Finite Group Databasebeta

← #25131103

Subgroups — #25131103

Order 11008.

1022 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583512C211yesmaximal, characteristic
2C2×#14583203C211yesmaximal, characteristic, complemented
3C2×#14583110C211yesmaximal, characteristic, complemented
4#19137278C211yesmaximal, characteristic
5#19135801C211yesmaximal, characteristic
6#19135794C211yesmaximal, characteristic
7#19135606C211yesmaximal, characteristic
8#19135606C211yesmaximal, characteristic
9#19135518C211yesmaximal, characteristic, complemented
10#19134937C211yesmaximal, characteristic, complemented
11#19134937C211yesmaximal, characteristic, complemented
12#19134864C211yesmaximal, characteristic, complemented
13#19134850C211yesmaximal, characteristic, complemented
14#19134576C211yesmaximal, characteristic
15C43×#1989C211yesmaximal, characteristic, complemented
16C22×Dic172C2211yescharacteristic
17C22×Dic172C2211yescharacteristic
18#14583512—24no—
19#14583512—24no—
20C2×#13063325C2211yescharacteristic
21C2×#13063320—21no—
22C2×#13063320—21no—
23C2×#13063316—21no—
24C2×#13063316—21no—
25C2×#13063305C2211yescharacteristic
26C2×#13063305C2211yescharacteristic
27C2×#13063305C2211yescharacteristic
28C2×C4×Dic86C2211yescharacteristic
29C2×#13063290C2211yescharacteristic
30C2×#13063290C2211yescharacteristic
31C2×#13063282C2211yescharacteristic
32#14583203—24no—
33#14583110—24no—
34C2×#13063258C2211yescharacteristic
35C2×#13063256C2211yescharacteristic
36C2×C4⋊(C43⋊C8)C2211yescharacteristic
37C2×#13063250C2211yescharacteristic
38C2×#13063250C2211yescharacteristic
39C2×#13063248C2211yescharacteristic
40C2×#13063248C2211yescharacteristic
41C2×#13063248C2211yescharacteristic
42C2×#13063248C2211yescharacteristic
43C2×#13063248C2211yescharacteristic
44C2×#13063248C2211yescharacteristic
45C2×(C2×D43)⋊C8C2211yescharacteristic
46C2×(C2×D43)⋊C8C2211yescharacteristic
47#14582804C2211yescharacteristic
48#14582798C2211yescharacteristic
49#14582797—21no—
50#14582791—21no—
51#14582790—21no—
52#14582784—21no—
53#14582756C2211yescharacteristic
54#14582653C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×#125×C43C2211yescharacteristic
57C2×C4⋊C8×C43C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C43×#340C2211yescharacteristic
63C22×Dic86C2311yescharacteristic
64C22×Dic86C2311yescharacteristic
65C22×Dic86C2311yescharacteristic
66C22×C4×D43C2311yescharacteristic
67C2×Dic172—22no—
68C2×Dic172—22no—
69C2×Dic172—22no—
70C2×Dic172—22no—
71C2×Dic172—41no—
72C2×Dic172—41no—
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75#13063325—22no—
76#13063325D414yes—
77#13063320—28no—
78#13063320—28no—
79#13063316—28no—
80#13063316—28no—
81#13063305—22no—
82#13063305—22no—
83#13063305—22no—
84#13063305—22no—
85#13063305—22no—
86#13063305—22no—
87C4×Dic86—22no—
88C4×Dic86—22no—
89#13063290—22no—
90#13063290—22no—
91#13063290—22no—
92#13063290—22no—
93#13063282—22no—
94#13063282—22no—
95C2×C4⋊Dic43—21no—
96C2×C4⋊Dic43—21no—
97C2×C4⋊Dic43—21no—
98C2×C4⋊Dic43—21no—
99C2×C4⋊Dic43—21no—
100C2×C4⋊Dic43—21no—
101C2×C4⋊Dic43C2311yescharacteristic
102C2×Dic43⋊C4—21no—
103C2×Dic43⋊C4—21no—
104C2×Dic43⋊C4—21no—
105C2×Dic43⋊C4D411yescharacteristic
106C2×Dic43⋊C4D411yescharacteristic
107C2×(C2×D43)⋊C4—21no—
108C2×(C2×D43)⋊C4—21no—
109C2×(C2×D43)⋊C4—21no—
110C2×(C2×D43)⋊C4—21no—
111C2×(C2×D43)⋊C4D411yescharacteristic
112C2×(C2×D43)⋊C4D411yescharacteristic
113C2×C4×Dic43—21no—
114C2×C4×Dic43—21no—
115C2×C4×Dic43—21no—
116C2×C4×Dic43C2311yescharacteristic
117#13063258—22no—
118#13063256—22no—
119C4⋊(C43⋊C8)—22no—
120#13063250—22no—
121#13063250—22no—
122#13063248—22no—
123#13063248—22no—
124#13063248—22no—
125#13063248—22no—
126#13063248—41no—
127#13063248—41no—
128(C2×D43)⋊C8—41no—
129(C2×D43)⋊C8—41no—
130C43⋊#95—21no—
131C43⋊#95—21no—
132C43⋊#95—21no—
133C43⋊#95—21no—
134C22×C8×C43C2311yescharacteristic
135C22×C8×C43C2311yescharacteristic
136#125×C43—22no—
137#125×C43D414yes—
138C2×C4⋊C4×C43—21no—
139C2×C4⋊C4×C43—21no—
140C2×C4⋊C4×C43C2311yescharacteristic
141C2×C4⋊C4×C43C2311yescharacteristic
142C2×C4⋊C4×C43C2311yescharacteristic
143C2×C4⋊C4×C43C2311yescharacteristic
144C2×C42×C43—21no—
145C2×C42×C43C2311yescharacteristic
146C4⋊C8×C43—22no—
147#95×C43—21no—
148C23×D43C2×D411yescharacteristic
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2411yescharacteristic
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C22×Dic43C2×D411yescharacteristic
163C2×C4×D43—21no—
164C2×C4×D43—21no—
165C2×C4×D43—21no—
166C2×C4×D43—21no—
167C2×C4×D43—21no—
168C2×C4×D43—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
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×Dic86—21no—
184C2×Dic86C2×D412yes—
185C2×Dic86C2×D412yes—
186C2×Dic86#4112yes—
187C2×C8×C43—21no—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43—21no—
192C2×C8×C43—21no—
193C4⋊C4×C43—24no—
194C4⋊C4×C43—24no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43—21no—
197C4⋊C4×C43—21no—
198C4⋊C4×C43—21no—
199C4⋊C4×C43—21no—
200C4⋊C4×C43C2×D412yes—
201C4⋊C4×C43C2×D412yes—
202C4⋊C4×C43#4112yes—
203C42×C43—81no—
204C42×C43C2×D412yes—
205C42×C43C2×D412yes—
206(C2×D43)⋊C4—24no—
207(C2×D43)⋊C4—24no—
208(C2×D43)⋊C4—24no—
209(C2×D43)⋊C4—22no—
210(C2×D43)⋊C4—22no—
211(C2×D43)⋊C4—81no—
212C4⋊Dic43—24no—
213C4⋊Dic43—24no—
214C4⋊Dic43—24no—
215C4⋊Dic43—24no—
216C4⋊Dic43—24no—
217C4⋊Dic43—21no—
218C4⋊Dic43—21no—
219C4⋊Dic43—81no—
220Dic43⋊C4—24no—
221Dic43⋊C4—24no—
222Dic43⋊C4—22no—
223Dic43⋊C4—22no—
224Dic43⋊C4—42no—
225C4×Dic43—24no—
226C4×Dic43—24no—
227C4×Dic43—21no—
228C4×Dic43—21no—
229C4×Dic43—42no—
230C2×C43⋊C8—21no—
231C2×C43⋊C8—21no—
232C2×C43⋊C8—21no—
233C2×C43⋊C8—21no—
234C2×C43⋊C8—21no—
235C2×C43⋊C8—21no—
236Dic172—44no—
237Dic172—44no—
238C23×C43C22×D411yescharacteristic
239C22×D43—21no—
240C22×D43—21no—
241C22×D43—21no—
242C22×D43—21no—
243C22×D43—21no—
244C22×D43—21no—
245C22×D43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—22no—
251C2×C4×C43—22no—
252C2×C4×C43—22no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43#12012yes—
263C2×C4×C43#12012yes—
264C2×C4×C43C22×D411yescharacteristic
265C2×C4×C43C22×D411yescharacteristic
266C2×C4×C43C2×#4111yescharacteristic
267C2×C4×C43C2×#4111yescharacteristic
268C2×C4×C43H(5,𝔽2)11yescharacteristic
269C2×C4×C432−1+411yescharacteristic
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—22no—
273C2×Dic43—22no—
274C2×Dic43—21no—
275C2×Dic43—21no—
276C2×Dic43—22no—
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—
291C4×D43—41no—
292C4×D43—41no—
293C4×D43—41no—
294C4×D43—41no—
295Dic86—22no—
296Dic86—22no—
297Dic86—22no—
298Dic86—22no—
299Dic86—22no—
300Dic86—22no—
301Dic86—41no—
302Dic86—41no—
303Dic86—41no—
304Dic86—41no—
305Dic86—41no—
306Dic86—41no—
307C8×C43—41no—
308C8×C43—41no—
309C43⋊C8—41no—
310C43⋊C8—41no—
311#24165—431nomaximal, Sylow
312C22×C43C2×#12011yescharacteristic
313C22×C43#52511yescharacteristic
314C22×C43#53711yescharacteristic
315C22×C43#57711yescharacteristic
316C22×C43#57811yescharacteristic
317C22×C43#57811yescharacteristic
318C22×C43#57811yescharacteristic
319C2×D43—41no—
320C2×D43—41no—
321C2×D43—41no—
322C2×D43—41no—
323C2×D43—41no—
324C2×D43—41no—
325C2×D43—41no—
326C4×C43—22no—
327C4×C43—22no—
328C4×C43—42no—
329C4×C43—42no—
330C4×C43—42no—
331C4×C43—81no—
332C4×C43#54611yescharacteristic
333C4×C43#54711yescharacteristic
334C4×C43#54811yescharacteristic
335C4×C43#55011yescharacteristic
336Dic43—42no—
337Dic43—42no—
338Dic43—42no—
339Dic43—41no—
340Dic43—41no—
341Dic43—41no—
342Dic43—81no—
343C2×#541—431no—
344C2×#479—431no—
345C2×#462—431no—
346#2323—431no—
347#2036—431no—
348#2035—431no—
349#2001—431no—
350#2001—431no—
351#1989D4311yescharacteristic, complemented
352#1981—431no—
353#1878—431no—
354#1878—431no—
355#1865—431no—
356#1863—431no—
357#1798—431no—
358C2×C43#240711yescharacteristic
359C2×C43#301111yescharacteristic
360C2×C43#301311yescharacteristic
361C2×C43#312011yescharacteristic
362C2×C43#312711yescharacteristic
363C2×C43#325811yescharacteristic
364C2×C43#326111yescharacteristic
365D43—81no—
366C22×Q16—431no—
367C22×Q16—431no—
368#541—864no—
369#541—864no—
370C2×C4⋊Q8—431no—
371C2×#126—861no—
372C2×#126—861no—
373C2×#125—861no—
374C2×#125—861no—
375C2×#125C2×D4311yescharacteristic
376C2×C22⋊Q8—431no—
377C2×C22⋊Q8—431no—
378C2×C22⋊Q8—431no—
379C2×C4×Q8—431no—
380C2×C4×Q8—431no—
381C2×C4×Q8—431no—
382C2×#117—431no—
383#479—864no—
384#462—864no—
385C2×#107—431no—
386C2×#106—431no—
387C2×C4⋊C8—431no—
388C2×C4⋊C8C2×D4311yescharacteristic
389C2×#103—431no—
390C2×#103—431no—
391C2×#103—431no—
392C2×#103—431no—
393C2×#103—431no—
394C2×#103—431no—
395C2×#103—431no—
396C2×#103—431no—
397C2×C22⋊C8—431no—
398C2×C22⋊C8—431no—
399#391—431no—
400#389—431no—
401#389—861no—
402#388—861no—
403#388—861no—
404#387—861no—
405#380—431no—
406C4×C4⋊C4C2×D4311yescharacteristic
407#340C2×D4311yescharacteristic
408#340C2×D4311yescharacteristic
409#340C2×D4311yescharacteristic
410#340C2×D4311yescharacteristic
411#336—431no—
412#336—431no—
413C43#2416511yescharacteristic, complemented, Sylow
414C22×Q8—431no—
415C22×Q8—431no—
416C22×Q8—431no—
417C23×C4—431no—
418C2×Q16—862no—
419C2×Q16—862no—
420C2×Q16—862no—
421C2×Q16—862no—
422C2×Q16—1721no—
423C2×Q16—1721no—
424C22×C8—431no—
425C22×C8—431no—
426C22×C8C22×D4311yescharacteristic
427C22×C8C22×D4311yescharacteristic
428C4⋊Q8—434no—
429C4⋊Q8—862no—
430#126—868no—
431#126—868no—
432#125—22no—
433#125—868no—
434#125—868no—
435#125D17214yes—
436C22⋊Q8—862no—
437C22⋊Q8—862no—
438C22⋊Q8—862no—
439C22⋊Q8—862no—
440C22⋊Q8—862no—
441C22⋊Q8—862no—
442C4×Q8—862no—
443C4×Q8—862no—
444C4×Q8—862no—
445C4×Q8—862no—
446C4×Q8—862no—
447C4×Q8—862no—
448#117—862no—
449#117—862no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—431no—
453C2×C4⋊C4—431no—
454C2×C4⋊C4—431no—
455C2×C4⋊C4—861no—
456C2×C4⋊C4—861no—
457C2×C4⋊C4—861no—
458C2×C4⋊C4—861no—
459C2×C4⋊C4—861no—
460C2×C4⋊C4—861no—
461C2×C4⋊C4—861no—
462C2×C4⋊C4—861no—
463C2×C4⋊C4—861no—
464C2×C4⋊C4C22×D4311yescharacteristic
465C2×C4⋊C4C22×D4311yescharacteristic
466C2×C4⋊C4C22×D4311yescharacteristic
467C2×C4⋊C4C22×D4311yescharacteristic
468C2×C22⋊C4—431no—
469C2×C22⋊C4—431no—
470C2×C22⋊C4—861no—
471C2×C22⋊C4—861no—
472C2×C22⋊C4—861no—
473C2×C22⋊C4—861no—
474C2×C42—21no—
475C2×C42—431no—
476C2×C42—861no—
477C2×C42—861no—
478C2×C42—861no—
479C2×C42C22×D4311yescharacteristic
480#107—862no—
481#106—862no—
482C4⋊C8—22no—
483C4⋊C8—862no—
484#103—862no—
485#103—862no—
486#103—862no—
487#103—862no—
488#103—862no—
489#103—862no—
490#103—1721no—
491#103—1721no—
492C22⋊C8—1721no—
493C22⋊C8—1721no—
494#95—21no—
495#95—861no—
496#95—861no—
497#95—861no—
498#95—861no—
499C24—431no—
500C2×Q8—432no—
501C2×Q8—432no—
502C2×Q8—432no—
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×Q8—861no—
512C2×Q8—861no—
513C2×Q8—861no—
514C2×Q8—861no—
515C2×Q8—861no—
516C2×Q8—861no—
517C2×Q8—861no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—431no—
523C22×C4—431no—
524C22×C4—431no—
525C22×C4—861no—
526C22×C4—861no—
527C22×C4—861no—
528C22×C4—861no—
529C22×C4—861no—
530C22×C4—861no—
531C22×C4—861no—
532C22×C4—861no—
533C22×C4—861no—
534C22×C4—861no—
535C22×C4D4×D4311yescharacteristic
536C22×C4D4×D4311yescharacteristic
537C22×C4C23×D4311yescharacteristic
538Q16—1724no—
539Q16—1724no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—861no—
547C2×C8—861no—
548C2×C8—861no—
549C2×C8—861no—
550C2×C8—861no—
551C2×C8—861no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—864no—
560C4⋊C4—864no—
561C4⋊C4—864no—
562C4⋊C4—864no—
563C4⋊C4—864no—
564C4⋊C4—862no—
565C4⋊C4—862no—
566C4⋊C4—864no—
567C4⋊C4—864no—
568C4⋊C4—861no—
569C4⋊C4—861no—
570C4⋊C4—3441no—
571C4⋊C4—1722no—
572C4⋊C4C2×D17212yes—
573C4⋊C4C2×D17212yes—
574C4⋊C4#1062190212yes—
575C22⋊C4—864no—
576C22⋊C4—864no—
577C22⋊C4—864no—
578C22⋊C4—862no—
579C22⋊C4—862no—
580C22⋊C4—3441no—
581C42—81no—
582C42—864no—
583C42—864no—
584C42—861no—
585C42—861no—
586C42—1722no—
587C42C2×D17212yes—
588C42D4×D4312yes—
589C23—861no—
590C23—861no—
591C23—861no—
592C23—861no—
593C23—861no—
594C23—861no—
595C23—861no—
596C23C2×D4×D4311yescharacteristic
597Q8—862no—
598Q8—862no—
599Q8—862no—
600Q8—862no—
601Q8—862no—
602Q8—862no—
603Q8—1721no—
604Q8—1721no—
605Q8—1721no—
606Q8—1721no—
607Q8—1721no—
608Q8—1721no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—861no—
626C2×C4—861no—
627C2×C4—862no—
628C2×C4—862no—
629C2×C4—861no—
630C2×C4—862no—
631C2×C4—861no—
632C2×C4—861no—
633C2×C4—861no—
634C2×C4—861no—
635C2×C4—861no—
636C2×C4—861no—
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×C4—1721no—
647C2×C4—1721no—
648C2×C4—1721no—
649C2×C4—1721no—
650C2×C4#1306329512yes—
651C2×C4#1306329512yes—
652C2×C4C2×D4×D4311yescharacteristic
653C2×C4C22×D17211yescharacteristic
654C2×C4#41×D4311yescharacteristic
655C2×C4C2×#1062190211yescharacteristic
656C2×C4#1306337111yescharacteristic
657C2×C4#1306337311yescharacteristic
658C8—41no—
659C8—41no—
660C8—1721no—
661C8—1721no—
662C22—1721no—
663C22—1721no—
664C22—1721no—
665C22—1721no—
666C22—1721no—
667C22—1721no—
668C22—1721no—
669C22C2×#1306329511yescharacteristic
670C22#1458341211yescharacteristic
671C22#1458347911yescharacteristic
672C22#1458370911yescharacteristic
673C22#1458371311yescharacteristic
674C22#1458371311yescharacteristic
675C22#1458371511yescharacteristic
676C4—22no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—3441no—
683C4—1721no—
684C4—1722no—
685C4—1722no—
686C4—1722no—
687C4—1721no—
688C4—1721no—
689C4#1458354211yescharacteristic
690C4#1458355211yescharacteristic
691C4#1458356111yescharacteristic
692C4#1458357111yescharacteristic
693C2—3441no—
694C2#1913794311yescharacteristic
695C2#1914389311yescharacteristic
696C2#1914391211yescharacteristic
697C2#1914481411yescharacteristic
698C2#1914489711yescharacteristic
699C2#1914602511yescharacteristic
700C2#1914604811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1989C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenter#1989D43
hypocenterC43#24165
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#14583203C2semidirect
3. C2×#14583110C2semidirect
9. #19135518C2semidirect
10. #19134937C2semidirect
11. #19134937C2semidirect
12. #19134864C2semidirect
13. #19134850C2semidirect
15. C43×#1989C2semidirect
351. #1989D43semidirect
413. C43#24165semidirect

Maximal subgroups

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

subgroupindex
C2×#145835122
C2×#145832032
C2×#145831102
#191372782
#191358012
#191357942
#191356062
#191355182
#191349372
#191348642
#191348502
#191345762
C43×#19892
#2416543