Finite Group Databasebeta

← #25098305

Subgroups — #25098305

Order 11008.

1894 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D43C211yesmaximal, characteristic
2C2×C22⋊Q8×D43C211yesmaximal, characteristic
3C2×#14582906C212yesmaximal, complemented
4#19136977C211yesmaximal, characteristic
5#19135029C211yesmaximal, characteristic, complemented
6#19135028C211yesmaximal, characteristic, complemented
7#19135026C211yesmaximal, characteristic, complemented
8#19135017C212yesmaximal, complemented
9#19135015C212yesmaximal, complemented
10#19134313C212yesmaximal, complemented
11C43×#1894C211yesmaximal, characteristic, complemented
12C22×Q8×D43C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C22×#10621877C2211yescharacteristic
15C22×#10621872C2212yes—
16C2×M16×D43—24no—
17C2×M16×D43—28no—
18C22×C8×D43C2212yes—
19C2×#13063305C2211yescharacteristic
20C2×#13063304C2211yescharacteristic
21C2×#13063303—21no—
22C2×#13063302—21no—
23C2×#13063301C2211yescharacteristic
24C22⋊Q8×D43—24no—
25C22⋊Q8×D43C418yescomplemented
26C2×#13063283C2212yes—
27C2×C4⋊C4×D43—21no—
28C2×C4⋊C4×D43C2211yescharacteristic
29C2×C22⋊C4×D43—21no—
30#14582906—28no—
31C2×#13063250C2212yescomplemented
32C2×#13063249C2212yescomplemented
33C2×#13063248C2212yescomplemented
34#14582761—21no—
35#14582759C2211yescharacteristic
36#14582756C2212yes—
37#14582754—21no—
38C22⋊C4×Dic43—21no—
39#14582746—21no—
40#14582662C2212yescomplemented
41#14582661C2212yescomplemented
42#14582661C2212yescomplemented
43C22×M16×C43C2211yescharacteristic
44C2×C22⋊Q8×C43C2211yescharacteristic
45C2×#103×C43C2212yescomplemented
46C43×#380C2211yescharacteristic
47C43×#340C2212yescomplemented
48C24×D43D411yescharacteristic
49C22×Dic86—21no—
50C22×Dic86C2311yescharacteristic
51C2×Q8×D43—22no—
52C2×Q8×D43—24no—
53C2×Q8×D43—24no—
54C2×Q8×D43C2×C414yes—
55C23×Dic43D411yescharacteristic
56C22×C4×D43—24no—
57C22×C4×D43—24no—
58C22×C4×D43—21no—
59C22×C4×D43—21no—
60C22×C4×D43C2×C414yes—
61C22×C4×D43D411yescharacteristic
62C22×C4×D43D411yescharacteristic
63C22×C4×D43D414yes—
64C22×C4×D43C2311yescharacteristic
65C22×C4×D43C2311yescharacteristic
66C2×#10621877—22no—
67C2×#10621877—24no—
68C2×#10621872—24no—
69C2×#10621872—24no—
70C2×#10621872D418yes—
71M16×D43—232no—
72C22×C43⋊C8C2312yes—
73C2×C8×D43—24no—
74C2×C8×D43—24no—
75C2×C8×D43—24no—
76#13063305—22no—
77#13063305C2×C414yes—
78#13063304—22no—
79#13063304C2×C414yes—
80#13063303—28no—
81#13063302—28no—
82#13063301—22no—
83#13063301C2×C414yes—
84#13063283—24no—
85#13063283—24no—
86C2×C4⋊Dic43—21no—
87C2×C4⋊Dic43C2312yes—
88C2×C4⋊Dic43C2311yescharacteristic
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C4⋊C4×D43—28no—
93C4⋊C4×D43—22no—
94C4⋊C4×D43C2×C414yes—
95C2×C22⋊Dic43—21no—
96C2×C22⋊Dic43—21no—
97C2×(C2×D43)⋊C4—21no—
98C2×(C2×D43)⋊C4—21no—
99C2×(C2×D43)⋊C4—22no—
100C2×(C2×D43)⋊C4—22no—
101C22⋊C4×D43—28no—
102C2×C4×Dic43—22no—
103#13063250—24no—
104#13063249—24no—
105#13063248—24no—
106C43⋊#95—21no—
107C43⋊#95—22no—
108C22×Q8×C43C2311yescharacteristic
109C23×C4×C43C2311yescharacteristic
110C2×M16×C43—22no—
111C2×M16×C43—24no—
112C22×C8×C43C2312yes—
113C22⋊Q8×C43—22no—
114C22⋊Q8×C43C2×C414yes—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2312yes—
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C22⋊C4×C43—21no—
119C2×C22⋊C4×C43—21no—
120#103×C43—24no—
121#95×C43—21no—
122C24×C43C2×D411yescharacteristic
123C23×D43—24no—
124C23×D43—24no—
125C23×D43—41no—
126C23×D43—41no—
127C23×D43C22⋊C414yes—
128C23×D43C2×D411yescharacteristic
129C23×D43C2×D411yescharacteristic
130C2×Q8×C43—21no—
131C2×Q8×C43—22no—
132C2×Q8×C43—22no—
133C2×Q8×C43C22×C412yes—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—22no—
139C22×C4×C43C22×C412yes—
140C22×C4×C43C2×D412yes—
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2411yescharacteristic
143C22×Dic43—24no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—22no—
147C22×Dic43C22⋊C414yes—
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150Q8×D43—216no—
151Q8×D43—44no—
152Q8×D43—44no—
153C2×C4×D43—22no—
154C2×C4×D43—22no—
155C2×C4×D43—24no—
156C2×C4×D43—216no—
157C2×C4×D43—28no—
158C2×C4×D43—28no—
159C2×C4×D43—24no—
160C2×C4×D43—24no—
161C2×C4×D43—24no—
162C2×C4×D43—42no—
163C2×C4×D43—42no—
164C2×C4×D43—44no—
165C2×C4×D43C22⋊C414yes—
166C2×C4×D43C22⋊C414yes—
167C2×C4×D43C22×C412yes—
168C2×C4×D43C22×C412yes—
169C2×C4×D43C2×D412yes—
170C2×C4×D43C2×D412yes—
171C2×C4×D43C2×D414yes—
172C2×C4×D43#4114yes—
173C2×Dic86—22no—
174C2×Dic86—22no—
175C2×Dic86—21no—
176C2×Dic86—24no—
177C2×Dic86—44no—
178C2×Dic86C22×C412yes—
179M16×C43—28no—
180C2×C8×C43—22no—
181C2×C8×C43—22no—
182C2×C8×C43C2×D414yes—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—22no—
186C4⋊C4×C43—22no—
187C4⋊C4×C43C22×C412yes—
188C22⋊C4×C43—24no—
189C22⋊C4×C43—42no—
190C22⋊Dic43—24no—
191C22⋊Dic43—42no—
192(C2×D43)⋊C4—24no—
193(C2×D43)⋊C4—24no—
194(C2×D43)⋊C4—44no—
195(C2×D43)⋊C4—44no—
196C4⋊Dic43—21no—
197C4⋊Dic43—24no—
198C4⋊Dic43—22no—
199C4⋊Dic43—22no—
200C4⋊Dic43C22×C412yes—
201Dic43⋊C4—24no—
202Dic43⋊C4—24no—
203Dic43⋊C4—24no—
204C4×Dic43—44no—
205#10621877—28no—
206C2×C43⋊C8—22no—
207C2×C43⋊C8—22no—
208C2×C43⋊C8C2×D414yes—
209#10621872—216no—
210C8×D43—216no—
211C23×C43—22no—
212C23×C43—22no—
213C23×C43—21no—
214C23×C43—21no—
215C23×C43C2×C22⋊C412yes—
216C23×C43C22×D411yescharacteristic
217C22×D43—22no—
218C22×D43—22no—
219C22×D43—21no—
220C22×D43—21no—
221C22×D43—21no—
222C22×D43—21no—
223C22×D43—28no—
224C22×D43—28no—
225C22×D43—44no—
226C22×D43—48no—
227C22×D43—48no—
228C22×D43C2×C22⋊C412yes—
229C22×D43C2×C22⋊C412yes—
230C22×D43#12114yes—
231C22×D43#13714yes—
232Q8×C43—24no—
233Q8×C43—41no—
234Q8×C43—41no—
235C2×C4×C43—22no—
236C2×C4×C43—24no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—22no—
240C2×C4×C43—22no—
241C2×C4×C43—42no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—42no—
247C2×C4×C43C2×C22⋊C412yes—
248C2×C4×C43C23×C411yescharacteristic
249C2×C4×C43C22×D412yes—
250C2×C4×C43C22×D411yescharacteristic
251C2×C4×C43C2×#4112yes—
252C2×Dic43—22no—
253C2×Dic43—28no—
254C2×Dic43—22no—
255C2×Dic43—22no—
256C2×Dic43—42no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43C2×C22⋊C412yes—
263C2×Dic43C2×C22⋊C412yes—
264C2×Dic43#12114yes—
265C4×D43—28no—
266C4×D43—24no—
267C4×D43—22no—
268C4×D43—22no—
269C4×D43—28no—
270C4×D43—48no—
271C4×D43—48no—
272C4×D43C4×D418yes—
273C4×D43#12014yes—
274C4×D43#12314yes—
275Dic86—24no—
276Dic86—41no—
277Dic86—41no—
278Dic86—48no—
279C8×C43—24no—
280C43⋊C8—24no—
281#20972—431nomaximal, Sylow
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—22no—
285C22×C43—42no—
286C22×C43—42no—
287C22×C43C22×C22⋊C411yescharacteristic
288C22×C43C2×#12112yes—
289C22×C43C2×#13712yes—
290C22×C43#57511yescharacteristic
291C22×C43#57511yescharacteristic
292C2×D43—22no—
293C2×D43—22no—
294C2×D43—22no—
295C2×D43—22no—
296C2×D43—24no—
297C2×D43—24no—
298C2×D43—24no—
299C2×D43—24no—
300C2×D43—416no—
301C2×D43#38614yes—
302C2×D43#41914yescomplemented
303C4×C43—22no—
304C4×C43—42no—
305C4×C43—42no—
306C4×C43—82no—
307C4×C43C2×C4×D412yes—
308C4×C43C2×#12011yescharacteristic
309C4×C43C2×#12311yescharacteristic
310Dic43—22no—
311Dic43—42no—
312Dic43—42no—
313Dic43—82no—
314Dic43#38614yes—
315C23×M16—431no—
316C22×C22⋊Q8—431no—
317C2×#419—432no—
318#2280—431no—
319#1894—431no—
320#1894—431no—
321#1894—431no—
322#1894D4311yescharacteristic, complemented
323#1892—432no—
324#1892—432no—
325#1736—432no—
326C2×C43—42no—
327C2×C43C2×#38611yescharacteristic
328C2×C43C2×#41911yescharacteristic
329C2×C43#289611yescharacteristic
330C2×C43#305612yes—
331C2×C43#305612yes—
332D43—24no—
333D43—24no—
334C23×Q8—431no—
335C24×C4—431no—
336C22×M16—431no—
337C22×M16—432no—
338C22×M16—868no—
339C22×M16—864no—
340C22×M16C2×D4311yescharacteristic
341C23×C8—432no—
342C2×C22⋊Q8—431no—
343C2×C22⋊Q8—431no—
344C2×C22⋊Q8—438no—
345C2×C22⋊Q8—431no—
346C2×C22⋊Q8—861no—
347C2×C22⋊Q8—861no—
348C2×C22⋊Q8—864no—
349C2×C22⋊Q8C2×D4311yescharacteristic
350C2×#117—432no—
351C22×C4⋊C4—431no—
352C22×C4⋊C4—861no—
353C22×C22⋊C4—861no—
354#419—868no—
355C2×#103—432no—
356C2×#103—432no—
357C2×#103—432no—
358C2×#103C2×D4312yescomplemented
359#381—861no—
360#380—431no—
361#380—432no—
362#380C2×D4311yescharacteristic
363#379—861no—
364C4×C22⋊C4—861no—
365C4×C22⋊C4—861no—
366#340—432no—
367#340—432no—
368#340—432no—
369#340C2×D4312yescomplemented
370C43#2097211yescharacteristic, complemented, Sylow
371C25—431no—
372C22×Q8—434no—
373C22×Q8—431no—
374C22×Q8—861no—
375C22×Q8—862no—
376C22×Q8—864no—
377C22×Q8—864no—
378C22×Q8C22×D4311yescharacteristic
379C23×C4—431no—
380C23×C4—431no—
381C23×C4—434no—
382C23×C4—431no—
383C23×C4—434no—
384C23×C4—431no—
385C23×C4—431no—
386C23×C4—864no—
387C23×C4—864no—
388C23×C4—861no—
389C23×C4—861no—
390C23×C4C22×D4311yescharacteristic
391C2×M16—22no—
392C2×M16—24no—
393C2×M16—438no—
394C2×M16—864no—
395C2×M16—864no—
396C2×M16—864no—
397C2×M16—8632no—
398C2×M16—862no—
399C22×C8—432no—
400C22×C8—864no—
401C22×C8—864no—
402C22×C8—864no—
403C22×C8C22×D4312yes—
404C22⋊Q8—22no—
405C22⋊Q8—434no—
406C22⋊Q8—434no—
407C22⋊Q8—434no—
408C22⋊Q8—868no—
409C22⋊Q8—868no—
410C22⋊Q8—862no—
411C22⋊Q8—862no—
412C22⋊Q8—862no—
413C22⋊Q8C4×D4314yes—
414#117—864no—
415#117—864no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—434no—
419C2×C4⋊C4—432no—
420C2×C4⋊C4—862no—
421C2×C4⋊C4—868no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4C22×D4312yes—
427C2×C4⋊C4C22×D4311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—868no—
431C2×C22⋊C4—861no—
432C2×C22⋊C4—861no—
433C2×C22⋊C4—861no—
434C2×C22⋊C4—862no—
435C2×C22⋊C4—862no—
436C2×C22⋊C4—861no—
437C2×C42—862no—
438#103—24no—
439#103—864no—
440#103—864no—
441#103—864no—
442#95—21no—
443#95—862no—
444#95—861no—
445C24—431no—
446C24—431no—
447C24—434no—
448C24—864no—
449C24—864no—
450C24—1721no—
451C24—1721no—
452C24D4×D4311yescharacteristic
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—432no—
457C2×Q8—864no—
458C2×Q8—8616no—
459C2×Q8—861no—
460C2×Q8—862no—
461C2×Q8—862no—
462C2×Q8—1724no—
463C2×Q8—1724no—
464C2×Q8—1724no—
465C2×Q8C2×C4×D4312yes—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—431no—
472C22×C4—432no—
473C22×C4—432no—
474C22×C4—434no—
475C22×C4—434no—
476C22×C4—434no—
477C22×C4—431no—
478C22×C4—434no—
479C22×C4—434no—
480C22×C4—432no—
481C22×C4—432no—
482C22×C4—868no—
483C22×C4—864no—
484C22×C4—8616no—
485C22×C4—868no—
486C22×C4—864no—
487C22×C4—864no—
488C22×C4—864no—
489C22×C4—864no—
490C22×C4—862no—
491C22×C4—862no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—862no—
495C22×C4—1724no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4C2×C4×D4312yes—
499C22×C4D4×D4311yescharacteristic
500C22×C4D4×D4312yes—
501C22×C4C23×D4311yescharacteristic
502M16—28no—
503M16—8616no—
504M16—868no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—434no—
508C2×C8—8616no—
509C2×C8—862no—
510C2×C8—862no—
511C2×C8D4×D4314yes—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—432no—
517C4⋊C4—864no—
518C4⋊C4—864no—
519C4⋊C4—864no—
520C4⋊C4—864no—
521C4⋊C4—861no—
522C4⋊C4—862no—
523C4⋊C4—862no—
524C4⋊C4C2×C4×D4312yes—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—864no—
528C22⋊C4—864no—
529C22⋊C4—864no—
530C22⋊C4—1724no—
531C22⋊C4—1724no—
532C22⋊C4—1722no—
533C42—1724no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—432no—
539C23—432no—
540C23—434no—
541C23—434no—
542C23—868no—
543C23—868no—
544C23—862no—
545C23—862no—
546C23—861no—
547C23—861no—
548C23—861no—
549C23—861no—
550C23—1724no—
551C23—1728no—
552C23—1728no—
553C23C22⋊C4×D4312yes—
554C23C2×D4×D4311yescharacteristic
555Q8—24no—
556Q8—41no—
557Q8—41no—
558Q8—864no—
559Q8—1728no—
560Q8—1721no—
561Q8—1721no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—432no—
575C2×C4—434no—
576C2×C4—438no—
577C2×C4—434no—
578C2×C4—434no—
579C2×C4—432no—
580C2×C4—868no—
581C2×C4—868no—
582C2×C4—868no—
583C2×C4—862no—
584C2×C4—864no—
585C2×C4—862no—
586C2×C4—862no—
587C2×C4—862no—
588C2×C4—862no—
589C2×C4—1728no—
590C2×C4—1728no—
591C2×C4—1722no—
592C2×C4—1722no—
593C2×C4—1722no—
594C2×C4—1722no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4C22⋊C4×D4312yes—
598C2×C4C22×C4×D4311yescharacteristic
599C2×C4C2×D4×D4312yes—
600C2×C4C2×D4×D4311yescharacteristic
601C2×C4#41×D4312yes—
602C8—24no—
603C8—864no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—434no—
610C22—434no—
611C22—864no—
612C22—864no—
613C22—864no—
614C22—864no—
615C22—862no—
616C22—862no—
617C22—862no—
618C22—862no—
619C22—17216no—
620C22C2×C22⋊C4×D4311yescharacteristic
621C22#121×D4312yes—
622C22#137×D4312yes—
623C22#1458369511yescharacteristic
624C22#1458369811yescharacteristic
625C4—22no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—434no—
630C4—862no—
631C4—3442no—
632C4—1722no—
633C4—1722no—
634C4C4×D4×D4312yes—
635C4#120×D4311yescharacteristic
636C4#123×D4311yescharacteristic
637C2—42no—
638C2—864no—
639C2—864no—
640C2#386×D4311yescharacteristic
641C2#419×D4311yescharacteristic
642C2#1914311111yescharacteristic
643C2#1914422012yes—
644C2#1914422412yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1894C2
socleC23×C43C22×D4
prime radicalC24×D43D4
cosocleC22×C4×C43C24
hypercenter#1894D43
hypocenterC43#20972
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×#14582906C2semidirect
5. #19135029C2semidirect
6. #19135028C2semidirect
7. #19135026C2semidirect
8. #19135017C2semidirect
9. #19135015C2semidirect
10. #19134313C2semidirect
11. C43×#1894C2semidirect
25. C22⋊Q8×D43C4semidirect
31. C2×#13063250C22semidirect
32. C2×#13063249C22semidirect
33. C2×#13063248C22semidirect
40. #14582662C22semidirect
41. #14582661C22semidirect
42. #14582661C22semidirect
45. C2×#103×C43C22semidirect
47. C43×#340C22semidirect
302. C2×D43#419semidirect
322. #1894D43semidirect
358. C2×#103C2×D43semidirect
369. #340C2×D43semidirect
370. C43#20972semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D432
C2×C22⋊Q8×D432
C2×#145829062
#191369772
#191350292
#191350282
#191350262
#191350172
#191350152
#191343132
C43×#18942
#2097243