Finite Group Databasebeta

← #25138154

Subgroups — #25138154

Order 11008.

946 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583120C211yesmaximal, characteristic
2C2×#14583118C211yesmaximal, characteristic
3C2×#14583108C211yesmaximal, characteristic, complemented
4C2×#14582990C211yesmaximal, characteristic
5#19139588C211yesmaximal, characteristic
6#19137509C211yesmaximal, characteristic, complemented
7#19135973C211yesmaximal, characteristic, complemented
8#19135972C211yesmaximal, characteristic, complemented
9#19135835C211yesmaximal, characteristic
10#19135395C211yesmaximal, characteristic, complemented
11#19135312C211yesmaximal, characteristic, complemented
12#19134935C211yesmaximal, characteristic, complemented
13#19134933C211yesmaximal, characteristic, complemented
14#19134593C211yesmaximal, characteristic
15C43×#2042C211yesmaximal, characteristic
16C22×Dic172C2211yescharacteristic
17C22×#10621885C2211yescharacteristic
18C22×#10621883C2211yescharacteristic
19C2×#13063323C2211yescharacteristic
20C2×#13063312C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×#13063290C2211yescharacteristic
23C2×#13063290C2211yescharacteristic
24C2×D4×Dic43C2211yescharacteristic
25#14583120—24no—
26#14583118—24no—
27#14583108—24no—
28#14582990—24no—
29C2×#13063256C2211yescharacteristic
30C2×C4⋊(C43⋊C8)C2211yescharacteristic
31C2×Dic43⋊C8C2211yescharacteristic
32C2×Dic43⋊C8C2211yescharacteristic
33C2×#13063250C2211yescharacteristic
34C2×#13063250C2211yescharacteristic
35C2×#13063249C2211yescharacteristic
36C2×#13063248C2211yescharacteristic
37C2×#13063248C2211yescharacteristic
38C2×#13063246C2211yescharacteristic
39C2×C8×Dic43C2211yescharacteristic
40#14582827—21no—
41#14582827—21no—
42#14582806C2211yescharacteristic
43#14582805C2211yescharacteristic
44#14582798C2211yescharacteristic
45#14582796—21no—
46#14582794—21no—
47#14582768—21no—
48#14582768C2211yescharacteristic
49#14582757—21no—
50#14582757—21no—
51C4×Dic43⋊C4C2211yescharacteristic
52#14582750—21no—
53#14582662C2211yescharacteristic
54#14582661C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×#124×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C2×#102×C43C2211yescharacteristic
61C43×#389C2211yescharacteristic
62C43×#336C2211yescharacteristic
63C22×Dic86C2311yescharacteristic
64C22×Dic86C2311yescharacteristic
65C23×Dic43—21no—
66C2×Dic172—22no—
67C2×Dic172—22no—
68C2×Dic172—22no—
69C2×#10621885—22no—
70C2×#10621885—22no—
71C2×#10621885—22no—
72C2×#10621883—22no—
73C2×#10621883—22no—
74C2×#10621883D414yes—
75C22×C43⋊C8C2311yescharacteristic
76C22×C43⋊C8C2311yescharacteristic
77#13063323—22no—
78#13063323—22no—
79#13063312—22no—
80#13063312—41no—
81C4×Dic86—22no—
82C4×Dic86—41no—
83#13063290—22no—
84#13063290—22no—
85#13063290—41no—
86#13063290D414yes—
87D4×Dic43—22no—
88D4×Dic43—22no—
89C2×C4⋊Dic43—21no—
90C2×C4⋊Dic43C2311yescharacteristic
91C2×C4⋊Dic43C2311yescharacteristic
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×Dic43⋊C4—21no—
96C2×Dic43⋊C4—21no—
97C2×Dic43⋊C4—21no—
98C2×Dic43⋊C4—21no—
99C2×Dic43⋊C4D411yescharacteristic
100C2×Dic43⋊C4D411yescharacteristic
101C2×C22⋊Dic43—21no—
102C2×C22⋊Dic43—21no—
103C2×C22⋊Dic43—21no—
104C2×C4×Dic43—21no—
105C2×C4×Dic43—21no—
106C2×C4×Dic43—21no—
107C2×C4×Dic43C2311yescharacteristic
108C2×C4×Dic43C2311yescharacteristic
109#13063256—22no—
110C4⋊(C43⋊C8)—22no—
111Dic43⋊C8—22no—
112Dic43⋊C8—22no—
113#13063250—22no—
114#13063250—41no—
115#13063249—22no—
116#13063248—22no—
117#13063248—22no—
118#13063246—22no—
119C8×Dic43—22no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C43⋊#95—21no—
124C43⋊#95—21no—
125C43⋊#95D411yescharacteristic
126C43⋊#95D411yescharacteristic
127C22×Q8×C43C2311yescharacteristic
128C22×D4×C43C2311yescharacteristic
129C22×C8×C43C2311yescharacteristic
130C22×C8×C43C2311yescharacteristic
131#124×C43—22no—
132#124×C43—41no—
133C2×C4⋊C4×C43—21no—
134C2×C4⋊C4×C43C2311yescharacteristic
135C2×C4⋊C4×C43C2311yescharacteristic
136C2×C22⋊C4×C43—21no—
137C2×C22⋊C4×C43—21no—
138C2×C42×C43C2311yescharacteristic
139#103×C43—22no—
140#103×C43—41no—
141#102×C43—22no—
142#102×C43—22no—
143#95×C43—21no—
144C24×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43—21no—
151C2×D4×C43—21no—
152C2×D4×C43—21no—
153C2×D4×C43—21no—
154C2×D4×C43—21no—
155C2×D4×C43—21no—
156C2×D4×C43C2×D412yes—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D411yescharacteristic
162C22×C4×C43C2411yescharacteristic
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—21no—
166C22×Dic43—21no—
167C22×Dic43—42no—
168C22×Dic43—42no—
169C22×Dic43—42no—
170C22×Dic43C2×D411yescharacteristic
171C22×Dic43C2×D411yescharacteristic
172C22×Dic43C2×D411yescharacteristic
173C22×Dic43C2×D411yescharacteristic
174C2×Dic86—21no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C2×Dic86—21no—
178C2×Dic86—21no—
179C2×Dic86—21no—
180C2×Dic86—21no—
181C2×Dic86—21no—
182C2×Dic86—21no—
183C2×Dic86C2×D412yes—
184C2×Dic86C2×D412yes—
185C2×Dic86#4112yes—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43—21no—
192C4⋊C4×C43—21no—
193C4⋊C4×C43—21no—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—81no—
196C4⋊C4×C43C2×D412yes—
197C22⋊C4×C43—42no—
198C22⋊C4×C43—42no—
199C42×C43—21no—
200C42×C43—21no—
201C22⋊Dic43—42no—
202C22⋊Dic43—42no—
203C22⋊Dic43—42no—
204#10621885—44no—
205#10621883—28no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—81no—
211Dic43⋊C4—24no—
212Dic43⋊C4—22no—
213Dic43⋊C4—22no—
214Dic43⋊C4—42no—
215Dic43⋊C4—42no—
216Dic43⋊C4—42no—
217Dic43⋊C4—42no—
218Dic43⋊C4—42no—
219Dic43⋊C4—81no—
220C4×Dic43—21no—
221C4×Dic43—21no—
222C4×Dic43—24no—
223C4×Dic43—42no—
224C4×Dic43—42no—
225C4×Dic43C2×D412yes—
226C4×Dic43#4112yes—
227C2×C43⋊C8—21no—
228C2×C43⋊C8—21no—
229C2×C43⋊C8—21no—
230C2×C43⋊C8—21no—
231C2×C43⋊C8—21no—
232C2×C43⋊C8C2×D412yes—
233Dic172—44no—
234C23×C43—22no—
235C23×C43—41no—
236C23×C43—41no—
237C23×C43—41no—
238C23×C43—41no—
239C23×C43—41no—
240C23×C43—41no—
241C23×C43C22×D411yescharacteristic
242Q8×C43—41no—
243Q8×C43—41no—
244Q8×C43—41no—
245Q8×C43—41no—
246D4×C43—22no—
247D4×C43—22no—
248D4×C43—41no—
249D4×C43—41no—
250C2×C4×C43—22no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—21no—
255C2×C4×C43—21no—
256C2×C4×C43—21no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43C22×D411yescharacteristic
266C2×C4×C43C22×D411yescharacteristic
267C2×C4×C43C2×#4111yescharacteristic
268C2×C4×C43C2×#4111yescharacteristic
269C2×C4×C43H(5,𝔽2)11yescharacteristic
270C2×C4×C432−1+411yescharacteristic
271C2×Dic43—22no—
272C2×Dic43—22no—
273C2×Dic43—22no—
274C2×Dic43—21no—
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—21no—
278C2×Dic43—21no—
279C2×Dic43—21no—
280C2×Dic43—21no—
281C2×Dic43—21no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289C2×Dic43—41no—
290C2×Dic43—41no—
291C2×Dic43—84no—
292C2×Dic43#12112yes—
293C2×Dic43#12112yes—
294C2×Dic43#13512yes—
295C2×Dic43#13712yes—
296Dic86—22no—
297Dic86—22no—
298Dic86—22no—
299Dic86—41no—
300Dic86—41no—
301Dic86—41no—
302Dic86—41no—
303Dic86#12114yes—
304C8×C43—41no—
305C8×C43—41no—
306C43⋊C8—22no—
307C43⋊C8—41no—
308#24729—431nomaximal, Sylow
309C22×C43—42no—
310C22×C43—42no—
311C22×C43—42no—
312C22×C43—81no—
313C22×C43—81no—
314C22×C43—81no—
315C22×C43—81no—
316C22×C43C2×#12111yescharacteristic
317C22×C43D4211yescharacteristic
318C22×C43#54811yescharacteristic
319C22×C43C2×#13511yescharacteristic
320C22×C43C2×#13711yescharacteristic
321C22×C43#57711yescharacteristic
322C22×C43#57811yescharacteristic
323C4×C43—41no—
324C4×C43—41no—
325C4×C43—42no—
326C4×C43—81no—
327C4×C43—81no—
328C4×C43C2×#12111yescharacteristic
329C4×C43#53711yescharacteristic
330C4×C43#54711yescharacteristic
331C4×C43#55311yescharacteristic
332Dic43—24no—
333Dic43—22no—
334Dic43—41no—
335Dic43—41no—
336Dic43—81no—
337Dic43—81no—
338Dic43—81no—
339Dic43#46414yescomplemented
340C2×#464—431no—
341C2×#464—431no—
342C2×#462—431no—
343C2×C4×SD16—431no—
344#2547—431no—
345Q8×C4⋊C4—431no—
346#2066—431no—
347#2066—431no—
348#2042—431no—
349#2042D4311yescharacteristic
350#1956—431no—
351#1939—431no—
352#1878—431no—
353#1878—431no—
354#1802—431no—
355C2×C43—82no—
356C2×C43#241611yescharacteristic
357C2×C43C2×#46411yescharacteristic
358C2×C43#304611yescharacteristic
359C2×C43#327611yescharacteristic
360C2×C43#328411yescharacteristic
361C2×C43#336411yescharacteristic
362C2×C43#336911yescharacteristic
363C22×Q16—431no—
364C22×Q16—431no—
365C22×SD16—431no—
366C2×C4⋊Q8—431no—
367C2×#124—431no—
368C2×#124C2×D4311yescharacteristic
369C2×C4×Q8—431no—
370C2×C4×Q8—431no—
371C2×C4×Q8—431no—
372C2×C4×D4—431no—
373#464—864no—
374#464—864no—
375#462—864no—
376C4×SD16—864no—
377C2×#106—431no—
378C2×C4⋊C8—431no—
379C2×C4⋊C8—431no—
380C2×C4⋊C8—431no—
381C2×#103—431no—
382C2×#103—431no—
383C2×#103—431no—
384C2×#103—431no—
385C2×#103—431no—
386C2×#103C2×D4311yescharacteristic
387C2×#103C2×D4311yescharacteristic
388C2×#102—431no—
389C2×#102C2×D4311yescharacteristic
390C2×#102C2×D4311yescharacteristic
391C2×C4×C8—431no—
392#397—861no—
393#397—861no—
394#391—431no—
395#391—431no—
396#389—431no—
397#389—861no—
398#389—861no—
399#389C2×D4311yescharacteristic
400#384—431no—
401#384—861no—
402#380—861no—
403#380—861no—
404C4×C4⋊C4—431no—
405C4×C4⋊C4—861no—
406#340—431no—
407#340—431no—
408#336—431no—
409#336C2×D4311yescharacteristic
410C43#2472911yescharacteristic, complemented, Sylow
411C22×Q8—431no—
412C22×Q8—431no—
413C22×Q8C22×D4311yescharacteristic
414C22×D4C22×D4311yescharacteristic
415C23×C4—861no—
416C2×Q16—862no—
417C2×Q16—862no—
418C2×Q16—862no—
419C2×Q16—862no—
420C2×Q16—862no—
421C2×Q16—862no—
422C2×SD16—434no—
423C2×SD16—862no—
424C2×SD16—862no—
425C22×C8—431no—
426C22×C8—431no—
427C22×C8C22×D4311yescharacteristic
428C22×C8C22×D4311yescharacteristic
429C4⋊Q8—862no—
430C4⋊Q8—862no—
431#124—22no—
432#124—41no—
433#124—862no—
434#124—1721no—
435C4×Q8—434no—
436C4×Q8—862no—
437C4×Q8—862no—
438C4×Q8—862no—
439C4×Q8—1721no—
440C4×Q8—1721no—
441C4×D4—862no—
442C4×D4—862no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—431no—
445C2×C4⋊C4—431no—
446C2×C4⋊C4—431no—
447C2×C4⋊C4—431no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4—861no—
450C2×C4⋊C4—861no—
451C2×C4⋊C4—861no—
452C2×C4⋊C4—861no—
453C2×C4⋊C4—861no—
454C2×C4⋊C4—861no—
455C2×C4⋊C4—861no—
456C2×C4⋊C4C22×D4311yescharacteristic
457C2×C4⋊C4C22×D4311yescharacteristic
458C2×C22⋊C4—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—861no—
461C2×C22⋊C4—861no—
462C2×C22⋊C4—861no—
463C2×C42—431no—
464C2×C42—431no—
465C2×C42—861no—
466C2×C42—861no—
467C2×C42—861no—
468C2×C42C22×D4311yescharacteristic
469#106—862no—
470C4⋊C8—862no—
471C4⋊C8—862no—
472C4⋊C8—862no—
473#103—22no—
474#103—41no—
475#103—862no—
476#103—862no—
477#103—862no—
478#103—862no—
479#103—1721no—
480#102—22no—
481#102—22no—
482#102—862no—
483C4×C8—862no—
484#95—21no—
485#95—431no—
486#95—431no—
487#95—861no—
488#95—861no—
489#95—861no—
490#95—861no—
491#95—861no—
492C24—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—432no—
500C2×Q8—432no—
501C2×Q8—432no—
502C2×Q8—861no—
503C2×Q8—861no—
504C2×Q8—861no—
505C2×Q8—861no—
506C2×Q8—861no—
507C2×Q8—861no—
508C2×Q8—861no—
509C2×Q8—861no—
510C2×Q8—861no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4D4×D4312yes—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—431no—
521C22×C4—431no—
522C22×C4—431no—
523C22×C4—431no—
524C22×C4—861no—
525C22×C4—861no—
526C22×C4—861no—
527C22×C4—861no—
528C22×C4—1722no—
529C22×C4—1722no—
530C22×C4—1722no—
531C22×C4D4×D4311yescharacteristic
532C22×C4D4×D4311yescharacteristic
533C22×C4C23×D4311yescharacteristic
534Q16—1724no—
535Q16—1724no—
536SD16—868no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—432no—
544C2×C8—861no—
545C2×C8—861no—
546C2×C8—861no—
547C2×C8—861no—
548C2×C8—861no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—81no—
553C4⋊C4—862no—
554C4⋊C4—862no—
555C4⋊C4—864no—
556C4⋊C4—861no—
557C4⋊C4—861no—
558C4⋊C4—861no—
559C4⋊C4—861no—
560C4⋊C4—3441no—
561C4⋊C4—3441no—
562C4⋊C4—1722no—
563C4⋊C4—1722no—
564C4⋊C4—1722no—
565C4⋊C4—1722no—
566C4⋊C4—1722no—
567C4⋊C4D4×D4312yes—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—1722no—
571C22⋊C4—1722no—
572C22⋊C4—1722no—
573C42—21no—
574C42—21no—
575C42—432no—
576C42—432no—
577C42—864no—
578C42—861no—
579C42—861no—
580C42—1722no—
581C42—1722no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23C2×D4×D4311yescharacteristic
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—434no—
595Q8—862no—
596Q8—862no—
597Q8—862no—
598Q8—1721no—
599Q8—1721no—
600Q8—1721no—
601Q8—1721no—
602D4—22no—
603D4—22no—
604D4—41no—
605D4—41no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—432no—
622C2×C4—432no—
623C2×C4—432no—
624C2×C4—432no—
625C2×C4—862no—
626C2×C4—862no—
627C2×C4—861no—
628C2×C4—861no—
629C2×C4—862no—
630C2×C4—861no—
631C2×C4—861no—
632C2×C4—861no—
633C2×C4—861no—
634C2×C4—861no—
635C2×C4—861no—
636C2×C4—3444no—
637C2×C4—1721no—
638C2×C4—1721no—
639C2×C4—1721no—
640C2×C4—1721no—
641C2×C4—1721no—
642C2×C4—1721no—
643C2×C4—1721no—
644C2×C4—1721no—
645C2×C4—1721no—
646C2×C4C2×D4×D4311yescharacteristic
647C2×C4C2×D4×D4311yescharacteristic
648C2×C4#41×D4311yescharacteristic
649C2×C4#41×D4311yescharacteristic
650C2×C4#1306337011yescharacteristic
651C2×C4#1306337211yescharacteristic
652C8—41no—
653C8—41no—
654C8—862no—
655C8—1721no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#121×D4311yescharacteristic
664C22#1458353111yescharacteristic
665C22#1458355711yescharacteristic
666C22#135×D4311yescharacteristic
667C22#137×D4311yescharacteristic
668C22#1458370911yescharacteristic
669C22#1458371311yescharacteristic
670C4—41no—
671C4—42no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—434no—
676C4—862no—
677C4—864no—
678C4—3441no—
679C4—3441no—
680C4—3441no—
681C4—1721no—
682C4—1721no—
683C4#121×D4311yescharacteristic
684C4#1458347811yescharacteristic
685C4#1458354711yescharacteristic
686C4#1458359411yescharacteristic
687C2—82no—
688C2#1913802811yescharacteristic
689C2#464×D4311yescharacteristic
690C2#1914414611yescharacteristic
691C2#1914623911yescharacteristic
692C2#1914632311yescharacteristic
693C2#1914727711yescharacteristic
694C2#1914733211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2042C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#2042D43
hypocenterC43#24729
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
3. C2×#14583108C2semidirect
6. #19137509C2semidirect
7. #19135973C2semidirect
8. #19135972C2semidirect
10. #19135395C2semidirect
11. #19135312C2semidirect
12. #19134935C2semidirect
13. #19134933C2semidirect
339. Dic43#464semidirect
410. C43#24729semidirect

Maximal subgroups

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

subgroupindex
C2×#145831202
C2×#145831182
C2×#145831082
C2×#145829902
#191395882
#191375092
#191359732
#191359722
#191358352
#191353952
#191353122
#191349352
#191349332
#191345932
C43×#20422
#2472943