Finite Group Databasebeta

← #25105676

Subgroups — #25105676

Order 11008.

1466 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D43C211yesmaximal, characteristic
2C2×#14582861C211yesmaximal, characteristic
3#19136974C211yesmaximal, characteristic
4#19135027C212yesmaximal, complemented
5#19135025C212yesmaximal, complemented
6#19134908C211yesmaximal, characteristic, complemented
7#19134907C211yesmaximal, characteristic, complemented
8#19134905C211yesmaximal, characteristic, complemented
9#19134684C212yesmaximal, complemented
10#19134528C212yesmaximal, complemented
11C43×#1873C211yesmaximal, characteristic, complemented
12C22×Q8×D43C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C22×#10621872C2212yes—
15C2×#13063305C2211yescharacteristic
16C2×#13063304C2211yescharacteristic
17C2×#13063303—21no—
18C2×#13063302—21no—
19C2×#13063301C2211yescharacteristic
20C22⋊Q8×D43—24no—
21C22⋊Q8×D43C418yescomplemented
22C2×#13063284C2211yescharacteristic
23C2×C4⋊C4×D43—21no—
24C2×C4⋊C4×D43C2211yescharacteristic
25C2×C22⋊C4×D43—21no—
26C2×#13063250C2212yescomplemented
27C2×#13063249C2212yescomplemented
28C2×#13063248C2212yescomplemented
29#14582861—24no—
30C2×C22⋊(C43⋊C8)C2211yescharacteristic
31C2×(C2×D43)⋊C8C2212yes—
32#14582760—21no—
33#14582756C2212yes—
34#14582756C2212yes—
35#14582754—21no—
36C4×(C2×D43)⋊C4—21no—
37C4×(C2×D43)⋊C4—21no—
38#14582662C2212yescomplemented
39#14582661C2212yescomplemented
40#14582661C2212yescomplemented
41C2×C22⋊Q8×C43C2211yescharacteristic
42C2×#117×C43C2211yescharacteristic
43C2×#103×C43C2212yescomplemented
44C2×C22⋊C8×C43C2211yescharacteristic
45C43×#340C2212yescomplemented
46C24×D43D411yescharacteristic
47C22×Dic86—21no—
48C22×Dic86C2311yescharacteristic
49C2×Q8×D43—22no—
50C2×Q8×D43—42no—
51C2×Q8×D43—42no—
52C2×Q8×D43C2×C414yes—
53C23×Dic43D411yescharacteristic
54C22×C4×D43—21no—
55C22×C4×D43—21no—
56C22×C4×D43—42no—
57C22×C4×D43—42no—
58C22×C4×D43C2×C414yes—
59C22×C4×D43D411yescharacteristic
60C22×C4×D43D411yescharacteristic
61C22×C4×D43D414yes—
62C22×C4×D43C2311yescharacteristic
63C22×C4×D43C2311yescharacteristic
64C2×#10621872—24no—
65C2×#10621872—42no—
66C2×#10621872—42no—
67C22×C43⋊C8C2312yes—
68#13063305—22no—
69#13063305C2×C414yes—
70#13063304—22no—
71#13063304C2×C414yes—
72#13063303—28no—
73#13063302—28no—
74#13063301—22no—
75#13063301C2×C414yes—
76#13063284—42no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43C2312yes—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C4⋊C4×D43—28no—
84C4⋊C4×D43—22no—
85C4⋊C4×D43C2×C414yes—
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—22no—
91C2×(C2×D43)⋊C4—22no—
92C22⋊C4×D43—28no—
93C2×C4×Dic43—21no—
94#13063250—42no—
95#13063249—42no—
96#13063248—42no—
97C22⋊(C43⋊C8)—22no—
98(C2×D43)⋊C8—24no—
99C43⋊#95—22no—
100C43⋊#95—22no—
101C22×Q8×C43C2311yescharacteristic
102C23×C4×C43C2311yescharacteristic
103C22×C8×C43C2312yes—
104C22⋊Q8×C43—22no—
105C22⋊Q8×C43C2×C414yes—
106#117×C43—42no—
107C2×C4⋊C4×C43—21no—
108C2×C4⋊C4×C43C2312yes—
109C2×C4⋊C4×C43C2311yescharacteristic
110C2×C22⋊C4×C43—21no—
111C2×C22⋊C4×C43—21no—
112C2×C42×C43—21no—
113#103×C43—42no—
114C22⋊C8×C43—22no—
115C24×C43C2×D411yescharacteristic
116C23×D43—24no—
117C23×D43—41no—
118C23×D43—41no—
119C23×D43—42no—
120C23×D43C22⋊C414yes—
121C23×D43C2×D411yescharacteristic
122C23×D43C2×D411yescharacteristic
123C2×Q8×C43—21no—
124C2×Q8×C43—22no—
125C2×Q8×C43—22no—
126C2×Q8×C43C22×C412yes—
127C22×C4×C43—22no—
128C22×C4×C43—22no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—22no—
132C22×C4×C43C22×C412yes—
133C22×C4×C43C2×D412yes—
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2411yescharacteristic
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—22no—
139C22×Dic43—41no—
140C22×Dic43—41no—
141C22×Dic43C22⋊C414yes—
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144Q8×D43—48no—
145Q8×D43—48no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—22no—
149C2×C4×D43—22no—
150C2×C4×D43—24no—
151C2×C4×D43—24no—
152C2×C4×D43—41no—
153C2×C4×D43—41no—
154C2×C4×D43—41no—
155C2×C4×D43—41no—
156C2×C4×D43—44no—
157C2×C4×D43—42no—
158C2×C4×D43—42no—
159C2×C4×D43—48no—
160C2×C4×D43—48no—
161C2×C4×D43C22⋊C414yes—
162C2×C4×D43C22⋊C414yes—
163C2×C4×D43C22×C412yes—
164C2×C4×D43C22×C412yes—
165C2×C4×D43C2×D412yes—
166C2×C4×D43C2×D412yes—
167C2×Dic86—22no—
168C2×Dic86—22no—
169C2×Dic86—21no—
170C2×Dic86—24no—
171C2×Dic86—42no—
172C2×Dic86—42no—
173C2×Dic86C22×C412yes—
174C2×C8×C43—22no—
175C2×C8×C43—22no—
176C2×C8×C43—22no—
177C4⋊C4×C43—24no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—22no—
180C4⋊C4×C43—22no—
181C4⋊C4×C43C22×C412yes—
182C22⋊C4×C43—24no—
183C22⋊C4×C43—42no—
184C42×C43—42no—
185C22⋊Dic43—24no—
186C22⋊Dic43—42no—
187(C2×D43)⋊C4—24no—
188(C2×D43)⋊C4—24no—
189(C2×D43)⋊C4—44no—
190(C2×D43)⋊C4—44no—
191C4⋊Dic43—21no—
192C4⋊Dic43—24no—
193C4⋊Dic43—22no—
194C4⋊Dic43—22no—
195C4⋊Dic43C22×C412yes—
196Dic43⋊C4—24no—
197Dic43⋊C4—24no—
198Dic43⋊C4—24no—
199C4×Dic43—42no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—22no—
203#10621872—48no—
204C23×C43—21no—
205C23×C43—21no—
206C23×C43—22no—
207C23×C43—22no—
208C23×C43C2×C22⋊C412yes—
209C23×C43C22×D411yescharacteristic
210C22×D43—21no—
211C22×D43—21no—
212C22×D43—22no—
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—22no—
216C22×D43—22no—
217C22×D43—28no—
218C22×D43—44no—
219C22×D43—44no—
220C22×D43—48no—
221C22×D43—48no—
222C22×D43C2×C22⋊C412yes—
223C22×D43C2×C22⋊C412yes—
224C22×D43#13714yes—
225Q8×C43—24no—
226Q8×C43—42no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—24no—
230C2×C4×C43—24no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—42no—
239C2×C4×C43C2×C22⋊C412yes—
240C2×C4×C43C23×C411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43H(5,𝔽2)12yes—
243C2×C4×C43H(5,𝔽2)12yes—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—21no—
248C2×Dic43—22no—
249C2×Dic43—22no—
250C2×Dic43—44no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—42no—
254C2×Dic43—42no—
255C2×Dic43—42no—
256C2×Dic43—42no—
257C2×Dic43C2×C22⋊C412yes—
258C2×Dic43C2×C22⋊C412yes—
259C4×D43—24no—
260C4×D43—24no—
261C4×D43—24no—
262C4×D43—24no—
263C4×D43—44no—
264C4×D43—44no—
265C4×D43—48no—
266C4×D43—48no—
267Dic86—24no—
268Dic86—48no—
269Dic86—42no—
270C8×C43—42no—
271C43⋊C8—42no—
272#21768—431nomaximal, Sylow
273C22×C43—22no—
274C22×C43—22no—
275C22×C43—22no—
276C22×C43—42no—
277C22×C43—42no—
278C22×C43C22×C22⋊C411yescharacteristic
279C22×C43#53411yescharacteristic
280C22×C43#53411yescharacteristic
281C22×C43C2×#13712yes—
282C22×C43#57512yes—
283C2×D43—22no—
284C2×D43—22no—
285C2×D43—24no—
286C2×D43—24no—
287C2×D43—22no—
288C2×D43—22no—
289C2×D43—22no—
290C2×D43—416no—
291C2×D43—42no—
292C2×D43—42no—
293C2×D43#41914yes—
294C4×C43—22no—
295C4×C43—42no—
296C4×C43—42no—
297C4×C43—82no—
298C4×C43#51812yes—
299C4×C43#53412yes—
300Dic43—22no—
301Dic43—22no—
302Dic43—42no—
303Dic43—42no—
304Dic43—82no—
305C22×C22⋊Q8—431no—
306C2×#407—431no—
307#2280—431no—
308#1894—432no—
309#1894—432no—
310#1873—431no—
311#1873—431no—
312#1873—431no—
313#1873D4311yescharacteristic, complemented
314#1824—432no—
315#1787—432no—
316C2×C43—42no—
317C2×C43#230811yescharacteristic
318C2×C43C2×#41911yescharacteristic
319C2×C43#289611yescharacteristic
320C2×C43#319612yes—
321C2×C43#319612yes—
322D43—24no—
323D43—42no—
324C23×Q8—431no—
325C24×C4—431no—
326C22×M16—432no—
327C2×C22⋊Q8—431no—
328C2×C22⋊Q8—431no—
329C2×C22⋊Q8—438no—
330C2×C22⋊Q8—431no—
331C2×C22⋊Q8—861no—
332C2×C22⋊Q8—861no—
333C2×C22⋊Q8—864no—
334C2×C22⋊Q8C2×D4311yescharacteristic
335C2×#117—431no—
336C2×#117C2×D4311yescharacteristic
337C22×C4⋊C4—431no—
338C22×C4⋊C4—861no—
339C22×C22⋊C4—861no—
340C2×#103—432no—
341C2×#103—432no—
342C2×#103—432no—
343C2×#103C2×D4312yescomplemented
344#407—864no—
345C2×C22⋊C8—431no—
346C2×C22⋊C8—432no—
347C2×C22⋊C8C2×D4311yescharacteristic
348#381—861no—
349#380—432no—
350#380—432no—
351#379—861no—
352C4×C22⋊C4—861no—
353C4×C22⋊C4—861no—
354#340—432no—
355#340—432no—
356#340—432no—
357#340C2×D4312yescomplemented
358C43#2176811yescharacteristic, complemented, Sylow
359C25—431no—
360C22×Q8—434no—
361C22×Q8—431no—
362C22×Q8—861no—
363C22×Q8—862no—
364C22×Q8—1722no—
365C22×Q8—1722no—
366C22×Q8C22×D4311yescharacteristic
367C23×C4—431no—
368C23×C4—431no—
369C23×C4—431no—
370C23×C4—434no—
371C23×C4—431no—
372C23×C4—431no—
373C23×C4—434no—
374C23×C4—861no—
375C23×C4—861no—
376C23×C4—1722no—
377C23×C4—1722no—
378C23×C4C22×D4311yescharacteristic
379C2×M16—864no—
380C2×M16—1722no—
381C2×M16—1722no—
382C22×C8—432no—
383C22×C8C22×D4312yes—
384C22⋊Q8—22no—
385C22⋊Q8—434no—
386C22⋊Q8—434no—
387C22⋊Q8—434no—
388C22⋊Q8—868no—
389C22⋊Q8—868no—
390C22⋊Q8—862no—
391C22⋊Q8—862no—
392C22⋊Q8—862no—
393C22⋊Q8C4×D4314yes—
394#117—42no—
395#117—1722no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—431no—
398C2×C4⋊C4—434no—
399C2×C4⋊C4—432no—
400C2×C4⋊C4—862no—
401C2×C4⋊C4—868no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4—861no—
404C2×C4⋊C4—861no—
405C2×C4⋊C4—861no—
406C2×C4⋊C4C22×D4312yes—
407C2×C4⋊C4C22×D4311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—868no—
411C2×C22⋊C4—861no—
412C2×C22⋊C4—861no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—862no—
415C2×C22⋊C4—862no—
416C2×C22⋊C4—861no—
417C2×C42—21no—
418C2×C42—861no—
419#103—42no—
420#103—1722no—
421#103—1722no—
422#103—1722no—
423C22⋊C8—22no—
424C22⋊C8—864no—
425C22⋊C8—862no—
426#95—862no—
427#95—862no—
428C24—431no—
429C24—431no—
430C24—434no—
431C24—864no—
432C24—1721no—
433C24—1721no—
434C24—1722no—
435C24D4×D4311yescharacteristic
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—432no—
440C2×Q8—864no—
441C2×Q8—861no—
442C2×Q8—862no—
443C2×Q8—862no—
444C2×Q8—1722no—
445C2×Q8—1722no—
446C2×Q8—1728no—
447C2×Q8—1728no—
448C2×Q8C2×C4×D4312yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—431no—
455C22×C4—431no—
456C22×C4—434no—
457C22×C4—432no—
458C22×C4—432no—
459C22×C4—432no—
460C22×C4—432no—
461C22×C4—434no—
462C22×C4—434no—
463C22×C4—864no—
464C22×C4—864no—
465C22×C4—862no—
466C22×C4—862no—
467C22×C4—862no—
468C22×C4—862no—
469C22×C4—861no—
470C22×C4—861no—
471C22×C4—862no—
472C22×C4—1728no—
473C22×C4—1728no—
474C22×C4—1721no—
475C22×C4—1721no—
476C22×C4—1721no—
477C22×C4—1721no—
478C22×C4—1722no—
479C22×C4—1722no—
480C22×C4—1724no—
481C22×C4—1721no—
482C22×C4—1721no—
483C22×C4C2×C4×D4312yes—
484C22×C4D4×D4311yescharacteristic
485C22×C4D4×D4312yes—
486C22×C4C23×D4311yescharacteristic
487M16—1728no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—862no—
492C2×C8—862no—
493C2×C8—862no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—432no—
499C4⋊C4—864no—
500C4⋊C4—864no—
501C4⋊C4—864no—
502C4⋊C4—864no—
503C4⋊C4—861no—
504C4⋊C4—862no—
505C4⋊C4—862no—
506C4⋊C4C2×C4×D4312yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—864no—
510C22⋊C4—864no—
511C22⋊C4—864no—
512C22⋊C4—1724no—
513C22⋊C4—1724no—
514C22⋊C4—1722no—
515C42—42no—
516C42—1722no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—432no—
522C23—434no—
523C23—432no—
524C23—868no—
525C23—862no—
526C23—861no—
527C23—861no—
528C23—862no—
529C23—862no—
530C23—861no—
531C23—861no—
532C23—1724no—
533C23—1724no—
534C23—1728no—
535C23—1728no—
536C23C22⋊C4×D4312yes—
537C23C2×D4×D4311yescharacteristic
538Q8—24no—
539Q8—42no—
540Q8—864no—
541Q8—1728no—
542Q8—1722no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—432no—
556C2×C4—432no—
557C2×C4—861no—
558C2×C4—861no—
559C2×C4—861no—
560C2×C4—861no—
561C2×C4—864no—
562C2×C4—864no—
563C2×C4—864no—
564C2×C4—864no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—1724no—
568C2×C4—1724no—
569C2×C4—1728no—
570C2×C4—1728no—
571C2×C4—1724no—
572C2×C4—1722no—
573C2×C4—1722no—
574C2×C4—1722no—
575C2×C4—1722no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4C22⋊C4×D4312yes—
579C2×C4C22×C4×D4311yescharacteristic
580C2×C4C2×D4×D4311yescharacteristic
581C2×C4#1306337012yes—
582C2×C4#1306337112yes—
583C8—42no—
584C8—1722no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—42no—
589C22—42no—
590C22—434no—
591C22—862no—
592C22—862no—
593C22—864no—
594C22—864no—
595C22—862no—
596C22—862no—
597C22—862no—
598C22—17216no—
599C22—1722no—
600C22—1722no—
601C22C2×C22⋊C4×D4311yescharacteristic
602C22#1458345911yescharacteristic
603C22#1458346011yescharacteristic
604C22#137×D4312yes—
605C22#1458369612yes—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—862no—
611C4—862no—
612C4—3442no—
613C4—1722no—
614C4—1722no—
615C4#1458337112yes—
616C4#1458345712yes—
617C2—42no—
618C2—864no—
619C2—1722no—
620C2#1913716911yescharacteristic
621C2#419×D4311yescharacteristic
622C2#1914311011yescharacteristic
623C2#1914542312yes—
624C2#1914542912yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1873C2
socleC23×C43C22×D4
prime radicalC24×D43D4
cosocleC22×C4×C43C24
hypercenter#1873D43
hypocenterC43#21768
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #19135027C2semidirect
5. #19135025C2semidirect
6. #19134908C2semidirect
7. #19134907C2semidirect
8. #19134905C2semidirect
9. #19134684C2semidirect
10. #19134528C2semidirect
11. C43×#1873C2semidirect
21. C22⋊Q8×D43C4semidirect
26. C2×#13063250C22semidirect
27. C2×#13063249C22semidirect
28. C2×#13063248C22semidirect
38. #14582662C22semidirect
39. #14582661C22semidirect
40. #14582661C22semidirect
43. C2×#103×C43C22semidirect
45. C43×#340C22semidirect
313. #1873D43semidirect
343. C2×#103C2×D43semidirect
357. #340C2×D43semidirect
358. C43#21768semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D432
C2×#145828612
#191369742
#191350272
#191350252
#191349082
#191349072
#191349052
#191346842
#191345282
C43×#18732
#2176843