Finite Group Databasebeta

← #25130948

Subgroups — #25130948

Order 11008.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583449C211yesmaximal, characteristic
2C2×#14583184C211yesmaximal, characteristic, complemented
3C2×#14583099C211yesmaximal, characteristic, complemented
4#19135784C212yesmaximal, complemented
5#19135596C212yesmaximal, complemented
6#19135547C211yesmaximal, characteristic, complemented
7#19135522C211yesmaximal, characteristic, complemented
8#19134932C212yesmaximal, complemented
9#19134855C212yesmaximal, complemented
10#19134575C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C43C211yesmaximal, characteristic, complemented
12C22×#10621898—21no—
13C22×#10621884C2212yes—
14#14583449—24no—
15#14583449—24no—
16C2×#13063325C2211yescharacteristic
17C2×#13063322C2211yescharacteristic
18C2×#13063313—21no—
19C2×#13063299—21no—
20C2×#13063299C2212yes—
21C2×C4×D172—21no—
22C2×C4×D172C2212yes—
23#14583184—24no—
24#14583099—24no—
25C2×#13063256C2212yes—
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28C2×#13063249C2212yes—
29C2×#13063245C2212yes—
30C2×#13063245C2212yes—
31C2×#13063245C2212yes—
32C2×C22⋊(C43⋊C8)C2212yes—
33#14582661C2212yes—
34#14582661C2212yes—
35#14582654C2212yes—
36C2×C22⋊Q8×C43C2211yescharacteristic
37C2×C4×Q8×C43C2211yescharacteristic
38C22×C42×C43C2211yescharacteristic
39C43×#391C2211yescharacteristic
40C43×#389C2211yescharacteristic
41C43×#387—21no—
42C43×#385—21no—
43C43×#380C2211yescharacteristic
44C4×C4⋊C4×C43—21no—
45C4×C4⋊C4×C43C2211yescharacteristic
46C4×C22⋊C4×C43—21no—
47C2×#10621898—28no—
48C2×#10621898—44no—
49C2×#10621898—44no—
50C22×Dic86—21no—
51C22×D172—21no—
52C22×D172—21no—
53C22×D172C2312yes—
54C22×C22⋊D43—22no—
55C22×C4×D43—22no—
56C2×#10621884—24no—
57C2×#10621884—24no—
58C2×#10621884—42no—
59C22×C43⋊C8C2312yes—
60C22×C43⋊C8C2312yes—
61#13063325—22no—
62#13063325D414yes—
63#13063322—22no—
64#13063322D414yes—
65#13063313—28no—
66#13063299—28no—
67#13063299—24no—
68#13063299—24no—
69C4×D172—28no—
70C4×D172—24no—
71C4×D172—24no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43C2312yes—
74C2×(C2×D43)⋊C4—22no—
75C2×(C2×D43)⋊C4—22no—
76#13063256—24no—
77C4⋊(C43⋊C8)—22no—
78C4⋊(C43⋊C8)—22no—
79#13063249—24no—
80#13063245—24no—
81#13063245—24no—
82#13063245—42no—
83C22⋊(C43⋊C8)—42no—
84C22×Q8×C43C2311yescharacteristic
85C23×C4×C43D411yescharacteristic
86C23×C4×C43D411yescharacteristic
87C23×C4×C43C2311yescharacteristic
88C22⋊Q8×C43—22no—
89C22⋊Q8×C43—22no—
90C4×Q8×C43—22no—
91C4×Q8×C43—22no—
92C2×C4⋊C4×C43—21no—
93C2×C4⋊C4×C43—21no—
94C2×C4⋊C4×C43—21no—
95C2×C4⋊C4×C43C2311yescharacteristic
96C2×C4⋊C4×C43C2311yescharacteristic
97C2×C4⋊C4×C43C2311yescharacteristic
98C2×C22⋊C4×C43—21no—
99C2×C22⋊C4×C43—21no—
100C2×C42×C43—22no—
101C2×C42×C43—22no—
102C2×C42×C43—21no—
103C2×C42×C43—21no—
104C2×C42×C43D411yescharacteristic
105C2×C42×C43D411yescharacteristic
106C2×C42×C43C2311yescharacteristic
107C2×C42×C43C2311yescharacteristic
108#95×C43—21no—
109#95×C43—21no—
110#95×C43—21no—
111#95×C43—21no—
112#95×C43—21no—
113C24×C43C2×D411yescharacteristic
114C23×D43—22no—
115C2×Q8×C43—21no—
116C2×Q8×C43—21no—
117C2×Q8×C43—21no—
118C2×Q8×C43—21no—
119C2×Q8×C43—21no—
120C2×Q8×C43#4112yescomplemented
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—41no—
135C22×C4×C43—41no—
136C22×C4×C43SD1614yescomplemented
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2×D411yescharacteristic
140C22×C4×C43C2×D411yescharacteristic
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2411yescharacteristic
143C2×C22⋊D43—28no—
144C2×C22⋊D43—44no—
145C2×C22⋊D43—44no—
146C22×Dic43—22no—
147#10621898—432no—
148C2×D172—24no—
149C2×D172—24no—
150C2×D172—22no—
151C2×D172—22no—
152C2×D172—22no—
153C2×D172—22no—
154C2×D172—22no—
155C2×D172—42no—
156C2×D172—42no—
157C2×D172—42no—
158C2×D172—42no—
159C2×D172C2×D414yes—
160C2×C4×D43—28no—
161C2×C4×D43—44no—
162C2×C4×D43—44no—
163C2×Dic86—24no—
164C2×Dic86—42no—
165C2×Dic86—42no—
166C4⋊C4×C43—24no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—21no—
172C4⋊C4×C43—42no—
173C4⋊C4×C43—42no—
174C4⋊C4×C43#4112yescomplemented
175C22⋊C4×C43—24no—
176C22⋊C4×C43—42no—
177C42×C43—22no—
178C42×C43—22no—
179C42×C43—21no—
180C42×C43—21no—
181C42×C43—42no—
182C42×C43—42no—
183C42×C43C2×D412yes—
184C42×C43C2×D412yes—
185#10621884—48no—
186(C2×D43)⋊C4—28no—
187(C2×D43)⋊C4—28no—
188C4⋊Dic43—24no—
189C4⋊Dic43—22no—
190C4⋊Dic43C2×D414yes—
191C2×C43⋊C8—22no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—22no—
194C2×C43⋊C8—22no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—22no—
197C23×C43—21no—
198C23×C43—21no—
199C23×C43—21no—
200C23×C43—21no—
201C23×C43—21no—
202C23×C43—21no—
203C23×C43C2×SD1612yes—
204C23×C43C22×D411yescharacteristic
205C22×D43—24no—
206C22×D43—42no—
207C22×D43—42no—
208C22×D43—42no—
209C22×D43—42no—
210C22×D43—42no—
211C22×D43—42no—
212Q8×C43—22no—
213Q8×C43—22no—
214Q8×C43—41no—
215Q8×C43—41no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—28no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—22no—
224C2×C4×C43—22no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—44no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
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#12012yes—
245C2×C4×C43#12012yes—
246C2×C4×C43C2×SD1612yes—
247C2×C4×C43C2×SD1612yes—
248C2×C4×C43AGL(1,ℤ/8)12yes—
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43C22×D411yescharacteristic
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43H(5,𝔽2)11yescharacteristic
254C2×C4×C432−1+411yescharacteristic
255C22⋊D43—416no—
256C2×Dic43—24no—
257C2×Dic43—42no—
258C2×Dic43—42no—
259D172—24no—
260D172—24no—
261D172—48no—
262D172—42no—
263D172—42no—
264D172—48no—
265C4×D43—416no—
266Dic86—48no—
267C43⋊C8—42no—
268C43⋊C8—42no—
269#24154—431nomaximal, Sylow
270C22×C43—22no—
271C22×C43—22no—
272C22×C43—22no—
273C22×C43—41no—
274C22×C43—41no—
275C22×C43—41no—
276C22×C43—41no—
277C22×C43C2×#12011yescharacteristic
278C22×C43#52511yescharacteristic
279C22×C43#53711yescharacteristic
280C22×C43C22×SD1611yescharacteristic
281C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C43#57511yescharacteristic
283C22×C43#57511yescharacteristic
284C2×D43—44no—
285C2×D43—44no—
286C2×D43—44no—
287C2×D43—82no—
288C2×D43—82no—
289C2×D43—82no—
290C2×D43—82no—
291C4×C43—22no—
292C4×C43—22no—
293C4×C43—42no—
294C4×C43—41no—
295C4×C43—42no—
296C4×C43—41no—
297C4×C43—81no—
298C4×C43—81no—
299C4×C43#45014yes—
300C4×C43#54611yescharacteristic
301C4×C43#54711yescharacteristic
302C4×C43#54811yescharacteristic
303C4×C43#55011yescharacteristic
304Dic43—44no—
305C2×#532—431no—
306C2×#475—431no—
307C2×#460—431no—
308C4×C22⋊Q8D4311yescharacteristic, complemented
309#2034—432no—
310#2000—432no—
311#1988—431no—
312#1982—431no—
313#1877—432no—
314#1864—432no—
315#1797—431no—
316C2×C43—42no—
317C2×C43#240711yescharacteristic
318C2×C43C2×#45011yescharacteristic
319C2×C43#300511yescharacteristic
320C2×C43#310311yescharacteristic
321C2×C43#311511yescharacteristic
322C2×C43#324611yescharacteristic
323C2×C43#325111yescharacteristic
324D43—84no—
325C22×#41—861no—
326C22×SD16—432no—
327#532—864no—
328#532—864no—
329C2×C4⋊Q8—431no—
330C2×#127—431no—
331C2×#124—861no—
332C2×C22⋊Q8C2×D4311yescharacteristic
333C2×#121—432no—
334C2×#121—861no—
335C2×C4×Q8C2×D4311yescharacteristic
336C2×C4×D4—432no—
337C2×C4×D4—861no—
338C22×C42C2×D4311yescharacteristic
339#475—864no—
340#460—864no—
341C2×#106—432no—
342C2×C4⋊C8—431no—
343C2×C4⋊C8—431no—
344C2×#103—432no—
345C2×#102—432no—
346C2×#102—432no—
347C2×#102—432no—
348C2×C22⋊C8—432no—
349#391C2×D4311yescharacteristic
350#389C2×D4311yescharacteristic
351#387—21no—
352#385—21no—
353#380C2×D4311yescharacteristic
354C4×C4⋊C4—21no—
355C4×C4⋊C4C2×D4311yescharacteristic
356C4×C22⋊C4—21no—
357#340—432no—
358#340—432no—
359#336—432no—
360C43#2415411yescharacteristic, complemented, Sylow
361C2×#41—868no—
362C2×#41—1724no—
363C2×#41—1724no—
364C22×Q8—861no—
365C22×Q8C22×D4311yescharacteristic
366C22×D4—432no—
367C22×D4—862no—
368C22×D4—861no—
369C22×D4—861no—
370C23×C4—862no—
371C23×C4C22⋊D4311yescharacteristic
372C23×C4C22⋊D4311yescharacteristic
373C23×C4C22×D4311yescharacteristic
374C2×SD16—864no—
375C2×SD16—864no—
376C2×SD16—1722no—
377C22×C8—432no—
378C22×C8—432no—
379C4⋊Q8—434no—
380C4⋊Q8—862no—
381#127—434no—
382#127—862no—
383#124—868no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—864no—
387#121—864no—
388#121—868no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—864no—
392C4×D4—864no—
393C4×D4—868no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—432no—
398C2×C4⋊C4—861no—
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C4⋊C4C22×D4311yescharacteristic
401C2×C4⋊C4C22×D4311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—862no—
405C2×C22⋊C4—862no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22⋊D4311yescharacteristic
411C2×C42C22⋊D4311yescharacteristic
412C2×C42C22×D4311yescharacteristic
413C2×C42C22×D4311yescharacteristic
414#106—864no—
415C4⋊C8—862no—
416C4⋊C8—862no—
417#103—864no—
418#102—864no—
419#102—864no—
420#102—1722no—
421C22⋊C8—1722no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C24—862no—
428C24C2×C22⋊D4311yescharacteristic
429#41—17232no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—864no—
436C2×Q8—1722no—
437C2×Q8—1722no—
438C2×Q8#1062189812yescomplemented
439C2×D4—434no—
440C2×D4—868no—
441C2×D4—864no—
442C2×D4—864no—
443C2×D4—862no—
444C2×D4—862no—
445C2×D4—862no—
446C2×D4—862no—
447C2×D4—862no—
448C2×D4—1722no—
449C2×D4—1722no—
450C2×D4—1724no—
451C2×D4—1724no—
452C2×D4—1722no—
453C2×D4—1722no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—868no—
470C22×C4—862no—
471C22×C4—1724no—
472C22×C4—1724no—
473C22×C4#1062188414yescomplemented
474C22×C4C2×C22⋊D4311yescharacteristic
475C22×C4C2×C22⋊D4311yescharacteristic
476C22×C4C2×C22⋊D4311yescharacteristic
477C22×C4C2×C22⋊D4311yescharacteristic
478C22×C4C2×C22⋊D4311yescharacteristic
479C22×C4C23×D4311yescharacteristic
480SD16—1728no—
481C2×C8—862no—
482C2×C8—862no—
483C2×C8—862no—
484C2×C8—862no—
485C2×C8—862no—
486C2×C8—862no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—434no—
496C4⋊C4—864no—
497C4⋊C4—862no—
498C4⋊C4#1062189812yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—868no—
502C22⋊C4—868no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42D4×D4312yes—
510C42D4×D4312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—864no—
518C23—1722no—
519C23—1722no—
520C23—1722no—
521C23—1722no—
522C23—1722no—
523C23—1722no—
524C23C2×#1062188412yes—
525C23C22×C22⋊D4311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—1728no—
531D4—864no—
532D4—864no—
533D4—1728no—
534D4—17216no—
535D4—1722no—
536D4—1728no—
537D4—1722no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—864no—
567C2×C4—17216no—
568C2×C4—1722no—
569C2×C4—1722no—
570C2×C4#1306329312yes—
571C2×C4#1306329312yes—
572C2×C4C2×#1062188412yes—
573C2×C4C2×#1062188412yes—
574C2×C4#1306335012yes—
575C2×C4C2×D4×D4311yescharacteristic
576C2×C4C2×D4×D4311yescharacteristic
577C2×C4C2×#1062189811yescharacteristic
578C2×C4C2×#1062189811yescharacteristic
579C2×C4#1306337111yescharacteristic
580C2×C4#1306337211yescharacteristic
581C8—1722no—
582C8—1722no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—3442no—
591C22—3442no—
592C22—3442no—
593C22—3442no—
594C22—1724no—
595C22—1724no—
596C22—1724no—
597C22C2×#1306329311yescharacteristic
598C22#1458341011yescharacteristic
599C22#1458347711yescharacteristic
600C22C22×#1062188411yescharacteristic
601C22C2×#1306335011yescharacteristic
602C22#1458369711yescharacteristic
603C22#1458369711yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—1724no—
613C4#1458304914yes—
614C4#1458354511yescharacteristic
615C4#1458355411yescharacteristic
616C4#1458355811yescharacteristic
617C4#1458357011yescharacteristic
618C2—42no—
619C2—3444no—
620C2#1913794011yescharacteristic
621C2C2×#1458304911yescharacteristic
622C2#1914383511yescharacteristic
623C2#1914462611yescharacteristic
624C2#1914474611yescharacteristic
625C2#1914590711yescharacteristic
626C2#1914595711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC4×C22⋊Q8×C43C2
socleC23×C43C22×D4
prime radicalC2×#14583449C2
hypercenterC4×C22⋊Q8D43
hypocenterC43#24154
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14583184C2semidirect
3. C2×#14583099C2semidirect
4. #19135784C2semidirect
5. #19135596C2semidirect
6. #19135547C2semidirect
7. #19135522C2semidirect
8. #19134932C2semidirect
9. #19134855C2semidirect
10. #19134575C2semidirect
11. C4×C22⋊Q8×C43C2semidirect
120. C2×Q8×C43#41semidirect
136. C22×C4×C43SD16semidirect
174. C4⋊C4×C43#41semidirect
308. C4×C22⋊Q8D43semidirect
360. C43#24154semidirect
438. C2×Q8#10621898semidirect
473. C22×C4#10621884semidirect
498. C4⋊C4#10621898semidirect

Maximal subgroups

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

subgroupindex
C2×#145834492
C2×#145831842
C2×#145830992
#191357842
#191355962
#191355472
#191355222
#191349322
#191348552
#191345752
C4×C22⋊Q8×C432
#2415443