Finite Group Databasebeta

← #25156684

Subgroups — #25156684

Order 11008.

832 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583122C211yesmaximal, characteristic
2C2×C4×#10621873C211yesmaximal, characteristic
3#19138382C211yesmaximal, characteristic
4#19137515C211yesmaximal, characteristic, complemented
5#19136142C211yesmaximal, characteristic, complemented
6#19136093C211yesmaximal, characteristic, complemented
7#19136010C211yesmaximal, characteristic
8#19135837C211yesmaximal, characteristic
9#19135271C211yesmaximal, characteristic, complemented
10#19134949C211yesmaximal, characteristic
11#19134935C211yesmaximal, characteristic, complemented
12#19134935C211yesmaximal, characteristic, complemented
13#19134852C211yesmaximal, characteristic, complemented
14#19134845C211yesmaximal, characteristic
15C43×#1916C211yesmaximal, characteristic, complemented
16C22×Dic172C2211yescharacteristic
17C22×#10621873C2211yescharacteristic
18C2×#13063313C2211yescharacteristic
19C2×C4×Dic86C2211yescharacteristic
20C2×C4×Dic86C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×C4×D172C2211yescharacteristic
23#14583122—24no—
24C4×#10621873—24no—
25C2×#13063259C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063250C2211yescharacteristic
31C2×#13063250C2211yescharacteristic
32C2×#13063247C2211yescharacteristic
33C2×#13063247C2211yescharacteristic
34C2×#13063247C2211yescharacteristic
35#14582828—21no—
36#14582825—21no—
37#14582805C2211yescharacteristic
38#14582805C2211yescharacteristic
39#14582801C2211yescharacteristic
40#14582797C2211yescharacteristic
41#14582797C2211yescharacteristic
42#14582797C2211yescharacteristic
43#14582794—21no—
44#14582794—21no—
45#14582770—21no—
46#14582765C2211yescharacteristic
47#14582763—21no—
48#14582761—21no—
49#14582750—21no—
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53#14582662C2211yescharacteristic
54C2×C4⋊C8×C43C2211yescharacteristic
55C2×C4×C8×C43C2211yescharacteristic
56C4×C4⋊C4×C43C2211yescharacteristic
57C43×#336C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C2×Dic172—22no—
66C2×Dic172—22no—
67C2×Dic172—22no—
68C2×#10621873—22no—
69C2×#10621873—22no—
70C2×#10621873D414yes—
71#13063313—22no—
72#13063313—41no—
73C4×Dic86—22no—
74C4×Dic86—22no—
75C4×Dic86—22no—
76C4×Dic86—22no—
77C4×Dic86—41no—
78C4×Dic86—41no—
79C4×D172—22no—
80C4×D172—41no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×C4×Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43—21no—
100#13063259—41no—
101#13063257—22no—
102#13063250—22no—
103#13063250—22no—
104#13063250—22no—
105#13063250—22no—
106#13063250—41no—
107#13063247—22no—
108#13063247—22no—
109#13063247—22no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C22×C8×C43C2311yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C22×C8×C43C2311yescharacteristic
122C22×C8×C43C2311yescharacteristic
123C2×C4⋊C4×C43D411yescharacteristic
124C2×C4⋊C4×C43D411yescharacteristic
125C2×C42×C43C2311yescharacteristic
126C2×C42×C43C2311yescharacteristic
127C2×C42×C43C2311yescharacteristic
128C4⋊C8×C43—22no—
129C4×C8×C43—22no—
130#95×C43D411yescharacteristic
131#95×C43D411yescharacteristic
132C23×D43—21no—
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2411yescharacteristic
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172—21no—
151C2×D172—21no—
152C2×D172C2×D412yes—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×C4×D43—81no—
156C2×Dic86—21no—
157C2×Dic86—21no—
158C2×Dic86—21no—
159C2×Dic86—21no—
160C2×Dic86—21no—
161C2×Dic86—21no—
162C2×Dic86—21no—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86C2×D412yes—
166C2×Dic86#4112yes—
167C2×Dic86#4112yes—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43C2×D412yes—
180C4⋊C4×C43—22no—
181C4⋊C4×C43—41no—
182C42×C43—21no—
183C42×C43—21no—
184C42×C43—21no—
185C42×C43—21no—
186C42×C43—21no—
187C42×C43—21no—
188(C2×D43)⋊C4—42no—
189(C2×D43)⋊C4—42no—
190(C2×D43)⋊C4—42no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—81no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43#4112yes—
203Dic43⋊C4—24no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206Dic43⋊C4—42no—
207Dic43⋊C4—42no—
208Dic43⋊C4—81no—
209C4×Dic43—24no—
210C4×Dic43—42no—
211C4×Dic43—42no—
212Dic172—44no—
213#10621873—28no—
214C23×C43C22×D411yescharacteristic
215C22×D43—22no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43#13512yes—
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43C2×#4111yescharacteristic
242C2×C4×C43C2×#4111yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C43H(5,𝔽2)11yescharacteristic
245C2×C4×C432−1+411yescharacteristic
246C2×Dic43—22no—
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—22no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267D172—22no—
268D172—22no—
269D172—41no—
270D172—41no—
271C4×D43—84no—
272Dic86—22no—
273Dic86—22no—
274Dic86—22no—
275Dic86—41no—
276Dic86—41no—
277Dic86—41no—
278Dic86—41no—
279Dic86#12314yes—
280C8×C43—22no—
281C8×C43—41no—
282C8×C43—41no—
283C8×C43—41no—
284#26165—431nomaximal, Sylow
285C22×C43#52511yescharacteristic
286C22×C43D4211yescharacteristic
287C22×C43#54811yescharacteristic
288C22×C43C2×#13511yescharacteristic
289C22×C43#57511yescharacteristic
290C22×C43#57711yescharacteristic
291C22×C43#57811yescharacteristic
292C2×D43—42no—
293C2×D43—42no—
294C2×D43—42no—
295C2×D43—81no—
296C2×D43—81no—
297C2×D43—81no—
298C2×D43—81no—
299C4×C43—22no—
300C4×C43—41no—
301C4×C43—41no—
302C4×C43—41no—
303C4×C43—41no—
304C4×C43—41no—
305C4×C43C2×#12311yescharacteristic
306C4×C43#53911yescharacteristic
307C4×C43#54711yescharacteristic
308C4×C43#55311yescharacteristic
309Dic43—24no—
310Dic43—42no—
311Dic43—81no—
312Dic43—81no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316C2×#464—431no—
317C2×C4×SD16—431no—
318#2447—431no—
319#2351—431no—
320#2097—431no—
321#2088—431no—
322#2073—431no—
323#2042—431no—
324#1933—431no—
325#1916D4311yescharacteristic, complemented
326#1880—431no—
327#1878—431no—
328#1878—431no—
329#1863—431no—
330#1862—431no—
331C2×C43#244311yescharacteristic
332C2×C43#310211yescharacteristic
333C2×C43#312111yescharacteristic
334C2×C43#326911yescharacteristic
335C2×C43#328211yescharacteristic
336C2×C43#336411yescharacteristic
337C2×C43#337111yescharacteristic
338D43—82no—
339C22×Q16—431no—
340C22×SD16—431no—
341C2×#124—431no—
342C2×C4×Q8—431no—
343C2×C4×Q8—431no—
344C2×C4×Q8—431no—
345C2×C4×D4—431no—
346#464—864no—
347C4×SD16—864no—
348C2×#107—431no—
349C2×#106—431no—
350C2×C4⋊C8C2×D4311yescharacteristic
351C2×#103—431no—
352C2×#103—431no—
353C2×#103—431no—
354C2×#103—431no—
355C2×#103—431no—
356C2×#102—431no—
357C2×#102—431no—
358C2×#102—431no—
359C2×C4×C8C2×D4311yescharacteristic
360#397—861no—
361#397—861no—
362#391—431no—
363#391—431no—
364#390—431no—
365#389—431no—
366#389—431no—
367#389—431no—
368#389—861no—
369#389—861no—
370#385—861no—
371#382—431no—
372#382—861no—
373#381—861no—
374C4×C4⋊C4—861no—
375C4×C4⋊C4C2×D4311yescharacteristic
376#340—431no—
377#340—431no—
378#340—431no—
379#340—431no—
380#336C2×D4311yescharacteristic
381#336C2×D4311yescharacteristic
382#336C2×D4311yescharacteristic
383#336C2×D4311yescharacteristic
384C43#2616511yescharacteristic, complemented, Sylow
385C22×Q8—431no—
386C22×Q8—431no—
387C22×D4—431no—
388C23×C4—861no—
389C2×Q16—862no—
390C2×Q16—862no—
391C2×Q16—862no—
392C2×SD16—434no—
393C2×SD16—862no—
394C2×SD16—862no—
395C22×C8C22×D4311yescharacteristic
396C22×C8C22×D4311yescharacteristic
397C22×C8C22×D4311yescharacteristic
398C22×C8C22×D4311yescharacteristic
399#124—862no—
400#124—1721no—
401C4×Q8—862no—
402C4×Q8—862no—
403C4×Q8—862no—
404C4×Q8—862no—
405C4×Q8—1721no—
406C4×Q8—1721no—
407C4×D4—862no—
408C4×D4—1721no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4D17211yescharacteristic
421C2×C4⋊C4D17211yescharacteristic
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4—861no—
425C2×C22⋊C4—861no—
426C2×C22⋊C4—861no—
427C2×C42—861no—
428C2×C42—861no—
429C2×C42—861no—
430C2×C42C22×D4311yescharacteristic
431C2×C42C22×D4311yescharacteristic
432C2×C42C22×D4311yescharacteristic
433#107—1721no—
434#106—862no—
435C4⋊C8—22no—
436#103—862no—
437#103—862no—
438#103—862no—
439#103—862no—
440#103—1721no—
441#102—862no—
442#102—862no—
443#102—862no—
444C4×C8—22no—
445#95—861no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95—861no—
454#95D17211yescharacteristic
455#95D17211yescharacteristic
456C24—861no—
457C2×Q8—432no—
458C2×Q8—432no—
459C2×Q8—432no—
460C2×Q8—861no—
461C2×Q8—861no—
462C2×Q8—861no—
463C2×Q8—861no—
464C2×Q8—861no—
465C2×Q8—861no—
466C2×Q8—861no—
467C2×Q8—861no—
468C2×Q8—861no—
469C2×D4—432no—
470C2×D4—861no—
471C2×D4—861no—
472C2×D4—861no—
473C2×D4—861no—
474C2×D4—861no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—861no—
479C22×C4—861no—
480C22×C4—861no—
481C22×C4—861no—
482C22×C4—3441no—
483C22×C4—1722no—
484C22×C4—1722no—
485C22×C4C2×D17211yescharacteristic
486C22×C4C2×D17211yescharacteristic
487C22×C4C2×D17211yescharacteristic
488C22×C4C2×D17211yescharacteristic
489C22×C4C2×D17211yescharacteristic
490C22×C4C2×D17211yescharacteristic
491C22×C4C23×D4311yescharacteristic
492Q16—1724no—
493SD16—868no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8D4×D4312yes—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—432no—
509C4⋊C4—864no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—3441no—
520C4⋊C4—1722no—
521C4⋊C4—1722no—
522C4⋊C4—1722no—
523C4⋊C4—1722no—
524C22⋊C4—3441no—
525C22⋊C4—1722no—
526C22⋊C4—1722no—
527C22⋊C4—1722no—
528C22⋊C4—1722no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—864no—
536C42—1722no—
537C42—1722no—
538C23—862no—
539C23—1721no—
540C23—1721no—
541C23—1721no—
542C23—1721no—
543C23—1721no—
544C23—1721no—
545C23C22×D17211yescharacteristic
546Q8—434no—
547Q8—862no—
548Q8—862no—
549Q8—862no—
550Q8—1721no—
551Q8—1721no—
552Q8—1721no—
553Q8—1721no—
554D4—862no—
555D4—862no—
556D4—1721no—
557D4—1721no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—862no—
576C2×C4—862no—
577C2×C4—862no—
578C2×C4—862no—
579C2×C4—3444no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4#1306334712yes—
598C2×C4C2×D4×D4311yescharacteristic
599C2×C4#41×D4311yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4#41×D4311yescharacteristic
602C2×C4#1306337111yescharacteristic
603C2×C4#1306337311yescharacteristic
604C8—22no—
605C8—41no—
606C8—41no—
607C8—41no—
608C22—3441no—
609C22—3441no—
610C22—3441no—
611C22—3441no—
612C22—1722no—
613C22—1722no—
614C22—1722no—
615C22#1458341111yescharacteristic
616C22D4×D17211yescharacteristic
617C22#1458356111yescharacteristic
618C22C2×#1306334711yescharacteristic
619C22#1458369911yescharacteristic
620C22#1458371111yescharacteristic
621C22#1458371611yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—864no—
629C4—3441no—
630C4—3441no—
631C4—3441no—
632C4—3441no—
633C4—3441no—
634C4—1722no—
635C4#123×D4311yescharacteristic
636C4#1458349711yescharacteristic
637C4#1458355111yescharacteristic
638C4#1458360011yescharacteristic
639C2—3442no—
640C2#1913833511yescharacteristic
641C2#1914461511yescharacteristic
642C2#1914482611yescharacteristic
643C2#1914615511yescharacteristic
644C2#1914630711yescharacteristic
645C2#1914728211yescharacteristic
646C2#1914736511yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1916C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#1916D43
hypocenterC43#26165
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
4. #19137515C2semidirect
5. #19136142C2semidirect
6. #19136093C2semidirect
9. #19135271C2semidirect
11. #19134935C2semidirect
12. #19134935C2semidirect
13. #19134852C2semidirect
15. C43×#1916C2semidirect
325. #1916D43semidirect
384. C43#26165semidirect

Maximal subgroups

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

subgroupindex
C2×#145831222
C2×C4×#106218732
#191383822
#191375152
#191361422
#191360932
#191360102
#191358372
#191352712
#191349492
#191349352
#191348522
#191348452
C43×#19162
#2616543