Finite Group Databasebeta

← #25105082

Subgroups — #25105082

Order 11008.

1110 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
1C2×#14583280C211yesmaximal, characteristic
2C2×#14583009C211yesmaximal, characteristic
3C2×#14583004C211yesmaximal, characteristic, complemented
4C2×C4×#10621873C211yesmaximal, characteristic, complemented
5#19137307C211yesmaximal, characteristic, complemented
6#19137302C211yesmaximal, characteristic, complemented
7#19135468C211yesmaximal, characteristic
8#19135463C211yesmaximal, characteristic, complemented
9#19135451C211yesmaximal, characteristic, complemented
10#19135447C211yesmaximal, characteristic, complemented
11#19135410C211yesmaximal, characteristic, complemented
12#19135408C211yesmaximal, characteristic, complemented
13#19134750C211yesmaximal, characteristic, complemented
14#19134281C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C43C211yesmaximal, characteristic, complemented
16C22×#10621885C2211yescharacteristic
17C22×#10621873C2211yescharacteristic
18C22×#10621882C2211yescharacteristic
19C2×#13063314C2211yescharacteristic
20C2×#13063312C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×C4×D172C2211yescharacteristic
23#14583280C418yescomplemented
24#14583009—24no—
25#14583004—24no—
26C4×#10621873—24no—
27C2×#13063257C2211yescharacteristic
28C2×#13063256C2211yescharacteristic
29C2×#13063256C2211yescharacteristic
30C2×#13063250C2211yescharacteristic
31C2×#13063249C2211yescharacteristic
32C2×#13063248C2211yescharacteristic
33C2×#13063247C2211yescharacteristic
34C2×#13063246C2211yescharacteristic
35C2×#13063245C2211yescharacteristic
36C2×#13063236C2211yescharacteristic
37C2×#13063235C2211yescharacteristic
38C2×#13063235C2211yescharacteristic
39C2×C8×Dic43C2211yescharacteristic
40C2×C8×Dic43C2211yescharacteristic
41#14582806C2211yescharacteristic
42#14582804C2211yescharacteristic
43#14582802C2211yescharacteristic
44#14582800C2211yescharacteristic
45#14582790—22no—
46#14582787—22no—
47#14582772—22no—
48#14582770—22no—
49#14582768C2211yescharacteristic
50#14582768C2211yescharacteristic
51#14582745C2211yescharacteristic
52C22×SD16×C43C2211yescharacteristic
53C2×C4×Q8×C43C2211yescharacteristic
54C2×C4×D4×C43C2211yescharacteristic
55C4×SD16×C43—24no—
56C2×#106×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C2×C4×C8×C43C2211yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C2×#10621885—22no—
64C2×#10621885—22no—
65C2×#10621885C2×C414yescomplemented
66C2×#10621873—22no—
67C2×#10621873—22no—
68C2×#10621873C2×C414yescomplemented
69C2×#10621882—22no—
70C2×#10621882—22no—
71C2×#10621882C2×C414yes—
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74#13063314—22no—
75#13063314C2×C414yescomplemented
76#13063312—22no—
77#13063312C2×C414yescomplemented
78C4×Dic86—22no—
79C4×Dic86—22no—
80C4×D172—22no—
81C4×D172—22no—
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—22no—
86C2×Dic43⋊C4—22no—
87C2×C22⋊Dic43—22no—
88C2×C22⋊Dic43—22no—
89C2×(C2×D43)⋊C4—22no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—22no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43C2311yescharacteristic
94C2×C4×Dic43C2311yescharacteristic
95#13063257—22no—
96#13063256—22no—
97#13063256—22no—
98#13063250—22no—
99#13063249—22no—
100#13063248—22no—
101#13063247—22no—
102#13063246—22no—
103#13063245—22no—
104#13063236D414yes—
105#13063235—22no—
106#13063235—22no—
107C8×Dic43—22no—
108C8×Dic43C2×C414yescomplemented
109C43⋊#95—22no—
110C43⋊#95—22no—
111C43⋊#95D412yes—
112C43⋊#95D412yes—
113C22×Q8×C43C2311yescharacteristic
114C22×D4×C43C2311yescharacteristic
115C23×C4×C43—21no—
116C2×SD16×C43—22no—
117C2×SD16×C43—22no—
118C2×SD16×C43C2×C414yescomplemented
119C22×C8×C43C2311yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C4×Q8×C43—22no—
122C4×Q8×C43—22no—
123C4×D4×C43—22no—
124C4×D4×C43—22no—
125C2×C4⋊C4×C43—21no—
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C22⋊C4×C43—21no—
129C2×C42×C43—21no—
130C2×C42×C43C2311yescharacteristic
131#106×C43—22no—
132#103×C43—22no—
133#102×C43—22no—
134C4×C8×C43D414yes—
135C24×C43—21no—
136C23×D43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43C22×C412yes—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C2×D4×C43C22×C412yes—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—42no—
153C22×C4×C43—82no—
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2411yescharacteristic
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43C2×D412yes—
161C22×Dic43C2×D412yes—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172C22×C412yes—
168C2×C4×D43—42no—
169C2×C4×D43—82no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86C22×C412yes—
176SD16×C43—28no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43C22×C412yes—
180C2×C8×C43C2×D412yes—
181C2×C8×C43C2×D412yes—
182C2×C8×C43#4112yes—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—42no—
188C22⋊C4×C43—42no—
189C42×C43—42no—
190C42×C43C2×D412yes—
191C42×C43C2×D412yes—
192C22⋊Dic43—28no—
193C22⋊Dic43—82no—
194#10621885—28no—
195#10621882—28no—
196(C2×D43)⋊C4—28no—
197(C2×D43)⋊C4—42no—
198(C2×D43)⋊C4—82no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203Dic43⋊C4—42no—
204Dic43⋊C4—82no—
205Dic43⋊C4—82no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—42no—
209C4×Dic43C22×C412yes—
210C4×Dic43#4112yes—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8C22×C412yes—
214C2×C43⋊C8C2×D412yes—
215C2×C43⋊C8C2×D412yes—
216C2×C43⋊C8#4112yes—
217#10621873—28no—
218C23×C43—22no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43—42no—
222C23×C43—42no—
223C23×C43C22×D411yescharacteristic
224C22×D43—22no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—42no—
228C22×D43—42no—
229Q8×C43—22no—
230Q8×C43—22no—
231Q8×C43—41no—
232Q8×C43—41no—
233D4×C43—22no—
234D4×C43—22no—
235D4×C43—41no—
236D4×C43—41no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—22no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—84no—
247C2×C4×C43#12712yes—
248C2×C4×C43#12712yes—
249C2×C4×C43C23×C411yescharacteristic
250C2×C4×C43C22×D411yescharacteristic
251C2×C4×C43C22×D411yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×Dic43—22no—
256C2×Dic43—22no—
257C2×Dic43—22no—
258C2×Dic43—22no—
259C2×Dic43—22no—
260C2×Dic43—22no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—42no—
266C2×Dic43—42no—
267C2×Dic43C4×D414yes—
268D172—22no—
269D172—22no—
270D172—41no—
271D172—41no—
272C4×D43—84no—
273Dic86—22no—
274Dic86—22no—
275Dic86—41no—
276Dic86—41no—
277C8×C43—22no—
278C8×C43C4×D414yes—
279C43⋊C8—22no—
280C43⋊C8C4×D414yes—
281#21694—431nomaximal, Sylow
282C22×C43—42no—
283C22×C43—44no—
284C22×C43—81no—
285C22×C43—81no—
286C22×C43—82no—
287C22×C43C2×C4×D411yescharacteristic
288C22×C43C2×#12711yescharacteristic
289C22×C43#53211yescharacteristic
290C22×C43#57712yes—
291C22×C43#57712yes—
292C2×D43—42no—
293C2×D43—44no—
294C2×D43—81no—
295C2×D43—81no—
296C2×D43—82no—
297C4×C43—22no—
298C4×C43—22no—
299C4×C43—42no—
300C4×C43—81no—
301C4×C43—81no—
302C4×C43C2×C4×D411yescharacteristic
303C4×C43C2×C4×D411yescharacteristic
304C4×C43#53211yescharacteristic
305C4×C43#53211yescharacteristic
306Dic43—24no—
307Dic43—42no—
308Dic43—42no—
309Dic43—81no—
310Dic43—81no—
311C2×#495—431no—
312C2×#442—431no—
313C2×#441—431no—
314C2×C4×SD16—431no—
315C2×C4×SD16D4311yescharacteristic, complemented
316#2326—431no—
317#2326—431no—
318#1972—431no—
319#1971—431no—
320#1968—431no—
321#1968—431no—
322#1959—431no—
323#1959—431no—
324#1844—431no—
325#1726—431no—
326C2×C43—82no—
327C2×C43C4×#12711yescharacteristic
328C2×C43#295212yes—
329C2×C43#315411yescharacteristic
330C2×C43#315411yescharacteristic
331C2×C43#317312yes—
332D43—82no—
333C22×Q16—431no—
334C22×SD16—431no—
335C22×SD16C2×D4311yescharacteristic
336C22×D8—431no—
337C2×#124—431no—
338C2×#124—431no—
339C2×C4×Q8—431no—
340C2×C4×Q8C2×D4311yescharacteristic
341C2×C4×D4—431no—
342C2×C4×D4C2×D4311yescharacteristic
343#495—438no—
344#442—864no—
345#441—864no—
346C4×SD16—24no—
347C4×SD16—864no—
348C2×#106—431no—
349C2×#106—431no—
350C2×#106—431no—
351C2×#106C2×D4311yescharacteristic
352C2×#103—431no—
353C2×#103—431no—
354C2×#103—431no—
355C2×#103C2×D4311yescharacteristic
356C2×#102—431no—
357C2×#102—431no—
358C2×#102—431no—
359C2×#102C2×D4311yescharacteristic
360C2×#97—431no—
361C2×#97—431no—
362C2×#97—431no—
363C2×C4×C8—431no—
364C2×C4×C8—431no—
365C2×C4×C8C2×D4311yescharacteristic
366#391—431no—
367#391—431no—
368#390—431no—
369#390—431no—
370#388—862no—
371#388—862no—
372#385—862no—
373#385—862no—
374#384—431no—
375#384—431no—
376#376—431no—
377C43#2169411yescharacteristic, complemented, Sylow
378C22×Q8—431no—
379C22×Q8C22×D4311yescharacteristic
380C22×D4—431no—
381C22×D4C22×D4311yescharacteristic
382C23×C4—21no—
383C23×C4—861no—
384C2×Q16—434no—
385C2×Q16—862no—
386C2×Q16—862no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16—434no—
390C2×SD16—862no—
391C2×SD16—862no—
392C2×SD16C4×D4314yescomplemented
393C2×D8—434no—
394C2×D8—862no—
395C2×D8—862no—
396C22×C8—431no—
397C22×C8—431no—
398C22×C8C22×D4311yescharacteristic
399C22×C8C22×D4311yescharacteristic
400#124—434no—
401#124—434no—
402#124—862no—
403#124—862no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—862no—
407C4×Q8—862no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—862no—
411C4×D4—862no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4—862no—
416C2×C4⋊C4—862no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4C22×D4311yescharacteristic
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—862no—
422C2×C22⋊C4—862no—
423C2×C22⋊C4—862no—
424C2×C22⋊C4—862no—
425C2×C22⋊C4—861no—
426C2×C42—21no—
427C2×C42—431no—
428C2×C42—431no—
429C2×C42—861no—
430C2×C42C22×D4311yescharacteristic
431#106—22no—
432#106—862no—
433#106—862no—
434#106—862no—
435#103—22no—
436#103—862no—
437#103—862no—
438#103—862no—
439#102—22no—
440#102—862no—
441#102—862no—
442#102—862no—
443#97—434no—
444#97—862no—
445#97—862no—
446C4×C8—434no—
447C4×C8—862no—
448C4×C8C22⋊D4314yes—
449#95—432no—
450#95—432no—
451#95—862no—
452#95—862no—
453C24—21no—
454C24—861no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—432no—
461C2×Q8—861no—
462C2×Q8—861no—
463C2×Q8—861no—
464C2×Q8—861no—
465C2×Q8—861no—
466C2×Q8C2×C4×D4312yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—432no—
473C2×D4—861no—
474C2×D4—861no—
475C2×D4—861no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4C2×C4×D4312yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—432no—
485C22×C4—432no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—3442no—
490C22×C4—1722no—
491C22×C4D4×D4311yescharacteristic
492C22×C4D4×D4311yescharacteristic
493C22×C4C23×D4311yescharacteristic
494Q16—868no—
495SD16—28no—
496SD16—868no—
497D8—868no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—432no—
501C2×C8—432no—
502C2×C8—432no—
503C2×C8—432no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8C2×C4×D4312yes—
507C2×C8#1062189812yes—
508C2×C8C2×C22⋊D4312yes—
509C2×C8C2×C22⋊D4312yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—3442no—
520C4⋊C4—3442no—
521C4⋊C4—1722no—
522C22⋊C4—42no—
523C22⋊C4—868no—
524C22⋊C4—868no—
525C22⋊C4—3442no—
526C22⋊C4—3442no—
527C22⋊C4—1722no—
528C42—42no—
529C42—432no—
530C42—432no—
531C42—861no—
532C42—861no—
533C42—1722no—
534C42D4×D4312yes—
535C42C2×C22⋊D4312yes—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—862no—
542C23—1721no—
543C23—1721no—
544C23—1722no—
545C23—1722no—
546C23C2×D4×D4311yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—862no—
552Q8—862no—
553Q8—1721no—
554Q8—1721no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—862no—
560D4—862no—
561D4—1721no—
562D4—1721no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—84no—
573C2×C4—434no—
574C2×C4—862no—
575C2×C4—862no—
576C2×C4—862no—
577C2×C4—862no—
578C2×C4—862no—
579C2×C4—862no—
580C2×C4—3444no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1722no—
584C2×C4—1722no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4#1306332112yes—
588C2×C4#1306332112yes—
589C2×C4C22×C4×D4311yescharacteristic
590C2×C4C2×D4×D4311yescharacteristic
591C2×C4C22×C22⋊D4311yescharacteristic
592C2×C4#41×D4311yescharacteristic
593C2×C4#41×D4311yescharacteristic
594C2×C4C2×#1062189811yescharacteristic
595C8—22no—
596C8—434no—
597C8—862no—
598C8C4×C22⋊D4314yes—
599C22—44no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—3441no—
605C22—3441no—
606C22—3442no—
607C22—1724no—
608C22—1722no—
609C22C4×D4×D4311yescharacteristic
610C22C2×#1306332111yescharacteristic
611C22#1458344611yescharacteristic
612C22#1458370512yes—
613C22#1458370612yes—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—864no—
620C4—3441no—
621C4—3441no—
622C4—1722no—
623C4—1722no—
624C4C4×D4×D4311yescharacteristic
625C4C2×C4×C22⋊D4311yescharacteristic
626C4#1458344511yescharacteristic
627C4#1458344611yescharacteristic
628C2—82no—
629C2—3442no—
630C2C4×#1306332111yescharacteristic
631C2#1914347312yes—
632C2#1914506511yescharacteristic
633C2#1914507211yescharacteristic
634C2#1914520812yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C4×SD16×C43C2
socleC23×C43C22×D4
prime radicalC22×#10621882C22
cosocleC22×C4×C43C24
hypercenterC2×C4×SD16D43
hypocenterC43#21694
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. C2×#14583004C2semidirect
4. C2×C4×#10621873C2semidirect
5. #19137307C2semidirect
6. #19137302C2semidirect
8. #19135463C2semidirect
9. #19135451C2semidirect
10. #19135447C2semidirect
11. #19135410C2semidirect
12. #19135408C2semidirect
13. #19134750C2semidirect
14. #19134281C2semidirect
15. C2×C4×SD16×C43C2semidirect
23. #14583280C4semidirect
65. C2×#10621885C2×C4semidirect
68. C2×#10621873C2×C4semidirect
75. #13063314C2×C4semidirect
77. #13063312C2×C4semidirect
108. C8×Dic43C2×C4semidirect
118. C2×SD16×C43C2×C4semidirect
315. C2×C4×SD16D43semidirect
377. C43#21694semidirect
392. C2×SD16C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832802
C2×#145830092
C2×#145830042
C2×C4×#106218732
#191373072
#191373022
#191354682
#191354632
#191354512
#191354472
#191354102
#191354082
#191347502
#191342812
C2×C4×SD16×C432
#2169443