Finite Group Databasebeta

← #25156612

Subgroups — #25156612

Order 11008.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583118C211yesmaximal, characteristic
2C2×#14582990C211yesmaximal, characteristic
3#19138378C211yesmaximal, characteristic
4#19137509C211yesmaximal, characteristic, complemented
5#19136136C211yesmaximal, characteristic, complemented
6#19136096C211yesmaximal, characteristic, complemented
7#19136001C211yesmaximal, characteristic
8#19135273C211yesmaximal, characteristic, complemented
9#19135155C211yesmaximal, characteristic, complemented
10#19134947C211yesmaximal, characteristic
11#19134933C211yesmaximal, characteristic, complemented
12#19134933C211yesmaximal, characteristic, complemented
13#19134866C211yesmaximal, characteristic, complemented
14#19134858C211yesmaximal, characteristic
15C43×#2042C211yesmaximal, characteristic
16C22×#10621885C2211yescharacteristic
17C22×#10621883C2211yescharacteristic
18C2×C4×Dic86C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20C2×#13063290C2211yescharacteristic
21C2×D4×Dic43C2211yescharacteristic
22#14583118—24no—
23#14582990—24no—
24C2×#13063257C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063246C2211yescharacteristic
31C2×C8×Dic43C2211yescharacteristic
32#14582827—21no—
33#14582827—21no—
34#14582806C2211yescharacteristic
35#14582806C2211yescharacteristic
36#14582802C2211yescharacteristic
37#14582798C2211yescharacteristic
38#14582797C2211yescharacteristic
39#14582796—21no—
40#14582794—21no—
41#14582771—21no—
42#14582768—21no—
43#14582768C2211yescharacteristic
44#14582760—21no—
45C4×Dic43⋊C4C2211yescharacteristic
46#14582750—21no—
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#124×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C23×Dic43—21no—
63C2×#10621885—22no—
64C2×#10621885—22no—
65C2×#10621885—22no—
66C2×#10621883—22no—
67C2×#10621883—22no—
68C2×#10621883D414yes—
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71C4×Dic86—22no—
72C4×Dic86—41no—
73#13063290—22no—
74#13063290—22no—
75#13063290—22no—
76#13063290—41no—
77D4×Dic43—22no—
78D4×Dic43—41no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4D411yescharacteristic
90C2×Dic43⋊C4D411yescharacteristic
91C2×C22⋊Dic43—21no—
92C2×C22⋊Dic43—21no—
93C2×C22⋊Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43C2311yescharacteristic
98C2×C4×Dic43C2311yescharacteristic
99#13063257—41no—
100#13063256—22no—
101C4⋊(C43⋊C8)—22no—
102#13063250—22no—
103#13063250—22no—
104#13063249—22no—
105#13063246—22no—
106C8×Dic43—22no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95D411yescharacteristic
114C43⋊#95D411yescharacteristic
115C22×Q8×C43C2311yescharacteristic
116C22×D4×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C22×C8×C43C2311yescharacteristic
119#124×C43—22no—
120#124×C43—41no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C22⋊C4×C43—21no—
125C2×C22⋊C4×C43—21no—
126C2×C42×C43C2311yescharacteristic
127#103×C43—22no—
128#103×C43—41no—
129#102×C43—22no—
130#102×C43—22no—
131#95×C43—21no—
132C24×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×D4×C43—21no—
140C2×D4×C43—21no—
141C2×D4×C43—21no—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43C2×D412yes—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2411yescharacteristic
151C22×Dic43—21no—
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—42no—
156C22×Dic43—42no—
157C22×Dic43—81no—
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
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—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—81no—
178C4⋊C4×C43#4112yes—
179C22⋊C4×C43—42no—
180C22⋊C4×C43—42no—
181C42×C43—21no—
182C42×C43—21no—
183C22⋊Dic43—42no—
184C22⋊Dic43—42no—
185C22⋊Dic43—81no—
186#10621885—44no—
187#10621883—28no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—81no—
195Dic43⋊C4—24no—
196Dic43⋊C4—22no—
197Dic43⋊C4—22no—
198Dic43⋊C4—42no—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201Dic43⋊C4—81no—
202Dic43⋊C4—81no—
203C4×Dic43—24no—
204C4×Dic43—21no—
205C4×Dic43—21no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—42no—
209C4×Dic43—42no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8C2×D412yes—
216C23×C43—22no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43C22×D411yescharacteristic
224Q8×C43—41no—
225Q8×C43—41no—
226Q8×C43—41no—
227Q8×C43—41no—
228D4×C43—22no—
229D4×C43—22no—
230D4×C43—41no—
231D4×C43—41no—
232C2×C4×C43—22no—
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—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43C22×D411yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43H(5,𝔽2)11yescharacteristic
252C2×C4×C432−1+411yescharacteristic
253C2×Dic43—21no—
254C2×Dic43—22no—
255C2×Dic43—21no—
256C2×Dic43—22no—
257C2×Dic43—22no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—84no—
276C2×Dic43#13512yes—
277C2×Dic43#13712yes—
278Dic86—22no—
279Dic86—22no—
280Dic86—22no—
281Dic86#12314yes—
282C8×C43—41no—
283C8×C43—41no—
284C43⋊C8—22no—
285C43⋊C8—41no—
286#26161—431nomaximal, Sylow
287C22×C43—42no—
288C22×C43—42no—
289C22×C43—42no—
290C22×C43—81no—
291C22×C43—81no—
292C22×C43—81no—
293C22×C43—81no—
294C22×C43#52511yescharacteristic
295C22×C43D4211yescharacteristic
296C22×C43#54811yescharacteristic
297C22×C43C2×#13511yescharacteristic
298C22×C43C2×#13711yescharacteristic
299C22×C43#57711yescharacteristic
300C22×C43#57711yescharacteristic
301C4×C43—42no—
302C4×C43—41no—
303C4×C43—41no—
304C4×C43—81no—
305C4×C43—81no—
306C4×C43C2×#12311yescharacteristic
307C4×C43#53911yescharacteristic
308C4×C43#54711yescharacteristic
309C4×C43#55311yescharacteristic
310Dic43—22no—
311Dic43—24no—
312Dic43—22no—
313Dic43—41no—
314Dic43—41no—
315Dic43—81no—
316Dic43—81no—
317Dic43—81no—
318C2×#464—431no—
319C2×C4×SD16—431no—
320#2447—431no—
321Q8×C4⋊C4—431no—
322#2096—431no—
323#2088—431no—
324#2071—431no—
325#2042D4311yescharacteristic
326#1933—431no—
327#1916—431no—
328#1880—431no—
329#1878—431no—
330#1878—431no—
331#1865—431no—
332#1864—431no—
333C2×C43—82no—
334C2×C43#244311yescharacteristic
335C2×C43#310811yescharacteristic
336C2×C43#312511yescharacteristic
337C2×C43#326611yescharacteristic
338C2×C43#328411yescharacteristic
339C2×C43#336511yescharacteristic
340C2×C43#336911yescharacteristic
341C22×Q16—431no—
342C22×SD16—431no—
343C2×#124C2×D4311yescharacteristic
344C2×C4×Q8—431no—
345C2×C4×Q8—431no—
346C2×C4×Q8—431no—
347C2×C4×D4—431no—
348#464—864no—
349C4×SD16—864no—
350C2×#106—431no—
351C2×#106—431no—
352C2×C4⋊C8—431no—
353C2×#103—431no—
354C2×#103—431no—
355C2×#103—431no—
356C2×#103C2×D4311yescharacteristic
357C2×#103C2×D4311yescharacteristic
358C2×#102—431no—
359C2×#102C2×D4311yescharacteristic
360C2×#102C2×D4311yescharacteristic
361C2×C4×C8—431no—
362#397—861no—
363#397—861no—
364#391—431no—
365#391—431no—
366#390—431no—
367#389—431no—
368#389—431no—
369#389—861no—
370#389—861no—
371#389C2×D4311yescharacteristic
372#385—861no—
373#384—431no—
374#384—861no—
375#381—861no—
376C4×C4⋊C4—431no—
377C4×C4⋊C4—861no—
378#340—431no—
379#340—431no—
380#340—431no—
381#340—431no—
382#336—431no—
383#336—431no—
384#336—431no—
385#336C2×D4311yescharacteristic
386C43#2616111yescharacteristic, complemented, Sylow
387C22×Q8—431no—
388C22×Q8C22×D4311yescharacteristic
389C22×D4C22×D4311yescharacteristic
390C23×C4—861no—
391C2×Q16—862no—
392C2×Q16—862no—
393C2×Q16—862no—
394C2×SD16—434no—
395C2×SD16—862no—
396C2×SD16—862no—
397C22×C8—431no—
398C22×C8—431no—
399C22×C8C22×D4311yescharacteristic
400C22×C8C22×D4311yescharacteristic
401#124—22no—
402#124—41no—
403C4×Q8—862no—
404C4×Q8—862no—
405C4×Q8—862no—
406C4×Q8—862no—
407C4×Q8—1721no—
408C4×Q8—1721no—
409C4×D4—862no—
410C4×D4—1721no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4—431no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4C22×D4311yescharacteristic
425C2×C4⋊C4C22×D4311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—861no—
429C2×C22⋊C4—861no—
430C2×C22⋊C4—861no—
431C2×C42—431no—
432C2×C42—431no—
433C2×C42—861no—
434C2×C42—861no—
435C2×C42—861no—
436C2×C42C22×D4311yescharacteristic
437#106—862no—
438#106—1721no—
439C4⋊C8—862no—
440#103—22no—
441#103—41no—
442#103—862no—
443#103—862no—
444#103—862no—
445#102—22no—
446#102—22no—
447#102—862no—
448C4×C8—862no—
449#95—21no—
450#95—431no—
451#95—431no—
452#95—861no—
453#95—861no—
454#95—861no—
455#95—861no—
456#95—861no—
457#95—861no—
458C24—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—432no—
466C2×Q8—432no—
467C2×Q8—432no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4D4×D4312yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—431no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—431no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—3441no—
489C22×C4—1722no—
490C22×C4—1722no—
491C22×C4D4×D4311yescharacteristic
492C22×C4D4×D4311yescharacteristic
493C22×C4C23×D4311yescharacteristic
494Q16—1724no—
495SD16—868no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—432no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8—861no—
507C2×C8—861no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—81no—
512C4⋊C4—862no—
513C4⋊C4—862no—
514C4⋊C4—864no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4—3441no—
522C4⋊C4—3441no—
523C4⋊C4—3441no—
524C4⋊C4—1722no—
525C4⋊C4—1722no—
526C4⋊C4—1722no—
527C4⋊C4#1062190012yes—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—3441no—
531C22⋊C4—1722no—
532C22⋊C4—1722no—
533C42—21no—
534C42—21no—
535C42—864no—
536C42—861no—
537C42—861no—
538C42—861no—
539C42—861no—
540C42—1722no—
541C42—1722no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D4311yescharacteristic
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—434no—
555Q8—862no—
556Q8—862no—
557Q8—862no—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—432no—
578C2×C4—432no—
579C2×C4—862no—
580C2×C4—861no—
581C2×C4—862no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—862no—
586C2×C4—861no—
587C2×C4—861no—
588C2×C4—861no—
589C2×C4—861no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—3444no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4C2×D4×D4311yescharacteristic
603C2×C4#41×D4311yescharacteristic
604C2×C4#41×D4311yescharacteristic
605C2×C4C2×#1062190011yescharacteristic
606C2×C4#1306337011yescharacteristic
607C2×C4#1306337211yescharacteristic
608C8—41no—
609C8—41no—
610C8—862no—
611C8—1721no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1458340911yescharacteristic
620C22#1458353111yescharacteristic
621C22#1458355711yescharacteristic
622C22#135×D4311yescharacteristic
623C22#137×D4311yescharacteristic
624C22#1458370611yescharacteristic
625C22#1458370911yescharacteristic
626C4—41no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—864no—
632C4—862no—
633C4—862no—
634C4—3441no—
635C4—3441no—
636C4—3441no—
637C4—1721no—
638C4—1721no—
639C4#123×D4311yescharacteristic
640C4#1458349211yescharacteristic
641C4#1458354911yescharacteristic
642C4#1458359611yescharacteristic
643C2—82no—
644C2#1913832311yescharacteristic
645C2#1914467411yescharacteristic
646C2#1914486911yescharacteristic
647C2#1914610311yescharacteristic
648C2#1914632311yescharacteristic
649C2#1914728811yescharacteristic
650C2#1914733211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2042C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#2042D43
hypocenterC43#26161
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
4. #19137509C2semidirect
5. #19136136C2semidirect
6. #19136096C2semidirect
8. #19135273C2semidirect
9. #19135155C2semidirect
11. #19134933C2semidirect
12. #19134933C2semidirect
13. #19134866C2semidirect
386. C43#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#145831182
C2×#145829902
#191383782
#191375092
#191361362
#191360962
#191360012
#191352732
#191351552
#191349472
#191349332
#191348662
#191348582
C43×#20422
#2616143