Finite Group Databasebeta

← #25112755

Subgroups — #25112755

Order 11008.

1322 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583552C211yesmaximal, characteristic
2M16×D172C211yesmaximal, characteristic, complemented
3#19143330C211yesmaximal, characteristic, complemented
4#19135585C211yesmaximal, characteristic, complemented
5#19135057C212yesmaximal, complemented
6#19135057C212yesmaximal, complemented
7#19134674C212yesmaximal, complemented
8#19134584C211yesmaximal, characteristic, complemented
9C4×#13063241C211yesmaximal, characteristic, complemented
10#343×D43C212yesmaximal, complemented
11C43×#1997C211yesmaximal, characteristic, complemented
12C22×#10621902C412yes—
13C2×M16×D43C2212yes—
14C2×M16×D43C2212yes—
15#14583552—21no—
16#14583552—21no—
17#14583552—22no—
18#14583552—22no—
19#14583552—21no—
20#14583552—21no—
21C2×#13063310C414yes—
22C2×#13063301C412yes—
23C2×#13063300C2212yes—
24C2×C4×D172C2211yescharacteristic
25C2×C4×D172C2211yescharacteristic
26C2×C4⋊C4×D43C2212yes—
27#14582978—21no—
28#14582976—21no—
29#14582972—22no—
30#14582969—22no—
31C8×D172—21no—
32#14582964—21no—
33#14582923C2211yescharacteristic
34C2×#13063241C2211yescharacteristic
35C2×#13063241C2211yescharacteristic
36C2×#13063241C2212yes—
37#14582862C2212yes—
38#14582862C2212yes—
39C4×#10621877C2211yescharacteristic
40#14582668C2212yes—
41#14582668C2212yes—
42#14582667C2212yes—
43#14582667C2211yescharacteristic
44#14582667C2211yescharacteristic
45C2×C4⋊Q8×C43C2211yescharacteristic
46C43×#423C2211yescharacteristic
47C4×M16×C43C2211yescharacteristic
48C43×#343C2212yes—
49C43×#343C2212yes—
50C2×#10621902—24no—
51C2×#10621902—22no—
52C2×#10621902—22no—
53C2×#10621902—48no—
54C22×D172C2×C411yescharacteristic
55C22×D172C2×C414yes—
56C22×D172C2311yescharacteristic
57C22×C4×D43C2×C414yes—
58C22×C4×D43C2×C412yes—
59C22×C4×D43C2312yes—
60C2×#10621877C2311yescharacteristic
61C2×#10621877C2312yes—
62C2×#10621877C2311yescharacteristic
63C2×#10621872—22no—
64C2×#10621872—22no—
65M16×D43—24no—
66M16×D43—24no—
67M16×D43—42no—
68M16×D43—42no—
69C2×C8×D43—22no—
70C2×C8×D43—22no—
71#13063310—24no—
72#13063310—24no—
73#13063310—24no—
74#13063310—24no—
75#13063301—24no—
76#13063301—22no—
77#13063301—22no—
78#13063300—22no—
79#13063300—22no—
80#13063300—22no—
81#13063300—22no—
82C4×D172—21no—
83C4×D172—21no—
84C4×D172—21no—
85C4×D172—21no—
86C4×D172D411yescharacteristic
87C4×D172D411yescharacteristic
88C4×D172D412yes—
89C4×D172D412yes—
90C4×D172D411yescharacteristic
91C4×D172D411yescharacteristic
92C2×C4⋊Dic43C2×C411yescharacteristic
93C2×C4⋊Dic43C2311yescharacteristic
94C2×Dic43⋊C4C2×C414yes—
95C4⋊C4×D43—22no—
96C4⋊C4×D43—22no—
97C4⋊C4×D43—22no—
98C4⋊C4×D43—22no—
99C2×(C2×D43)⋊C4C2×C412yes—
100C2×(C2×D43)⋊C4C2×C414yes—
101C2×(C2×D43)⋊C4C2×C412yes—
102C4⋊(C43⋊C8)—21no—
103C4⋊(C43⋊C8)—21no—
104#13063241—22no—
105#13063241—22no—
106#13063241—42no—
107(C2×D43)⋊C8—22no—
108(C2×D43)⋊C8—22no—
109(C2×D43)⋊C8—22no—
110(C2×D43)⋊C8—22no—
111#13063236—21no—
112C4×C43⋊C8—21no—
113C22×Q8×C43C2×C412yes—
114C2×M16×C43C2311yescharacteristic
115C2×M16×C43C2312yes—
116C2×M16×C43C2311yescharacteristic
117C4⋊Q8×C43—21no—
118C4⋊Q8×C43—21no—
119C4⋊Q8×C43—21no—
120C4⋊Q8×C43—21no—
121C2×C4⋊C4×C43C2×C412yes—
122C2×C4⋊C4×C43C2312yes—
123C2×C42×C43C2311yescharacteristic
124C4⋊C8×C43—21no—
125C4⋊C8×C43—21no—
126#97×C43—21no—
127C4×C8×C43—21no—
128C23×D43C22×C412yes—
129C23×D43C22×C412yes—
130C2×Q8×C43—22no—
131C2×Q8×C43—22no—
132C2×Q8×C43—44no—
133C22×C4×C43C22×C411yescharacteristic
134C22×C4×C43C22×C414yes—
135C22×C4×C43C22×C411yescharacteristic
136C22×C4×C43C2411yescharacteristic
137C22×Dic43C22×C412yes—
138C22×Dic43C22×C412yes—
139#10621902—416no—
140#10621902—48no—
141#10621902—48no—
142C2×D172—24no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—24no—
150C2×D172—24no—
151C2×D172—48no—
152C2×D172C22⋊C411yescharacteristic
153C2×D172C22⋊C412yes—
154C2×D172C22⋊C411yescharacteristic
155C2×C4×D43—22no—
156C2×C4×D43—22no—
157C2×C4×D43—22no—
158C2×C4×D43—22no—
159C2×C4×D43—22no—
160C2×C4×D43—22no—
161C2×C4×D43—24no—
162C2×C4×D43—24no—
163C2×C4×D43—48no—
164C2×C4×D43—44no—
165C2×C4×D43C2×D412yes—
166C2×C4×D43C2×D412yes—
167C2×C4×D43C2×D412yes—
168C2×C4×D43C2×D412yes—
169M16×C43—21no—
170M16×C43—21no—
171M16×C43—22no—
172M16×C43—22no—
173M16×C43—21no—
174M16×C43#4112yescomplemented
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—22no—
178C4⋊C4×C43—22no—
179C4⋊C4×C43—22no—
180C4⋊C4×C43—22no—
181C4⋊C4×C43—22no—
182C42×C43C2×D411yescharacteristic
183C42×C43C2×D411yescharacteristic
184C42×C43C2×D411yescharacteristic
185C42×C43C2×D411yescharacteristic
186(C2×D43)⋊C4—22no—
187(C2×D43)⋊C4—22no—
188(C2×D43)⋊C4—24no—
189(C2×D43)⋊C4—24no—
190(C2×D43)⋊C4C22⋊C412yes—
191(C2×D43)⋊C4C22⋊C412yes—
192(C2×D43)⋊C4C22⋊C412yes—
193(C2×D43)⋊C4C22⋊C412yes—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43C22⋊C412yes—
197C4⋊Dic43C22⋊C411yescharacteristic
198C4⋊Dic43C22⋊C411yescharacteristic
199Dic43⋊C4—24no—
200Dic43⋊C4—24no—
201#10621877—21no—
202#10621877—21no—
203#10621877—22no—
204#10621877—22no—
205#10621877—21no—
206#10621877#4112yescomplemented
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—22no—
210#10621872—44no—
211#10621872—44no—
212C8×D43—44no—
213C8×D43—44no—
214C23×C43C23×C411yescharacteristic
215C22×D43—22no—
216C22×D43—22no—
217C22×D43—24no—
218C22×D43—22no—
219C22×D43—22no—
220C22×D43—22no—
221C22×D43—42no—
222C22×D43—42no—
223C22×D43—42no—
224C22×D43C2×C22⋊C412yes—
225C22×D43C2×C22⋊C412yes—
226Q8×C43—44no—
227Q8×C43—44no—
228C2×C4×C43—22no—
229C2×C4×C43—24no—
230C2×C4×C43—41no—
231C2×C4×C43—44no—
232C2×C4×C43C2×C22⋊C411yescharacteristic
233C2×C4×C43C2×C22⋊C411yescharacteristic
234C2×C4×C43C2×C22⋊C411yescharacteristic
235C2×C4×C43C2×C22⋊C411yescharacteristic
236C2×C4×C43C22×D411yescharacteristic
237C2×C4×C43C22×D411yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×C4×C43C2×#4111yescharacteristic
240C2×C4×C43H(5,𝔽2)11yescharacteristic
241C2×C4×C432−1+411yescharacteristic
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—22no—
245C2×Dic43—42no—
246C2×Dic43C2×C22⋊C412yes—
247C2×Dic43C2×C22⋊C412yes—
248D172—24no—
249D172—22no—
250D172—48no—
251D172—48no—
252D172—41no—
253D172—41no—
254D172—41no—
255D172—41no—
256D172#10014yescomplemented
257C4×D43—22no—
258C4×D43—22no—
259C4×D43—22no—
260C4×D43—22no—
261C4×D43—22no—
262C4×D43—22no—
263C4×D43—22no—
264C4×D43—22no—
265C4×D43—44no—
266C4×D43—48no—
267C4×D43—48no—
268C4×D43—44no—
269C8×C43—22no—
270C8×C43—42no—
271C8×C43—41no—
272C43⋊C8—22no—
273C43⋊C8—42no—
274C43⋊C8—41no—
275#22479—431nomaximal, Sylow
276C22×C43—21no—
277C22×C43—21no—
278C22×C43C22×C22⋊C411yescharacteristic
279C22×C43C4×#4111yescharacteristic
280C22×C43#52011yescharacteristic
281C2×D43—22no—
282C2×D43—22no—
283C2×D43—22no—
284C2×D43—22no—
285C2×D43—22no—
286C2×D43—44no—
287C2×D43—44no—
288C2×D43—42no—
289C2×D43—42no—
290C2×D43—42no—
291C2×D43—82no—
292C2×D43—82no—
293C2×D43C2×#10014yes—
294C4×C43—21no—
295C4×C43—21no—
296C4×C43—21no—
297C4×C43—44no—
298C4×C43C2×#10012yes—
299C4×C43#54611yescharacteristic
300C4×C43#54611yescharacteristic
301C4×C43#54711yescharacteristic
302C4×C43#54711yescharacteristic
303Dic43—22no—
304Dic43—22no—
305Dic43—42no—
306C2×#547—431no—
307D4×M16—431no—
308D4×M16—431no—
309#1997—431no—
310#1997D4311yescharacteristic, complemented
311#1900—432no—
312#1900—432no—
313#1821—432no—
314#1800—431no—
315C4×#100—431no—
316C2×#343—432no—
317C2×C43—21no—
318C2×C43—21no—
319C2×C43#233911yescharacteristic
320C2×C43C22×#10011yescharacteristic
321C2×C43#288911yescharacteristic
322D43—24no—
323D43—42no—
324D43—82no—
325C22×#41—432no—
326C22×M16—432no—
327C22×M16—432no—
328#547—861no—
329#547—861no—
330#547—862no—
331#547—862no—
332#547—861no—
333#547—861no—
334C2×C4⋊Q8C2×D4311yescharacteristic
335C2×#123—434no—
336C2×C22⋊Q8—432no—
337C2×#121—432no—
338C2×C4×D4—431no—
339C2×C4×D4—431no—
340C22×C4⋊C4—432no—
341#436—861no—
342#436—861no—
343#435—862no—
344#435—862no—
345C8×D4—861no—
346C8×D4—861no—
347#423—431no—
348#423C2×D4311yescharacteristic
349C2×#100—431no—
350C2×#100—431no—
351C2×#100—432no—
352#407—432no—
353#407—432no—
354C4×M16—431no—
355C4×M16C2×D4311yescharacteristic
356#343—432no—
357#343—431no—
358#343—431no—
359#343—432no—
360#343—432no—
361#343C2×D4312yes—
362#343C2×D4312yes—
363C43#2247911yescharacteristic, complemented, Sylow
364C2×#41—862no—
365C2×#41—864no—
366C2×#41—862no—
367C2×#41—1728no—
368C22×Q8C4×D4312yes—
369C22×D4—431no—
370C22×D4—434no—
371C22×D4—431no—
372C23×C4—432no—
373C23×C4—434no—
374C23×C4—432no—
375C2×M16—431no—
376C2×M16—432no—
377C2×M16—431no—
378C2×M16—862no—
379C2×M16—862no—
380C2×M16—864no—
381C2×M16—864no—
382C2×M16—1722no—
383C2×M16—1722no—
384C2×M16C22×D4312yes—
385C2×M16C22×D4311yescharacteristic
386C2×M16C22×D4311yescharacteristic
387C22×C8—862no—
388C22×C8—862no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393#123—864no—
394#123—864no—
395#123—864no—
396#123—864no—
397C22⋊Q8—864no—
398C22⋊Q8—862no—
399C22⋊Q8—862no—
400#121—862no—
401#121—862no—
402#121—862no—
403#121—862no—
404C4×D4—431no—
405C4×D4—431no—
406C4×D4—432no—
407C4×D4—432no—
408C4×D4—431no—
409C4×D4—431no—
410C4×D4—861no—
411C4×D4—861no—
412C4×D4—861no—
413C4×D4—861no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4—434no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—862no—
418C2×C4⋊C4—862no—
419C2×C4⋊C4—862no—
420C2×C4⋊C4—862no—
421C2×C4⋊C4C4×D4312yes—
422C2×C4⋊C4C22×D4312yes—
423C2×C22⋊C4—432no—
424C2×C22⋊C4—432no—
425C2×C22⋊C4—434no—
426C2×C42C22×D4311yescharacteristic
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8—861no—
430C4⋊C8—861no—
431#100—862no—
432#100—862no—
433#100—1722no—
434C22⋊C8—862no—
435C22⋊C8—862no—
436C22⋊C8—862no—
437C22⋊C8—862no—
438#97—21no—
439#97—861no—
440C4×C8—21no—
441C4×C8—861no—
442C24—432no—
443C24—432no—
444#41—1728no—
445#41—17216no—
446#41—1728no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×D4—431no—
451C2×D4—432no—
452C2×D4—431no—
453C2×D4—864no—
454C2×D4—864no—
455C2×D4—864no—
456C2×D4—861no—
457C2×D4—861no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—1728no—
463C22×C4—432no—
464C22×C4—432no—
465C22×C4—432no—
466C22×C4—432no—
467C22×C4—432no—
468C22×C4—432no—
469C22×C4—864no—
470C22×C4—864no—
471C22×C4—862no—
472C22×C4—862no—
473C22×C4—862no—
474C22×C4—862no—
475C22×C4—862no—
476C22×C4—862no—
477C22×C4—1724no—
478C22×C4—1728no—
479C22×C4C2×C4×D4311yescharacteristic
480C22×C4C2×C4×D4314yes—
481C22×C4C2×C4×D4311yescharacteristic
482C22×C4C23×D4311yescharacteristic
483M16—21no—
484M16—21no—
485M16—22no—
486M16—22no—
487M16—21no—
488M16—432no—
489M16—861no—
490M16—861no—
491M16—862no—
492M16—862no—
493M16—861no—
494M16—1724no—
495M16—1724no—
496M16#1062190212yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—861no—
501C2×C8—861no—
502C2×C8—862no—
503C2×C8—1724no—
504C2×C8—1724no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—432no—
510C4⋊C4—431no—
511C4⋊C4—431no—
512C4⋊C4—864no—
513C4⋊C4—864no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C22⋊C4—432no—
517C22⋊C4—432no—
518C22⋊C4—432no—
519C22⋊C4—432no—
520C22⋊C4—864no—
521C22⋊C4—864no—
522C22⋊C4—862no—
523C22⋊C4—862no—
524C42D4×D4311yescharacteristic
525C42D4×D4311yescharacteristic
526C42D4×D4311yescharacteristic
527C42D4×D4311yescharacteristic
528C23—432no—
529C23—432no—
530C23—864no—
531C23—862no—
532C23—862no—
533C23—862no—
534C23—862no—
535C23—862no—
536C23—1722no—
537C23—1722no—
538C23—1722no—
539C23C22×C4×D4311yescharacteristic
540Q8—44no—
541Q8—44no—
542D4—434no—
543D4—864no—
544D4—862no—
545D4—1728no—
546D4—1728no—
547D4—1721no—
548D4—1721no—
549D4—1721no—
550D4—1721no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—432no—
556C2×C4—432no—
557C2×C4—862no—
558C2×C4—862no—
559C2×C4—862no—
560C2×C4—862no—
561C2×C4—862no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—862no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—862no—
568C2×C4—1724no—
569C2×C4—1728no—
570C2×C4—1728no—
571C2×C4—1724no—
572C2×C4—1722no—
573C2×C4C22⋊C4×D4311yescharacteristic
574C2×C4C22⋊C4×D4311yescharacteristic
575C2×C4C22⋊C4×D4311yescharacteristic
576C2×C4C22⋊C4×D4311yescharacteristic
577C2×C4C2×D4×D4311yescharacteristic
578C2×C4C2×D4×D4311yescharacteristic
579C2×C4#41×D4311yescharacteristic
580C2×C4C2×#1062190211yescharacteristic
581C2×C4#1306337111yescharacteristic
582C2×C4#1306337311yescharacteristic
583C8—22no—
584C8—42no—
585C8—41no—
586C8—862no—
587C8—1721no—
588C8—1722no—
589C22—21no—
590C22—21no—
591C22—434no—
592C22—862no—
593C22—862no—
594C22—862no—
595C22—862no—
596C22—862no—
597C22—3442no—
598C22—3442no—
599C22—1724no—
600C22—1724no—
601C22—1722no—
602C22—1722no—
603C22—1722no—
604C22C2×C22⋊C4×D4311yescharacteristic
605C22C4×#1062190211yescharacteristic
606C22#1458338111yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—44no—
611C4—862no—
612C4—862no—
613C4—1722no—
614C4#100×D4312yes—
615C4#1458354011yescharacteristic
616C4#1458354211yescharacteristic
617C4#1458355111yescharacteristic
618C4#1458355211yescharacteristic
619C2—21no—
620C2—21no—
621C2—864no—
622C2—3442no—
623C2—1722no—
624C2#1913742411yescharacteristic
625C2C2×#100×D4311yescharacteristic
626C2#1914307511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC23×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1997C2
socleC22×C43C22×C22⋊C4
prime radicalC2×#14583552C2
cosocleC22×C4×C43C24
hypercenter#1997D43
hypocenterC43#22479
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. M16×D172C2semidirect
3. #19143330C2semidirect
4. #19135585C2semidirect
5. #19135057C2semidirect
6. #19135057C2semidirect
7. #19134674C2semidirect
8. #19134584C2semidirect
9. C4×#13063241C2semidirect
10. #343×D43C2semidirect
11. C43×#1997C2semidirect
174. M16×C43#41semidirect
206. #10621877#41semidirect
256. D172#100semidirect
310. #1997D43semidirect
363. C43#22479semidirect
496. M16#10621902semidirect

Maximal subgroups

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

subgroupindex
C2×#145835522
M16×D1722
#191433302
#191355852
#191350572
#191346742
#191345842
C4×#130632412
#343×D432
C43×#19972
#2247943