Finite Group Databasebeta

← #25104276

Subgroups — #25104276

Order 11008.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583202C211yesmaximal, characteristic
2#19137154C211yesmaximal, characteristic, complemented
3C4×#13063287C211yesmaximal, characteristic
4#19135549C211yesmaximal, characteristic, complemented
5#19135362C211yesmaximal, characteristic, complemented
6#19135180C211yesmaximal, characteristic, complemented
7#19134859C211yesmaximal, characteristic
8#19134841C211yesmaximal, characteristic
9#19134604C211yesmaximal, characteristic
10#19134494C211yesmaximal, characteristic, complemented
11#19134491C211yesmaximal, characteristic, complemented
12#19134447C211yesmaximal, characteristic, complemented
13#19134419C211yesmaximal, characteristic
14#19134419C211yesmaximal, characteristic
15C43×#1918C211yesmaximal, characteristic, complemented
16C2×#13063316C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×#13063287C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20C2×C42×D43—21no—
21#14583202C418yescomplemented
22C2×#13063258C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×Dic43⋊C8C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063245C2211yescharacteristic
30C2×#13063245C2211yescharacteristic
31C2×#13063235C2211yescharacteristic
32C2×#13063235C2211yescharacteristic
33#14582797C2211yescharacteristic
34#14582784—21no—
35#14582780—21no—
36#14582764C2211yescharacteristic
37C4⋊C4×Dic43—21no—
38C4⋊C4×Dic43C2211yescharacteristic
39C4⋊C4×Dic43C2211yescharacteristic
40C4⋊C4×Dic43C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42#14582750—21no—
43C4×(C2×D43)⋊C4—21no—
44#14582746—21no—
45#14582746—21no—
46C42×Dic43C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C43×#389C2211yescharacteristic
55C4×C4⋊C4×C43C2211yescharacteristic
56C43×#340C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66#13063316—22no—
67#13063316C2×C414yescomplemented
68C4×D172—22no—
69C4×D172—22no—
70#13063287—22no—
71#13063287—22no—
72#13063287C2×C414yes—
73#13063287Q814yes—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43D411yescharacteristic
88C2×C4×Dic43D411yescharacteristic
89C2×C4×Dic43D411yescharacteristic
90C2×C4×Dic43D411yescharacteristic
91C2×C4×Dic43C2311yescharacteristic
92C2×C4×Dic43C2311yescharacteristic
93C42×D43—44no—
94#13063258—22no—
95#13063257—22no—
96#13063256C2×C414yescomplemented
97Dic43⋊C8—22no—
98Dic43⋊C8C2×C414yescomplemented
99#13063247—22no—
100#13063247C2×C414yes—
101#13063245—22no—
102#13063245C2×C414yes—
103#13063235—22no—
104#13063235—22no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C42×C43—21no—
120C2×C42×C43C2311yescharacteristic
121#107×C43C2×C414yescomplemented
122#95×C43—21no—
123#95×C43—21no—
124C23×D43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2411yescharacteristic
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43C2×D411yescharacteristic
136C22×Dic43C2×D411yescharacteristic
137C22×Dic43C2×D411yescharacteristic
138C22×Dic43C2×D411yescharacteristic
139C2×D172—21no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172C22×C412yes—
143C2×D172C2×Q812yes—
144C2×D172#4112yes—
145C2×C4×D43—24no—
146C2×C4×D43—24no—
147C2×C4×D43—24no—
148C2×C4×D43—42no—
149C2×C4×D43—42no—
150C2×C4×D43—42no—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C2×C8×C43—21no—
155C2×C8×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43C22×C412yes—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—42no—
166C4⋊C4×C43C22×C412yes—
167C4⋊C4×C43C22×C412yes—
168C4⋊C4×C43C2×Q812yes—
169C42×C43—21no—
170C42×C43—21no—
171C42×C43—42no—
172(C2×D43)⋊C4—24no—
173(C2×D43)⋊C4—24no—
174(C2×D43)⋊C4—24no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—24no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43C22×C412yes—
181C4⋊Dic43#4112yes—
182Dic43⋊C4—24no—
183Dic43⋊C4—42no—
184C4×Dic43—22no—
185C4×Dic43—41no—
186C4×Dic43—41no—
187C4×Dic43—41no—
188C4×Dic43—42no—
189C4×Dic43—42no—
190C4×Dic43—42no—
191C4×Dic43—42no—
192C4×Dic43C22×C412yes—
193C4×Dic43C2×Q812yes—
194C4×Dic43#4112yes—
195C4×Dic43#4112yes—
196C2×C43⋊C8—21no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8C22×C412yes—
202C23×C43C22×D411yescharacteristic
203C22×D43—22no—
204C22×D43—22no—
205C22×D43—22no—
206C22×D43—41no—
207C22×D43—41no—
208C22×D43—41no—
209C22×D43—41no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—22no—
213C2×C4×C43—22no—
214C2×C4×C43—22no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
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—41no—
227C2×C4×C43AGL(1,ℤ/8)12yes—
228C2×C4×C43C23×C411yescharacteristic
229C2×C4×C43C22×Q811yescharacteristic
230C2×C4×C43C2×#4111yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C2×C4×C43C2×#4111yescharacteristic
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43C4×D412yes—
252C2×Dic43C4×D412yes—
253C2×Dic43C22⋊Q812yes—
254C2×Dic43C22⋊Q812yes—
255D172—22no—
256D172—22no—
257D172—22no—
258D172C4×Q814yes—
259C4×D43—48no—
260C4×D43—48no—
261C4×D43—48no—
262C8×C43—22no—
263C8×C43—41no—
264C43⋊C8—22no—
265C43⋊C8—41no—
266#21617—431nomaximal, Sylow
267C22×C43C2×C4×D411yescharacteristic
268C22×C43C2×C22⋊Q811yescharacteristic
269C22×C43#52511yescharacteristic
270C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C43#57511yescharacteristic
272C22×C43#57511yescharacteristic
273C22×C43#57511yescharacteristic
274C2×D43—24no—
275C2×D43—42no—
276C2×D43—42no—
277C2×D43—42no—
278C2×D43—42no—
279C2×D43—42no—
280C2×D43—42no—
281C4×C43—22no—
282C4×C43—42no—
283C4×C43—41no—
284C4×C43—42no—
285C4×C43—42no—
286C4×C43—81no—
287C4×C43C2×C4×Q811yescharacteristic
288C4×C43C4×#4111yescharacteristic
289C4×C43#52911yescharacteristic
290C4×C43#53311yescharacteristic
291Dic43—22no—
292Dic43—22no—
293Dic43—22no—
294Dic43—22no—
295Dic43—42no—
296Dic43—42no—
297Dic43—81no—
298C2×#478—431no—
299C4×#125—431no—
300C42×D4—431no—
301#1989—431no—
302#1949—431no—
303#1920—431no—
304#1918D4311yescharacteristic, complemented
305#1864—431no—
306#1862—431no—
307#1804—431no—
308#1776—431no—
309#1776—431no—
310#1767—431no—
311#1762—431no—
312#1762—431no—
313C2×C43C4×C22⋊Q811yescharacteristic
314C2×C43#294311yescharacteristic
315C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C43#307711yescharacteristic
317C2×C43#308011yescharacteristic
318C2×C43#311111yescharacteristic
319C2×C43#311411yescharacteristic
320D43—44no—
321C2×#125—431no—
322C2×C4×D4—431no—
323C2×C4×D4—431no—
324C2×C4×D4—431no—
325C22×C42—861no—
326#478—438no—
327C2×#107—431no—
328C2×#107C2×D4311yescharacteristic
329C2×#106—431no—
330C2×#106—431no—
331C2×C4⋊C8—431no—
332C2×C4⋊C8—431no—
333C2×#102—431no—
334C2×#102—431no—
335C2×#102—431no—
336C2×#102—431no—
337C2×#97—431no—
338C2×#97—431no—
339#389—431no—
340#389C2×D4311yescharacteristic
341#387—861no—
342#387—861no—
343#382—431no—
344C4×C4⋊C4—431no—
345C4×C4⋊C4—431no—
346C4×C4⋊C4—431no—
347C4×C4⋊C4—431no—
348C4×C4⋊C4—861no—
349C4×C4⋊C4—861no—
350C4×C4⋊C4C2×D4311yescharacteristic
351C4×C22⋊C4—861no—
352C4×C22⋊C4—861no—
353C4×C22⋊C4—861no—
354C43—431no—
355#340—431no—
356#340—431no—
357#340—431no—
358#340—431no—
359#340—431no—
360#340C2×D4311yescharacteristic
361#340C2×D4311yescharacteristic
362#340C2×D4311yescharacteristic
363#336—431no—
364#336C2×D4311yescharacteristic
365C43#2161711yescharacteristic, complemented, Sylow
366C22×D4—431no—
367C23×C4—861no—
368C23×C4—861no—
369C23×C4—861no—
370C22×C8—431no—
371C22×C8—431no—
372C22×C8C22×D4311yescharacteristic
373C22×C8C22×D4311yescharacteristic
374#125—434no—
375#125—862no—
376C4×D4—434no—
377C4×D4—434no—
378C4×D4—862no—
379C4×D4—862no—
380C4×D4—862no—
381C4×D4—862no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—431no—
384C2×C4⋊C4—431no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—861no—
387C2×C4⋊C4—861no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4C22×D4311yescharacteristic
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C4⋊C4C22×D4311yescharacteristic
392C2×C4⋊C4C22×D4311yescharacteristic
393C2×C22⋊C4—861no—
394C2×C22⋊C4—861no—
395C2×C22⋊C4—861no—
396C2×C42—21no—
397C2×C42—431no—
398C2×C42—431no—
399C2×C42—431no—
400C2×C42—431no—
401C2×C42—431no—
402C2×C42—431no—
403C2×C42—861no—
404C2×C42—861no—
405C2×C42—861no—
406C2×C42—861no—
407C2×C42—1724no—
408C2×C42C22×D4311yescharacteristic
409#107—862no—
410#107C4×D4314yescomplemented
411#106—434no—
412#106—862no—
413C4⋊C8—434no—
414C4⋊C8—862no—
415#102—434no—
416#102—434no—
417#102—862no—
418#102—862no—
419#97—862no—
420#97—862no—
421#95—21no—
422#95—21no—
423#95—861no—
424#95—861no—
425#95—861no—
426#95—861no—
427#95—861no—
428#95—861no—
429#95—861no—
430C24—861no—
431C2×D4—432no—
432C2×D4—432no—
433C2×D4—432no—
434C2×D4—861no—
435C2×D4—861no—
436C2×D4—861no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—431no—
442C22×C4—431no—
443C22×C4—431no—
444C22×C4—431no—
445C22×C4—864no—
446C22×C4—864no—
447C22×C4—864no—
448C22×C4—861no—
449C22×C4—861no—
450C22×C4—861no—
451C22×C4—1722no—
452C22×C4—1722no—
453C22×C4—1722no—
454C22×C4—1722no—
455C22×C4—1722no—
456C22×C4—1722no—
457C22×C4D4×D4311yescharacteristic
458C22×C4D4×D4311yescharacteristic
459C22×C4C23×D4311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—432no—
466C2×C8—861no—
467C2×C8—861no—
468C2×C8—861no—
469C2×C8—861no—
470C2×C8—861no—
471C2×C8C2×C4×D4312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—432no—
479C4⋊C4—432no—
480C4⋊C4—864no—
481C4⋊C4—864no—
482C4⋊C4—861no—
483C4⋊C4—861no—
484C4⋊C4—861no—
485C4⋊C4—861no—
486C4⋊C4—1722no—
487C4⋊C4C2×C4×D4312yes—
488C4⋊C4C2×C4×D4312yes—
489C4⋊C4Q8×D4312yes—
490C22⋊C4—864no—
491C22⋊C4—864no—
492C22⋊C4—864no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—432no—
497C42—432no—
498C42—432no—
499C42—432no—
500C42—862no—
501C42—1721no—
502C42—1721no—
503C42—1721no—
504C42—1722no—
505C42—1722no—
506C42—1722no—
507C42—1722no—
508C23—862no—
509C23—862no—
510C23—862no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23—1721no—
515C23C2×D4×D4311yescharacteristic
516D4—434no—
517D4—862no—
518D4—862no—
519D4—862no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—432no—
538C2×C4—432no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—861no—
542C2×C4—861no—
543C2×C4—861no—
544C2×C4—861no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—861no—
548C2×C4—861no—
549C2×C4—862no—
550C2×C4—862no—
551C2×C4—1728no—
552C2×C4—1728no—
553C2×C4—1728no—
554C2×C4—1721no—
555C2×C4—1721no—
556C2×C4—1721no—
557C2×C4—1721no—
558C2×C4—1721no—
559C2×C4—1721no—
560C2×C4—1721no—
561C2×C4#1306334912yes—
562C2×C4C22×C4×D4311yescharacteristic
563C2×C4C2×Q8×D4311yescharacteristic
564C2×C4#41×D4311yescharacteristic
565C2×C4#41×D4311yescharacteristic
566C2×C4#41×D4311yescharacteristic
567C2×C4#41×D4311yescharacteristic
568C8—22no—
569C8—41no—
570C8—862no—
571C8—1721no—
572C22—864no—
573C22—1722no—
574C22—1722no—
575C22—1722no—
576C22—1722no—
577C22—1722no—
578C22—1722no—
579C22C4×D4×D4311yescharacteristic
580C22C22⋊Q8×D4311yescharacteristic
581C22#1458340711yescharacteristic
582C22C2×#1306334911yescharacteristic
583C22#1458369611yescharacteristic
584C22#1458369811yescharacteristic
585C22#1458369811yescharacteristic
586C4—22no—
587C4—41no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—862no—
593C4—862no—
594C4—862no—
595C4—862no—
596C4—3441no—
597C4—1722no—
598C4—1722no—
599C4C4×Q8×D4311yescharacteristic
600C4#1458336711yescharacteristic
601C4#1458343211yescharacteristic
602C4#1458345511yescharacteristic
603C2—1724no—
604C2#1913712911yescharacteristic
605C2#1914339411yescharacteristic
606C2C4×#1306334911yescharacteristic
607C2#1914438711yescharacteristic
608C2#1914441411yescharacteristic
609C2#1914470711yescharacteristic
610C2#1914473011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1918C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1918D43
hypocenterC43#21617
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #19137154C2semidirect
4. #19135549C2semidirect
5. #19135362C2semidirect
6. #19135180C2semidirect
10. #19134494C2semidirect
11. #19134491C2semidirect
12. #19134447C2semidirect
15. C43×#1918C2semidirect
21. #14583202C4semidirect
67. #13063316C2×C4semidirect
96. #13063256C2×C4semidirect
98. Dic43⋊C8C2×C4semidirect
121. #107×C43C2×C4semidirect
304. #1918D43semidirect
365. C43#21617semidirect
410. #107C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832022
#191371542
C4×#130632872
#191355492
#191353622
#191351802
#191348592
#191348412
#191346042
#191344942
#191344912
#191344472
#191344192
C43×#19182
#2161743