Finite Group Databasebeta

← #25105198

Subgroups — #25105198

Order 11008.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583288C211yesmaximal, characteristic
2C2×#14583012C211yesmaximal, characteristic, complemented
3C2×#14583001C211yesmaximal, characteristic, complemented
4C2×C4×#10621873C211yesmaximal, characteristic, complemented
5C2×C4×D344C211yesmaximal, characteristic, complemented
6#19137320C211yesmaximal, characteristic, complemented
7#19137306C211yesmaximal, characteristic, complemented
8#19135476C211yesmaximal, characteristic, complemented
9#19135471C211yesmaximal, characteristic, complemented
10#19135461C211yesmaximal, characteristic, complemented
11#19135449C211yesmaximal, characteristic, complemented
12#19135406C211yesmaximal, characteristic, complemented
13#19135401C211yesmaximal, characteristic, complemented
14#19134763C211yesmaximal, characteristic, complemented
15C43×#1726C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C22×#10621873C2211yescharacteristic
18C22×D344C2211yescharacteristic
19C22×D344C2211yescharacteristic
20C2×#13063322C2211yescharacteristic
21C2×#13063313C2211yescharacteristic
22C2×C4×Dic86C2211yescharacteristic
23C2×C4×D172C2211yescharacteristic
24C2×C4×D172C2211yescharacteristic
25C2×C4×D172C2211yescharacteristic
26#14583288C418yescomplemented
27#14583012—24no—
28#14583001—24no—
29C4×#10621873—24no—
30C4×D344—24no—
31C2×#13063259C2211yescharacteristic
32C2×#13063259C2211yescharacteristic
33C2×#13063257C2211yescharacteristic
34C2×#13063257C2211yescharacteristic
35C2×#13063250C2211yescharacteristic
36C2×#13063250C2211yescharacteristic
37C2×#13063247C2211yescharacteristic
38C2×#13063247C2211yescharacteristic
39C2×#13063247C2211yescharacteristic
40C2×#13063247C2211yescharacteristic
41C2×#13063247C2211yescharacteristic
42C2×#13063247C2211yescharacteristic
43#14582805C2211yescharacteristic
44#14582801C2211yescharacteristic
45#14582801C2211yescharacteristic
46#14582801C2211yescharacteristic
47#14582791—22no—
48#14582777—22no—
49#14582777—22no—
50#14582773—22no—
51#14582769C2211yescharacteristic
52#14582765C2211yescharacteristic
53C2×#97×C43C2211yescharacteristic
54C2×#97×C43C2211yescharacteristic
55C2×#97×C43C2211yescharacteristic
56C2×C4×C8×C43C2211yescharacteristic
57C2×C4×C8×C43C2211yescharacteristic
58C2×C4×C8×C43C2211yescharacteristic
59C43×#376C2211yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×D172—22no—
62C22×D172C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×D172C2311yescharacteristic
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C22×C4×D43—21no—
68C2×#10621873—22no—
69C2×#10621873—22no—
70C2×#10621873—22no—
71C2×#10621873—22no—
72C2×#10621873C2×C414yescomplemented
73C2×#10621873C2×C414yescomplemented
74C2×D344—22no—
75C2×D344—22no—
76C2×D344—22no—
77C2×D344—22no—
78C2×D344C2×C414yescomplemented
79C2×D344C2×C414yescomplemented
80#13063322—22no—
81#13063322C2×C414yescomplemented
82#13063313—22no—
83#13063313C2×C414yescomplemented
84C4×Dic86—22no—
85C4×Dic86—22no—
86C4×D172—22no—
87C4×D172—22no—
88C4×D172—22no—
89C4×D172—22no—
90C4×D172—22no—
91C4×D172—22no—
92C2×C4⋊Dic43—22no—
93C2×C4⋊Dic43C2311yescharacteristic
94C2×C4⋊Dic43C2311yescharacteristic
95C2×C4⋊Dic43C2311yescharacteristic
96C2×C4⋊Dic43C2311yescharacteristic
97C2×Dic43⋊C4—21no—
98C2×(C2×D43)⋊C4—22no—
99C2×(C2×D43)⋊C4—21no—
100C2×(C2×D43)⋊C4—21no—
101C2×(C2×D43)⋊C4—21no—
102C2×(C2×D43)⋊C4—22no—
103C2×(C2×D43)⋊C4—22no—
104C2×(C2×D43)⋊C4—22no—
105C2×C4×Dic43—21no—
106#13063259—22no—
107#13063259—22no—
108#13063257—22no—
109#13063257—22no—
110#13063250—22no—
111#13063250—22no—
112#13063247—22no—
113#13063247—22no—
114#13063247—22no—
115#13063247—22no—
116#13063247—22no—
117#13063247—22no—
118C43⋊#95—22no—
119C43⋊#95—22no—
120C22×C8×C43C2311yescharacteristic
121C22×C8×C43C2311yescharacteristic
122C22×C8×C43C2311yescharacteristic
123C22×C8×C43C2311yescharacteristic
124C2×C42×C43C2311yescharacteristic
125C2×C42×C43C2311yescharacteristic
126C2×C42×C43C2311yescharacteristic
127#97×C43—22no—
128#97×C43C2×C414yescomplemented
129#97×C43D414yescomplemented
130C4×C8×C43—22no—
131C4×C8×C43—22no—
132C4×C8×C43D414yescomplemented
133#95×C43D412yes—
134#95×C43D412yes—
135C23×D43—21no—
136C23×D43—21no—
137C23×D43—21no—
138C22×C4×C43C2×D412yes—
139C22×C4×C43C2×D412yes—
140C22×C4×C43C2×D411yescharacteristic
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2411yescharacteristic
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172—21no—
151C2×D172—28no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×D172—21no—
156C2×D172—21no—
157C2×D172—21no—
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—82no—
165C2×D172—82no—
166C2×D172C22×C412yes—
167C2×D172C22×C412yes—
168C2×D172C22×C412yes—
169C2×C4×D43—42no—
170C2×C4×D43—42no—
171C2×C4×D43—42no—
172C2×C4×D43—82no—
173C2×C4×D43—82no—
174C2×C4×D43—82no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C2×Dic86—21no—
178C2×Dic86—21no—
179C2×Dic86—21no—
180C2×Dic86C22×C412yes—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43C22×C412yes—
186C2×C8×C43C22×C412yes—
187C2×C8×C43C2×D412yes—
188C2×C8×C43C2×D412yes—
189C2×C8×C43C2×D412yes—
190C2×C8×C43C2×D412yes—
191C2×C8×C43#4112yes—
192C2×C8×C43#4112yes—
193C42×C43—21no—
194C42×C43—21no—
195C42×C43C22×C412yes—
196C42×C43C2×D412yes—
197C42×C43C2×D412yes—
198C42×C43#4112yes—
199(C2×D43)⋊C4—28no—
200(C2×D43)⋊C4—42no—
201(C2×D43)⋊C4—42no—
202(C2×D43)⋊C4—42no—
203(C2×D43)⋊C4—82no—
204(C2×D43)⋊C4—82no—
205(C2×D43)⋊C4—82no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214C4⋊Dic43—82no—
215Dic43⋊C4—42no—
216C4×Dic43—42no—
217D344—28no—
218D344—28no—
219#10621873—28no—
220#10621873—28no—
221C23×C43C22×D411yescharacteristic
222C22×D43—22no—
223C22×D43—22no—
224C22×D43—22no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231C22×D43—42no—
232C22×D43—42no—
233C22×D43—42no—
234C22×D43—42no—
235C22×D43—42no—
236C22×D43—42no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—22no—
240C2×C4×C43—22no—
241C2×C4×C43—22no—
242C2×C4×C43—22no—
243C2×C4×C43—22no—
244C2×C4×C43C4×D414yes—
245C2×C4×C43#12712yes—
246C2×C4×C43#12712yes—
247C2×C4×C43C23×C411yescharacteristic
248C2×C4×C43C22×D411yescharacteristic
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×Dic43—22no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43—42no—
263C2×Dic43—42no—
264D172—22no—
265D172—22no—
266D172—22no—
267D172—22no—
268D172—22no—
269D172—22no—
270D172—41no—
271D172—41no—
272D172—41no—
273D172—41no—
274D172—41no—
275D172—41no—
276D172—88no—
277C4×D43—84no—
278C4×D43—84no—
279C4×D43—84no—
280Dic86—22no—
281Dic86—22no—
282Dic86—41no—
283Dic86—41no—
284C8×C43—22no—
285C8×C43—22no—
286C8×C43C4×D414yes—
287C8×C43C4×D414yes—
288#21704—431nomaximal, Sylow
289C22×C43C2×C4×D411yescharacteristic
290C22×C43C2×#12711yescharacteristic
291C22×C43#53211yescharacteristic
292C22×C43#57612yes—
293C22×C43#57712yes—
294C2×D43—42no—
295C2×D43—42no—
296C2×D43—42no—
297C2×D43—44no—
298C2×D43—44no—
299C2×D43—44no—
300C2×D43—81no—
301C2×D43—81no—
302C2×D43—81no—
303C2×D43—81no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—82no—
307C2×D43—82no—
308C2×D43—82no—
309C4×C43—22no—
310C4×C43—22no—
311C4×C43—24no—
312C4×C43—42no—
313C4×C43C2×C4×D411yescharacteristic
314C4×C43C2×C4×D411yescharacteristic
315C4×C43#53211yescharacteristic
316C4×C43#53211yescharacteristic
317Dic43—42no—
318Dic43—81no—
319Dic43—81no—
320Dic43—81no—
321Dic43—81no—
322C2×#496—431no—
323C2×#442—431no—
324C2×#440—431no—
325C2×C4×SD16—431no—
326C2×C4×D8—431no—
327#2328—431no—
328#2326—431no—
329#1973—431no—
330#1972—431no—
331#1970—431no—
332#1968—431no—
333#1958—431no—
334#1957—431no—
335#1847—431no—
336#1726D4311yescharacteristic, complemented
337C2×C43C4×#12711yescharacteristic
338C2×C43#294612yes—
339C2×C43#315411yescharacteristic
340C2×C43#315511yescharacteristic
341C2×C43#317912yes—
342D43—82no—
343D43—82no—
344D43—82no—
345C22×SD16—431no—
346C22×SD16—431no—
347C22×D8—431no—
348C22×D8—431no—
349C2×#127—431no—
350C2×#124—431no—
351C2×C4×Q8—431no—
352C2×C4×D4—431no—
353C2×C4×D4—431no—
354C2×C4×D4—431no—
355#496—438no—
356#442—864no—
357#440—864no—
358C4×SD16—864no—
359C4×D8—864no—
360C2×#107—431no—
361C2×#107—431no—
362C2×#106—431no—
363C2×#106—431no—
364C2×#103—431no—
365C2×#103—431no—
366C2×#102—431no—
367C2×#102—431no—
368C2×#102—431no—
369C2×#102—431no—
370C2×#102—431no—
371C2×#102—431no—
372C2×#97C2×D4311yescharacteristic
373C2×#97C2×D4311yescharacteristic
374C2×#97C2×D4311yescharacteristic
375C2×C4×C8C2×D4311yescharacteristic
376C2×C4×C8C2×D4311yescharacteristic
377C2×C4×C8C2×D4311yescharacteristic
378#391—431no—
379#390—431no—
380#390—431no—
381#390—431no—
382#388—862no—
383#386—862no—
384#386—862no—
385#385—862no—
386#384—431no—
387#382—431no—
388#376C2×D4311yescharacteristic
389C43#2170411yescharacteristic, complemented, Sylow
390C22×Q8—431no—
391C22×D4—431no—
392C22×D4—431no—
393C22×D4—431no—
394C22×D4—862no—
395C23×C4—861no—
396C23×C4—861no—
397C23×C4—861no—
398C2×SD16—434no—
399C2×SD16—434no—
400C2×SD16—862no—
401C2×SD16—862no—
402C2×SD16—862no—
403C2×SD16—862no—
404C2×D8—434no—
405C2×D8—434no—
406C2×D8—862no—
407C2×D8—862no—
408C2×D8—862no—
409C2×D8—862no—
410C22×C8C22×D4311yescharacteristic
411C22×C8C22×D4311yescharacteristic
412C22×C8C22×D4311yescharacteristic
413C22×C8C22×D4311yescharacteristic
414#127—434no—
415#127—862no—
416#124—434no—
417#124—862no—
418C4×Q8—862no—
419C4×Q8—862no—
420C4×D4—862no—
421C4×D4—862no—
422C4×D4—862no—
423C4×D4—862no—
424C4×D4—862no—
425C4×D4—862no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—431no—
430C2×C4⋊C4—862no—
431C2×C4⋊C4—861no—
432C2×C22⋊C4—862no—
433C2×C22⋊C4—862no—
434C2×C22⋊C4—862no—
435C2×C22⋊C4—862no—
436C2×C22⋊C4—861no—
437C2×C22⋊C4—861no—
438C2×C22⋊C4—861no—
439C2×C42—861no—
440C2×C42C22×D4311yescharacteristic
441C2×C42C22×D4311yescharacteristic
442C2×C42C22×D4311yescharacteristic
443#107—862no—
444#107—862no—
445#106—862no—
446#106—862no—
447#103—862no—
448#103—862no—
449#102—862no—
450#102—862no—
451#102—862no—
452#102—862no—
453#102—862no—
454#102—862no—
455#97—22no—
456#97C4×D4314yescomplemented
457#97D17214yescomplemented
458C4×C8—22no—
459C4×C8—22no—
460C4×C8D17214yescomplemented
461#95—862no—
462#95—862no—
463#95D17212yes—
464#95D17212yes—
465C24—861no—
466C24—861no—
467C24—861no—
468C2×Q8—432no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×Q8—861no—
472C2×Q8—861no—
473C2×Q8—861no—
474C2×D4—432no—
475C2×D4—432no—
476C2×D4—432no—
477C2×D4—868no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4—861no—
481C2×D4—861no—
482C2×D4—861no—
483C2×D4—861no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C2×D4—861no—
489C2×D4—861no—
490C2×D4—861no—
491C2×D4—861no—
492C2×D4—861no—
493C2×D4—3442no—
494C2×D4—3442no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—861no—
499C22×C4—861no—
500C22×C4—3442no—
501C22×C4—3442no—
502C22×C4—3442no—
503C22×C4—1722no—
504C22×C4—1722no—
505C22×C4—1722no—
506C22×C4C2×D17212yes—
507C22×C4C2×D17212yes—
508C22×C4C2×D17211yescharacteristic
509C22×C4C2×D17211yescharacteristic
510C22×C4C23×D4311yescharacteristic
511SD16—868no—
512SD16—868no—
513D8—868no—
514D8—868no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8C2×C4×D4312yes—
520C2×C8C2×C4×D4312yes—
521C2×C8C2×D17212yes—
522C2×C8C2×D17212yes—
523C2×C8C2×D17212yes—
524C2×C8C2×D17212yes—
525C2×C8#1062189812yes—
526C2×C8#1062189812yes—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—3442no—
536C4⋊C4—1722no—
537C22⋊C4—868no—
538C22⋊C4—3442no—
539C22⋊C4—3442no—
540C22⋊C4—3442no—
541C22⋊C4—1722no—
542C22⋊C4—1722no—
543C22⋊C4—1722no—
544C42—21no—
545C42—21no—
546C42—1722no—
547C42C2×C4×D4312yes—
548C42C2×D17212yes—
549C42C2×D17212yes—
550C42#1062189812yes—
551C23—862no—
552C23—862no—
553C23—862no—
554C23—1721no—
555C23—1721no—
556C23—1721no—
557C23—1722no—
558C23—1722no—
559C23—1722no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23—1722no—
564C23—1722no—
565C23—1722no—
566C23C22×D17211yescharacteristic
567Q8—862no—
568Q8—862no—
569Q8—1721no—
570Q8—1721no—
571D4—862no—
572D4—862no—
573D4—862no—
574D4—862no—
575D4—862no—
576D4—862no—
577D4—3448no—
578D4—1721no—
579D4—1721no—
580D4—1721no—
581D4—1721no—
582D4—1721no—
583D4—1721no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—862no—
592C2×C4—3444no—
593C2×C4—3444no—
594C2×C4—3444no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1722no—
600C2×C4—1722no—
601C2×C4—1722no—
602C2×C4—1722no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4C4×D17214yes—
606C2×C4#1306332212yes—
607C2×C4#1306332212yes—
608C2×C4C22×C4×D4311yescharacteristic
609C2×C4C22×D17211yescharacteristic
610C2×C4C22×D17211yescharacteristic
611C2×C4C2×#1062189811yescharacteristic
612C2×C4C2×#1062189811yescharacteristic
613C2×C4C2×#1062189811yescharacteristic
614C8—22no—
615C8—22no—
616C8C4×D17214yes—
617C8C4×D17214yes—
618C22—3442no—
619C22—3442no—
620C22—3442no—
621C22—3441no—
622C22—3441no—
623C22—3441no—
624C22—3441no—
625C22—3441no—
626C22—3441no—
627C22—1724no—
628C22—1724no—
629C22—1724no—
630C22—1722no—
631C22—1722no—
632C22—1722no—
633C22C2×C4×D17211yescharacteristic
634C22C2×#1306332211yescharacteristic
635C22#1458344911yescharacteristic
636C22#1458370412yes—
637C22#1458371112yes—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—3441no—
643C4—3441no—
644C4—3441no—
645C4—3441no—
646C4—1722no—
647C4C2×C4×D17211yescharacteristic
648C4C2×C4×D17211yescharacteristic
649C4#1458344911yescharacteristic
650C4#1458344911yescharacteristic
651C2—3442no—
652C2—3442no—
653C2—3442no—
654C2C4×#1306332211yescharacteristic
655C2#1914342912yes—
656C2#1914507111yescharacteristic
657C2#1914508111yescharacteristic
658C2#1914527812yes—
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1726C2
socleC23×C43C22×D4
prime radicalC2×#14583288C2
cosocleC22×C4×C43C24
hypercenter#1726D43
hypocenterC43#21704
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. C2×#14583012C2semidirect
3. C2×#14583001C2semidirect
4. C2×C4×#10621873C2semidirect
5. C2×C4×D344C2semidirect
6. #19137320C2semidirect
7. #19137306C2semidirect
8. #19135476C2semidirect
9. #19135471C2semidirect
10. #19135461C2semidirect
11. #19135449C2semidirect
12. #19135406C2semidirect
13. #19135401C2semidirect
14. #19134763C2semidirect
15. C43×#1726C2semidirect
26. #14583288C4semidirect
72. C2×#10621873C2×C4semidirect
73. C2×#10621873C2×C4semidirect
78. C2×D344C2×C4semidirect
79. C2×D344C2×C4semidirect
81. #13063322C2×C4semidirect
83. #13063313C2×C4semidirect
128. #97×C43C2×C4semidirect
129. #97×C43D4semidirect
132. C4×C8×C43D4semidirect
336. #1726D43semidirect
389. C43#21704semidirect
456. #97C4×D43semidirect
457. #97D172semidirect
460. C4×C8D172semidirect

Maximal subgroups

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

subgroupindex
C2×#145832882
C2×#145830122
C2×#145830012
C2×C4×#106218732
C2×C4×D3442
#191373202
#191373062
#191354762
#191354712
#191354612
#191354492
#191354062
#191354012
#191347632
C43×#17262
#2170443