Finite Group Databasebeta

← #25103749

Subgroups — #25103749

Order 11008.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583226C211yesmaximal, characteristic
2#19137103C211yesmaximal, characteristic
3C4×#13063283C211yesmaximal, characteristic, complemented
4#19135222C211yesmaximal, characteristic, complemented
5#19135216C211yesmaximal, characteristic, complemented
6#19135177C211yesmaximal, characteristic, complemented
7#19134916C211yesmaximal, characteristic
8#19134797C211yesmaximal, characteristic, complemented
9#19134689C211yesmaximal, characteristic, complemented
10#107×Dic43C211yesmaximal, characteristic, complemented
11#19134483C211yesmaximal, characteristic, complemented
12#19134409C211yesmaximal, characteristic, complemented
13#19134409C211yesmaximal, characteristic, complemented
14#19134352C211yesmaximal, characteristic, complemented
15C43×#1921C211yesmaximal, characteristic, complemented
16C2×#13063300C2211yescharacteristic
17C2×#13063287—21no—
18C2×#13063287C2211yescharacteristic
19C2×#13063283C2211yescharacteristic
20C2×#13063283C2211yescharacteristic
21C2×C42×D43C2211yescharacteristic
22#14583226C418yescomplemented
23C2×#13063259C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×#13063245C2211yescharacteristic
30C2×(C2×D43)⋊C8C2211yescharacteristic
31C2×(C2×D43)⋊C8C2211yescharacteristic
32C2×C8×Dic43C2211yescharacteristic
33C2×C8×Dic43C2211yescharacteristic
34#14582801C2211yescharacteristic
35#14582789—21no—
36#14582777—21no—
37#14582756C2211yescharacteristic
38C4⋊C4×Dic43C2211yescharacteristic
39C4⋊C4×Dic43C2211yescharacteristic
40C4⋊C4×Dic43C2211yescharacteristic
41C4×C4⋊Dic43C2211yescharacteristic
42C4×(C2×D43)⋊C4—21no—
43#14582746—21no—
44#14582746—21no—
45#14582744—21no—
46C42×Dic43—21no—
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C43×#389C2211yescharacteristic
55C4×C4⋊C4×C43C2211yescharacteristic
56C43×#340C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172—21no—
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C22×C4×D43D411yescharacteristic
64C22×C4×D43D411yescharacteristic
65C22×C4×D43C2311yescharacteristic
66C22×C43⋊C8C2311yescharacteristic
67C22×C43⋊C8C2311yescharacteristic
68#13063300—22no—
69#13063300C2×C414yes—
70#13063287—22no—
71#13063287—22no—
72#13063287—44no—
73#13063283—22no—
74#13063283—22no—
75#13063283—22no—
76#13063283C2×C414yescomplemented
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43D411yescharacteristic
94C2×C4×Dic43D411yescharacteristic
95C2×C4×Dic43C2311yescharacteristic
96C42×D43—22no—
97C42×D43—22no—
98#13063259—22no—
99#13063256—22no—
100#13063256C2×C414yescomplemented
101#13063247—22no—
102#13063247C2×C414yescomplemented
103#13063245—22no—
104#13063245C2×C414yescomplemented
105(C2×D43)⋊C8—22no—
106(C2×D43)⋊C8C2×C414yescomplemented
107C8×Dic43—22no—
108C8×Dic43—22no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C2×C4⋊C4×C43—21no—
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C4⋊C4×C43C2311yescharacteristic
121C2×C42×C43—21no—
122C2×C42×C43C2311yescharacteristic
123#107×C43C2×C414yescomplemented
124#95×C43—21no—
125#95×C43—21no—
126C23×D43—21no—
127C23×D43C2×D411yescharacteristic
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2411yescharacteristic
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C22×Dic43C2×D411yescharacteristic
141C2×D172—21no—
142C2×D172—24no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—42no—
148C2×D172—42no—
149C2×D172C22×C412yes—
150C2×C4×D43—21no—
151C2×C4×D43—21no—
152C2×C4×D43—21no—
153C2×C4×D43—21no—
154C2×C4×D43—22no—
155C2×C4×D43—22no—
156C2×C4×D43—22no—
157C2×C4×D43—22no—
158C2×C4×D43—22no—
159C2×C4×D43—41no—
160C2×C4×D43—42no—
161C2×C4×D43—42no—
162C2×C4×D43—42no—
163C2×C4×D43C22×C412yes—
164C2×C4×D43#4112yes—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43C22×C412yes—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—42no—
177C4⋊C4×C43C22×C412yes—
178C4⋊C4×C43C22×C412yes—
179C4⋊C4×C43#4112yes—
180C42×C43—21no—
181C42×C43—21no—
182C42×C43—42no—
183(C2×D43)⋊C4—24no—
184(C2×D43)⋊C4—24no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—42no—
187(C2×D43)⋊C4—42no—
188(C2×D43)⋊C4—42no—
189(C2×D43)⋊C4—42no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43C22×C412yes—
195C4⋊Dic43#4112yes—
196C4×Dic43—24no—
197C4×Dic43—24no—
198C4×Dic43—22no—
199C4×Dic43—22no—
200C4×Dic43—42no—
201C4×Dic43—42no—
202C4×Dic43—42no—
203C4×Dic43—42no—
204C4×Dic43#4112yes—
205C4×Dic43#4112yes—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8C22×C412yes—
212C23×C43C22×D411yescharacteristic
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—21no—
216C22×D43—21no—
217C22×D43—21no—
218C22×D43—21no—
219C22×D43—22no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43C4×D412yes—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—21no—
232C2×C4×C43—22no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43#13512yes—
245C2×C4×C43C23×C411yescharacteristic
246C2×C4×C43C2×#4111yescharacteristic
247C2×C4×C43C2×#4111yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—22no—
256C2×Dic43—22no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43C4×D412yes—
265C2×Dic43#12312yes—
266C2×Dic43#12312yes—
267C2×Dic43#13512yes—
268C2×Dic43AGL(1,ℤ/8)12yes—
269D172—22no—
270D172—22no—
271D172—48no—
272D172—41no—
273D172—41no—
274C4×D43—22no—
275C4×D43—22no—
276C4×D43—22no—
277C4×D43—22no—
278C4×D43—44no—
279C4×D43—44no—
280C4×D43—84no—
281C8×C43—22no—
282C8×C43—41no—
283C43⋊C8—22no—
284C43⋊C8—41no—
285C4×#482—431nomaximal, Sylow
286C22×C43C2×C4×D411yescharacteristic
287C22×C43C2×#12311yescharacteristic
288C22×C43#52511yescharacteristic
289C22×C43C2×#13511yescharacteristic
290C22×C43C2×#13511yescharacteristic
291C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C43#57511yescharacteristic
293C2×D43—22no—
294C2×D43—22no—
295C2×D43—22no—
296C2×D43—41no—
297C2×D43—41no—
298C2×D43—41no—
299C2×D43—41no—
300C2×D43—42no—
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—81no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C4×C43—22no—
308C4×C43—41no—
309C4×C43—42no—
310C4×C43—42no—
311C4×C43—42no—
312C4×C43—81no—
313C4×C43C4×#4111yescharacteristic
314C4×C43C4×#4111yescharacteristic
315C4×C43#52911yescharacteristic
316C4×C43#52911yescharacteristic
317Dic43—22no—
318Dic43—22no—
319Dic43—42no—
320Dic43—42no—
321Dic43—81no—
322Dic43#48214yescomplemented
323C2×#482—431no—
324C4×#121—431no—
325C4×#117—431no—
326#1926—431no—
327#1925—431no—
328#1921D4311yescharacteristic, complemented
329#1920—431no—
330#1875—431no—
331#1854—431no—
332#1825—431no—
333C4×#107—431no—
334C4×#106—431no—
335C4×#102—431no—
336C4×#102—431no—
337C4×C22⋊C8—431no—
338C2×C43C4×#12311yescharacteristic
339C2×C43C4×#13511yescharacteristic
340C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C43C2×#48211yescharacteristic
342C2×C43#309111yescharacteristic
343C2×C43#310211yescharacteristic
344C2×C43#310611yescharacteristic
345D43—42no—
346D43—82no—
347C2×#121—431no—
348C2×C4×D4—431no—
349C2×C4×D4—861no—
350C2×#117—431no—
351C2×#117—431no—
352C22×C42—431no—
353#482—438no—
354C2×#107—431no—
355C2×#107C2×D4311yescharacteristic
356C2×#106—431no—
357C2×#106—431no—
358C2×#102—431no—
359C2×#102—431no—
360C2×#102—431no—
361C2×#102—431no—
362C2×C22⋊C8—431no—
363C2×C22⋊C8—431no—
364C2×C4×C8—431no—
365C2×C4×C8—431no—
366#390—431no—
367#389C2×D4311yescharacteristic
368#388—861no—
369#386—861no—
370#380—431no—
371C4×C4⋊C4—431no—
372C4×C4⋊C4—431no—
373C4×C4⋊C4—431no—
374C4×C4⋊C4—431no—
375C4×C4⋊C4C2×D4311yescharacteristic
376C4×C22⋊C4—861no—
377C4×C22⋊C4—861no—
378C4×C22⋊C4—861no—
379#376—861no—
380C43—861no—
381#340—431no—
382#340—431no—
383#340—431no—
384#340—431no—
385#340—431no—
386#340C2×D4311yescharacteristic
387#340C2×D4311yescharacteristic
388#340C2×D4311yescharacteristic
389#336—431no—
390#336C2×D4311yescharacteristic
391C43C4×#48211yescharacteristic, complemented, Sylow
392C22×D4—431no—
393C22×D4—861no—
394C23×C4—431no—
395C23×C4—431no—
396C23×C4—431no—
397C23×C4—861no—
398C22×C8—431no—
399C22×C8—431no—
400C22×C8C22×D4311yescharacteristic
401C22×C8C22×D4311yescharacteristic
402#121—434no—
403#121—862no—
404C4×D4—862no—
405C4×D4—862no—
406C4×D4—1724no—
407#117—434no—
408#117—862no—
409#117—862no—
410#117—862no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4C22×D4311yescharacteristic
416C2×C4⋊C4C22×D4311yescharacteristic
417C2×C4⋊C4C22×D4311yescharacteristic
418C2×C4⋊C4C22×D4311yescharacteristic
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4—861no—
425C2×C22⋊C4—861no—
426C2×C42—21no—
427C2×C42—431no—
428C2×C42—431no—
429C2×C42—431no—
430C2×C42—862no—
431C2×C42—862no—
432C2×C42—861no—
433C2×C42—861no—
434C2×C42—861no—
435C2×C42—861no—
436C2×C42—861no—
437C2×C42—861no—
438C2×C42C22×D4311yescharacteristic
439#107—862no—
440#107C4×D4314yescomplemented
441#106—434no—
442#106—862no—
443#102—434no—
444#102—434no—
445#102—862no—
446#102—862no—
447C22⋊C8—434no—
448C22⋊C8—862no—
449C4×C8—862no—
450C4×C8—862no—
451#95—21no—
452#95—21no—
453#95—861no—
454#95—861no—
455#95—861no—
456#95—861no—
457#95—861no—
458C24—431no—
459C24—861no—
460C2×D4—432no—
461C2×D4—864no—
462C2×D4—861no—
463C2×D4—861no—
464C2×D4—861no—
465C2×D4—861no—
466C2×D4—861no—
467C2×D4—1722no—
468C2×D4—1722no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—431no—
474C22×C4—431no—
475C22×C4—431no—
476C22×C4—432no—
477C22×C4—432no—
478C22×C4—862no—
479C22×C4—862no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—861no—
483C22×C4—861no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—862no—
489C22×C4—861no—
490C22×C4—1721no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1722no—
494C22×C4D4×D4311yescharacteristic
495C22×C4D4×D4311yescharacteristic
496C22×C4C23×D4311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—432no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8—861no—
507C2×C8—861no—
508C2×C8C2×C4×D4312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—432no—
516C4⋊C4—432no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4C2×C4×D4312yes—
522C4⋊C4C2×C4×D4312yes—
523C4⋊C4#1062190212yes—
524C22⋊C4—864no—
525C22⋊C4—864no—
526C22⋊C4—864no—
527C22⋊C4—1722no—
528C22⋊C4—1722no—
529C22⋊C4—1722no—
530C22⋊C4—1722no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—432no—
535C42—432no—
536C42—862no—
537C42—862no—
538C42—864no—
539C42—864no—
540C42—1722no—
541C42—1722no—
542C42—1722no—
543C42—1722no—
544C23—432no—
545C23—861no—
546C23—861no—
547C23—861no—
548C23—861no—
549C23—861no—
550C23—861no—
551C23—862no—
552C23—1721no—
553C23—1721no—
554C23—1721no—
555C23—1721no—
556C23—1721no—
557C23—1721no—
558C23C2×D4×D4311yescharacteristic
559D4—862no—
560D4—862no—
561D4—1728no—
562D4—1721no—
563D4—1721no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—432no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—432no—
585C2×C4—432no—
586C2×C4—861no—
587C2×C4—861no—
588C2×C4—861no—
589C2×C4—861no—
590C2×C4—862no—
591C2×C4—862no—
592C2×C4—862no—
593C2×C4—862no—
594C2×C4—862no—
595C2×C4—862no—
596C2×C4—3444no—
597C2×C4—1724no—
598C2×C4—1724no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4#1306334812yes—
607C2×C4C22×C4×D4311yescharacteristic
608C2×C4#41×D4311yescharacteristic
609C2×C4#41×D4311yescharacteristic
610C2×C4#41×D4311yescharacteristic
611C2×C4#41×D4311yescharacteristic
612C2×C4C2×#1062190211yescharacteristic
613C8—22no—
614C8—41no—
615C8—862no—
616C8—1721no—
617C22—862no—
618C22—862no—
619C22—862no—
620C22—3441no—
621C22—3441no—
622C22—3441no—
623C22—3441no—
624C22—1722no—
625C22—1722no—
626C22—1722no—
627C22—1721no—
628C22—1721no—
629C22—1721no—
630C22—1721no—
631C22C4×D4×D4311yescharacteristic
632C22#123×D4311yescharacteristic
633C22#1458341211yescharacteristic
634C22#135×D4311yescharacteristic
635C22C2×#1306334811yescharacteristic
636C22AGL(1,ℤ/8)×D4311yescharacteristic
637C22#1458369511yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—434no—
645C4—862no—
646C4—862no—
647C4—3441no—
648C4—1722no—
649C4—1722no—
650C4#1458336711yescharacteristic
651C4C4×#1062190211yescharacteristic
652C4#1458343211yescharacteristic
653C4#1458343411yescharacteristic
654C2—3442no—
655C2—1722no—
656C2#1913711611yescharacteristic
657C2C4×#1306334811yescharacteristic
658C2#1914340611yescharacteristic
659C2#482×D4311yescharacteristic
660C2#1914451111yescharacteristic
661C2#1914461811yescharacteristic
662C2#1914465811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1921C2
socleC23×C43C22×D4
prime radicalC2×#13063300C22
cosocleC22×C4×C43C24
hypercenter#1921D43
hypocenterC43C4×#482
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
3. C4×#13063283C2semidirect
4. #19135222C2semidirect
5. #19135216C2semidirect
6. #19135177C2semidirect
8. #19134797C2semidirect
9. #19134689C2semidirect
10. #107×Dic43C2semidirect
11. #19134483C2semidirect
12. #19134409C2semidirect
13. #19134409C2semidirect
14. #19134352C2semidirect
15. C43×#1921C2semidirect
22. #14583226C4semidirect
76. #13063283C2×C4semidirect
100. #13063256C2×C4semidirect
102. #13063247C2×C4semidirect
104. #13063245C2×C4semidirect
106. (C2×D43)⋊C8C2×C4semidirect
123. #107×C43C2×C4semidirect
322. Dic43#482semidirect
328. #1921D43semidirect
391. C43C4×#482semidirect
440. #107C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832262
#191371032
C4×#130632832
#191352222
#191352162
#191351772
#191349162
#191347972
#191346892
#107×Dic432
#191344832
#191344092
#191343522
C43×#19212
C4×#48243