Finite Group Databasebeta

← #25106269

Subgroups — #25106269

Order 11008.

1502 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583444C211yesmaximal, characteristic
2C2×#14582949C211yesmaximal, characteristic, complemented
3C2×#14582768C211yesmaximal, characteristic, complemented
4#19135554C212yesmaximal, complemented
5#19135399C212yesmaximal, complemented
6#19134913C211yesmaximal, characteristic, complemented
7#19134912C211yesmaximal, characteristic, complemented
8#19134910C211yesmaximal, characteristic, complemented
9#19134594C212yesmaximal, complemented
10#19134593C212yesmaximal, complemented
11C43×#1874C211yesmaximal, characteristic, complemented
12C22×#10621900—21no—
13#14583444—24no—
14#14583444C418yescomplemented
15C2×#13063324C2211yescharacteristic
16C2×#13063321C2211yescharacteristic
17C2×#13063311—21no—
18C2×#13063299C2211yescharacteristic
19C2×#13063297—21no—
20C2×D4×Dic43C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22C2×(C22⋊D43)⋊C4—21no—
23C22×C4⋊Dic43C2211yescharacteristic
24C22×Dic43⋊C4—21no—
25C22×Dic43⋊C4—21no—
26C22×C4×Dic43C2211yescharacteristic
27#14582949—24no—
28C2×Dic43⋊C8C2212yes—
29C2×#13063247C2212yes—
30C2×#13063246C2212yes—
31C2×#13063245C2212yes—
32C2×C22⋊(C43⋊C8)C2211yescharacteristic
33C2×C8×Dic43C2212yes—
34#14582768—21no—
35#14582768—21no—
36#14582768C2212yes—
37#14582768C2212yes—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×#121×C43C2211yescharacteristic
42C22×C4⋊C4×C43C2211yescharacteristic
43C2×#102×C43C2212yes—
44C2×C22⋊C8×C43C2211yescharacteristic
45C43×#340C2212yes—
46C2×#10621900—28no—
47C2×#10621900—44no—
48C2×#10621900—44no—
49C22×Dic86—21no—
50C22×D172C2311yescharacteristic
51C22×C22⋊D43—21no—
52C22×C22⋊D43—21no—
53C22×C22⋊D43—21no—
54C23×Dic43—21no—
55C23×Dic43—21no—
56C23×Dic43D411yescharacteristic
57C23×Dic43D411yescharacteristic
58C22×C4×D43—21no—
59C22×C43⋊C8C2312yes—
60#13063324—22no—
61#13063324C2×C414yescomplemented
62#13063321—22no—
63#13063321C2×C414yescomplemented
64#13063311—28no—
65#13063299—22no—
66#13063299C2×C414yes—
67#13063297—28no—
68D4×Dic43—22no—
69D4×Dic43C2×C414yescomplemented
70#13063287—22no—
71#13063287C2×C414yescomplemented
72(C22⋊D43)⋊C4—28no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—42no—
75C2×C4⋊Dic43C2312yes—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—22no—
79C2×Dic43⋊C4—22no—
80C2×Dic43⋊C4—22no—
81C2×Dic43⋊C4—22no—
82C2×Dic43⋊C4—44no—
83C2×Dic43⋊C4—44no—
84C2×C22⋊Dic43—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×C4×Dic43C2×C414yes—
88C2×C4×Dic43D411yescharacteristic
89C2×C4×Dic43D411yescharacteristic
90C2×C4×Dic43D414yes—
91C2×C4×Dic43C2311yescharacteristic
92C2×C4×Dic43C2311yescharacteristic
93Dic43⋊C8—24no—
94#13063247—42no—
95#13063246—42no—
96#13063245—42no—
97C22⋊(C43⋊C8)—22no—
98C8×Dic43—24no—
99C22×D4×C43—21no—
100C22×D4×C43C2311yescharacteristic
101C23×C4×C43—21no—
102C23×C4×C43C2311yescharacteristic
103C22×C8×C43C2312yes—
104#121×C43—22no—
105#121×C43C2×C414yes—
106C2×C4⋊C4×C43—21no—
107C2×C4⋊C4×C43—42no—
108C2×C4⋊C4×C43C2312yes—
109C2×C4⋊C4×C43C2311yescharacteristic
110C2×C22⋊C4×C43—21no—
111#102×C43—42no—
112C22⋊C8×C43—22no—
113C24×C43—21no—
114C24×C43C2×D411yescharacteristic
115C23×D43—21no—
116C2×D4×C43—21no—
117C2×D4×C43—24no—
118C2×D4×C43—22no—
119C2×D4×C43—22no—
120C2×D4×C43—42no—
121C2×D4×C43—42no—
122C2×D4×C43C22×C412yes—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—21no—
126C22×C4×C43—22no—
127C22×C4×C43—42no—
128C22×C4×C43—42no—
129C22×C4×C43—42no—
130C22×C4×C43C22×C412yes—
131C22×C4×C43C2×D412yes—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2411yescharacteristic
134C2×C22⋊D43—24no—
135C2×C22⋊D43—24no—
136C2×C22⋊D43—24no—
137C2×C22⋊D43—42no—
138C2×C22⋊D43—42no—
139C2×C22⋊D43—42no—
140C2×C22⋊D43—42no—
141C2×C22⋊D43—42no—
142C2×C22⋊D43—42no—
143C22×Dic43—22no—
144C22×Dic43—22no—
145C22×Dic43—24no—
146C22×Dic43—21no—
147C22×Dic43—22no—
148C22×Dic43—42no—
149C22×Dic43—42no—
150C22×Dic43—42no—
151C22×Dic43—42no—
152C22×Dic43—42no—
153C22×Dic43C22⋊C414yes—
154C22×Dic43C22⋊C414yes—
155C22×Dic43D814yes—
156C22×Dic43SD1614yes—
157C22×Dic43C2×D411yescharacteristic
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161#10621900—432no—
162C2×D172—21no—
163C2×D172—22no—
164C2×D172—22no—
165C2×D172C22×C412yes—
166C2×C4×D43—24no—
167C2×C4×D43—42no—
168C2×C4×D43—42no—
169C2×Dic86—24no—
170C2×Dic86—42no—
171C2×Dic86—42no—
172C2×C8×C43—22no—
173C2×C8×C43—22no—
174C2×C8×C43—22no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—22no—
177C4⋊C4×C43—22no—
178C4⋊C4×C43—42no—
179C4⋊C4×C43C22×C412yes—
180C22⋊C4×C43—24no—
181C22⋊Dic43—24no—
182(C2×D43)⋊C4—24no—
183(C2×D43)⋊C4—24no—
184C4⋊Dic43—21no—
185C4⋊Dic43—22no—
186C4⋊Dic43—22no—
187C4⋊Dic43—42no—
188C4⋊Dic43C22×C412yes—
189Dic43⋊C4—24no—
190Dic43⋊C4—44no—
191Dic43⋊C4—44no—
192Dic43⋊C4—44no—
193Dic43⋊C4—44no—
194C4×Dic43C22⋊C414yes—
195C4×Dic43C22⋊C414yes—
196C4×Dic43C22×C412yes—
197C4×Dic43C22×C412yes—
198C4×Dic43C2×D412yes—
199C4×Dic43C2×D412yes—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—22no—
203C23×C43—21no—
204C23×C43—21no—
205C23×C43—21no—
206C23×C43—21no—
207C23×C43—22no—
208C23×C43—41no—
209C23×C43—41no—
210C23×C43—41no—
211C23×C43—41no—
212C23×C43—41no—
213C23×C43—41no—
214C23×C43C2×C22⋊C412yes—
215C23×C43C2×D812yes—
216C23×C43C2×SD1612yes—
217C23×C43C22×D411yescharacteristic
218C22×D43—22no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225D4×C43—24no—
226D4×C43—48no—
227D4×C43—42no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—24no—
231C2×C4×C43—24no—
232C2×C4×C43—22no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—84no—
239C2×C4×C43C2×C22⋊C412yes—
240C2×C4×C43C23×C411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43H(5,𝔽2)12yes—
243C2×C4×C432−1+412yes—
244C22⋊D43—48no—
245C22⋊D43—48no—
246C22⋊D43—48no—
247C2×Dic43—21no—
248C2×Dic43—21no—
249C2×Dic43—28no—
250C2×Dic43—22no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—48no—
254C2×Dic43—42no—
255C2×Dic43—42no—
256C2×Dic43—42no—
257C2×Dic43—84no—
258C2×Dic43#102116yescomplemented
259C2×Dic43C2×C22⋊C412yes—
260C2×Dic43C2×C22⋊C412yes—
261C2×Dic43C2×C22⋊C412yes—
262C2×Dic43C2×C22⋊C412yes—
263C2×Dic43C2×D812yes—
264C2×Dic43C2×D812yes—
265C2×Dic43C2×SD1612yes—
266C2×Dic43C2×SD1612yes—
267C2×Dic43AGL(1,ℤ/8)12yes—
268C2×Dic43#13712yes—
269D172—24no—
270D172—42no—
271C4×D43—48no—
272Dic86—48no—
273C8×C43—42no—
274C43⋊C8—42no—
275#21859—431nomaximal, Sylow
276C22×C43—22no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43—22no—
280C22×C43—22no—
281C22×C43—22no—
282C22×C43—42no—
283C22×C43—42no—
284C22×C43—42no—
285C22×C43—81no—
286C22×C43—81no—
287C22×C43—81no—
288C22×C43—81no—
289C22×C43C2×#10214yes—
290C22×C43C22×C22⋊C411yescharacteristic
291C22×C43#53511yescharacteristic
292C22×C43#53611yescharacteristic
293C22×C43C22×D811yescharacteristic
294C22×C43C22×SD1611yescharacteristic
295C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C43C2×#13711yescharacteristic
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—42no—
300C2×D43—81no—
301C2×D43—81no—
302C2×D43—81no—
303C2×D43—81no—
304C4×C43—22no—
305C4×C43—42no—
306C4×C43—82no—
307C4×C43#52012yes—
308C4×C43#53712yes—
309Dic43—22no—
310Dic43—42no—
311Dic43—82no—
312Dic43C2×#10214yes—
313Dic43C2×#10214yes—
314Dic43#41714yes—
315C2×#532—431no—
316C2×#431—431no—
317C2×#384—431no—
318#1990—432no—
319#1957—432no—
320#1874—431no—
321#1874—431no—
322#1874—431no—
323#1874D4311yescharacteristic, complemented
324#1802—432no—
325#1802—432no—
326C2×C43—24no—
327C2×C43—82no—
328C2×C43#231111yescharacteristic
329C2×C43C22×#10211yescharacteristic
330C2×C43C2×#41711yescharacteristic
331C2×C43#320511yescharacteristic
332C2×C43#320611yescharacteristic
333C2×C43#322611yescharacteristic
334C2×C43#322711yescharacteristic
335D43—82no—
336C22×#41—861no—
337#532—438no—
338#532—864no—
339C2×C4⋊Q8—431no—
340C2×#127—431no—
341C2×#124—861no—
342C2×#121—431no—
343C2×#121—861no—
344C2×#121C2×D4311yescharacteristic
345C2×C4×D4—431no—
346C2×C4×D4—431no—
347C2×C4×D4—861no—
348C22×C4⋊C4—431no—
349C22×C4⋊C4—861no—
350C22×C4⋊C4—861no—
351C22×C4⋊C4C2×D4311yescharacteristic
352C22×C42—431no—
353#431—864no—
354C2×C4⋊C8—432no—
355C2×#102—432no—
356C2×#102—432no—
357C2×#102—432no—
358C2×#102C2×D4312yes—
359C2×C22⋊C8—431no—
360C2×C22⋊C8C2×D4311yescharacteristic
361C2×C4×C8—432no—
362#384—432no—
363#384—432no—
364#384—861no—
365#384—861no—
366#340—432no—
367#340—432no—
368#340—432no—
369#340C2×D4312yes—
370C43#2185911yescharacteristic, complemented, Sylow
371C2×#41—868no—
372C2×#41—1724no—
373C2×#41—1724no—
374C22×Q8—861no—
375C22×D4—21no—
376C22×D4—431no—
377C22×D4—861no—
378C22×D4—861no—
379C22×D4—861no—
380C22×D4C22×D4311yescharacteristic
381C23×C4—21no—
382C23×C4—431no—
383C23×C4—431no—
384C23×C4—861no—
385C23×C4—861no—
386C23×C4—861no—
387C23×C4C22×D4311yescharacteristic
388C22×C8—432no—
389C22×C8C22×D4312yes—
390C4⋊Q8—434no—
391C4⋊Q8—862no—
392#127—434no—
393#127—862no—
394#124—868no—
395#121—22no—
396#121—434no—
397#121—868no—
398#121—862no—
399#121C4×D4314yes—
400C4×D4—434no—
401C4×D4—434no—
402C4×D4—868no—
403C4×D4—862no—
404C4×D4—862no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—432no—
409C2×C4⋊C4—861no—
410C2×C4⋊C4—862no—
411C2×C4⋊C4—862no—
412C2×C4⋊C4—862no—
413C2×C4⋊C4—862no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—1722no—
416C2×C4⋊C4—1724no—
417C2×C4⋊C4—1724no—
418C2×C4⋊C4C22×D4312yes—
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C42—434no—
425C2×C42—431no—
426C2×C42—431no—
427C2×C42—434no—
428C2×C42—431no—
429C2×C42—431no—
430C4⋊C8—864no—
431#102—42no—
432#102—1722no—
433#102—1722no—
434#102—1722no—
435C22⋊C8—22no—
436C22⋊C8—862no—
437C4×C8—864no—
438C24—21no—
439C24—861no—
440C24D4×D4311yescharacteristic
441#41—17232no—
442C2×Q8—864no—
443C2×Q8—1722no—
444C2×Q8—1722no—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—432no—
452C2×D4—864no—
453C2×D4—864no—
454C2×D4—864no—
455C2×D4—861no—
456C2×D4—862no—
457C2×D4—862no—
458C2×D4—1722no—
459C2×D4—1722no—
460C2×D4—1722no—
461C2×D4—1722no—
462C2×D4—1722no—
463C2×D4—1722no—
464C2×D4C2×C4×D4312yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—434no—
473C22×C4—434no—
474C22×C4—434no—
475C22×C4—431no—
476C22×C4—431no—
477C22×C4—431no—
478C22×C4—434no—
479C22×C4—431no—
480C22×C4—864no—
481C22×C4—862no—
482C22×C4—862no—
483C22×C4—864no—
484C22×C4—861no—
485C22×C4—862no—
486C22×C4—1722no—
487C22×C4—1722no—
488C22×C4—1722no—
489C22×C4—1722no—
490C22×C4—1722no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4C2×C4×D4312yes—
494C22×C4D4×D4311yescharacteristic
495C22×C4D4×D4312yes—
496C22×C4C23×D4311yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—862no—
501C2×C8—862no—
502C2×C8—862no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—432no—
508C4⋊C4—864no—
509C4⋊C4—861no—
510C4⋊C4—862no—
511C4⋊C4—862no—
512C4⋊C4—1722no—
513C4⋊C4—1724no—
514C4⋊C4—1724no—
515C4⋊C4—1724no—
516C4⋊C4—1724no—
517C4⋊C4C2×C4×D4312yes—
518C22⋊C4—24no—
519C22⋊C4—864no—
520C22⋊C4—864no—
521C22⋊C4—864no—
522C42—434no—
523C42—434no—
524C42—432no—
525C42—432no—
526C42—432no—
527C42—432no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—862no—
540C23—1721no—
541C23—1721no—
542C23—1721no—
543C23—1721no—
544C23—1721no—
545C23—1721no—
546C23C22⋊C4×D4312yes—
547C23D8×D4312yes—
548C23SD16×D4312yes—
549C23C2×D4×D4311yescharacteristic
550Q8—1728no—
551D4—24no—
552D4—48no—
553D4—42no—
554D4—864no—
555D4—1728no—
556D4—1728no—
557D4—1728no—
558D4—1722no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—84no—
570C2×C4—432no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—4316no—
574C2×C4—432no—
575C2×C4—432no—
576C2×C4—432no—
577C2×C4—432no—
578C2×C4—432no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—868no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—862no—
585C2×C4—3444no—
586C2×C4—1728no—
587C2×C4—1728no—
588C2×C4—1722no—
589C2×C4—1722no—
590C2×C4—1722no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4C22⋊C4×D4312yes—
594C2×C4C22×C4×D4311yescharacteristic
595C2×C4C2×D4×D4311yescharacteristic
596C2×C4#1306337012yes—
597C2×C4#1306337212yes—
598C8—42no—
599C8—1722no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—3441no—
614C22—3441no—
615C22—3441no—
616C22—3441no—
617C22—1722no—
618C22—1722no—
619C22—1722no—
620C22#102×D4314yes—
621C22C2×C22⋊C4×D4311yescharacteristic
622C22#1458346411yescharacteristic
623C22#1458347111yescharacteristic
624C22C2×D8×D4311yescharacteristic
625C22C2×SD16×D4311yescharacteristic
626C22AGL(1,ℤ/8)×D4311yescharacteristic
627C22#137×D4311yescharacteristic
628C4—22no—
629C4—42no—
630C4—82no—
631C4—434no—
632C4—434no—
633C4—434no—
634C4—862no—
635C4—3442no—
636C4—1722no—
637C4#1458337812yes—
638C4#1458347612yes—
639C2—24no—
640C2—82no—
641C2—3442no—
642C2#1913718211yescharacteristic
643C2C2×#102×D4311yescharacteristic
644C2#417×D4311yescharacteristic
645C2#1914551111yescharacteristic
646C2#1914551811yescharacteristic
647C2#1914571411yescharacteristic
648C2#1914572111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1874C2
socleC23×C43C22×D4
prime radicalC2×#14583444C2
cosocleC22×C4×C43C24
hypercenter#1874D43
hypocenterC43#21859
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#14582949C2semidirect
3. C2×#14582768C2semidirect
4. #19135554C2semidirect
5. #19135399C2semidirect
6. #19134913C2semidirect
7. #19134912C2semidirect
8. #19134910C2semidirect
9. #19134594C2semidirect
10. #19134593C2semidirect
11. C43×#1874C2semidirect
14. #14583444C4semidirect
61. #13063324C2×C4semidirect
63. #13063321C2×C4semidirect
69. D4×Dic43C2×C4semidirect
71. #13063287C2×C4semidirect
258. C2×Dic43#102semidirect
323. #1874D43semidirect
370. C43#21859semidirect

Maximal subgroups

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

subgroupindex
C2×#145834442
C2×#145829492
C2×#145827682
#191355542
#191353992
#191349132
#191349122
#191349102
#191345942
#191345932
C43×#18742
#2185943