Finite Group Databasebeta

← #25199722

Subgroups — #25199722

Order 11008.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19142963C211yesmaximal, characteristic
2#19137070C211yesmaximal, characteristic, complemented
3#19136097C211yesmaximal, characteristic, complemented
4#19135982C211yesmaximal, characteristic
5#19135910C211yesmaximal, characteristic
6#19135446C211yesmaximal, characteristic
7#19135326C211yesmaximal, characteristic, complemented
8#19135276C211yesmaximal, characteristic
9#19135268C211yesmaximal, characteristic, complemented
10#19135214C211yesmaximal, characteristic
11#19135183C211yesmaximal, characteristic, complemented
12#19134948C211yesmaximal, characteristic
13#19134851C211yesmaximal, characteristic, complemented
14C4⋊C8×Dic43C211yesmaximal, characteristic, complemented
15C43×#2068C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C2×#13063320—21no—
18C2×Q8×Dic43C2211yescharacteristic
19C2×#13063259C2211yescharacteristic
20C2×#13063256C2211yescharacteristic
21C2×C4⋊(C43⋊C8)C2211yescharacteristic
22C2×#13063249C2211yescharacteristic
23C2×#13063247C2211yescharacteristic
24C2×#13063247C2211yescharacteristic
25C2×#13063245C2211yescharacteristic
26C2×C8×Dic43C2211yescharacteristic
27C2×C8×Dic43C2211yescharacteristic
28#14582806C2211yescharacteristic
29#14582801C2211yescharacteristic
30#14582800C2211yescharacteristic
31#14582800C2211yescharacteristic
32#14582798C2211yescharacteristic
33#14582797C2211yescharacteristic
34#14582795C2211yescharacteristic
35#14582795C2211yescharacteristic
36#14582791—21no—
37#14582789—21no—
38#14582788—21no—
39#14582784—21no—
40#14582782—21no—
41C4⋊C4×Dic43C2211yescharacteristic
42#14582750—21no—
43#14582744—21no—
44C42×Dic43C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52C2×#107×C43C2211yescharacteristic
53C2×C4⋊C8×C43C2211yescharacteristic
54C2×#103×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C43×#391C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C22×D172C2311yescharacteristic
60C2×#10621884—22no—
61C2×#10621884—22no—
62C2×#10621884—41no—
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65#13063320—44no—
66Q8×Dic43—22no—
67Q8×Dic43—22no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43C2311yescharacteristic
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—21no—
76C2×(C2×D43)⋊C4—21no—
77C2×(C2×D43)⋊C4—21no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×C4×Dic43—21no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43D411yescharacteristic
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43D411yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89C2×C4×Dic43C2311yescharacteristic
90#13063259—22no—
91#13063256—22no—
92C4⋊(C43⋊C8)—22no—
93#13063249—22no—
94#13063247—22no—
95#13063247—22no—
96#13063245—41no—
97C8×Dic43—41no—
98C8×Dic43—41no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C22×Q8×C43C2311yescharacteristic
107C22×C8×C43C2311yescharacteristic
108C22×C8×C43C2311yescharacteristic
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43C2311yescharacteristic
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C42×C43C2311yescharacteristic
114#107×C43—41no—
115C4⋊C8×C43—22no—
116#103×C43—22no—
117#103×C43—41no—
118#95×C43—21no—
119#95×C43—21no—
120#95×C43—21no—
121C23×D43—21no—
122C2×Q8×C43—21no—
123C2×Q8×C43—21no—
124C2×Q8×C43—21no—
125C2×Q8×C43—21no—
126C2×Q8×C43—21no—
127C2×Q8×C43#4112yes—
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
141C22×Dic43C2×D411yescharacteristic
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×C8×C43—21no—
149C2×C8×C43—21no—
150C2×C8×C43—21no—
151C2×C8×C43—21no—
152C2×C8×C43—21no—
153C2×C8×C43#4112yes—
154C4⋊C4×C43—21no—
155C4⋊C4×C43—21no—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—42no—
161C42×C43—21no—
162C42×C43—21no—
163#10621884—44no—
164(C2×D43)⋊C4—42no—
165(C2×D43)⋊C4—42no—
166(C2×D43)⋊C4—42no—
167(C2×D43)⋊C4—42no—
168(C2×D43)⋊C4—42no—
169(C2×D43)⋊C4—42no—
170C4⋊Dic43—24no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—42no—
177C4⋊Dic43#4112yes—
178Dic43⋊C4—42no—
179Dic43⋊C4—42no—
180Dic43⋊C4—42no—
181C4×Dic43—24no—
182C4×Dic43—22no—
183C4×Dic43—21no—
184C4×Dic43—21no—
185C4×Dic43—21no—
186C4×Dic43—21no—
187C4×Dic43—22no—
188C4×Dic43—22no—
189C4×Dic43—22no—
190C4×Dic43—42no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8—21no—
193C2×C43⋊C8—21no—
194C2×C43⋊C8—21no—
195C2×C43⋊C8—21no—
196C2×C43⋊C8#4112yes—
197C23×C43C22×D411yescharacteristic
198C22×D43—41no—
199C22×D43—41no—
200C22×D43—41no—
201C22×D43—41no—
202C22×D43—41no—
203C22×D43—41no—
204C22×D43—41no—
205Q8×C43—22no—
206Q8×C43—22no—
207Q8×C43—41no—
208Q8×C43—41no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—22no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43#13512yes—
227C2×C4×C43C2×#4111yescharacteristic
228C2×C4×C43C2×#4111yescharacteristic
229C2×C4×C43C2×#4111yescharacteristic
230C2×C4×C43C2×#4111yescharacteristic
231C2×C4×C43H(5,𝔽2)11yescharacteristic
232C2×C4×C432−1+411yescharacteristic
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—22no—
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43#13512yes—
252C2×Dic43AGL(1,ℤ/8)12yes—
253C2×Dic43#13712yes—
254D172—41no—
255D172—41no—
256D172—41no—
257D172—41no—
258C8×C43—22no—
259C8×C43—41no—
260C43⋊C8—22no—
261C43⋊C8—41no—
262#29435—431nomaximal, Sylow
263C22×C43#52511yescharacteristic
264C22×C43#53211yescharacteristic
265C22×C43#53711yescharacteristic
266C22×C43C2×#13511yescharacteristic
267C22×C43C2×#13511yescharacteristic
268C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C43C2×#13711yescharacteristic
270C2×D43—81no—
271C2×D43—81no—
272C2×D43—81no—
273C2×D43—81no—
274C2×D43—81no—
275C2×D43—81no—
276C2×D43—81no—
277C4×C43—22no—
278C4×C43—42no—
279C4×C43—41no—
280C4×C43—81no—
281C4×C43—81no—
282C4×C43—81no—
283C4×C43#55111yescharacteristic
284C4×C43#55211yescharacteristic
285C4×C43#55311yescharacteristic
286C4×C43#55611yescharacteristic
287Dic43—22no—
288Dic43—22no—
289Dic43—22no—
290Dic43—41no—
291Dic43—42no—
292Dic43—81no—
293Dic43—81no—
294#2860—431no—
295#2295—431no—
296#2088—431no—
297#2068D4311yescharacteristic, complemented
298#2067—431no—
299#2055—431no—
300#1968—431no—
301#1942—431no—
302#1934—431no—
303#1933—431no—
304#1925—431no—
305#1920—431no—
306#1880—431no—
307#1863—431no—
308C4×C4⋊C8—431no—
309C2×C43#260111yescharacteristic
310C2×C43#310611yescharacteristic
311C2×C43#310811yescharacteristic
312C2×C43#316811yescharacteristic
313C2×C43#318211yescharacteristic
314C2×C43#324711yescharacteristic
315C2×C43#324811yescharacteristic
316D43—161no—
317C22×SD16—431no—
318C2×#126—861no—
319C2×C4×Q8—431no—
320C2×#107—431no—
321C2×#107C2×D4311yescharacteristic
322C2×#106—431no—
323C2×C4⋊C8—431no—
324C2×C4⋊C8C2×D4311yescharacteristic
325C2×#103—431no—
326C2×#103C2×D4311yescharacteristic
327C2×#103C2×D4311yescharacteristic
328C2×#102—431no—
329C2×#102—431no—
330C2×#102—431no—
331C2×C4×C8—431no—
332C2×C4×C8—431no—
333#391—431no—
334#391C2×D4311yescharacteristic
335#390—431no—
336#390—431no—
337#390—431no—
338#389—431no—
339#389—431no—
340#389—431no—
341#389—431no—
342#389C2×D4311yescharacteristic
343#388—861no—
344#388—861no—
345#388—861no—
346#387—861no—
347#387—861no—
348C4×C4⋊C4—431no—
349C4×C4⋊C4—861no—
350#376—861no—
351C43—431no—
352#340—431no—
353#340—431no—
354#340—431no—
355#340—431no—
356#340—431no—
357#340C2×D4311yescharacteristic
358#336—431no—
359#336—431no—
360C43#2943511yescharacteristic, complemented, Sylow
361C22×Q8C22×D4311yescharacteristic
362C22×D4—431no—
363C2×SD16—862no—
364C2×SD16—862no—
365C2×SD16—1721no—
366C22×C8—431no—
367C22×C8—431no—
368C22×C8C22×D4311yescharacteristic
369C22×C8C22×D4311yescharacteristic
370#126—1724no—
371C4×Q8—862no—
372C4×Q8—862no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—431no—
376C2×C4⋊C4—431no—
377C2×C4⋊C4—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—861no—
382C2×C4⋊C4C22×D4311yescharacteristic
383C2×C4⋊C4C22×D4311yescharacteristic
384C2×C4⋊C4C22×D4311yescharacteristic
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4—861no—
390C2×C22⋊C4—861no—
391C2×C42—431no—
392C2×C42—431no—
393C2×C42—431no—
394C2×C42—431no—
395C2×C42—431no—
396C2×C42—431no—
397C2×C42—861no—
398C2×C42—861no—
399C2×C42C22×D4311yescharacteristic
400#107—41no—
401#107—862no—
402#106—862no—
403C4⋊C8—22no—
404C4⋊C8—862no—
405#103—22no—
406#103—41no—
407#103—862no—
408#102—862no—
409#102—862no—
410#102—1721no—
411C4×C8—1721no—
412C4×C8—1721no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—861no—
417#95—861no—
418#95—861no—
419#95—861no—
420#95—861no—
421#95—861no—
422#95—861no—
423C24—861no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1062190212yes—
430C2×D4—861no—
431C2×D4—861no—
432C2×D4—861no—
433C2×D4—861no—
434C2×D4—861no—
435C2×D4—861no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—431no—
441C22×C4—431no—
442C22×C4—431no—
443C22×C4—431no—
444C22×C4—861no—
445C22×C4—861no—
446C22×C4—861no—
447C22×C4D4×D4311yescharacteristic
448C22×C4D4×D4311yescharacteristic
449C22×C4C23×D4311yescharacteristic
450SD16—1724no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—432no—
457C2×C8—861no—
458C2×C8—861no—
459C2×C8—861no—
460C2×C8—861no—
461C2×C8—861no—
462C2×C8#1062190012yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—432no—
471C4⋊C4—864no—
472C4⋊C4—861no—
473C4⋊C4—861no—
474C4⋊C4—861no—
475C4⋊C4—861no—
476C4⋊C4—861no—
477C4⋊C4—1722no—
478C4⋊C4—1722no—
479C4⋊C4—1722no—
480C4⋊C4—1722no—
481C22⋊C4—1722no—
482C22⋊C4—1722no—
483C22⋊C4—1722no—
484C22⋊C4—1722no—
485C22⋊C4—1722no—
486C22⋊C4—1722no—
487C42—21no—
488C42—21no—
489C42—862no—
490C42—862no—
491C42—862no—
492C42—864no—
493C42—862no—
494C42—861no—
495C42—861no—
496C42—861no—
497C42—861no—
498C42—1722no—
499C23—1721no—
500C23—1721no—
501C23—1721no—
502C23—1721no—
503C23—1721no—
504C23—1721no—
505C23—1721no—
506C23C2×D4×D4311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—1721no—
512D4—1721no—
513D4—1721no—
514D4—1721no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—432no—
533C2×C4—432no—
534C2×C4—432no—
535C2×C4—861no—
536C2×C4—861no—
537C2×C4—861no—
538C2×C4—861no—
539C2×C4—861no—
540C2×C4—861no—
541C2×C4—862no—
542C2×C4—861no—
543C2×C4—861no—
544C2×C4—861no—
545C2×C4—1721no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4—1721no—
553C2×C4#1306334812yes—
554C2×C4#41×D4311yescharacteristic
555C2×C4#41×D4311yescharacteristic
556C2×C4C2×#1062190011yescharacteristic
557C2×C4C2×#1062190211yescharacteristic
558C2×C4#1306337011yescharacteristic
559C2×C4#1306337211yescharacteristic
560C8—22no—
561C8—41no—
562C8—862no—
563C8—1721no—
564C22—3441no—
565C22—3441no—
566C22—3441no—
567C22—3441no—
568C22—3441no—
569C22—3441no—
570C22—3441no—
571C22#1458341211yescharacteristic
572C22#1458344411yescharacteristic
573C22#1458347811yescharacteristic
574C22#135×D4311yescharacteristic
575C22C2×#1306334811yescharacteristic
576C22AGL(1,ℤ/8)×D4311yescharacteristic
577C22#137×D4311yescharacteristic
578C4—22no—
579C4—41no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—862no—
585C4—862no—
586C4—862no—
587C4—3441no—
588C4—3441no—
589C4—1721no—
590C4—1722no—
591C4#1458357611yescharacteristic
592C4#1458359111yescharacteristic
593C4#1458359211yescharacteristic
594C4#1458361611yescharacteristic
595C2—6881no—
596C2#1914019311yescharacteristic
597C2#1914465811yescharacteristic
598C2#1914468011yescharacteristic
599C2#1914516611yescharacteristic
600C2#1914530811yescharacteristic
601C2#1914591811yescharacteristic
602C2#1914593011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2068C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2068D43
hypocenterC43#29435
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
2. #19137070C2semidirect
3. #19136097C2semidirect
7. #19135326C2semidirect
9. #19135268C2semidirect
11. #19135183C2semidirect
13. #19134851C2semidirect
14. C4⋊C8×Dic43C2semidirect
15. C43×#2068C2semidirect
297. #2068D43semidirect
360. C43#29435semidirect

Maximal subgroups

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

subgroupindex
#191429632
#191370702
#191360972
#191359822
#191359102
#191354462
#191353262
#191352762
#191352682
#191352142
#191351832
#191349482
#191348512
C4⋊C8×Dic432
C43×#20682
#2943543