—42no—
289C2×Dic43—44no—
290C2×Dic43—44no—
291D172—22no—
292D172—22no—
293D172—42no—
294C4×D43—42no—
295C4×D43—42no—
296C4×D43—44no—
297Dic86—22no—
298Dic86—22no—
299Dic86—42no—
300Dic86—42no—
301Dic86—42no—
302Dic86—42no—
303Dic86—44no—
304Dic86—44no—
305Dic86—42no—
306C8×C43—42no—
307C43⋊C8—24no—
308C2×#2005—431nomaximal, Sylow
309C22×C43—22no—
310C22×C43—22no—
311C22×C43—24no—
312C22×C43—24no—
313C22×C43#39212yes—
314C22×C43#39212yes—
315C22×C43#45614yes—
316C22×C43C2×#12011yescharacteristic
317C22×C43C2×#12011yescharacteristic
318C22×C43C2×#12111yescharacteristic
319C2×D43—42no—
320C2×D43—44no—
321C2×D43—84no—
322C4×C43—44no—
323C4×C43—44no—
324C4×C43#39212yes—
325C4×C43#39212yes—
326C4×C43C2×#12011yescharacteristic
327C4×C43C2×#12111yescharacteristic
328C4×C43C2×#12111yescharacteristic
329C4×C43C2×#12711yescharacteristic
330Dic43—28no—
331Dic43—42no—
332Dic43—44no—
333C22×#137—431no—
334C2×#536—431no—
335C22×C4⋊Q8—431no—
336C22×#104—431no—
337C2×#419D4311yescharacteristic, complemented
338C22×#101—431no—
339#2005—438no—
340C2×#339—431no—
341C2×C43—21no—
342C2×C43—21no—
343C2×C43—22no—
344C2×C43#200514yescomplemented
345C2×C43C2×#39211yescharacteristic
346C2×C43C2×#45612yes—
347D43—82no—
348C22×#41—431no—
349C23×Q8—431no—
350C23×Q8C2×D4311yescharacteristic
351C2×#137—434no—
352C2×#137—864no—
353C2×#137—862no—
354C2×#137—864no—
355C22×Q16—861no—
356C22×SD16—861no—
357C22×M16—431no—
358C22×M16C2×D4311yescharacteristic
359#536—434no—
360#536—862no—
361#536—862no—
362#536—862no—
363C2×C4⋊Q8—434no—
364C2×C4⋊Q8—861no—
365C2×C4⋊Q8—864no—
366C2×C4⋊Q8—862no—
367C2×C4⋊Q8—864no—
368C2×#124—861no—
369C2×#123—862no—
370C2×C22⋊Q8—862no—
371C2×#117C2×D4311yescharacteristic
372C22×C4⋊C4—862no—
373C22×C42—431no—
374C2×#104—434no—
375C2×#104—864no—
376#419—22no—
377#419C2×D4314yescomplemented
378C2×#103—22no—
379C2×#101—434no—
380C2×#101—864no—
381#339—434no—
382C43C2×#200511yescharacteristic, complemented, Sylow
383C2×#41—432no—
384C2×#41—432no—
385C2×#41—861no—
386C2×#41—861no—
387C2×#41—862no—
388C2×#41—862no—
389C2×#41—862no—
390C2×#41—862no—
391C22×Q8—22no—
392C22×Q8—21no—
393C22×Q8—22no—
394C22×Q8—21no—
395C22×Q8—21no—
396C22×Q8—22no—
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—432no—
400C22×Q8—432no—
401C22×Q8—431no—
402C22×Q8—862no—
403C22×Q8—862no—
404C22×Q8—861no—
405C22×Q8—862no—
406C22×Q8—861no—
407C22×Q8—862no—
408C22×Q8—861no—
409C22×Q8—861no—
410C22×Q8D17214yes—
411C22×Q8C22×D4312yes—
412C22×D4—431no—
413C22×D4—861no—
414C23×C4—21no—
415C23×C4—432no—
416C23×C4—861no—
417C23×C4—861no—
418C23×C4C22×D4311yescharacteristic
419#137—868no—
420#137—8616no—
421#137—868no—
422C2×Q16—864no—
423C2×Q16—868no—
424C2×SD16—864no—
425C2×SD16—868no—
426C2×M16—21no—
427C2×M16—22no—
428C2×M16—22no—
429C2×M16—434no—
430C2×M16—432no—
431C2×M16—862no—
432C2×M16—861no—
433C2×M16C22×D4312yes—
434C22×C8—21no—
435C22×C8—861no—
436C4⋊Q8—864no—
437C4⋊Q8—864no—
438C4⋊Q8—868no—
439C4⋊Q8—8616no—
440C4⋊Q8—868no—
441#124—864no—
442#124—864no—
443#123—868no—
444#123—868no—
445C22⋊Q8—868no—
446C22⋊Q8—868no—
447#117—21no—
448#117—21no—
449#117C22⋊D4312yes—
450#117C22×D4312yes—
451C2×C4⋊C4—868no—
452C2×C4⋊C4—8616no—
453C2×C4⋊C4—864no—
454C2×C4⋊C4C22⋊D4312yes—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—864no—
457C2×C42—21no—
458C2×C42—434no—
459C2×C42—432no—
460C2×C42—862no—
461C2×C42—861no—
462#104—868no—
463#103—28no—
464#101—868no—
465C24—861no—
466C24D4×D4311yescharacteristic
467#41—864no—
468#41—864no—
469#41—862no—
470#41—862no—
471#41—862no—
472#41—862no—
473#41—1722no—
474#41—1722no—
475#41—1722no—
476#41—1722no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—24no—
480C2×Q8—24no—
481C2×Q8—24no—
482C2×Q8—24no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—41no—
488C2×Q8—41no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—432no—
495C2×Q8—862no—
496C2×Q8—862no—
497C2×Q8—861no—
498C2×Q8—864no—
499C2×Q8—864no—
500C2×Q8—862no—
501C2×Q8—862no—
502C2×Q8—862no—
503C2×Q8—862no—
504C2×Q8—862no—
505C2×Q8—862no—
506C2×Q8—862no—
507C2×Q8—862no—
508C2×Q8—1722no—
509C2×Q8—1722no—
510C2×Q8—1722no—
511C2×Q8—1722no—
512C2×Q8—1722no—
513C2×Q8—1722no—
514C2×Q8—1722no—
515C2×Q8—1722no—
516C2×Q8C2×D17214yes—
517C2×D4—432no—
518C2×D4—862no—
519C2×D4—862no—
520C2×D4—861no—
521C2×D4—862no—
522C2×D4—862no—
523C2×D4—1722no—
524C2×D4—1722no—
525C22×C4—22no—
526C22×C4—24no—
527C22×C4—22no—
528C22×C4—41no—
529C22×C4—41no—
530C22×C4—42no—
531C22×C4—434no—
532C22×C4—868no—
533C22×C4—864no—
534C22×C4—862no—
535C22×C4—862no—
536C22×C4—862no—
537C22×C4—861no—
538C22×C4—864no—
539C22×C4—864no—
540C22×C4—862no—
541C22×C4—862no—
542C22×C4—1722no—
543C22×C4—1722no—
544C22×C4C2×D17212yes—
545C22×C4#1062189812yes—
546C22×C4D4×D4311yescharacteristic
547C22×C4D4×D4311yescharacteristic
548C22×C4D4×D4311yescharacteristic
549C22×C4D4×D4312yes—
550C22×C4D4×D4312yes—
551C22×C4C2×C22⋊D4311yescharacteristic
552C22×C4C2×C22⋊D4311yescharacteristic
553C22×C4C23×D4311yescharacteristic
554Q16—8616no—
555SD16—8616no—
556M16—22no—
557M16—22no—
558M16—42no—
559M16—434no—
560M16—864no—
561M16—862no—
562C2×C8—22no—
563C2×C8—42no—
564C2×C8—864no—
565C2×C8—862no—
566C4⋊C4—22no—
567C4⋊C4—8632no—
568C4⋊C4—8616no—
569C4⋊C4C2×C22⋊D4314yes—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—8616no—
573C42—22no—
574C42—22no—
575C42—434no—
576C42—864no—
577C42—862no—
578C23—24no—
579C23—862no—
580C23—1722no—
581C23—1724no—
582C23#1306329311yescharacteristic
583C23#1306329311yescharacteristic
584C23#1306329512yes—
585C23#1306329812yes—
586C23C2×D4×D4311yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—44no—
590Q8—44no—
591Q8—42no—
592Q8—42no—
593Q8—862no—
594Q8—862no—
595Q8—1724no—
596Q8—1724no—
597Q8—1722no—
598Q8—1722no—
599Q8—1722no—
600Q8—1722no—
601Q8—1722no—
602D4—862no—
603D4—862no—
604D4—1724no—
605D4—1722no—
606D4—1722no—
607D4—1722no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—8616no—
616C2×C4—868no—
617C2×C4—864no—
618C2×C4—864no—
619C2×C4—862no—
620C2×C4—864no—
621C2×C4—864no—
622C2×C4—1724no—
623C2×C4—1724no—
624C2×C4—1722no—
625C2×C4—1724no—
626C2×C4—1722no—
627C2×C4—1722no—
628C2×C4#1306329311yescharacteristic
629C2×C4#1306329311yescharacteristic
630C2×C4#1306329512yes—
631C2×C4#1306329512yes—
632C2×C4#1306329512yes—
633C2×C4#1306329812yes—
634C2×C4#1306329812yes—
635C2×C4#1306329812yes—
636C2×C4#1306329912yes—
637C2×C4#1306329912yes—
638C2×C4#1306332112yes—
639C2×C4#1306332112yes—
640C2×C4C2×D4×D4311yescharacteristic
641C2×C4C2×D4×D4311yescharacteristic
642C2×C4C2×D4×D4311yescharacteristic
643C2×C4C22×C22⋊D4311yescharacteristic
644C2×C4C22×D17211yescharacteristic
645C2×C4C2×#1062189811yescharacteristic
646C8—42no—
647C8—864no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—3444no—
653C22—1722no—
654C22—1724no—
655C22#1458280812yes—
656C22#1458280812yes—
657C22#1458308014yes—
658C22C2×#1306329311yescharacteristic
659C22C2×#1306329511yescharacteristic
660C22C2×#1306329811yescharacteristic
661C4—44no—
662C4—44no—
663C4—868no—
664C4—1724no—
665C4—1722no—
666C4#1458280812yes—
667C4#1458280812yes—
668C4C2×#1306329511yescharacteristic
669C4C2×#1306329811yescharacteristic
670C4C2×#1306329911yescharacteristic
671C4C2×#1306332111yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—3442no—
676C2#1913563214yesdirect product factor
677C2C2×#1458280811yescharacteristic
678C2C2×#1458308012yes—
subgroupG/H
centerC23#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#419C2
socleC23×C43#120
prime radicalC22×#10621898C22
hypercenterC2×#419D43
hypocenterC43C2×#2005
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C22×#13063323C2semidirect
5. C22×#13063243C2semidirect
6. #19135632C2direct (N × H)
7. C2×#14582658C2semidirect
8. C2×C43×#419C2semidirect
26. C2×#13063323C22semidirect
35. C2×#13063243C22semidirect
36. #14582658C22semidirect
41. C43×#419C22semidirect
337. C2×#419D43semidirect
344. C2×C43#2005semidirect
377. #419C2×D43semidirect
382. C43C2×#2005semidirect
676. C2#19135632direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130633562
C2×#145834752
C22×#130633232
C22×#130632512
C22×#130632432
#191356322
C2×#145826582
C2×C43×#4192
C2×#200543