Finite Group Databasebeta

← #25004080

Subgroups — #25004080

Order 11008.

1190 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063241C211yesmaximal, characteristic
2#19137175C211yesmaximal, characteristic
3C2×#14582800C211yesmaximal, characteristic
4#19134314C211yesmaximal, characteristic, complemented
5C2×#14582660C211yesmaximal, characteristic, complemented
6C2×#14582658C211yesmaximal, characteristic, complemented
7C2×C43×#428C211yesmaximal, characteristic, complemented
8C23×D172C2211yescharacteristic
9C22×#10621877C2211yescharacteristic
10C2×#13063284C2211yescharacteristic
11C22×(C2×D43)⋊C4—22no—
12C22×C4×Dic43C2211yescharacteristic
13C2×#13063241—22no—
14C2×#13063241—22no—
15C2×#13063241C414yescomplemented
16#14582801—22no—
17#14582800—22no—
18#14582800—22no—
19#14582800—21no—
20#14582800—21no—
21#14582754—22no—
22#14582661—22no—
23#14582660—22no—
24#14582658—22no—
25C22×M16×C43C2211yescharacteristic
26C2×#117×C43C2211yescharacteristic
27C22×C4⋊C4×C43C2211yescharacteristic
28C43×#428—22no—
29C43×#428—22no—
30C2×#107×C43—21no—
31C2×#106×C43—21no—
32C24×D43D412yes—
33C22×D172—21no—
34C22×D172—21no—
35C22×D172—21no—
36C22×D172—21no—
37C22×D172—21no—
38C22×D172—21no—
39C22×D172—21no—
40C22×D172—21no—
41C22×D172—21no—
42C22×D172—21no—
43C22×D172—21no—
44C22×D172—21no—
45C22×D172C2×C412yes—
46C22×D172Q812yes—
47C23×Dic43—21no—
48C2×#10621877—21no—
49C2×#10621877—21no—
50C2×#10621877—21no—
51C2×#10621877—21no—
52C2×#10621877—21no—
53C2×#10621877C2×C412yes—
54C22×C43⋊C8—21no—
55#13063284—21no—
56#13063284—21no—
57#13063284—21no—
58#13063284—21no—
59C2×C4⋊Dic43D412yes—
60C2×C22⋊Dic43—21no—
61C2×(C2×D43)⋊C4—28no—
62C2×(C2×D43)⋊C4—24no—
63C2×(C2×D43)⋊C4—24no—
64C2×(C2×D43)⋊C4—48no—
65C2×C4×Dic43—21no—
66C2×C4×Dic43—21no—
67C2×C4×Dic43—21no—
68C2×C4×Dic43—21no—
69C2×C4×Dic43—21no—
70C2×C4×Dic43—21no—
71C2×C4×Dic43D412yes—
72#13063241—28no—
73C23×C4×C43—21no—
74C23×C4×C43C2311yescharacteristic
75C2×M16×C43—21no—
76C2×M16×C43—21no—
77C2×M16×C43—21no—
78C2×M16×C43C2×C412yes—
79C2×M16×C43D412yes—
80C2×M16×C43D412yes—
81C22×C8×C43—21no—
82#117×C43—21no—
83#117×C43—21no—
84#117×C43—21no—
85#117×C43—21no—
86C2×C4⋊C4×C43—21no—
87C2×C4⋊C4×C43—21no—
88C2×C4⋊C4×C43—21no—
89C2×C4⋊C4×C43—21no—
90C2×C4⋊C4×C43—21no—
91C2×C4⋊C4×C43D412yes—
92C2×C4⋊C4×C43D411yescharacteristic
93C2×C4⋊C4×C43D411yescharacteristic
94C2×C22⋊C4×C43—21no—
95C2×C42×C43—21no—
96#107×C43—24no—
97#106×C43—24no—
98C24×C43C2×D411yescharacteristic
99C23×D43—22no—
100C23×D43—22no—
101C23×D43—22no—
102C23×D43—22no—
103C23×D43—22no—
104C23×D43—22no—
105C23×D43—44no—
106C23×D43—44no—
107C23×D43C4⋊C414yes—
108C22×C4×C43—21no—
109C22×C4×C43—21no—
110C22×C4×C43—22no—
111C22×C4×C43—44no—
112C22×C4×C43—41no—
113C22×C4×C43—41no—
114C22×C4×C43C22×C411yescharacteristic
115C22×C4×C43C2×D411yescharacteristic
116C22×C4×C43C2×D411yescharacteristic
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43C2×D411yescharacteristic
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×Q811yescharacteristic
122C22×C4×C43#4111yescharacteristic
123C22×C4×C43#4111yescharacteristic
124C22×C4×C43#4111yescharacteristic
125C22×C4×C43#4111yescharacteristic
126C22×C4×C43#4111yescharacteristic
127C22×C4×C43#4111yescharacteristic
128C22×Dic43—24no—
129C22×Dic43—22no—
130C22×Dic43—22no—
131C22×Dic43—42no—
132C22×Dic43—41no—
133C22×Dic43—41no—
134C2×D172—22no—
135C2×D172—22no—
136C2×D172—22no—
137C2×D172—22no—
138C2×D172—22no—
139C2×D172—22no—
140C2×D172—22no—
141C2×D172—22no—
142C2×D172—22no—
143C2×D172—22no—
144C2×D172—22no—
145C2×D172—22no—
146C2×D172—41no—
147C2×D172—41no—
148C2×D172—41no—
149C2×D172—41no—
150C2×D172—41no—
151C2×D172—41no—
152C2×D172—41no—
153C2×D172—41no—
154C2×D172—41no—
155C2×D172—41no—
156C2×D172—41no—
157C2×D172—41no—
158C2×D172—41no—
159C2×D172—41no—
160C2×D172—41no—
161C2×D172—41no—
162M16×C43—22no—
163M16×C43—22no—
164M16×C43—22no—
165M16×C43C22⋊C414yescomplemented
166C2×C8×C43—22no—
167C2×C8×C43—22no—
168C2×C8×C43—22no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—22no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—22no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—41no—
176C4⋊C4×C43—41no—
177C22⋊C4×C43—41no—
178C22⋊C4×C43—41no—
179C42×C43—41no—
180C42×C43—41no—
181C22⋊Dic43—41no—
182C22⋊Dic43—41no—
183(C2×D43)⋊C4—416no—
184(C2×D43)⋊C4—84no—
185(C2×D43)⋊C4—84no—
186C4⋊Dic43—22no—
187C4⋊Dic43—22no—
188C4×Dic43—22no—
189C4×Dic43—22no—
190C4×Dic43—41no—
191C4×Dic43—41no—
192C4×Dic43—41no—
193C4×Dic43—41no—
194#10621877—22no—
195#10621877—22no—
196#10621877—41no—
197#10621877—41no—
198C2×C43⋊C8—22no—
199C2×C43⋊C8—41no—
200C2×C43⋊C8—41no—
201C23×C43—22no—
202C23×C43—22no—
203C23×C43C2×C4⋊C411yescharacteristic
204C23×C43#12011yescharacteristic
205C23×C43#12011yescharacteristic
206C23×C43#12111yescharacteristic
207C23×C43#12111yescharacteristic
208C23×C43#12311yescharacteristic
209C23×C43#12311yescharacteristic
210C22×D43—24no—
211C22×D43—24no—
212C22×D43—24no—
213C22×D43—48no—
214C22×D43—44no—
215C22×D43—44no—
216C22×D43—48no—
217C22×D43—42no—
218C22×D43—42no—
219C22×D43—42no—
220C22×D43—42no—
221C22×D43—44no—
222C22×D43—44no—
223C22×D43—84no—
224C22×D43—82no—
225C22×D43—82no—
226C22×D43—84no—
227C2×C4×C43—44no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—42no—
231C2×C4×C43—42no—
232C2×C4×C43—42no—
233C2×C4×C43—82no—
234C2×C4×C43—82no—
235C2×C4×C43C2×C22⋊C411yescharacteristic
236C2×C4×C43#11711yescharacteristic
237C2×C4×C43C4×D411yescharacteristic
238C2×C4×C43C4×D411yescharacteristic
239C2×C4×C43C4×D411yescharacteristic
240C2×C4×C43C4×D411yescharacteristic
241C2×C4×C43#12011yescharacteristic
242C2×C4×C43#12111yescharacteristic
243C2×C4×C43#12111yescharacteristic
244C2×C4×C43#12111yescharacteristic
245C2×C4×C43#12111yescharacteristic
246C2×C4×C43#12111yescharacteristic
247C2×C4×C43#12111yescharacteristic
248C2×C4×C43C22⋊Q811yescharacteristic
249C2×C4×C43C22⋊Q811yescharacteristic
250C2×C4×C43C22⋊Q811yescharacteristic
251C2×C4×C43C22⋊Q811yescharacteristic
252C2×C4×C43C22⋊Q811yescharacteristic
253C2×C4×C43C22⋊Q811yescharacteristic
254C2×C4×C43#12311yescharacteristic
255C2×C4×C43#12311yescharacteristic
256C2×C4×C43#12311yescharacteristic
257C2×C4×C43#12411yescharacteristic
258C2×C4×C43#12411yescharacteristic
259C2×C4×C43#12411yescharacteristic
260C2×C4×C43#12611yescharacteristic
261C2×C4×C43#12611yescharacteristic
262C2×C4×C43#12711yescharacteristic
263C2×Dic43—24no—
264C2×Dic43—44no—
265C2×Dic43—42no—
266C2×Dic43—44no—
267C2×Dic43—42no—
268C2×Dic43—42no—
269C2×Dic43—42no—
270C2×Dic43—42no—
271D172—42no—
272D172—42no—
273D172—42no—
274D172—42no—
275D172—42no—
276D172—42no—
277D172—42no—
278D172—42no—
279C8×C43—24no—
280C43⋊C8—42no—
281#11135—431nomaximal, Sylow
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—21no—
285C22×C43—21no—
286C22×C43—21no—
287C22×C43—21no—
288C22×C43—22no—
289C22×C43—22no—
290C22×C43#38011yescharacteristic
291C22×C43#38511yescharacteristic
292C22×C43#38511yescharacteristic
293C22×C43#39211yescharacteristic
294C22×C43#39411yescharacteristic
295C22×C43#39411yescharacteristic
296C22×C43#39711yescharacteristic
297C22×C43#45312yes—
298C22×C43#45512yes—
299C2×D43—48no—
300C2×D43—48no—
301C2×D43—44no—
302C2×D43—84no—
303C2×D43—84no—
304C2×D43—82no—
305C2×D43—82no—
306C2×D43—84no—
307C2×D43—84no—
308C4×C43—81no—
309C4×C43—82no—
310C4×C43—81no—
311C4×C43#38611yescharacteristic
312C4×C43#38811yescharacteristic
313C4×C43#38811yescharacteristic
314C4×C43#39011yescharacteristic
315C4×C43#39311yescharacteristic
316C4×C43#39611yescharacteristic
317C4×C43#39611yescharacteristic
318C4×C43#39911yescharacteristic
319Dic43—44no—
320Dic43—82no—
321C2×#428D4311yescharacteristic, complemented
322C22×#100—431no—
323#2309—431no—
324C2×#390—431no—
325#1736—431no—
326C2×#339—431no—
327C2×#339—431no—
328C2×C43—21no—
329C2×C43—21no—
330C2×C43—21no—
331C2×C43—21no—
332C2×C43#143911yescharacteristic
333C2×C43#185612yes—
334C2×C43#200211yescharacteristic
335C2×C43#200411yescharacteristic
336C2×C43#203712yes—
337D43—84no—
338C23×D4—431no—
339C22×M16—431no—
340C22×M16C2×D4311yescharacteristic
341C2×#117—431no—
342C2×#117C2×D4311yescharacteristic
343C22×C4⋊C4C2×D4311yescharacteristic
344C22×C22⋊C4—862no—
345C22×C42—431no—
346#428—22no—
347#428—22no—
348C2×#107—21no—
349C2×#106—21no—
350C2×#100—434no—
351C2×#100—862no—
352C2×#100—862no—
353#390—862no—
354#390—862no—
355#390—861no—
356#390—861no—
357#390—862no—
358#379—862no—
359#340—862no—
360#339—862no—
361#339—862no—
362C43#1113511yescharacteristic, complemented, Sylow
363C25—432no—
364C22×D4—432no—
365C22×D4—432no—
366C22×D4—861no—
367C22×D4—861no—
368C22×D4—861no—
369C22×D4—861no—
370C22×D4—861no—
371C22×D4—861no—
372C22×D4—861no—
373C22×D4—861no—
374C22×D4—861no—
375C22×D4—861no—
376C22×D4—861no—
377C22×D4—861no—
378C23×C4—21no—
379C23×C4—861no—
380C23×C4C22×D4311yescharacteristic
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—432no—
385C2×M16—861no—
386C2×M16—861no—
387C2×M16—861no—
388C2×M16—861no—
389C2×M16—861no—
390C2×M16C4×D4312yes—
391C2×M16D17212yes—
392C2×M16C22⋊D4312yes—
393C22×C8—21no—
394C22×C8—861no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—861no—
400#117—861no—
401#117—861no—
402#117—861no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—432no—
409C2×C4⋊C4C22⋊D4312yes—
410C2×C4⋊C4C22⋊D4311yescharacteristic
411C2×C4⋊C4C22⋊D4311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—864no—
414C2×C22⋊C4—864no—
415C2×C22⋊C4—868no—
416C2×C22⋊C4—861no—
417C2×C22⋊C4—1728no—
418C2×C42—21no—
419C2×C42—432no—
420C2×C42—861no—
421C2×C42—861no—
422C2×C42—861no—
423C2×C42—861no—
424C2×C42—861no—
425C2×C42—861no—
426#107—24no—
427#106—24no—
428#100—868no—
429C24—434no—
430C24—862no—
431C24—862no—
432C24—862no—
433C24—862no—
434C24—862no—
435C24—862no—
436C24—1724no—
437C24—1724no—
438C24D4×D4311yescharacteristic
439C2×D4—862no—
440C2×D4—862no—
441C2×D4—862no—
442C2×D4—862no—
443C2×D4—862no—
444C2×D4—862no—
445C2×D4—862no—
446C2×D4—862no—
447C2×D4—862no—
448C2×D4—862no—
449C2×D4—862no—
450C2×D4—862no—
451C2×D4—1721no—
452C2×D4—1721no—
453C2×D4—1721no—
454C2×D4—1721no—
455C2×D4—1721no—
456C2×D4—1721no—
457C2×D4—1721no—
458C2×D4—1721no—
459C2×D4—1721no—
460C2×D4—1721no—
461C2×D4—1721no—
462C2×D4—1721no—
463C2×D4—1721no—
464C2×D4—1721no—
465C2×D4—1721no—
466C2×D4—1721no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—864no—
474C22×C4—862no—
475C22×C4—862no—
476C22×C4—1722no—
477C22×C4—1721no—
478C22×C4—1721no—
479C22×C4C2×C4×D4311yescharacteristic
480C22×C4C2×D17211yescharacteristic
481C22×C4#1062189811yescharacteristic
482C22×C4#1062189811yescharacteristic
483C22×C4D4×D4311yescharacteristic
484C22×C4D4×D4311yescharacteristic
485C22×C4#1062190011yescharacteristic
486C22×C4Q8×D4311yescharacteristic
487C22×C4#1062190211yescharacteristic
488C22×C4#1062190211yescharacteristic
489C22×C4#1062190211yescharacteristic
490C22×C4C2×C22⋊D4311yescharacteristic
491C22×C4C2×C22⋊D4311yescharacteristic
492C22×C4C2×C22⋊D4311yescharacteristic
493M16—22no—
494M16—22no—
495M16—22no—
496M16—862no—
497M16—862no—
498M16—1721no—
499M16—1721no—
500M16(C2×D43)⋊C414yescomplemented
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—862no—
505C2×C8—1721no—
506C2×C8—1721no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—862no—
516C4⋊C4—862no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—3444no—
520C22⋊C4—3444no—
521C22⋊C4—17216no—
522C22⋊C4—1721no—
523C22⋊C4—1721no—
524C42—41no—
525C42—41no—
526C42—862no—
527C42—862no—
528C42—1721no—
529C42—1721no—
530C42—1721no—
531C42—1721no—
532C23—22no—
533C23—22no—
534C23—864no—
535C23—864no—
536C23—864no—
537C23—3444no—
538C23—3442no—
539C23—3442no—
540C23—3444no—
541C23—1724no—
542C23—1728no—
543C23—1724no—
544C23—1728no—
545C23—1722no—
546C23—1724no—
547C23—1722no—
548C23—1722no—
549C23—1724no—
550C23—1722no—
551C23C4⋊C4×D4311yescharacteristic
552C23#1306329311yescharacteristic
553C23#1306329311yescharacteristic
554C23#1306329611yescharacteristic
555C23#1306330011yescharacteristic
556C23#1306331011yescharacteristic
557C23#1306331011yescharacteristic
558D4—1722no—
559D4—1722no—
560D4—1722no—
561D4—1722no—
562D4—1722no—
563D4—1722no—
564D4—1722no—
565D4—1722no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—864no—
575C2×C4—1722no—
576C2×C4—1722no—
577C2×C4—1722no—
578C2×C4—1724no—
579C2×C4—1724no—
580C2×C4—1722no—
581C2×C4—1722no—
582C2×C4C2×(C2×D43)⋊C411yescharacteristic
583C2×C4#1306328311yescharacteristic
584C2×C4C4×C22⋊D4311yescharacteristic
585C2×C4C4×C22⋊D4311yescharacteristic
586C2×C4#1306328711yescharacteristic
587C2×C4#1306328711yescharacteristic
588C2×C4#1306329411yescharacteristic
589C2×C4#1306329711yescharacteristic
590C2×C4#1306329711yescharacteristic
591C2×C4#1306329911yescharacteristic
592C2×C4#1306329911yescharacteristic
593C2×C4#1306330011yescharacteristic
594C2×C4#1306330011yescharacteristic
595C2×C4#1306330111yescharacteristic
596C2×C4#1306330111yescharacteristic
597C2×C4#1306330111yescharacteristic
598C2×C4#1306330311yescharacteristic
599C2×C4#1306330311yescharacteristic
600C2×C4#1306330511yescharacteristic
601C2×C4#1306330811yescharacteristic
602C2×C4#1306331011yescharacteristic
603C2×C4#1306331011yescharacteristic
604C2×C4#1306331411yescharacteristic
605C2×C4#1306331411yescharacteristic
606C2×C4#1306331411yescharacteristic
607C2×C4#1306332011yescharacteristic
608C2×C4#1306332011yescharacteristic
609C2×C4#1306332111yescharacteristic
610C8—24no—
611C8—1722no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—3444no—
621C22—3444no—
622C22—3444no—
623C22—3444no—
624C22—3442no—
625C22—3442no—
626C22—1728no—
627C22—1728no—
628C22—1724no—
629C22#1458275611yescharacteristic
630C22#1458277011yescharacteristic
631C22#1458277011yescharacteristic
632C22#1458280711yescharacteristic
633C22#1458281611yescharacteristic
634C22#1458281611yescharacteristic
635C22#1458282511yescharacteristic
636C22#1458306512yes—
637C22#1458307512yes—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—3442no—
642C4—1724no—
643C4#1458277811yescharacteristic
644C4#1458278911yescharacteristic
645C4#1458278911yescharacteristic
646C4#1458280011yescharacteristic
647C4#1458281011yescharacteristic
648C4#1458282111yescharacteristic
649C4#1458282111yescharacteristic
650C4#1458283411yescharacteristic
651C2—21no—
652C2—21no—
653C2—21no—
654C2—21no—
655C2—3444no—
656C2#1913283711yescharacteristic
657C2#1913480312yes—
658C2#1913560711yescharacteristic
659C2#1913562111yescharacteristic
660C2#1913580712yes—
subgroupG/H
centerC23#13063293
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC2×C43×#428C2
socleC23×C43#120
prime radicalC23×D172C22
cosocleC23×C4×C43C23
hypercenterC2×#428D43
hypocenterC43#11135
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #19134314C2semidirect
5. C2×#14582660C2semidirect
6. C2×#14582658C2semidirect
7. C2×C43×#428C2semidirect
15. C2×#13063241C4semidirect
165. M16×C43C22⋊C4semidirect
321. C2×#428D43semidirect
362. C43#11135semidirect
500. M16(C2×D43)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130632412
#191371752
C2×#145828002
#191343142
C2×#145826602
C2×#145826582
C2×C43×#4282
#1113543