Finite Group Databasebeta

← #25102783

Subgroups — #25102783

Order 11008.

1104 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
1C2×#14583103C211yesmaximal, characteristic
2C2×#14582986C211yesmaximal, characteristic
3C2×#14582984C211yesmaximal, characteristic
4#19137164C211yesmaximal, characteristic, complemented
5C4×#13063287C211yesmaximal, characteristic
6#19135516C211yesmaximal, characteristic, complemented
7#19135355C211yesmaximal, characteristic, complemented
8#19135229C211yesmaximal, characteristic, complemented
9#19134932C211yesmaximal, characteristic
10#19134928C211yesmaximal, characteristic
11#19134594C211yesmaximal, characteristic
12#19134443C211yesmaximal, characteristic, complemented
13#103×Dic43C211yesmaximal, characteristic, complemented
14#19134414C211yesmaximal, characteristic, complemented
15C43×#1927C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C22×#10621884C2211yescharacteristic
18C2×#13063323C2211yescharacteristic
19C2×Q8×Dic43C2211yescharacteristic
20C2×#13063290C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×#13063287C2211yescharacteristic
23C2×#13063287C2211yescharacteristic
24C2×C42×D43—21no—
25#14583103C418yescomplemented
26#14582986—24no—
27#14582984—24no—
28C2×#13063256C2211yescharacteristic
29C2×Dic43⋊C8C2211yescharacteristic
30C2×Dic43⋊C8C2211yescharacteristic
31C2×#13063250C2211yescharacteristic
32C2×#13063249C2211yescharacteristic
33C2×#13063248C2211yescharacteristic
34C2×#13063248C2211yescharacteristic
35C2×#13063247C2211yescharacteristic
36C2×#13063245C2211yescharacteristic
37C2×C8×Dic43C2211yescharacteristic
38C2×C8×Dic43C2211yescharacteristic
39#14582805C2211yescharacteristic
40#14582796—21no—
41#14582796—21no—
42#14582768C2211yescharacteristic
43C4⋊C4×Dic43C2211yescharacteristic
44C4⋊C4×Dic43C2211yescharacteristic
45#14582750—21no—
46#14582750—21no—
47C4×(C2×D43)⋊C4—21no—
48#14582746—21no—
49#14582746—21no—
50C42×Dic43C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54#14582654C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C4×C4⋊C4×C43C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#336C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C2×#10621873—22no—
68C2×#10621873—22no—
69C2×#10621873C2×C414yes—
70C2×#10621884—22no—
71C2×#10621884—22no—
72C2×#10621884C2×C414yes—
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75#13063323—22no—
76#13063323C2×C414yescomplemented
77Q8×Dic43—22no—
78Q8×Dic43—22no—
79#13063290—22no—
80#13063290—22no—
81C4×D172—22no—
82C4×D172—22no—
83#13063287—22no—
84#13063287—22no—
85#13063287C2×C414yes—
86#13063287D414yes—
87C2×C4⋊Dic43—21no—
88C2×C4⋊Dic43C2311yescharacteristic
89C2×C4⋊Dic43C2311yescharacteristic
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43—21no—
102C2×C4×Dic43D411yescharacteristic
103C2×C4×Dic43D411yescharacteristic
104C2×C4×Dic43D411yescharacteristic
105C2×C4×Dic43D411yescharacteristic
106C2×C4×Dic43C2311yescharacteristic
107C2×C4×Dic43C2311yescharacteristic
108C42×D43—44no—
109#13063256—22no—
110Dic43⋊C8—22no—
111Dic43⋊C8C2×C414yescomplemented
112#13063250—22no—
113#13063249—22no—
114#13063248—22no—
115#13063248C2×C414yescomplemented
116#13063247—22no—
117#13063245—22no—
118C8×Dic43—22no—
119C8×Dic43—22no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C43⋊#95—21no—
124C22×Q8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C22×C8×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C42×C43—21no—
131C2×C42×C43C2311yescharacteristic
132#106×C43—22no—
133#103×C43—22no—
134#103×C43C2×C414yescomplemented
135#95×C43—21no—
136#95×C43—21no—
137#95×C43—21no—
138C23×D43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43C22×C412yes—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2411yescharacteristic
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43C2×D411yescharacteristic
156C22×Dic43C2×D411yescharacteristic
157C22×Dic43C2×D411yescharacteristic
158C22×Dic43C2×D411yescharacteristic
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172C22×C412yes—
163C2×D172C2×D412yes—
164C2×D172#4112yes—
165C2×C4×D43—24no—
166C2×C4×D43—24no—
167C2×C4×D43—24no—
168C2×C4×D43—42no—
169C2×C4×D43—42no—
170C2×C4×D43—42no—
171C2×C4×D43—42no—
172C2×C4×D43—42no—
173C2×C4×D43—42no—
174C2×Dic86—21no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C2×Dic86—21no—
178C2×Dic86—21no—
179C2×Dic86C22×C412yes—
180C2×C8×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43C22×C412yes—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43C22×C412yes—
191C4⋊C4×C43C2×D412yes—
192C42×C43—21no—
193C42×C43—21no—
194C42×C43—42no—
195#10621884—28no—
196(C2×D43)⋊C4—24no—
197(C2×D43)⋊C4—24no—
198(C2×D43)⋊C4—24no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—42no—
203C4⋊Dic43#4112yes—
204Dic43⋊C4—24no—
205Dic43⋊C4—24no—
206Dic43⋊C4—42no—
207Dic43⋊C4—42no—
208Dic43⋊C4—42no—
209C4×Dic43—22no—
210C4×Dic43—22no—
211C4×Dic43—41no—
212C4×Dic43—42no—
213C4×Dic43—42no—
214C4×Dic43—42no—
215C4×Dic43—42no—
216C4×Dic43SD1614yescomplemented
217C4×Dic43C22×C412yes—
218C4×Dic43C2×D412yes—
219C4×Dic43#4112yes—
220C4×Dic43#4112yes—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8C22×C412yes—
227#10621873—28no—
228C23×C43C22×D411yescharacteristic
229C22×D43—22no—
230C22×D43—22no—
231C22×D43—22no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235C22×D43—41no—
236Q8×C43—22no—
237Q8×C43—22no—
238Q8×C43—41no—
239Q8×C43—41no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—22no—
243C2×C4×C43—22no—
244C2×C4×C43—22no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43C2×SD1612yes—
258C2×C4×C43C23×C411yescharacteristic
259C2×C4×C43C22×D411yescharacteristic
260C2×C4×C43C2×#4111yescharacteristic
261C2×C4×C43C2×#4111yescharacteristic
262C2×C4×C43C2×#4111yescharacteristic
263C2×C4×C43C2×#4111yescharacteristic
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—22no—
269C2×Dic43—22no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43C4×D412yes—
278C2×Dic43C4×D412yes—
279C2×Dic43#12112yes—
280C2×Dic43#12112yes—
281C2×Dic43C2×SD1612yes—
282C2×Dic43C2×SD1612yes—
283C2×Dic43#13512yes—
284C2×Dic43#13712yes—
285D172—22no—
286D172—22no—
287D172—22no—
288D172C4×D414yes—
289C4×D43—48no—
290C4×D43—48no—
291C4×D43—48no—
292Dic86—22no—
293Dic86—22no—
294Dic86—41no—
295Dic86—41no—
296C8×C43—22no—
297C8×C43—41no—
298C43⋊C8—22no—
299C43⋊C8—41no—
300C4×#461—431nomaximal, Sylow
301C22×C43C2×C4×D411yescharacteristic
302C22×C43C2×#12111yescharacteristic
303C22×C43#52511yescharacteristic
304C22×C43C22×SD1611yescharacteristic
305C22×C43C2×#13511yescharacteristic
306C22×C43C2×#13711yescharacteristic
307C22×C43#57511yescharacteristic
308C2×D43—24no—
309C2×D43—42no—
310C2×D43—42no—
311C2×D43—42no—
312C2×D43—42no—
313C2×D43—42no—
314C2×D43—42no—
315C4×C43—22no—
316C4×C43—41no—
317C4×C43—42no—
318C4×C43—42no—
319C4×C43—42no—
320C4×C43—81no—
321C4×C43C2×C4×D411yescharacteristic
322C4×C43C4×#4111yescharacteristic
323C4×C43#52911yescharacteristic
324C4×C43#53211yescharacteristic
325Dic43—22no—
326Dic43—22no—
327Dic43—42no—
328Dic43—42no—
329Dic43—81no—
330Dic43C4×SD1614yescomplemented
331Dic43#46114yescomplemented
332C2×#461—431no—
333C2×C4×SD16—431no—
334C2×C4×SD16—431no—
335C4×C4⋊Q8—431no—
336C42×D4—431no—
337#1981—431no—
338#1948—431no—
339#1927—431no—
340#1927D4311yescharacteristic, complemented
341#1877—431no—
342#1877—431no—
343#1802—431no—
344C4×C4⋊C8—431no—
345C4×#103—431no—
346C4×#103—431no—
347C2×C43C4×#12111yescharacteristic
348C2×C43C2×C4×SD1611yescharacteristic
349C2×C43C4×#13711yescharacteristic
350C2×C43C2×#46111yescharacteristic
351C2×C43#303511yescharacteristic
352C2×C43#310111yescharacteristic
353C2×C43#310711yescharacteristic
354D43—44no—
355C22×SD16—431no—
356C22×SD16—431no—
357C2×C4⋊Q8—431no—
358C2×C4×Q8—431no—
359C2×C4×Q8—431no—
360C2×C4×D4—431no—
361C2×C4×D4—431no—
362C2×C4×D4—431no—
363C22×C42—861no—
364#461—438no—
365C4×SD16—864no—
366C4×SD16—864no—
367C2×#106—431no—
368C2×#106C2×D4311yescharacteristic
369C2×C4⋊C8—431no—
370C2×C4⋊C8—431no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103—431no—
375C2×#103C2×D4311yescharacteristic
376C2×#103C2×D4311yescharacteristic
377C2×#102—431no—
378C2×#102—431no—
379C2×C4×C8—431no—
380C2×C4×C8—431no—
381#391—431no—
382#391C2×D4311yescharacteristic
383#389—861no—
384#389—861no—
385#384—431no—
386C4×C4⋊C4—431no—
387C4×C4⋊C4—431no—
388C4×C4⋊C4—861no—
389C4×C4⋊C4—861no—
390C4×C4⋊C4C2×D4311yescharacteristic
391C4×C22⋊C4—861no—
392C4×C22⋊C4—861no—
393C4×C22⋊C4—861no—
394C43—431no—
395#340—431no—
396#340—431no—
397#340—431no—
398#340C2×D4311yescharacteristic
399#336—431no—
400#336C2×D4311yescharacteristic
401C43C4×#46111yescharacteristic, complemented, Sylow
402C22×Q8—431no—
403C22×Q8C22×D4311yescharacteristic
404C22×D4—431no—
405C23×C4—861no—
406C23×C4—861no—
407C23×C4—861no—
408C2×SD16—434no—
409C2×SD16—434no—
410C2×SD16—862no—
411C2×SD16—862no—
412C2×SD16—862no—
413C2×SD16—862no—
414C22×C8—431no—
415C22×C8—431no—
416C22×C8C22×D4311yescharacteristic
417C22×C8C22×D4311yescharacteristic
418C4⋊Q8—434no—
419C4⋊Q8—862no—
420C4×Q8—862no—
421C4×Q8—862no—
422C4×Q8—862no—
423C4×Q8—862no—
424C4×D4—434no—
425C4×D4—434no—
426C4×D4—862no—
427C4×D4—862no—
428C4×D4—862no—
429C4×D4—862no—
430C2×C4⋊C4—431no—
431C2×C4⋊C4—431no—
432C2×C4⋊C4—861no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4—861no—
436C2×C4⋊C4—861no—
437C2×C4⋊C4—861no—
438C2×C4⋊C4C22×D4311yescharacteristic
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C4⋊C4C22×D4311yescharacteristic
441C2×C22⋊C4—861no—
442C2×C22⋊C4—861no—
443C2×C22⋊C4—861no—
444C2×C42—21no—
445C2×C42—431no—
446C2×C42—431no—
447C2×C42—431no—
448C2×C42—431no—
449C2×C42—431no—
450C2×C42—431no—
451C2×C42—861no—
452C2×C42—861no—
453C2×C42—861no—
454C2×C42—861no—
455C2×C42—1724no—
456C2×C42C22×D4311yescharacteristic
457#106—22no—
458#106—862no—
459C4⋊C8—434no—
460C4⋊C8—862no—
461#103—22no—
462#103—434no—
463#103—862no—
464#103—862no—
465#103—862no—
466#103C4×D4314yescomplemented
467#102—862no—
468#102—862no—
469C4×C8—862no—
470C4×C8—862no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478C24—861no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—432no—
485C2×Q8—861no—
486C2×Q8—861no—
487C2×Q8—861no—
488C2×Q8—861no—
489C2×Q8—861no—
490C2×Q8C2×C4×D4312yes—
491C2×D4—432no—
492C2×D4—432no—
493C2×D4—432no—
494C2×D4—861no—
495C2×D4—861no—
496C2×D4—861no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—431no—
502C22×C4—431no—
503C22×C4—431no—
504C22×C4—431no—
505C22×C4—864no—
506C22×C4—864no—
507C22×C4—864no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—861no—
511C22×C4—1722no—
512C22×C4—1722no—
513C22×C4—1722no—
514C22×C4—1722no—
515C22×C4—1722no—
516C22×C4—1722no—
517C22×C4D4×D4311yescharacteristic
518C22×C4D4×D4311yescharacteristic
519C22×C4C23×D4311yescharacteristic
520SD16—868no—
521SD16—868no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—432no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C2×C8C2×C4×D4312yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—432no—
539C4⋊C4—864no—
540C4⋊C4—864no—
541C4⋊C4—861no—
542C4⋊C4—861no—
543C4⋊C4—861no—
544C4⋊C4—1722no—
545C4⋊C4—1722no—
546C4⋊C4—1722no—
547C4⋊C4—1722no—
548C4⋊C4C2×C4×D4312yes—
549C4⋊C4D4×D4312yes—
550C22⋊C4—864no—
551C22⋊C4—864no—
552C22⋊C4—864no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—434no—
557C42—432no—
558C42—432no—
559C42—432no—
560C42—432no—
561C42—862no—
562C42—862no—
563C42—1721no—
564C42—1722no—
565C42—1722no—
566C42—1722no—
567C42—1722no—
568C23—862no—
569C23—862no—
570C23—862no—
571C23—1721no—
572C23—1721no—
573C23—1721no—
574C23—1721no—
575C23C2×D4×D4311yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—862no—
581Q8—862no—
582Q8—1721no—
583Q8—1721no—
584D4—434no—
585D4—862no—
586D4—862no—
587D4—862no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—432no—
606C2×C4—432no—
607C2×C4—432no—
608C2×C4—432no—
609C2×C4—432no—
610C2×C4—432no—
611C2×C4—432no—
612C2×C4—432no—
613C2×C4—861no—
614C2×C4—861no—
615C2×C4—861no—
616C2×C4—861no—
617C2×C4—862no—
618C2×C4—862no—
619C2×C4—1728no—
620C2×C4—1728no—
621C2×C4—1728no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4—1721no—
628C2×C4—1721no—
629C2×C4SD16×D4312yes—
630C2×C4C22×C4×D4311yescharacteristic
631C2×C4C2×D4×D4311yescharacteristic
632C2×C4#41×D4311yescharacteristic
633C2×C4#41×D4311yescharacteristic
634C2×C4#41×D4311yescharacteristic
635C2×C4#41×D4311yescharacteristic
636C8—22no—
637C8—41no—
638C8—862no—
639C8—1721no—
640C22—864no—
641C22—1722no—
642C22—1722no—
643C22—1722no—
644C22—1722no—
645C22—1722no—
646C22—1722no—
647C22C4×D4×D4311yescharacteristic
648C22#121×D4311yescharacteristic
649C22#1458340711yescharacteristic
650C22C2×SD16×D4311yescharacteristic
651C22#135×D4311yescharacteristic
652C22#137×D4311yescharacteristic
653C22#1458369811yescharacteristic
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—81no—
660C4—434no—
661C4—434no—
662C4—862no—
663C4—862no—
664C4—3441no—
665C4—1722no—
666C4—1722no—
667C4C4×D4×D4311yescharacteristic
668C4#1458336711yescharacteristic
669C4#1458343211yescharacteristic
670C4#1458344811yescharacteristic
671C2—1724no—
672C2#1913710711yescharacteristic
673C2C4×SD16×D4311yescharacteristic
674C2#1914341111yescharacteristic
675C2#461×D4311yescharacteristic
676C2#1914404911yescharacteristic
677C2#1914460011yescharacteristic
678C2#1914466111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1927C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1927D43
hypocenterC43C4×#461
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #19137164C2semidirect
6. #19135516C2semidirect
7. #19135355C2semidirect
8. #19135229C2semidirect
12. #19134443C2semidirect
13. #103×Dic43C2semidirect
14. #19134414C2semidirect
15. C43×#1927C2semidirect
25. #14583103C4semidirect
76. #13063323C2×C4semidirect
111. Dic43⋊C8C2×C4semidirect
115. #13063248C2×C4semidirect
134. #103×C43C2×C4semidirect
216. C4×Dic43SD16semidirect
330. Dic43C4×SD16semidirect
331. Dic43#461semidirect
340. #1927D43semidirect
401. C43C4×#461semidirect
466. #103C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145831032
C2×#145829862
C2×#145829842
#191371642
C4×#130632872
#191355162
#191353552
#191352292
#191349322
#191349282
#191345942
#191344432
#103×Dic432
#191344142
C43×#19272
C4×#46143