Finite Group Databasebeta

← #25159754

Subgroups — #25159754

Order 11008.

932 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582979C211yesmaximal, characteristic
2#19138463C211yesmaximal, characteristic, complemented
3D4×Dic43⋊C4C211yesmaximal, characteristic, complemented
4#19136125C211yesmaximal, characteristic, complemented
5#19136069C211yesmaximal, characteristic, complemented
6#19135968C211yesmaximal, characteristic, complemented
7#19135930C211yesmaximal, characteristic, complemented
8#19135321C211yesmaximal, characteristic, complemented
9#19135149C211yesmaximal, characteristic, complemented
10#19134943C211yesmaximal, characteristic
11#19134941C211yesmaximal, characteristic
12#19134858C211yesmaximal, characteristic, complemented
13#19134854C211yesmaximal, characteristic, complemented
14#19134839C211yesmaximal, characteristic, complemented
15C2×C43×#478C211yesmaximal, characteristic, complemented
16C22×#10621882C2211yescharacteristic
17C2×D4×Dic43C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20C22×Dic43⋊C4—21no—
21#14582979—24no—
22C2×#13063258C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063246C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×C8×Dic43C2211yescharacteristic
30#14582834—21no—
31#14582821—21no—
32#14582801C2211yescharacteristic
33#14582800C2211yescharacteristic
34#14582798C2211yescharacteristic
35#14582797C2211yescharacteristic
36#14582795C2211yescharacteristic
37#14582795C2211yescharacteristic
38#14582772—21no—
39#14582771—21no—
40#14582770—21no—
41#14582768C2211yescharacteristic
42#14582757—21no—
43#14582756—21no—
44C4×Dic43⋊C4C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C2×#125×C43C2211yescharacteristic
54C2×C4×D4×C43C2211yescharacteristic
55C43×#478—24no—
56C2×#107×C43C2211yescharacteristic
57C2×#106×C43C2211yescharacteristic
58C2×C4⋊C8×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C2×#102×C43C2211yescharacteristic
61C22×D172C2311yescharacteristic
62C23×Dic43—21no—
63C23×Dic43—21no—
64C22×C4×D43—21no—
65C2×#10621882—22no—
66C2×#10621882—22no—
67C2×#10621882Q814yes—
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70D4×Dic43—22no—
71D4×Dic43—22no—
72D4×Dic43—22no—
73D4×Dic43—41no—
74#13063287—22no—
75#13063287—41no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—44no—
86C2×Dic43⋊C4D411yescharacteristic
87C2×Dic43⋊C4D411yescharacteristic
88C2×C22⋊Dic43—21no—
89C2×C22⋊Dic43—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×C4×Dic43C2311yescharacteristic
96C2×C4×Dic43C2311yescharacteristic
97#13063258—22no—
98#13063257—22no—
99#13063256—41no—
100#13063247—22no—
101#13063247—22no—
102#13063246—22no—
103#13063245—41no—
104C8×Dic43—22no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95D411yescharacteristic
110C43⋊#95D411yescharacteristic
111C22×D4×C43C2311yescharacteristic
112C23×C4×C43—21no—
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115#125×C43—22no—
116#125×C43—41no—
117C4×D4×C43—22no—
118C4×D4×C43—41no—
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C22⋊C4×C43—21no—
125C2×C42×C43C2311yescharacteristic
126#107×C43—22no—
127#106×C43—22no—
128C4⋊C8×C43—22no—
129#102×C43—22no—
130#102×C43D414yes—
131C24×C43—21no—
132C23×D43—21no—
133C2×D4×C43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—21no—
136C2×D4×C43C2×D412yes—
137C2×D4×C43C2×Q812yes—
138C2×D4×C43#4112yes—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—42no—
143C22×C4×C43—42no—
144C22×C4×C43—81no—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C22×Dic43—24no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43—42no—
153C22×Dic43—42no—
154C22×Dic43—42no—
155C22×Dic43—42no—
156C22×Dic43—81no—
157C22×Dic43C2×D411yescharacteristic
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161C2×D172—21no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172C2×Q812yes—
167C2×C4×D43—42no—
168C2×C4×D43—42no—
169C2×C4×D43—81no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43C2×D412yes—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—42no—
182C4⋊C4×C43—42no—
183C4⋊C4×C43C2×D412yes—
184C22⋊C4×C43—42no—
185C42×C43—21no—
186C42×C43—21no—
187C22⋊Dic43—24no—
188C22⋊Dic43—42no—
189#10621882—28no—
190(C2×D43)⋊C4—42no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—81no—
193(C2×D43)⋊C4—81no—
194(C2×D43)⋊C4—81no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—81no—
201C4⋊Dic43#4112yes—
202Dic43⋊C4—22no—
203Dic43⋊C4—22no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206Dic43⋊C4—42no—
207Dic43⋊C4—81no—
208Dic43⋊C4—81no—
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—21no—
212C4×Dic43—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8C2×Q812yes—
219C23×C43—22no—
220C23×C43—22no—
221C23×C43—22no—
222C23×C43—41no—
223C23×C43—41no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43C22×D411yescharacteristic
227C22×D43—22no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—41no—
233C22×D43—41no—
234D4×C43—22no—
235D4×C43—22no—
236D4×C43—22no—
237D4×C43C22⋊Q814yes—
238C2×C4×C43—22no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—84no—
254C2×C4×C43C22×D411yescharacteristic
255C2×C4×C43C22×Q811yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C43H(5,𝔽2)11yescharacteristic
259C2×C4×C432−1+411yescharacteristic
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—22no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—48no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—84no—
281C2×Dic43#13512yes—
282C2×Dic43AGL(1,ℤ/8)12yes—
283D172—22no—
284D172—22no—
285D172—41no—
286D172—41no—
287C4×D43—84no—
288C8×C43—22no—
289C8×C43—41no—
290C43⋊C8—22no—
291C43⋊C8—41no—
292#26382—431nomaximal, Sylow
293C22×C43—24no—
294C22×C43—42no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—42no—
298C22×C43—42no—
299C22×C43—42no—
300C22×C43#52511yescharacteristic
301C22×C43D4211yescharacteristic
302C22×C43D4×Q811yescharacteristic
303C22×C43C2×#13511yescharacteristic
304C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C43#57611yescharacteristic
306C22×C43#57711yescharacteristic
307C2×D43—42no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—81no—
311C2×D43—81no—
312C2×D43—81no—
313C2×D43—81no—
314C4×C43—42no—
315C4×C43—41no—
316C4×C43—41no—
317C4×C43—81no—
318C4×C43—81no—
319C4×C43C2×C22⋊Q811yescharacteristic
320C4×C43#53911yescharacteristic
321C4×C43#54711yescharacteristic
322C4×C43#55411yescharacteristic
323Dic43—22no—
324Dic43—22no—
325Dic43—42no—
326Dic43—41no—
327Dic43—41no—
328Dic43—81no—
329Dic43—81no—
330C2×#478D4311yescharacteristic, complemented
331C2×C4×D8—431no—
332#2454—431no—
333D4×C4⋊C4—431no—
334#2094—431no—
335#2083—431no—
336#2065—431no—
337#2059—431no—
338#1941—431no—
339#1916—431no—
340#1879—431no—
341#1879—431no—
342#1864—431no—
343#1864—431no—
344#1862—431no—
345C2×C43—44no—
346C2×C43#244911yescharacteristic
347C2×C43#310511yescharacteristic
348C2×C43#311911yescharacteristic
349C2×C43#326411yescharacteristic
350C2×C43#328411yescharacteristic
351C2×C43#338011yescharacteristic
352C2×C43#338411yescharacteristic
353D43—82no—
354C22×D8—431no—
355C2×#125C2×D4311yescharacteristic
356C2×C4×D4—431no—
357C2×C4×D4—431no—
358C2×C4×D4—431no—
359C2×C4×D4C2×D4311yescharacteristic
360C22×C4⋊C4—861no—
361#478—24no—
362C4×D8—864no—
363C2×#107—431no—
364C2×#107C2×D4311yescharacteristic
365C2×#106—431no—
366C2×#106—431no—
367C2×#106C2×D4311yescharacteristic
368C2×C4⋊C8C2×D4311yescharacteristic
369C2×#102—431no—
370C2×#102—431no—
371C2×#102—431no—
372C2×#102—431no—
373C2×#102C2×D4311yescharacteristic
374C2×#102C2×D4311yescharacteristic
375C2×C4×C8—431no—
376#399—861no—
377#396—861no—
378#390—431no—
379#390—431no—
380#389—431no—
381#389—431no—
382#389—431no—
383#389—431no—
384#385—861no—
385#385—861no—
386#385—861no—
387#384—431no—
388#380—861no—
389#380—861no—
390C4×C4⋊C4—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#340—431no—
395#336—431no—
396#336—431no—
397#336—431no—
398#336—431no—
399C43#2638211yescharacteristic, complemented, Sylow
400C22×D4—431no—
401C22×D4C22×D4311yescharacteristic
402C23×C4—21no—
403C23×C4—861no—
404C23×C4—861no—
405C23×C4—861no—
406C2×D8—434no—
407C2×D8—862no—
408C2×D8—862no—
409C22×C8—431no—
410C22×C8—431no—
411C22×C8C22×D4311yescharacteristic
412C22×C8C22×D4311yescharacteristic
413#125—22no—
414#125—41no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—862no—
418C4×D4—862no—
419C4×D4—862no—
420C4×D4—862no—
421C4×D4—1721no—
422C4×D4—1721no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—431no—
430C2×C4⋊C4—861no—
431C2×C4⋊C4—861no—
432C2×C4⋊C4—861no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4—861no—
436C2×C4⋊C4—1724no—
437C2×C4⋊C4C22×D4311yescharacteristic
438C2×C4⋊C4C22×D4311yescharacteristic
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—861no—
442C2×C22⋊C4—861no—
443C2×C22⋊C4—861no—
444C2×C22⋊C4—861no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C22⋊C4—861no—
448C2×C42—431no—
449C2×C42—431no—
450C2×C42C22×D4311yescharacteristic
451#107—22no—
452#107—862no—
453#106—22no—
454#106—862no—
455#106—1721no—
456C4⋊C8—22no—
457#102—22no—
458#102—862no—
459#102—862no—
460#102—862no—
461#102—1721no—
462#102C22⋊D4314yes—
463C4×C8—862no—
464#95—431no—
465#95—431no—
466#95—861no—
467#95—861no—
468#95—861no—
469#95—861no—
470C24—21no—
471C24—861no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—432no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4—861no—
481C2×D4Q8×D4312yes—
482C2×D4#1062190212yes—
483C2×D4C2×C22⋊D4312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—431no—
491C22×C4—431no—
492C22×C4—431no—
493C22×C4—431no—
494C22×C4—864no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—3441no—
499C22×C4—3441no—
500C22×C4—1722no—
501C22×C4—1722no—
502C22×C4—1722no—
503C22×C4—1722no—
504C22×C4—1722no—
505C22×C4—1722no—
506C22×C4D4×D4311yescharacteristic
507C22×C4D4×D4311yescharacteristic
508C22×C4C23×D4311yescharacteristic
509D8—868no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—432no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8—861no—
521C2×C8C2×C22⋊D4312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—432no—
530C4⋊C4—862no—
531C4⋊C4—862no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—3441no—
538C4⋊C4—3441no—
539C4⋊C4—3441no—
540C4⋊C4—1722no—
541C4⋊C4—1722no—
542C4⋊C4—1722no—
543C4⋊C4C2×C22⋊D4312yes—
544C22⋊C4—42no—
545C22⋊C4—864no—
546C22⋊C4—3441no—
547C22⋊C4—3441no—
548C22⋊C4—3441no—
549C22⋊C4—1722no—
550C22⋊C4—1722no—
551C22⋊C4—1722no—
552C42—21no—
553C42—21no—
554C42—861no—
555C42—861no—
556C42—861no—
557C42—861no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—862no—
566C23—1721no—
567C23—1721no—
568C23—1721no—
569C23—1721no—
570C23—1721no—
571C23—1721no—
572C23C2×D4×D4311yescharacteristic
573D4—22no—
574D4—22no—
575D4—22no—
576D4—862no—
577D4—862no—
578D4—1721no—
579D4—1721no—
580D4#1306330114yes—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—432no—
598C2×C4—432no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—862no—
605C2×C4—861no—
606C2×C4—861no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—861no—
610C2×C4—3444no—
611C2×C4—3444no—
612C2×C4—1728no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4C22×C22⋊D4311yescharacteristic
622C2×C4C2×Q8×D4311yescharacteristic
623C2×C4#41×D4311yescharacteristic
624C2×C4C2×#1062190211yescharacteristic
625C2×C4#1306337011yescharacteristic
626C2×C4#1306337211yescharacteristic
627C8—22no—
628C8—41no—
629C8—862no—
630C8—1721no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—3441no—
639C22—3441no—
640C22—3441no—
641C22—3441no—
642C22—1722no—
643C22—1722no—
644C22—1722no—
645C22#1458341211yescharacteristic
646C22D4×C22⋊D4311yescharacteristic
647C22#1458356211yescharacteristic
648C22#135×D4311yescharacteristic
649C22AGL(1,ℤ/8)×D4311yescharacteristic
650C22#1458370311yescharacteristic
651C22#1458370611yescharacteristic
652C4—41no—
653C4—41no—
654C4—42no—
655C4—81no—
656C4—81no—
657C4—862no—
658C4—862no—
659C4—3441no—
660C4—3441no—
661C4—1721no—
662C4—1721no—
663C4—1722no—
664C4C2×#1306330111yescharacteristic
665C4#1458349811yescharacteristic
666C4#1458354811yescharacteristic
667C4#1458360311yescharacteristic
668C2—44no—
669C2—3442no—
670C2#1913840011yescharacteristic
671C2#1914464711yescharacteristic
672C2#1914479911yescharacteristic
673C2#1914606911yescharacteristic
674C2#1914631911yescharacteristic
675C2#1914745711yescharacteristic
676C2#1914749711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#478C2
socleC23×C43C22×D4
prime radicalC22×#10621882C22
hypercenterC2×#478D43
hypocenterC43#26382
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #19138463C2semidirect
3. D4×Dic43⋊C4C2semidirect
4. #19136125C2semidirect
5. #19136069C2semidirect
6. #19135968C2semidirect
7. #19135930C2semidirect
8. #19135321C2semidirect
9. #19135149C2semidirect
12. #19134858C2semidirect
13. #19134854C2semidirect
14. #19134839C2semidirect
15. C2×C43×#478C2semidirect
330. C2×#478D43semidirect
399. C43#26382semidirect

Maximal subgroups

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

subgroupindex
C2×#145829792
#191384632
D4×Dic43⋊C42
#191361252
#191360692
#191359682
#191359302
#191353212
#191351492
#191349432
#191349412
#191348582
#191348542
#191348392
C2×C43×#4782
#2638243