Finite Group Databasebeta

← #25169792

Subgroups — #25169792

Order 11008.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19139181C211yesmaximal, characteristic
2#19137651C211yesmaximal, characteristic, complemented
3#19136137C211yesmaximal, characteristic, complemented
4#19135913C211yesmaximal, characteristic
5#19135842C211yesmaximal, characteristic, complemented
6#19135680C211yesmaximal, characteristic
7#19135449C211yesmaximal, characteristic
8#19135230C211yesmaximal, characteristic, complemented
9#19135223C211yesmaximal, characteristic
10#19135174C211yesmaximal, characteristic, complemented
11#19134949C211yesmaximal, characteristic
12#19134949C211yesmaximal, characteristic
13#19134601C211yesmaximal, characteristic, complemented
14C4×#13063250C211yesmaximal, characteristic, complemented
15C43×#1916C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C22×#10621873C2211yescharacteristic
18C2×#13063325C2211yescharacteristic
19C2×#13063320—21no—
20C2×C4×Dic86C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×#13063257C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063250C2211yescharacteristic
25C2×#13063250C2211yescharacteristic
26C2×#13063250C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063247C2211yescharacteristic
32#14582826—21no—
33#14582821—21no—
34#14582805C2211yescharacteristic
35#14582805C2211yescharacteristic
36#14582801C2211yescharacteristic
37#14582801C2211yescharacteristic
38#14582801C2211yescharacteristic
39#14582797C2211yescharacteristic
40#14582795—21no—
41#14582791—21no—
42#14582782—21no—
43#14582782—21no—
44#14582769C2211yescharacteristic
45C4⋊C4×Dic43—21no—
46C4×C4⋊Dic43C2211yescharacteristic
47C4×C4⋊Dic43C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52C2×C4⋊C8×C43C2211yescharacteristic
53C2×C4×C8×C43C2211yescharacteristic
54C4×C4⋊C4×C43C2211yescharacteristic
55C43×#336C2211yescharacteristic
56C43×#336C2211yescharacteristic
57C43×#336C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C22×Dic86C2311yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×D172C2311yescharacteristic
62C2×#10621873—22no—
63C2×#10621873—22no—
64C2×#10621873—22no—
65C2×#10621873—41no—
66C2×#10621873—41no—
67C2×#10621873—41no—
68#13063325—22no—
69#13063325—22no—
70#13063320—44no—
71C4×Dic86—22no—
72C4×Dic86—22no—
73C4×Dic86—41no—
74C4×Dic86—41no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×C4×Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97#13063257—22no—
98#13063257—41no—
99#13063250—22no—
100#13063250—22no—
101#13063250—22no—
102#13063250—41no—
103#13063250—41no—
104#13063250—41no—
105#13063247—22no—
106#13063247—41no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C2×C4⋊C4×C43D411yescharacteristic
119C2×C4⋊C4×C43D411yescharacteristic
120C2×C42×C43C2311yescharacteristic
121C2×C42×C43C2311yescharacteristic
122C2×C42×C43C2311yescharacteristic
123C4⋊C8×C43—41no—
124C4×C8×C43—41no—
125#95×C43D411yescharacteristic
126#95×C43D411yescharacteristic
127C23×D43—21no—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2411yescharacteristic
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×Dic86—21no—
149C2×Dic86—21no—
150C2×Dic86—21no—
151C2×Dic86—21no—
152C2×Dic86—21no—
153C2×Dic86—21no—
154C2×Dic86—21no—
155C2×Dic86—21no—
156C2×Dic86—21no—
157C2×Dic86—21no—
158C2×Dic86#4112yes—
159C2×Dic86#4112yes—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43#4112yes—
172C4⋊C4×C43—41no—
173C4⋊C4×C43SD1614yescomplemented
174C42×C43—21no—
175C42×C43—21no—
176C42×C43—21no—
177C42×C43—21no—
178C42×C43—21no—
179C42×C43—21no—
180(C2×D43)⋊C4—42no—
181(C2×D43)⋊C4—42no—
182(C2×D43)⋊C4—42no—
183(C2×D43)⋊C4—42no—
184(C2×D43)⋊C4—81no—
185(C2×D43)⋊C4—81no—
186C4⋊Dic43—24no—
187C4⋊Dic43—24no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—42no—
198C4⋊Dic43—81no—
199C4⋊Dic43#4112yes—
200Dic43⋊C4—42no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203C4×Dic43—42no—
204C4×Dic43—42no—
205C4×Dic43—42no—
206C4×Dic43—42no—
207#10621873—44no—
208#10621873—44no—
209C23×C43C22×D411yescharacteristic
210C22×D43—41no—
211C22×D43—41no—
212C22×D43—41no—
213C22×D43—41no—
214C22×D43—41no—
215C22×D43—41no—
216C22×D43—41no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
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×C43C2×SD1612yes—
233C2×C4×C43C2×SD1612yes—
234C2×C4×C43C2×SD1612yes—
235C2×C4×C43C2×#4111yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×C4×C43H(5,𝔽2)11yescharacteristic
240C2×C4×C43H(5,𝔽2)11yescharacteristic
241C2×Dic43—22no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
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—
262D172—41no—
263D172—41no—
264D172—41no—
265D172—41no—
266Dic86—22no—
267Dic86—22no—
268Dic86—22no—
269Dic86—22no—
270Dic86—41no—
271Dic86—41no—
272Dic86—41no—
273Dic86—41no—
274C8×C43—22no—
275C8×C43—41no—
276C8×C43—41no—
277C8×C43—41no—
278#27066—431nomaximal, Sylow
279C22×C43#52511yescharacteristic
280C22×C43#54611yescharacteristic
281C22×C43#55011yescharacteristic
282C22×C43C22×SD1611yescharacteristic
283C22×C43#57511yescharacteristic
284C22×C43#57711yescharacteristic
285C22×C43#57811yescharacteristic
286C2×D43—81no—
287C2×D43—81no—
288C2×D43—81no—
289C2×D43—81no—
290C2×D43—81no—
291C2×D43—81no—
292C2×D43—81no—
293C4×C43—22no—
294C4×C43—22no—
295C4×C43—22no—
296C4×C43—41no—
297C4×C43—41no—
298C4×C43—41no—
299C4×C43#52911yescharacteristic
300C4×C43#53811yescharacteristic
301C4×C43#55111yescharacteristic
302C4×C43#55311yescharacteristic
303Dic43—42no—
304Dic43—42no—
305Dic43—42no—
306Dic43—81no—
307Dic43—81no—
308Dic43—81no—
309Dic43—81no—
310#2512—431no—
311#2365—431no—
312#2096—431no—
313#2055—431no—
314#2043—431no—
315#2013—431no—
316#1968—431no—
317#1927—431no—
318#1926—431no—
319#1919—431no—
320#1916D4311yescharacteristic, complemented
321#1880—431no—
322#1880—431no—
323#1803—431no—
324C4×#103—431no—
325C2×C43#246711yescharacteristic
326C2×C43#310311yescharacteristic
327C2×C43#312111yescharacteristic
328C2×C43#329811yescharacteristic
329C2×C43#330711yescharacteristic
330C2×C43#339211yescharacteristic
331C2×C43#340211yescharacteristic
332D43—161no—
333C22×SD16—431no—
334C22×SD16—431no—
335C2×C4⋊Q8—431no—
336C2×#126—861no—
337C2×C4×Q8—431no—
338C2×C4×Q8—431no—
339C2×#106—431no—
340C2×#106—431no—
341C2×C4⋊C8C2×D4311yescharacteristic
342C2×#103—431no—
343C2×#103—431no—
344C2×#103—431no—
345C2×#103—431no—
346C2×#103—431no—
347C2×#103—431no—
348C2×#102—431no—
349C2×#102—431no—
350C2×C4×C8C2×D4311yescharacteristic
351#397—861no—
352#396—861no—
353#391—431no—
354#391—431no—
355#390—431no—
356#390—431no—
357#390—431no—
358#389—431no—
359#389—861no—
360#388—861no—
361#387—861no—
362#387—861no—
363#384—431no—
364C4×C4⋊C4—431no—
365C4×C4⋊C4—431no—
366C4×C4⋊C4—861no—
367C4×C4⋊C4C2×D4311yescharacteristic
368#340—431no—
369#340—431no—
370#340—431no—
371#340—431no—
372#336C2×D4311yescharacteristic
373#336C2×D4311yescharacteristic
374#336C2×D4311yescharacteristic
375#336C2×D4311yescharacteristic
376C43#2706611yescharacteristic, complemented, Sylow
377C22×Q8—431no—
378C22×Q8—431no—
379C22×D4—431no—
380C2×SD16—862no—
381C2×SD16—862no—
382C2×SD16—862no—
383C2×SD16—1721no—
384C2×SD16—1721no—
385C2×SD16—1721no—
386C22×C8C22×D4311yescharacteristic
387C22×C8C22×D4311yescharacteristic
388C22×C8C22×D4311yescharacteristic
389C22×C8C22×D4311yescharacteristic
390C4⋊Q8—862no—
391C4⋊Q8—862no—
392#126—1724no—
393C4×Q8—862no—
394C4×Q8—862no—
395C4×Q8—1721no—
396C4×Q8—1721no—
397C2×C4⋊C4—431no—
398C2×C4⋊C4—431no—
399C2×C4⋊C4—431no—
400C2×C4⋊C4—431no—
401C2×C4⋊C4—431no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4—861no—
404C2×C4⋊C4—861no—
405C2×C4⋊C4—861no—
406C2×C4⋊C4—861no—
407C2×C4⋊C4—861no—
408C2×C4⋊C4—861no—
409C2×C4⋊C4D17211yescharacteristic
410C2×C4⋊C4D17211yescharacteristic
411C2×C22⋊C4—861no—
412C2×C22⋊C4—861no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—861no—
415C2×C22⋊C4—861no—
416C2×C22⋊C4—861no—
417C2×C42—861no—
418C2×C42—861no—
419C2×C42—861no—
420C2×C42—861no—
421C2×C42C22×D4311yescharacteristic
422C2×C42C22×D4311yescharacteristic
423C2×C42C22×D4311yescharacteristic
424#106—862no—
425#106—1721no—
426C4⋊C8—41no—
427#103—862no—
428#103—862no—
429#103—862no—
430#103—1721no—
431#103—1721no—
432#103—1721no—
433#102—862no—
434#102—1721no—
435C4×C8—41no—
436#95—861no—
437#95—861no—
438#95—861no—
439#95—861no—
440#95—861no—
441#95—861no—
442#95—861no—
443#95D17211yescharacteristic
444#95D17211yescharacteristic
445C24—861no—
446C2×Q8—432no—
447C2×Q8—432no—
448C2×Q8—861no—
449C2×Q8—861no—
450C2×Q8—861no—
451C2×Q8—861no—
452C2×Q8—861no—
453C2×Q8—861no—
454C2×Q8—861no—
455C2×Q8—861no—
456C2×Q8—861no—
457C2×Q8—861no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—861no—
463C2×D4—861no—
464C22×C4—861no—
465C22×C4—861no—
466C22×C4—861no—
467C22×C4—861no—
468C22×C4—861no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4C2×D17211yescharacteristic
472C22×C4C2×D17211yescharacteristic
473C22×C4C2×D17211yescharacteristic
474C22×C4C2×D17211yescharacteristic
475C22×C4C2×D17211yescharacteristic
476C22×C4C2×D17211yescharacteristic
477C22×C4C23×D4311yescharacteristic
478SD16—1724no—
479SD16—1724no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8#1062190212yes—
492C4⋊C4—41no—
493C4⋊C4—432no—
494C4⋊C4—864no—
495C4⋊C4—864no—
496C4⋊C4—861no—
497C4⋊C4—861no—
498C4⋊C4—861no—
499C4⋊C4—861no—
500C4⋊C4—861no—
501C4⋊C4—861no—
502C4⋊C4—861no—
503C4⋊C4—861no—
504C4⋊C4—861no—
505C4⋊C4—3441no—
506C4⋊C4—1722no—
507C4⋊C4—1722no—
508C4⋊C4—1722no—
509C4⋊C4—1722no—
510C4⋊C4#1062187314yescomplemented
511C22⋊C4—3441no—
512C22⋊C4—3441no—
513C22⋊C4—1722no—
514C22⋊C4—1722no—
515C22⋊C4—1722no—
516C22⋊C4—1722no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—1722no—
524C42—1722no—
525C42—1722no—
526C42—1722no—
527C23—1721no—
528C23—1721no—
529C23—1721no—
530C23—1721no—
531C23—1721no—
532C23—1721no—
533C23—1721no—
534C23C22×D17211yescharacteristic
535Q8—862no—
536Q8—862no—
537Q8—862no—
538Q8—862no—
539Q8—1721no—
540Q8—1721no—
541Q8—1721no—
542Q8—1721no—
543D4—1721no—
544D4—1721no—
545D4—1721no—
546D4—1721no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—862no—
565C2×C4—1721no—
566C2×C4—1721no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1721no—
572C2×C4—1721no—
573C2×C4—1721no—
574C2×C4—1721no—
575C2×C4—1721no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4C2×#1062187312yes—
584C2×C4C2×#1062187312yes—
585C2×C4C2×#1062187312yes—
586C2×C4#41×D4311yescharacteristic
587C2×C4#41×D4311yescharacteristic
588C2×C4#41×D4311yescharacteristic
589C2×C4C2×#1062190211yescharacteristic
590C2×C4#1306337011yescharacteristic
591C2×C4#1306337111yescharacteristic
592C8—22no—
593C8—41no—
594C8—41no—
595C8—41no—
596C22—3441no—
597C22—3441no—
598C22—3441no—
599C22—3441no—
600C22—3441no—
601C22—3441no—
602C22—3441no—
603C22#1458341111yescharacteristic
604C22#1458354411yescharacteristic
605C22#1458357111yescharacteristic
606C22C22×#1062187311yescharacteristic
607C22#1458369911yescharacteristic
608C22#1458371111yescharacteristic
609C22#1458371611yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—3441no—
617C4—3441no—
618C4—3441no—
619C4—3441no—
620C4—1722no—
621C4—1722no—
622C4—1722no—
623C4#1458343411yescharacteristic
624C4#1458348511yescharacteristic
625C4#1458357611yescharacteristic
626C4#1458360011yescharacteristic
627C2—6881no—
628C2#1913864411yescharacteristic
629C2#1914462711yescharacteristic
630C2#1914482611yescharacteristic
631C2#1914646111yescharacteristic
632C2#1914657311yescharacteristic
633C2#1914757711yescharacteristic
634C2#1914767611yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1916C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#1916D43
hypocenterC43#27066
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
2. #19137651C2semidirect
3. #19136137C2semidirect
5. #19135842C2semidirect
8. #19135230C2semidirect
10. #19135174C2semidirect
13. #19134601C2semidirect
14. C4×#13063250C2semidirect
15. C43×#1916C2semidirect
173. C4⋊C4×C43SD16semidirect
320. #1916D43semidirect
376. C43#27066semidirect
510. C4⋊C4#10621873semidirect

Maximal subgroups

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

subgroupindex
#191391812
#191376512
#191361372
#191359132
#191358422
#191356802
#191354492
#191352302
#191352232
#191351742
#191349492
#191346012
C4×#130632502
C43×#19162
#2706643