Finite Group Databasebeta

← #25108757

Subgroups — #25108757

Order 11008.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583202C211yesmaximal, characteristic
2#19137154C211yesmaximal, characteristic, complemented
3#19137053C211yesmaximal, characteristic
4#19135549C211yesmaximal, characteristic, complemented
5#19135364C211yesmaximal, characteristic, complemented
6#19135335C211yesmaximal, characteristic, complemented
7#19135292C211yesmaximal, characteristic
8#19135291C211yesmaximal, characteristic
9#19135221C211yesmaximal, characteristic
10#19135213C211yesmaximal, characteristic
11#19135176C211yesmaximal, characteristic, complemented
12#19135168C211yesmaximal, characteristic, complemented
13#19134604C211yesmaximal, characteristic
14#19134449C211yesmaximal, characteristic, complemented
15C43×#1943C211yesmaximal, characteristic, complemented
16C2×#13063316C2211yescharacteristic
17C2×#13063287C2211yescharacteristic
18C2×#13063283—21no—
19#14583202C418yescomplemented
20C2×#13063258C2211yescharacteristic
21C2×#13063256C2211yescharacteristic
22C2×Dic43⋊C8C2211yescharacteristic
23C2×Dic43⋊C8C2211yescharacteristic
24C2×#13063247C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063245C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×#13063236C2211yescharacteristic
29#14582801C2211yescharacteristic
30#14582800C2211yescharacteristic
31#14582798C2211yescharacteristic
32#14582797C2211yescharacteristic
33#14582791—21no—
34#14582789—21no—
35#14582784—21no—
36#14582780—21no—
37#14582764C2211yescharacteristic
38C4⋊C4×Dic43—21no—
39C4⋊C4×Dic43C2211yescharacteristic
40C4⋊C4×Dic43C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42#14582750—21no—
43#14582746—21no—
44C42×Dic43C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×#97×C43C2211yescharacteristic
56C43×#389C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C43⋊C8C2311yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64#13063316—22no—
65#13063316C2×C414yes—
66#13063287—22no—
67#13063287C2×C414yes—
68#13063283—44no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
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×C4×Dic43—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#13063258—41no—
91#13063256C2×C414yes—
92Dic43⋊C8—22no—
93Dic43⋊C8C2×C414yes—
94#13063247—41no—
95#13063247C2×C414yes—
96#13063245—41no—
97#13063245C2×C414yes—
98#13063236—41no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C22×C8×C43C2311yescharacteristic
107C22×C8×C43C2311yescharacteristic
108C2×C4⋊C4×C43—21no—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43C2311yescharacteristic
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C4⋊C4×C43C2311yescharacteristic
114C2×C42×C43C2311yescharacteristic
115#107×C43C2×C414yes—
116#106×C43—41no—
117#97×C43—41no—
118#95×C43—21no—
119#95×C43—21no—
120C23×D43—21no—
121C22×C4×C43—21no—
122C22×C4×C43—21no—
123C22×C4×C43—21no—
124C22×C4×C43—21no—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2411yescharacteristic
128C22×Dic43—21no—
129C22×Dic43—21no—
130C22×Dic43—21no—
131C22×Dic43C2×D411yescharacteristic
132C22×Dic43C2×D411yescharacteristic
133C22×Dic43C2×D411yescharacteristic
134C22×Dic43C2×D411yescharacteristic
135C2×D172—21no—
136C2×D172—21no—
137C2×D172—21no—
138C2×D172—21no—
139C2×D172—21no—
140C2×D172C22×C412yes—
141C2×C4×D43—24no—
142C2×C4×D43—42no—
143C2×C4×D43—42no—
144C2×C8×C43—21no—
145C2×C8×C43—21no—
146C2×C8×C43—21no—
147C2×C8×C43—21no—
148C2×C8×C43—21no—
149C2×C8×C43C22×C412yes—
150C4⋊C4×C43—21no—
151C4⋊C4×C43—21no—
152C4⋊C4×C43—21no—
153C4⋊C4×C43—21no—
154C4⋊C4×C43—21no—
155C4⋊C4×C43—21no—
156C4⋊C4×C43—42no—
157C4⋊C4×C43—42no—
158C4⋊C4×C43C22×C412yes—
159C4⋊C4×C43C22×C412yes—
160C42×C43—21no—
161C42×C43—21no—
162(C2×D43)⋊C4—24no—
163(C2×D43)⋊C4—42no—
164(C2×D43)⋊C4—42no—
165(C2×D43)⋊C4—42no—
166(C2×D43)⋊C4—42no—
167C4⋊Dic43—21no—
168C4⋊Dic43—24no—
169C4⋊Dic43—21no—
170C4⋊Dic43—21no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—42no—
174C4⋊Dic43C22×C412yes—
175Dic43⋊C4—24no—
176Dic43⋊C4—42no—
177C4×Dic43—21no—
178C4×Dic43—21no—
179C4×Dic43—22no—
180C4×Dic43—41no—
181C4×Dic43—41no—
182C4×Dic43—41no—
183C4×Dic43—42no—
184C4×Dic43—42no—
185C4×Dic43—42no—
186C4×Dic43C22×C412yes—
187C4×Dic43#4112yes—
188C2×C43⋊C8—21no—
189C2×C43⋊C8—21no—
190C2×C43⋊C8—21no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8—21no—
193C2×C43⋊C8C22×C412yes—
194C23×C43C22×D411yescharacteristic
195C22×D43—22no—
196C22×D43—41no—
197C22×D43—41no—
198C22×D43—41no—
199C22×D43—41no—
200C22×D43—41no—
201C22×D43—41no—
202C2×C4×C43—22no—
203C2×C4×C43—22no—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—21no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—21no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43C23×C411yescharacteristic
221C2×C4×C43C2×#4111yescharacteristic
222C2×C4×C43H(5,𝔽2)11yescharacteristic
223C2×C4×C432−1+411yescharacteristic
224C2×C4×C432−1+411yescharacteristic
225C2×C4×C432−1+411yescharacteristic
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—22no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43C4×D412yes—
245C2×Dic43C4×D412yes—
246C2×Dic43AGL(1,ℤ/8)12yes—
247D172—22no—
248D172—22no—
249D172—41no—
250D172—41no—
251C4×D43—48no—
252C8×C43—22no—
253C8×C43—41no—
254C43⋊C8—22no—
255C43⋊C8—41no—
256#22120—431nomaximal, Sylow
257C22×C43C2×C4×D411yescharacteristic
258C22×C43#53611yescharacteristic
259C22×C43#53711yescharacteristic
260C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C43#57511yescharacteristic
262C22×C43#57511yescharacteristic
263C22×C43#57511yescharacteristic
264C2×D43—42no—
265C2×D43—42no—
266C2×D43—42no—
267C2×D43—81no—
268C2×D43—81no—
269C2×D43—81no—
270C2×D43—81no—
271C4×C43—42no—
272C4×C43—42no—
273C4×C43—41no—
274C4×C43—41no—
275C4×C43—81no—
276C4×C43—81no—
277C4×C43#51911yescharacteristic
278C4×C43#52011yescharacteristic
279C4×C43#54111yescharacteristic
280C4×C43#54211yescharacteristic
281Dic43—22no—
282Dic43—22no—
283Dic43—22no—
284Dic43—42no—
285Dic43—41no—
286Dic43—81no—
287Dic43—81no—
288C2×#478—431no—
289C4×#125—431no—
290#2293—431no—
291#1989—431no—
292#1949—431no—
293#1944—431no—
294#1943D4311yescharacteristic, complemented
295#1936—431no—
296#1936—431no—
297#1926—431no—
298#1925—431no—
299#1919—431no—
300#1918—431no—
301#1804—431no—
302#1767—431no—
303C2×C43#232311yescharacteristic
304C2×C43#294311yescharacteristic
305C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C43#323411yescharacteristic
307C2×C43#324011yescharacteristic
308C2×C43#325111yescharacteristic
309C2×C43#325611yescharacteristic
310D43—82no—
311C2×#125—431no—
312C2×C4×D4—431no—
313C2×#117—861no—
314#478—438no—
315C2×#107—431no—
316C2×#107C2×D4311yescharacteristic
317C2×#106—431no—
318C2×#106C2×D4311yescharacteristic
319C2×C4⋊C8—431no—
320C2×C4⋊C8—431no—
321C2×#102—431no—
322C2×#102—431no—
323C2×#102—431no—
324C2×#102—431no—
325C2×#97—431no—
326C2×#97C2×D4311yescharacteristic
327#390—431no—
328#390—431no—
329#389—431no—
330#389—431no—
331#389C2×D4311yescharacteristic
332#389C2×D4311yescharacteristic
333#388—861no—
334#388—861no—
335#387—861no—
336#387—861no—
337#382—431no—
338C4×C4⋊C4—431no—
339C4×C4⋊C4—431no—
340C4×C4⋊C4—431no—
341C4×C4⋊C4—861no—
342C4×C4⋊C4—861no—
343C4×C22⋊C4—861no—
344C43—431no—
345#340—431no—
346#340—431no—
347#340—431no—
348#340—431no—
349#340—431no—
350#340—431no—
351#340C2×D4311yescharacteristic
352#340C2×D4311yescharacteristic
353#336—431no—
354#336—431no—
355C43#2212011yescharacteristic, complemented, Sylow
356C22×D4—431no—
357C23×C4—861no—
358C22×C8—431no—
359C22×C8—431no—
360C22×C8C22×D4311yescharacteristic
361C22×C8C22×D4311yescharacteristic
362#125—434no—
363#125—862no—
364C4×D4—434no—
365C4×D4—862no—
366#117—1724no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—431no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—861no—
373C2×C4⋊C4—861no—
374C2×C4⋊C4—861no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4C22×D4311yescharacteristic
377C2×C4⋊C4C22×D4311yescharacteristic
378C2×C4⋊C4C22×D4311yescharacteristic
379C2×C4⋊C4C22×D4311yescharacteristic
380C2×C22⋊C4—861no—
381C2×C22⋊C4—861no—
382C2×C22⋊C4—861no—
383C2×C22⋊C4—861no—
384C2×C22⋊C4—861no—
385C2×C42—431no—
386C2×C42—431no—
387C2×C42—431no—
388C2×C42—431no—
389C2×C42—431no—
390C2×C42—431no—
391C2×C42—861no—
392C2×C42—861no—
393C2×C42—861no—
394C2×C42C22×D4311yescharacteristic
395#107—1721no—
396#107C4×D4314yes—
397#106—41no—
398#106—434no—
399C4⋊C8—434no—
400C4⋊C8—862no—
401#102—434no—
402#102—434no—
403#102—1721no—
404#102—1721no—
405#97—41no—
406#97—1721no—
407#95—21no—
408#95—21no—
409#95—861no—
410#95—861no—
411#95—861no—
412#95—861no—
413#95—861no—
414#95—861no—
415#95—861no—
416C24—861no—
417C2×D4—432no—
418C2×D4—861no—
419C2×D4—861no—
420C2×D4—861no—
421C2×D4—861no—
422C2×D4—861no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—431no—
428C22×C4—431no—
429C22×C4—431no—
430C22×C4—431no—
431C22×C4—864no—
432C22×C4—861no—
433C22×C4—861no—
434C22×C4—861no—
435C22×C4—1722no—
436C22×C4—1722no—
437C22×C4D4×D4311yescharacteristic
438C22×C4D4×D4311yescharacteristic
439C22×C4C23×D4311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—432no—
446C2×C8—861no—
447C2×C8—861no—
448C2×C8—861no—
449C2×C8—861no—
450C2×C8—861no—
451C2×C8C2×C4×D4312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—432no—
461C4⋊C4—864no—
462C4⋊C4—864no—
463C4⋊C4—861no—
464C4⋊C4—861no—
465C4⋊C4—861no—
466C4⋊C4—861no—
467C4⋊C4—861no—
468C4⋊C4—1722no—
469C4⋊C4—1722no—
470C4⋊C4C2×C4×D4312yes—
471C4⋊C4C2×C4×D4312yes—
472C22⋊C4—864no—
473C22⋊C4—1722no—
474C22⋊C4—1722no—
475C22⋊C4—1722no—
476C22⋊C4—1722no—
477C42—21no—
478C42—21no—
479C42—432no—
480C42—432no—
481C42—862no—
482C42—861no—
483C42—861no—
484C42—1721no—
485C42—1721no—
486C42—1721no—
487C42—1722no—
488C42—1722no—
489C42—1722no—
490C23—862no—
491C23—1721no—
492C23—1721no—
493C23—1721no—
494C23—1721no—
495C23—1721no—
496C23—1721no—
497C23C2×D4×D4311yescharacteristic
498D4—862no—
499D4—862no—
500D4—1721no—
501D4—1721no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—432no—
521C2×C4—432no—
522C2×C4—432no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—861no—
526C2×C4—861no—
527C2×C4—861no—
528C2×C4—861no—
529C2×C4—862no—
530C2×C4—861no—
531C2×C4—861no—
532C2×C4—861no—
533C2×C4—1728no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4C22×C4×D4311yescharacteristic
543C2×C4#41×D4311yescharacteristic
544C2×C4#1306337011yescharacteristic
545C2×C4#1306337211yescharacteristic
546C2×C4#1306337311yescharacteristic
547C2×C4#1306337311yescharacteristic
548C8—22no—
549C8—41no—
550C8—862no—
551C8—1721no—
552C22—3441no—
553C22—3441no—
554C22—3441no—
555C22—3441no—
556C22—1722no—
557C22—1722no—
558C22—1722no—
559C22C4×D4×D4311yescharacteristic
560C22#1458347411yescharacteristic
561C22#1458347611yescharacteristic
562C22AGL(1,ℤ/8)×D4311yescharacteristic
563C22#1458369511yescharacteristic
564C22#1458369811yescharacteristic
565C22#1458369811yescharacteristic
566C4—41no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—862no—
573C4—862no—
574C4—862no—
575C4—3441no—
576C4—3441no—
577C4—1721no—
578C4—1722no—
579C4#1458337511yescharacteristic
580C4#1458338011yescharacteristic
581C4#1458350811yescharacteristic
582C4#1458351911yescharacteristic
583C2—3442no—
584C2#1913727211yescharacteristic
585C2#1914339411yescharacteristic
586C2#1914340611yescharacteristic
587C2#1914578611yescharacteristic
588C2#1914583611yescharacteristic
589C2#1914595211yescharacteristic
590C2#1914599211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1943C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1943D43
hypocenterC43#22120
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
2. #19137154C2semidirect
4. #19135549C2semidirect
5. #19135364C2semidirect
6. #19135335C2semidirect
11. #19135176C2semidirect
12. #19135168C2semidirect
14. #19134449C2semidirect
15. C43×#1943C2semidirect
19. #14583202C4semidirect
294. #1943D43semidirect
355. C43#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#145832022
#191371542
#191370532
#191355492
#191353642
#191353352
#191352922
#191352912
#191352212
#191352132
#191351762
#191351682
#191346042
#191344492
C43×#19432
#2212043