Finite Group Databasebeta

← #25149945

Subgroups — #25149945

Order 11008.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583265C211yesmaximal, characteristic, complemented
2C2×#14583099C211yesmaximal, characteristic, complemented
3#19139541C211yesmaximal, characteristic
4#19135596C212yesmaximal, complemented
5#19135596C212yesmaximal, complemented
6#19135553C211yesmaximal, characteristic, complemented
7#19135394C211yesmaximal, characteristic, complemented
8#19134917C212yesmaximal, complemented
9#19134912C212yesmaximal, complemented
10#19134425C211yesmaximal, characteristic, complemented
11C2×C43×#552C211yesmaximal, characteristic, complemented
12C22×#10621884C2212yes—
13C2×#13063322C2211yescharacteristic
14C2×#13063321—21no—
15C2×#13063299C2212yes—
16C2×#13063299C2212yes—
17#14583265—24no—
18#14583099—24no—
19C2×C4⋊(C43⋊C8)C2211yescharacteristic
20C2×#13063249C2212yes—
21C2×#13063245C2212yes—
22C2×#13063245C2212yes—
23C2×#13063245C2212yes—
24C2×C22⋊(C43⋊C8)C2212yes—
25C2×C22⋊(C43⋊C8)C2212yes—
26C2×#13063236C2211yescharacteristic
27#14582808—21no—
28#14582808—21no—
29#14582800—21no—
30C4×C4⋊Dic43C2211yescharacteristic
31#14582661C2212yes—
32#14582661C2212yes—
33#14582661C2212yes—
34C43×#552—24no—
35C43×#552—24no—
36C2×#126×C43—21no—
37C2×#125×C43—21no—
38C2×#125×C43C2211yescharacteristic
39C2×#123×C43—21no—
40C2×C22⋊Q8×C43C2211yescharacteristic
41C2×C22⋊Q8×C43C2211yescharacteristic
42C2×C4×Q8×C43C2211yescharacteristic
43C2×C4×D4×C43C2211yescharacteristic
44C2×#117×C43—21no—
45C2×#117×C43C2211yescharacteristic
46C22×C4⋊C4×C43C2211yescharacteristic
47C22×D172—21no—
48C22×D172—21no—
49C22×D172C2312yes—
50C22×C22⋊D43—22no—
51C22×C22⋊D43—22no—
52C2×#10621884—24no—
53C2×#10621884—42no—
54C2×#10621884—42no—
55C22×C43⋊C8C2312yes—
56C22×C43⋊C8C2312yes—
57#13063322—22no—
58#13063322—22no—
59#13063321—44no—
60#13063299—24no—
61#13063299—24no—
62#13063299—42no—
63#13063299—42no—
64C2×C4⋊Dic43C2312yes—
65C2×(C2×D43)⋊C4—22no—
66C2×(C2×D43)⋊C4—22no—
67C2×C4×Dic43—21no—
68C4⋊(C43⋊C8)—22no—
69#13063249—24no—
70#13063245—24no—
71#13063245—24no—
72#13063245—24no—
73C22⋊(C43⋊C8)—42no—
74C22⋊(C43⋊C8)—42no—
75#13063236—22no—
76C43⋊#95—21no—
77C22×Q8×C43C2311yescharacteristic
78C22×D4×C43D411yescharacteristic
79C23×C4×C43—21no—
80C23×C4×C43C2311yescharacteristic
81C23×C4×C43C2311yescharacteristic
82#126×C43—28no—
83#125×C43—28no—
84#125×C43—22no—
85#125×C43D414yescomplemented
86#123×C43—28no—
87C22⋊Q8×C43—22no—
88C22⋊Q8×C43—22no—
89C22⋊Q8×C43—22no—
90C22⋊Q8×C43—22no—
91C4×Q8×C43—22no—
92C4×Q8×C43D414yescomplemented
93C4×D4×C43—22no—
94C4×D4×C43—22no—
95#117×C43—28no—
96#117×C43—22no—
97#117×C43—22no—
98C2×C4⋊C4×C43—21no—
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—22no—
105C2×C4⋊C4×C43—22no—
106C2×C4⋊C4×C43D411yescharacteristic
107C2×C4⋊C4×C43C2311yescharacteristic
108C2×C4⋊C4×C43C2311yescharacteristic
109C2×C4⋊C4×C43C2311yescharacteristic
110C2×C22⋊C4×C43—21no—
111C2×C22⋊C4×C43—21no—
112C2×C22⋊C4×C43—21no—
113C2×C22⋊C4×C43D411yescharacteristic
114C2×C22⋊C4×C43D411yescharacteristic
115C2×C42×C43—21no—
116C2×C42×C43—21no—
117C2×C42×C43C2311yescharacteristic
118C24×C43C2×D411yescharacteristic
119C24×C43C2×D411yescharacteristic
120C23×D43—22no—
121C2×Q8×C43—21no—
122C2×Q8×C43—22no—
123C2×Q8×C43—22no—
124C2×Q8×C43C2×D412yes—
125C2×D4×C43—22no—
126C2×D4×C43—22no—
127C2×D4×C43—41no—
128C22×C4×C43—24no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—22no—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—42no—
142C22×C4×C43—42no—
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C2×C22⋊D43—44no—
149C2×C22⋊D43—44no—
150C2×C22⋊D43—44no—
151C2×C22⋊D43—44no—
152C2×C22⋊D43—82no—
153C2×C22⋊D43—82no—
154C22×Dic43—22no—
155C2×D172—22no—
156C2×D172—22no—
157C2×D172—22no—
158C2×D172—22no—
159C2×D172—22no—
160C2×D172—22no—
161C2×D172—42no—
162C2×D172—42no—
163C2×D172—42no—
164C2×D172—42no—
165C2×D172—82no—
166C4⋊C4×C43—24no—
167C4⋊C4×C43—24no—
168C4⋊C4×C43—24no—
169C4⋊C4×C43—24no—
170C4⋊C4×C43—24no—
171C4⋊C4×C43—24no—
172C4⋊C4×C43—22no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43C2×D412yes—
177C4⋊C4×C43C2×D412yes—
178C4⋊C4×C43C2×D412yes—
179C22⋊C4×C43—24no—
180C22⋊C4×C43—24no—
181C22⋊C4×C43—24no—
182C22⋊C4×C43—22no—
183C22⋊C4×C43—22no—
184C42×C43—24no—
185C42×C43—24no—
186C42×C43C2×D412yes—
187C42×C43C2×D412yes—
188#10621884—48no—
189(C2×D43)⋊C4—44no—
190(C2×D43)⋊C4—44no—
191C4⋊Dic43—22no—
192C4⋊Dic43#4114yes—
193C4×Dic43—42no—
194C2×C43⋊C8—22no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—22no—
198C2×C43⋊C8—22no—
199C2×C43⋊C8—22no—
200C23×C43—21no—
201C23×C43—22no—
202C23×C43—21no—
203C23×C43—22no—
204C23×C43—21no—
205C23×C43—22no—
206C23×C43—21no—
207C23×C43—21no—
208C23×C43—22no—
209C23×C43AGL(1,ℤ/8)12yes—
210C23×C43C22×D411yescharacteristic
211C22×D43—42no—
212C22×D43—42no—
213C22×D43—42no—
214C22×D43—42no—
215C22×D43—42no—
216C22×D43—42no—
217C22×D43—42no—
218Q8×C43—22no—
219Q8×C43—22no—
220Q8×C43—42no—
221D4×C43—44no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—22no—
225C2×C4×C43—21no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—22no—
229C2×C4×C43—21no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—48no—
234C2×C4×C43—42no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—42no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43#12012yes—
247C2×C4×C43#12012yes—
248C2×C4×C43C22×D411yescharacteristic
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43C2×#4112yes—
251C2×C4×C43H(5,𝔽2)12yes—
252C22⋊D43—88no—
253C22⋊D43—88no—
254C2×Dic43—24no—
255C2×Dic43—42no—
256C2×Dic43—42no—
257D172—42no—
258D172—42no—
259D172—42no—
260D172—42no—
261D172—84no—
262D172—84no—
263C43⋊C8—42no—
264C43⋊C8—42no—
265#25583—431nomaximal, Sylow
266C22×C43—22no—
267C22×C43—24no—
268C22×C43—41no—
269C22×C43—41no—
270C22×C43—42no—
271C22×C43—41no—
272C22×C43—41no—
273C22×C43—41no—
274C22×C43—42no—
275C22×C43—42no—
276C22×C43C2×#12011yescharacteristic
277C22×C43#54612yes—
278C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C43#57511yescharacteristic
280C22×C43#57712yes—
281C2×D43—82no—
282C2×D43—82no—
283C2×D43—82no—
284C2×D43—82no—
285C2×D43—82no—
286C2×D43—82no—
287C2×D43—82no—
288C4×C43—22no—
289C4×C43—22no—
290C4×C43—41no—
291C4×C43—41no—
292C4×C43—42no—
293C4×C43—42no—
294C4×C43—42no—
295C4×C43—42no—
296C4×C43#53211yescharacteristic
297C4×C43#53511yescharacteristic
298C4×C43#55012yes—
299Dic43—44no—
300C2×#552D4311yescharacteristic, complemented
301C2×#490—431no—
302C2×#460—431no—
303#2541—431no—
304#2000—432no—
305#2000—432no—
306#1989—431no—
307#1955—431no—
308#1875—432no—
309#1874—432no—
310#1763—431no—
311C2×C43—42no—
312C2×C43—81no—
313C2×C43#243111yescharacteristic
314C2×C43#300611yescharacteristic
315C2×C43#301011yescharacteristic
316C2×C43#331012yes—
317C2×C43#331712yes—
318D43—162no—
319C22×SD16—432no—
320#552—24no—
321#552—24no—
322C2×#127—431no—
323C2×#127—861no—
324C2×#126—21no—
325C2×#125—21no—
326C2×#125C2×D4311yescharacteristic
327C2×#123—21no—
328C2×C22⋊Q8C2×D4311yescharacteristic
329C2×C22⋊Q8C2×D4311yescharacteristic
330C2×#121—432no—
331C2×#121—432no—
332C2×C4×Q8C2×D4311yescharacteristic
333C2×C4×D4C2×D4311yescharacteristic
334C2×#117—21no—
335C2×#117C2×D4311yescharacteristic
336C22×C4⋊C4C2×D4311yescharacteristic
337#490—864no—
338#460—864no—
339C2×C4⋊C8—431no—
340C2×#103—432no—
341C2×#102—432no—
342C2×#102—432no—
343C2×#102—432no—
344C2×C22⋊C8—432no—
345C2×C22⋊C8—432no—
346C2×#97—431no—
347#392—861no—
348#392—861no—
349#390—861no—
350C4×C4⋊C4—431no—
351#340—432no—
352#340—432no—
353#340—432no—
354C43#2558311yescharacteristic, complemented, Sylow
355C22×Q8C22×D4311yescharacteristic
356C22×D4—432no—
357C22×D4—862no—
358C22×D4—862no—
359C22×D4—861no—
360C22×D4—861no—
361C22×D4C22⋊D4311yescharacteristic
362C23×C4—21no—
363C23×C4C22×D4311yescharacteristic
364C23×C4C22×D4311yescharacteristic
365C2×SD16—864no—
366C2×SD16—1722no—
367C2×SD16—1722no—
368C22×C8—432no—
369C22×C8—432no—
370#127—862no—
371#127—862no—
372#127—1724no—
373#126—28no—
374#125—28no—
375#125—22no—
376#125C22⋊D4314yescomplemented
377#123—28no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382#121—864no—
383#121—864no—
384#121—1722no—
385#121—1722no—
386C4×Q8—22no—
387C4×Q8C22⋊D4314yescomplemented
388C4×D4—22no—
389C4×D4—22no—
390#117—28no—
391#117—22no—
392#117—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—432no—
402C2×C4⋊C4C22⋊D4311yescharacteristic
403C2×C4⋊C4C22×D4311yescharacteristic
404C2×C4⋊C4C22×D4311yescharacteristic
405C2×C4⋊C4C22×D4311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—862no—
410C2×C22⋊C4—862no—
411C2×C22⋊C4C22⋊D4311yescharacteristic
412C2×C22⋊C4C22⋊D4311yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—861no—
416C2×C42C22×D4311yescharacteristic
417C4⋊C8—862no—
418#103—864no—
419#102—864no—
420#102—864no—
421#102—864no—
422C22⋊C8—1722no—
423C22⋊C8—1722no—
424#97—862no—
425#95—861no—
426C24—862no—
427C24C2×C22⋊D4311yescharacteristic
428C24C2×C22⋊D4311yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C22⋊D4312yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—862no—
437C2×D4—862no—
438C2×D4—862no—
439C2×D4—862no—
440C2×D4—862no—
441C2×D4—862no—
442C2×D4—3442no—
443C2×D4—3442no—
444C2×D4—3442no—
445C2×D4—1722no—
446C2×D4—1722no—
447C2×D4—1724no—
448C2×D4—1724no—
449C2×D4—1724no—
450C2×D4—1724no—
451C2×D4—1722no—
452C2×D4—1722no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—862no—
469C22×C4C2×C22⋊D4311yescharacteristic
470C22×C4C2×C22⋊D4311yescharacteristic
471C22×C4C2×C22⋊D4311yescharacteristic
472C22×C4C2×C22⋊D4311yescharacteristic
473C22×C4C23×D4311yescharacteristic
474SD16—1728no—
475C2×C8—862no—
476C2×C8—862no—
477C2×C8—862no—
478C2×C8—862no—
479C2×C8—862no—
480C2×C8—862no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—434no—
492C4⋊C4—862no—
493C4⋊C4C2×C22⋊D4312yes—
494C4⋊C4C2×C22⋊D4312yes—
495C4⋊C4C2×C22⋊D4312yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—1724no—
502C22⋊C4—1724no—
503C42—24no—
504C42—24no—
505C42—1722no—
506C42C2×C22⋊D4312yes—
507C42C2×C22⋊D4312yes—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—1722no—
518C23—1722no—
519C23—1722no—
520C23—1722no—
521C23—1722no—
522C23—1722no—
523C23—1722no—
524C23#1306335012yes—
525C23C22×C22⋊D4311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529D4—44no—
530D4—3444no—
531D4—3444no—
532D4—3448no—
533D4—3448no—
534D4—1722no—
535D4—1722no—
536D4—1722no—
537D4—1722no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—864no—
563C2×C4—1722no—
564C2×C4—1722no—
565C2×C4#1306329412yes—
566C2×C4#1306329412yes—
567C2×C4C22×C22⋊D4311yescharacteristic
568C2×C4C22×C22⋊D4311yescharacteristic
569C2×C4#41×D4312yes—
570C2×C4#1306337112yes—
571C8—1722no—
572C8—1722no—
573C22—24no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—3442no—
584C22—3442no—
585C22—3442no—
586C22—3442no—
587C22—3442no—
588C22—3442no—
589C22—3442no—
590C22C2×#1306329411yescharacteristic
591C22#1458353712yes—
592C22C2×#1306335011yescharacteristic
593C22#1458369711yescharacteristic
594C22#1458370712yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—1724no—
604C4#1458344511yescharacteristic
605C4#1458346311yescharacteristic
606C4#1458356812yes—
607C2—42no—
608C2—81no—
609C2—6882no—
610C2#1913821011yescharacteristic
611C2#1914384711yescharacteristic
612C2#1914388111yescharacteristic
613C2#1914661012yes—
614C2#1914671512yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#552C2
socleC23×C43C22×D4
prime radical#19139541C2
hypercenterC2×#552D43
hypocenterC43#25583
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#14583265C2semidirect
2. C2×#14583099C2semidirect
4. #19135596C2semidirect
5. #19135596C2semidirect
6. #19135553C2semidirect
7. #19135394C2semidirect
8. #19134917C2semidirect
9. #19134912C2semidirect
10. #19134425C2semidirect
11. C2×C43×#552C2semidirect
85. #125×C43D4semidirect
92. C4×Q8×C43D4semidirect
300. C2×#552D43semidirect
354. C43#25583semidirect
376. #125C22⋊D43semidirect
387. C4×Q8C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832652
C2×#145830992
#191395412
#191355962
#191355532
#191353942
#191349172
#191349122
#191344252
C2×C43×#5522
#2558343