Finite Group Databasebeta

← #25146312

Subgroups — #25146312

Order 11008.

802 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×#14583101C211yesmaximal, characteristic
2C2×#14583030C211yesmaximal, characteristic, complemented
3#19139953C211yesmaximal, characteristic
4#19135990C211yesmaximal, characteristic, complemented
5#19135976C211yesmaximal, characteristic, complemented
6#19135910C211yesmaximal, characteristic
7#19135910C211yesmaximal, characteristic
8#19135479C211yesmaximal, characteristic
9#19135452C211yesmaximal, characteristic
10#19135416C211yesmaximal, characteristic, complemented
11#19135363C211yesmaximal, characteristic, complemented
12#19135288C211yesmaximal, characteristic, complemented
13#19135274C211yesmaximal, characteristic, complemented
14#19134423C211yesmaximal, characteristic
15C43×#2568C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C22×#10621884C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19#14583101—24no—
20#14583030—24no—
21C2×#13063258C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×C4⋊(C43⋊C8)C2211yescharacteristic
24C2×C4⋊(C43⋊C8)C2211yescharacteristic
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063249C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063249C2211yescharacteristic
32C2×#13063245C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×#13063236C2211yescharacteristic
35C2×#13063236C2211yescharacteristic
36C2×C4×C43⋊C8C2211yescharacteristic
37#14582834—21no—
38#14582810—21no—
39#14582801C2211yescharacteristic
40#14582801C2211yescharacteristic
41#14582797C2211yescharacteristic
42#14582797C2211yescharacteristic
43#14582770—21no—
44#14582770—21no—
45#14582765C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50C2×C4⋊Q8×C43C2211yescharacteristic
51C2×C4×Q8×C43C2211yescharacteristic
52C43×#400—21no—
53C43×#400—21no—
54C43×#391—21no—
55C43×#391C2211yescharacteristic
56C43×#391C2211yescharacteristic
57C43×#391C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C43×#382—21no—
60C4×C4⋊C4×C43C2211yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C2×#10621884—22no—
64C2×#10621884—22no—
65C2×#10621884—22no—
66C2×#10621884—22no—
67C2×#10621884—41no—
68C2×#10621884—41no—
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C4×D172—22no—
74C4×D172—22no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85#13063258—41no—
86#13063256—41no—
87C4⋊(C43⋊C8)—22no—
88C4⋊(C43⋊C8)—22no—
89C4⋊(C43⋊C8)D414yes—
90#13063249—22no—
91#13063249—22no—
92#13063249—22no—
93#13063249—22no—
94#13063249—41no—
95#13063249—41no—
96#13063245—22no—
97#13063245—22no—
98#13063236—22no—
99#13063236—22no—
100C4×C43⋊C8—22no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C22×Q8×C43D411yescharacteristic
106C22×Q8×C43C2311yescharacteristic
107C22×Q8×C43C2311yescharacteristic
108C4⋊Q8×C43—22no—
109C4⋊Q8×C43—22no—
110C4×Q8×C43—22no—
111C4×Q8×C43—22no—
112C2×C4⋊C4×C43—21no—
113C2×C4⋊C4×C43—21no—
114C2×C4⋊C4×C43D411yescharacteristic
115C2×C4⋊C4×C43D411yescharacteristic
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C42×C43—21no—
120C2×C42×C43C2311yescharacteristic
121C2×C42×C43C2311yescharacteristic
122C2×C42×C43C2311yescharacteristic
123#95×C43—21no—
124#95×C43—21no—
125#95×C43—21no—
126#95×C43—21no—
127#95×C43—21no—
128#95×C43—21no—
129#95×C43—21no—
130#95×C43—21no—
131#95×C43—21no—
132C23×D43—21no—
133C2×Q8×C43—22no—
134C2×Q8×C43—22no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—41no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2411yescharacteristic
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172#4112yes—
168C2×C4×D43—42no—
169C2×C4×D43—81no—
170C2×C4×D43—81no—
171C4⋊C4×C43—22no—
172C4⋊C4×C43—22no—
173C4⋊C4×C43—22no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—42no—
179C4⋊C4×C43—42no—
180C42×C43—21no—
181C42×C43—21no—
182C42×C43—21no—
183C42×C43—21no—
184C42×C43—42no—
185C42×C43C2×D412yes—
186C42×C43#4112yes—
187#10621884—44no—
188#10621884—44no—
189(C2×D43)⋊C4—42no—
190(C2×D43)⋊C4—42no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—81no—
193(C2×D43)⋊C4—81no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200Dic43⋊C4—81no—
201Dic43⋊C4—81no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8C2×D412yes—
212C2×C43⋊C8C2×D412yes—
213C2×C43⋊C8#4112yes—
214C23×C43C22×D411yescharacteristic
215C22×D43—22no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222Q8×C43—44no—
223Q8×C43—41no—
224Q8×C43—41no—
225Q8×C43—41no—
226Q8×C43—41no—
227Q8×C43—41no—
228Q8×C43—41no—
229Q8×C43—41no—
230Q8×C43—41no—
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—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43#12112yes—
259C2×C4×C43#12112yes—
260C2×C4×C43#13712yes—
261C2×C4×C43C22×D411yescharacteristic
262C2×C4×C43C2×#4111yescharacteristic
263C2×C4×C43C2×#4111yescharacteristic
264C2×C4×C43C2×#4111yescharacteristic
265C2×C4×C43H(5,𝔽2)11yescharacteristic
266C2×C4×C432−1+411yescharacteristic
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—41no—
276D172—22no—
277D172—22no—
278D172—41no—
279D172—41no—
280C4×D43—84no—
281C43⋊C8—22no—
282C43⋊C8—22no—
283C43⋊C8—22no—
284C43⋊C8—41no—
285#25308—431nomaximal, Sylow
286C22×C43C2×#12111yescharacteristic
287C22×C43#54811yescharacteristic
288C22×C43#55011yescharacteristic
289C22×C43C2×#13711yescharacteristic
290C22×C43#57511yescharacteristic
291C22×C43#57711yescharacteristic
292C22×C43#57711yescharacteristic
293C2×D43—42no—
294C2×D43—42no—
295C2×D43—42no—
296C2×D43—81no—
297C2×D43—81no—
298C2×D43—81no—
299C2×D43—81no—
300C4×C43—22no—
301C4×C43—22no—
302C4×C43—22no—
303C4×C43—41no—
304C4×C43—41no—
305C4×C43—41no—
306C4×C43—81no—
307C4×C43—81no—
308C4×C43—81no—
309C4×C43—81no—
310C4×C43#52911yescharacteristic
311C4×C43#54211yescharacteristic
312C4×C43#54611yescharacteristic
313C4×C43#54811yescharacteristic
314Dic43—81no—
315Dic43—81no—
316Dic43—81no—
317C2×#461—431no—
318C2×#446—431no—
319#2580—431no—
320#2568D4311yescharacteristic, complemented
321#2068—431no—
322#2066—431no—
323#2055—431no—
324#2055—431no—
325#1973—431no—
326#1968—431no—
327#1960—431no—
328#1949—431no—
329#1935—431no—
330#1933—431no—
331#1762—431no—
332C2×C43#242611yescharacteristic
333C2×C43#305611yescharacteristic
334C2×C43#305711yescharacteristic
335C2×C43#337211yescharacteristic
336C2×C43#337611yescharacteristic
337C2×C43#340011yescharacteristic
338C2×C43#340111yescharacteristic
339D43—82no—
340C22×SD16—431no—
341C22×SD16—431no—
342C2×C4⋊Q8C2×D4311yescharacteristic
343C2×C4×Q8C2×D4311yescharacteristic
344C2×C4×D4—431no—
345#461—864no—
346#446—864no—
347C2×#107—431no—
348C2×#106—431no—
349C2×C4⋊C8—431no—
350C2×C4⋊C8—431no—
351C2×C4⋊C8—431no—
352C2×#103—431no—
353C2×#103—431no—
354C2×#103—431no—
355C2×#103—431no—
356C2×#103—431no—
357C2×#103—431no—
358C2×#102—431no—
359C2×#102—431no—
360C2×#97—431no—
361C2×#97—431no—
362C2×C4×C8—431no—
363#400—21no—
364#400—21no—
365#399—861no—
366#393—861no—
367#391—21no—
368#391C2×D4311yescharacteristic
369#391C2×D4311yescharacteristic
370#391C2×D4311yescharacteristic
371#391C2×D4311yescharacteristic
372#390—431no—
373#390—431no—
374#389—431no—
375#389—431no—
376#385—861no—
377#385—861no—
378#382—21no—
379#382—431no—
380C4×C4⋊C4C2×D4311yescharacteristic
381#340—431no—
382#340—431no—
383#340—431no—
384#340—431no—
385C43#2530811yescharacteristic, complemented, Sylow
386C22×Q8C22⋊D4311yescharacteristic
387C22×Q8C22×D4311yescharacteristic
388C22×Q8C22×D4311yescharacteristic
389C22×D4—431no—
390C23×C4—861no—
391C2×SD16—862no—
392C2×SD16—862no—
393C2×SD16—862no—
394C2×SD16—862no—
395C2×SD16—1721no—
396C2×SD16—1721no—
397C22×C8—431no—
398C22×C8—431no—
399C22×C8—431no—
400C22×C8—431no—
401C4⋊Q8—22no—
402C4⋊Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×D4—862no—
406C4×D4—862no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4C22⋊D4311yescharacteristic
415C2×C4⋊C4C22⋊D4311yescharacteristic
416C2×C4⋊C4C22⋊D4311yescharacteristic
417C2×C4⋊C4C22×D4311yescharacteristic
418C2×C4⋊C4C22×D4311yescharacteristic
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C42—21no—
425C2×C42C22×D4311yescharacteristic
426C2×C42C22×D4311yescharacteristic
427C2×C42C22×D4311yescharacteristic
428#107—1721no—
429#106—1721no—
430C4⋊C8—434no—
431C4⋊C8—862no—
432C4⋊C8—862no—
433#103—862no—
434#103—862no—
435#103—862no—
436#103—862no—
437#103—1721no—
438#103—1721no—
439#102—862no—
440#102—862no—
441#97—862no—
442#97—862no—
443C4×C8—862no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—861no—
454#95—861no—
455#95—861no—
456#95—861no—
457C24—861no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—41no—
473C2×D4—432no—
474C2×D4—861no—
475C2×D4—861no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—861no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—3441no—
487C22×C4—3441no—
488C22×C4—1722no—
489C22×C4C2×C22⋊D4311yescharacteristic
490C22×C4C2×C22⋊D4311yescharacteristic
491C22×C4C2×C22⋊D4311yescharacteristic
492C22×C4C2×C22⋊D4311yescharacteristic
493C22×C4C2×C22⋊D4311yescharacteristic
494C22×C4C2×C22⋊D4311yescharacteristic
495C22×C4C23×D4311yescharacteristic
496SD16—1724no—
497SD16—1724no—
498C2×C8—432no—
499C2×C8—432no—
500C2×C8—432no—
501C2×C8—861no—
502C2×C8—861no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8—861no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8—861no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4—861no—
522C4⋊C4—861no—
523C4⋊C4—861no—
524C4⋊C4—861no—
525C4⋊C4—3441no—
526C4⋊C4—3441no—
527C22⋊C4—3441no—
528C22⋊C4—3441no—
529C22⋊C4—1722no—
530C22⋊C4—1722no—
531C22⋊C4—1722no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42D4×D4312yes—
538C42#1062190012yes—
539C23—862no—
540C23—1721no—
541C23—1721no—
542C23—1721no—
543C23—1721no—
544C23—1721no—
545C23—1721no—
546C23C22×C22⋊D4311yescharacteristic
547Q8—44no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556D4—862no—
557D4—862no—
558D4—1721no—
559D4—1721no—
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—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—3444no—
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#1306329712yes—
598C2×C4#1306329712yes—
599C2×C4#1306335612yes—
600C2×C4C2×D4×D4311yescharacteristic
601C2×C4#41×D4311yescharacteristic
602C2×C4#41×D4311yescharacteristic
603C2×C4C2×#1062190011yescharacteristic
604C2×C4#1306337111yescharacteristic
605C2×C4#1306337211yescharacteristic
606C8—862no—
607C8—862no—
608C8—862no—
609C8—1721no—
610C22—3441no—
611C22—3441no—
612C22—3441no—
613C22—3441no—
614C22—1722no—
615C22—1722no—
616C22—1722no—
617C22C2×#1306329711yescharacteristic
618C22#1458355611yescharacteristic
619C22#1458356811yescharacteristic
620C22C2×#1306335611yescharacteristic
621C22#1458369711yescharacteristic
622C22#1458370711yescharacteristic
623C22#1458370711yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—3441no—
635C4—3441no—
636C4—3441no—
637C4#1458342811yescharacteristic
638C4#1458351811yescharacteristic
639C4#1458354211yescharacteristic
640C4#1458355711yescharacteristic
641C2—3442no—
642C2#1913816411yescharacteristic
643C2#1914422111yescharacteristic
644C2#1914423611yescharacteristic
645C2#1914737211yescharacteristic
646C2#1914741611yescharacteristic
647C2#1914764411yescharacteristic
648C2#1914766111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2568C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2568D43
hypocenterC43#25308
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
2. C2×#14583030C2semidirect
4. #19135990C2semidirect
5. #19135976C2semidirect
10. #19135416C2semidirect
11. #19135363C2semidirect
12. #19135288C2semidirect
13. #19135274C2semidirect
15. C43×#2568C2semidirect
320. #2568D43semidirect
385. C43#25308semidirect

Maximal subgroups

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

subgroupindex
C2×#145831012
C2×#145830302
#191399532
#191359902
#191359762
#191359102
#191354792
#191354522
#191354162
#191353632
#191352882
#191352742
#191344232
C43×#25682
#2530843