Finite Group Databasebeta

← #25105176

Subgroups — #25105176

Order 11008.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583293C211yesmaximal, characteristic
2C2×#14583005C211yesmaximal, characteristic, complemented
3C2×#14582998C211yesmaximal, characteristic
4C2×#14582993C211yesmaximal, characteristic, complemented
5C2×#14582986C211yesmaximal, characteristic
6#19137312C211yesmaximal, characteristic, complemented
7#19137303C211yesmaximal, characteristic
8#19135474C211yesmaximal, characteristic
9#19135464C211yesmaximal, characteristic, complemented
10#19135454C211yesmaximal, characteristic, complemented
11#19135447C211yesmaximal, characteristic
12#19135404C211yesmaximal, characteristic
13#19135396C211yesmaximal, characteristic, complemented
14#19134283C211yesmaximal, characteristic, complemented
15C43×#1847C211yesmaximal, characteristic, complemented
16C22×Dic172C2211yescharacteristic
17C22×Dic172C2211yescharacteristic
18C22×#10621873C2211yescharacteristic
19C22×#10621873C2211yescharacteristic
20C2×#13063325C2211yescharacteristic
21C2×#13063313C2211yescharacteristic
22C2×#13063290C2211yescharacteristic
23C2×#13063290C2211yescharacteristic
24C2×#13063290C2211yescharacteristic
25C2×#13063287C2211yescharacteristic
26#14583293C418yescomplemented
27#14583005—24no—
28#14582998—24no—
29#14582993—24no—
30#14582986—24no—
31C2×#13063248C2211yescharacteristic
32C2×#13063248C2211yescharacteristic
33C2×#13063248C2211yescharacteristic
34C2×#13063248C2211yescharacteristic
35C2×#13063248C2211yescharacteristic
36C2×#13063248C2211yescharacteristic
37C2×#13063245C2211yescharacteristic
38C2×#13063245C2211yescharacteristic
39C2×#13063235C2211yescharacteristic
40C2×#13063235C2211yescharacteristic
41C2×C4×C43⋊C8C2211yescharacteristic
42C2×C8×Dic43C2211yescharacteristic
43C2×C8×Dic43C2211yescharacteristic
44#14582804C2211yescharacteristic
45#14582804C2211yescharacteristic
46#14582804C2211yescharacteristic
47#14582800C2211yescharacteristic
48#14582789—22no—
49#14582782—22no—
50#14582782—22no—
51#14582770—22no—
52#14582745C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C2×#106×C43C2211yescharacteristic
57C2×#97×C43C2211yescharacteristic
58C43×#384C2211yescharacteristic
59C43×#382C2211yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C2×Dic172—22no—
66C2×Dic172—22no—
67C2×Dic172—22no—
68C2×Dic172—22no—
69C2×Dic172C2×C414yescomplemented
70C2×Dic172C2×C414yescomplemented
71C2×#10621873—22no—
72C2×#10621873—22no—
73C2×#10621873—22no—
74C2×#10621873—22no—
75C2×#10621873C2×C414yes—
76C2×#10621873C2×C414yes—
77C22×C43⋊C8C2311yescharacteristic
78C22×C43⋊C8C2311yescharacteristic
79#13063325—22no—
80#13063325C2×C414yescomplemented
81#13063313—22no—
82#13063313C2×C414yes—
83#13063290—22no—
84#13063290—22no—
85#13063290—22no—
86#13063290—22no—
87#13063290—22no—
88#13063290—22no—
89#13063287—22no—
90#13063287—22no—
91C2×C4⋊Dic43—22no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×(C2×D43)⋊C4—22no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4—22no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43C2311yescharacteristic
102C2×C4×Dic43C2311yescharacteristic
103#13063248—22no—
104#13063248—22no—
105#13063248—22no—
106#13063248—22no—
107#13063248—22no—
108#13063248—22no—
109#13063245—22no—
110#13063245—22no—
111#13063235—22no—
112#13063235D414yes—
113C4×C43⋊C8—22no—
114C8×Dic43—22no—
115C8×Dic43D414yes—
116C43⋊#95—22no—
117C43⋊#95—22no—
118C43⋊#95—22no—
119C43⋊#95D412yes—
120C43⋊#95D412yes—
121C22×C8×C43C2311yescharacteristic
122C22×C8×C43C2311yescharacteristic
123C2×C4⋊C4×C43—22no—
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C42×C43C2311yescharacteristic
129#107×C43—22no—
130#107×C43—22no—
131#106×C43—22no—
132#106×C43—22no—
133#97×C43C2×C414yescomplemented
134#95×C43—22no—
135C23×D43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43C2×D412yes—
141C22×C4×C43C2411yescharacteristic
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43C2×D412yes—
146C22×Dic43C2×D411yescharacteristic
147C22×Dic43C2×D411yescharacteristic
148C2×D172—21no—
149C2×D172—21no—
150C2×D172—21no—
151C2×D172—21no—
152C2×D172—21no—
153C2×D172C22×C412yes—
154C2×C4×D43—42no—
155C2×C4×D43—82no—
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×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86C22×C412yes—
172C2×Dic86C22×C412yes—
173C2×Dic86C22×C412yes—
174C2×C8×C43C22×C412yes—
175C2×C8×C43C22×C412yes—
176C2×C8×C43C2×D412yes—
177C2×C8×C43C2×D412yes—
178C2×C8×C43#4112yes—
179C2×C8×C43#4112yes—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—82no—
189C42×C43C22×C412yes—
190C42×C43#4112yes—
191(C2×D43)⋊C4—28no—
192(C2×D43)⋊C4—42no—
193(C2×D43)⋊C4—82no—
194C4⋊Dic43—28no—
195Dic43⋊C4—42no—
196Dic43⋊C4—42no—
197Dic43⋊C4—42no—
198C4×Dic43—21no—
199C4×Dic43—21no—
200C4×Dic43—42no—
201C4×Dic43—42no—
202C4×Dic43—42no—
203C4×Dic43C2×D412yes—
204C4×Dic43C2×D412yes—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8C2×D412yes—
210C2×C43⋊C8C2×D412yes—
211Dic172—28no—
212Dic172—28no—
213#10621873—28no—
214#10621873—28no—
215C23×C43C22×D411yescharacteristic
216C22×D43—22no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—42no—
220C22×D43—42no—
221C2×C4×C43—22no—
222C2×C4×C43—22no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—42no—
228C2×C4×C43—42no—
229C2×C4×C43—42no—
230C2×C4×C43—42no—
231C2×C4×C43C4×D414yes—
232C2×C4×C43C23×C411yescharacteristic
233C2×C4×C43C22×D411yescharacteristic
234C2×C4×C43C22×D411yescharacteristic
235C2×C4×C43C2×#4111yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×Dic43—22no—
239C2×Dic43—22no—
240C2×Dic43—22no—
241C2×Dic43—22no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—22no—
245C2×Dic43—22no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43#12712yes—
253C2×Dic43#12712yes—
254D172—22no—
255D172—22no—
256D172—41no—
257D172—41no—
258C4×D43—84no—
259Dic86—22no—
260Dic86—22no—
261Dic86—22no—
262Dic86—22no—
263Dic86—22no—
264Dic86—22no—
265Dic86—41no—
266Dic86—41no—
267Dic86—41no—
268Dic86—41no—
269Dic86—41no—
270Dic86—41no—
271C8×C43C4×D414yes—
272C8×C43C4×D414yes—
273C43⋊C8—22no—
274C43⋊C8—22no—
275#21703—431nomaximal, Sylow
276C22×C43C2×C4×D411yescharacteristic
277C22×C43C2×#12711yescharacteristic
278C22×C43#53211yescharacteristic
279C22×C43#57712yes—
280C22×C43#57812yes—
281C2×D43—42no—
282C2×D43—44no—
283C2×D43—81no—
284C2×D43—81no—
285C2×D43—82no—
286C4×C43—24no—
287C4×C43—81no—
288C4×C43—81no—
289C4×C43—81no—
290C4×C43—81no—
291C4×C43C2×C4×D411yescharacteristic
292C4×C43C2×C4×D411yescharacteristic
293C4×C43#53211yescharacteristic
294C4×C43#53211yescharacteristic
295Dic43—22no—
296Dic43—22no—
297Dic43—42no—
298Dic43—42no—
299Dic43—42no—
300Dic43—42no—
301C2×#497—431no—
302C2×#441—431no—
303C2×#440—431no—
304C2×C4×Q16—431no—
305C2×C4×SD16—431no—
306#2327—431no—
307#2326—431no—
308#1973—431no—
309#1971—431no—
310#1969—431no—
311#1968—431no—
312#1958—431no—
313#1956—431no—
314#1847D4311yescharacteristic, complemented
315#1726—431no—
316C2×C43C4×#12711yescharacteristic
317C2×C43#294712yes—
318C2×C43#315411yescharacteristic
319C2×C43#315611yescharacteristic
320C2×C43#317812yes—
321D43—82no—
322C22×Q16—431no—
323C22×Q16—431no—
324C22×SD16—431no—
325C22×SD16—431no—
326C2×C4⋊Q8—431no—
327C2×#124—431no—
328C2×C4×Q8—431no—
329C2×C4×Q8—431no—
330C2×C4×Q8—431no—
331C2×C4×D4—431no—
332#497—438no—
333#441—864no—
334#440—864no—
335C4×Q16—864no—
336C4×SD16—864no—
337C2×#107C2×D4311yescharacteristic
338C2×#107C2×D4311yescharacteristic
339C2×#106C2×D4311yescharacteristic
340C2×#106C2×D4311yescharacteristic
341C2×#103—431no—
342C2×#103—431no—
343C2×#103—431no—
344C2×#103—431no—
345C2×#103—431no—
346C2×#103—431no—
347C2×#102—431no—
348C2×#102—431no—
349C2×#97—431no—
350C2×#97—431no—
351C2×#97C2×D4311yescharacteristic
352C2×C4×C8—431no—
353C2×C4×C8—431no—
354C2×C4×C8—431no—
355#391—431no—
356#391—431no—
357#391—431no—
358#390—431no—
359#388—862no—
360#387—862no—
361#387—862no—
362#385—862no—
363#384C2×D4311yescharacteristic
364#382C2×D4311yescharacteristic
365#376—431no—
366C43#2170311yescharacteristic, complemented, Sylow
367C22×Q8—431no—
368C22×Q8—431no—
369C22×Q8—431no—
370C22×D4—431no—
371C23×C4—861no—
372C2×Q16—434no—
373C2×Q16—434no—
374C2×Q16—862no—
375C2×Q16—862no—
376C2×Q16—862no—
377C2×Q16—862no—
378C2×SD16—434no—
379C2×SD16—434no—
380C2×SD16—862no—
381C2×SD16—862no—
382C2×SD16—862no—
383C2×SD16—862no—
384C22×C8—431no—
385C22×C8—431no—
386C22×C8C22×D4311yescharacteristic
387C22×C8C22×D4311yescharacteristic
388C4⋊Q8—434no—
389C4⋊Q8—862no—
390#124—434no—
391#124—862no—
392C4×Q8—862no—
393C4×Q8—862no—
394C4×Q8—862no—
395C4×Q8—862no—
396C4×Q8—862no—
397C4×Q8—862no—
398C4×D4—862no—
399C4×D4—862no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—862no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4—861no—
404C2×C4⋊C4—861no—
405C2×C4⋊C4C22×D4311yescharacteristic
406C2×C4⋊C4C22×D4311yescharacteristic
407C2×C4⋊C4C22×D4311yescharacteristic
408C2×C4⋊C4C22×D4311yescharacteristic
409C2×C22⋊C4—862no—
410C2×C22⋊C4—862no—
411C2×C22⋊C4—861no—
412C2×C42—431no—
413C2×C42—431no—
414C2×C42—861no—
415C2×C42—861no—
416C2×C42—861no—
417C2×C42C22×D4311yescharacteristic
418#107—22no—
419#107—22no—
420#106—22no—
421#106—22no—
422#103—862no—
423#103—862no—
424#103—862no—
425#103—862no—
426#103—862no—
427#103—862no—
428#102—862no—
429#102—862no—
430#97—434no—
431#97—862no—
432#97C4×D4314yescomplemented
433C4×C8—434no—
434C4×C8—862no—
435C4×C8—862no—
436#95—22no—
437#95—432no—
438#95—432no—
439#95—862no—
440#95—862no—
441#95—862no—
442C24—861no—
443C2×Q8—432no—
444C2×Q8—432no—
445C2×Q8—432no—
446C2×Q8—861no—
447C2×Q8—861no—
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×Q8—861no—
459C2×Q8—861no—
460C2×Q8—861no—
461C2×D4—432no—
462C2×D4—861no—
463C2×D4—861no—
464C2×D4—861no—
465C2×D4—861no—
466C2×D4—861no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—431no—
472C22×C4—431no—
473C22×C4—432no—
474C22×C4—861no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—3442no—
478C22×C4—1722no—
479C22×C4D4×D4312yes—
480C22×C4C23×D4311yescharacteristic
481Q16—868no—
482Q16—868no—
483SD16—868no—
484SD16—868no—
485C2×C8—432no—
486C2×C8—432no—
487C2×C8—861no—
488C2×C8—861no—
489C2×C8—861no—
490C2×C8—861no—
491C2×C8C2×C4×D4312yes—
492C2×C8C2×C4×D4312yes—
493C2×C8D4×D4312yes—
494C2×C8D4×D4312yes—
495C2×C8#1062190212yes—
496C2×C8#1062190212yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—82no—
506C4⋊C4—868no—
507C4⋊C4—1722no—
508C4⋊C4—1722no—
509C4⋊C4—1722no—
510C22⋊C4—868no—
511C22⋊C4—3442no—
512C22⋊C4—1722no—
513C42—432no—
514C42—432no—
515C42—861no—
516C42—861no—
517C42—1722no—
518C42—1722no—
519C42—1722no—
520C42C2×C4×D4312yes—
521C42#1062190212yes—
522C23—862no—
523C23—1721no—
524C23—1721no—
525C23—1722no—
526C23—1722no—
527C23C2×D4×D4311yescharacteristic
528Q8—862no—
529Q8—862no—
530Q8—862no—
531Q8—862no—
532Q8—862no—
533Q8—862no—
534Q8—1721no—
535Q8—1721no—
536Q8—1721no—
537Q8—1721no—
538Q8—1721no—
539Q8—1721no—
540D4—862no—
541D4—862no—
542D4—1721no—
543D4—1721no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—432no—
555C2×C4—432no—
556C2×C4—862no—
557C2×C4—862no—
558C2×C4—862no—
559C2×C4—862no—
560C2×C4—862no—
561C2×C4—862no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—3444no—
565C2×C4—1721no—
566C2×C4—1721no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4#1306328714yes—
572C2×C4C22×C4×D4311yescharacteristic
573C2×C4C2×D4×D4311yescharacteristic
574C2×C4C2×D4×D4311yescharacteristic
575C2×C4C2×#1062190211yescharacteristic
576C2×C4C2×#1062190211yescharacteristic
577C2×C4C2×#1062190211yescharacteristic
578C8—862no—
579C8—862no—
580C8#1306328714yes—
581C8#1306328714yes—
582C22—3442no—
583C22—3441no—
584C22—3441no—
585C22—1724no—
586C22—1722no—
587C22C2×#1306328711yescharacteristic
588C22#127×D4311yescharacteristic
589C22#1458345011yescharacteristic
590C22#1458370912yes—
591C22#1458371512yes—
592C4—24no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—862no—
598C4—862no—
599C4—1722no—
600C4—1722no—
601C4—1722no—
602C4—1722no—
603C4C2×#1306328711yescharacteristic
604C4C2×#1306328711yescharacteristic
605C4#1458345011yescharacteristic
606C4#1458345011yescharacteristic
607C2—3442no—
608C2#1913715711yescharacteristic
609C2#1914343612yes—
610C2#1914506811yescharacteristic
611C2#1914508711yescharacteristic
612C2#1914526812yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1847C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1847D43
hypocenterC43#21703
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
2. C2×#14583005C2semidirect
4. C2×#14582993C2semidirect
6. #19137312C2semidirect
9. #19135464C2semidirect
10. #19135454C2semidirect
13. #19135396C2semidirect
14. #19134283C2semidirect
15. C43×#1847C2semidirect
26. #14583293C4semidirect
69. C2×Dic172C2×C4semidirect
70. C2×Dic172C2×C4semidirect
80. #13063325C2×C4semidirect
133. #97×C43C2×C4semidirect
314. #1847D43semidirect
366. C43#21703semidirect
432. #97C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832932
C2×#145830052
C2×#145829982
C2×#145829932
C2×#145829862
#191373122
#191373032
#191354742
#191354642
#191354542
#191354472
#191354042
#191353962
#191342832
C43×#18472
#2170343