Finite Group Databasebeta

← #25106521

Subgroups — #25106521

Order 11008.

1162 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583431C211yesmaximal, characteristic
2C2×#14582921C211yesmaximal, characteristic, complemented
3#19137000C211yesmaximal, characteristic, complemented
4#19135549C212yesmaximal, complemented
5#19135526C212yesmaximal, complemented
6#19135030C212yesmaximal
7#19135027C212yesmaximal, complemented
8#19134601C211yesmaximal, characteristic, complemented
9#19134597C211yesmaximal, characteristic, complemented
10#19134595C211yesmaximal, characteristic, complemented
11C43×#1802C211yesmaximal, characteristic, complemented
12C22×#10621872C2212yes—
13#14583431—24no—
14#14583431C418yescomplemented
15C2×#13063318—21no—
16C2×#13063316C2211yescharacteristic
17C2×#13063312C2211yescharacteristic
18C2×#13063307—21no—
19C2×#13063305C2211yescharacteristic
20C2×#13063296C2211yescharacteristic
21C2×C4×Dic86C2211yescharacteristic
22C2×C4×C22⋊D43—21no—
23C2×#13063284—21no—
24C2×#13063283—21no—
25C2×C42×D43C2211yescharacteristic
26#14582921—24no—
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28C2×Dic43⋊C8C2212yes—
29C2×#13063250C2212yes—
30C2×#13063248C2212yes—
31C2×#13063246C2212yes—
32#14582769C2211yescharacteristic
33#14582764—21no—
34#14582756—21no—
35#14582756C2212yes—
36C4⋊C4×Dic43—21no—
37C4⋊C4×Dic43C2212yes—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×C4×D4×C43C2211yescharacteristic
42C2×C4⋊C8×C43C2211yescharacteristic
43C2×#102×C43C2212yes—
44C43×#384C2211yescharacteristic
45C43×#340C2212yes—
46C22×Dic86C2311yescharacteristic
47C22×C22⋊D43—21no—
48C22×C4×D43D411yescharacteristic
49C22×C4×D43D411yescharacteristic
50C22×C4×D43C2311yescharacteristic
51C2×#10621872—24no—
52C2×#10621872—42no—
53C2×#10621872—42no—
54C22×C43⋊C8C2312yes—
55#13063318—28no—
56#13063316—22no—
57#13063316C2×C414yes—
58#13063312—22no—
59#13063312C2×C414yes—
60#13063307—28no—
61#13063305—22no—
62#13063305C2×C414yes—
63#13063296—22no—
64#13063296C2×C414yes—
65C4×Dic86—22no—
66C4×Dic86C2×C414yes—
67C4×C22⋊D43—28no—
68#13063284—28no—
69#13063283—44no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43—21no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—21no—
78C2×C22⋊Dic43—21no—
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—22no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43—22no—
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C42×D43C2×C414yes—
89C42×D43D414yes—
90C4⋊(C43⋊C8)—22no—
91Dic43⋊C8—24no—
92#13063250—42no—
93#13063248—42no—
94#13063246—42no—
95C43⋊#95—22no—
96C43⋊#95—22no—
97C22×D4×C43C2311yescharacteristic
98C23×C4×C43—21no—
99C22×C8×C43C2312yes—
100C4×D4×C43—22no—
101C4×D4×C43C2×C414yes—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43C2312yes—
105C2×C4⋊C4×C43C2311yescharacteristic
106C2×C22⋊C4×C43—21no—
107C2×C42×C43C2311yescharacteristic
108C4⋊C8×C43—22no—
109#102×C43—42no—
110C24×C43—21no—
111C23×D43C2×D411yescharacteristic
112C2×D4×C43—21no—
113C2×D4×C43—21no—
114C2×D4×C43—21no—
115C2×D4×C43—21no—
116C2×D4×C43—21no—
117C2×D4×C43C22×C412yes—
118C22×C4×C43—24no—
119C22×C4×C43—21no—
120C22×C4×C43—22no—
121C22×C4×C43—42no—
122C22×C4×C43—42no—
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2411yescharacteristic
126C2×C22⋊D43—24no—
127C2×C22⋊D43—42no—
128C2×C22⋊D43—42no—
129C22×Dic43—21no—
130C22×Dic43—21no—
131C22×Dic43—22no—
132C22×Dic43C2×D411yescharacteristic
133C22×Dic43C2×D411yescharacteristic
134C22×Dic43C2×D411yescharacteristic
135C2×C4×D43—22no—
136C2×C4×D43—22no—
137C2×C4×D43—21no—
138C2×C4×D43—21no—
139C2×C4×D43—21no—
140C2×C4×D43—21no—
141C2×C4×D43—41no—
142C2×C4×D43—41no—
143C2×C4×D43C22⋊C414yes—
144C2×C4×D43C22⋊C414yes—
145C2×C4×D43C22×C412yes—
146C2×C4×D43C2×D412yes—
147C2×Dic86—21no—
148C2×Dic86—21no—
149C2×Dic86—21no—
150C2×Dic86—21no—
151C2×Dic86—21no—
152C2×Dic86C22×C412yes—
153C2×C8×C43—22no—
154C2×C8×C43—22no—
155C2×C8×C43—22no—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—22no—
158C4⋊C4×C43—22no—
159C4⋊C4×C43—42no—
160C4⋊C4×C43—42no—
161C4⋊C4×C43C22×C412yes—
162C22⋊C4×C43—24no—
163C42×C43C22×C412yes—
164C42×C43C2×D412yes—
165C22⋊Dic43—24no—
166C22⋊Dic43—24no—
167C22⋊Dic43—24no—
168(C2×D43)⋊C4—24no—
169(C2×D43)⋊C4—44no—
170C4⋊Dic43—21no—
171C4⋊Dic43—24no—
172C4⋊Dic43—24no—
173C4⋊Dic43—22no—
174C4⋊Dic43—22no—
175C4⋊Dic43—42no—
176C4⋊Dic43—42no—
177C4⋊Dic43C22×C412yes—
178Dic43⋊C4—24no—
179Dic43⋊C4—24no—
180C4×Dic43—24no—
181C4×Dic43—44no—
182C4×Dic43C22⋊C414yes—
183C4×Dic43C22⋊C414yes—
184C4×Dic43C22×C412yes—
185C4×Dic43C2×D412yes—
186C2×C43⋊C8—22no—
187C2×C43⋊C8—22no—
188C2×C43⋊C8—22no—
189#10621872—48no—
190C23×C43—22no—
191C23×C43—41no—
192C23×C43—41no—
193C23×C43—41no—
194C23×C43—41no—
195C23×C43—41no—
196C23×C43—41no—
197C23×C43C22×D411yescharacteristic
198C22×D43—21no—
199C22×D43—21no—
200C22×D43—21no—
201C22×D43—21no—
202C22×D43—21no—
203C22×D43—21no—
204C22×D43C2×C22⋊C412yes—
205D4×C43—22no—
206D4×C43—22no—
207D4×C43—41no—
208D4×C43—41no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—22no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—48no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43—42no—
218C2×C4×C43C2×C22⋊C412yes—
219C2×C4×C43C2×C22⋊C412yes—
220C2×C4×C43AGL(1,ℤ/8)12yes—
221C2×C4×C43#13712yes—
222C2×C4×C43C23×C411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43H(5,𝔽2)11yescharacteristic
225C2×C4×C43H(5,𝔽2)11yescharacteristic
226C2×C4×C432−1+411yescharacteristic
227C2×C4×C432−1+411yescharacteristic
228C22⋊D43—48no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—22no—
236C2×Dic43—22no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—42no—
244C2×Dic43C2×C22⋊C412yes—
245C2×Dic43C2×C22⋊C412yes—
246C2×Dic43C2×C22⋊C412yes—
247C4×D43—22no—
248C4×D43—22no—
249C4×D43—22no—
250C4×D43—22no—
251C4×D43—28no—
252C4×D43—44no—
253Dic86—22no—
254Dic86—22no—
255Dic86—41no—
256Dic86—41no—
257C8×C43—42no—
258C43⋊C8—42no—
259#21890—431nomaximal, Sylow
260C22×C43—42no—
261C22×C43—42no—
262C22×C43—42no—
263C22×C43—81no—
264C22×C43—81no—
265C22×C43—81no—
266C22×C43—81no—
267C22×C43C22×C22⋊C411yescharacteristic
268C22×C43#53711yescharacteristic
269C22×C43#53711yescharacteristic
270C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C43C2×#13711yescharacteristic
272C22×C43#57511yescharacteristic
273C22×C43#57511yescharacteristic
274C2×D43—22no—
275C2×D43—22no—
276C2×D43—22no—
277C2×D43—41no—
278C2×D43—41no—
279C2×D43—41no—
280C2×D43—41no—
281C4×C43—22no—
282C4×C43—42no—
283C4×C43—82no—
284C4×C43#41714yes—
285C4×C43#52011yescharacteristic
286C4×C43#52011yescharacteristic
287C4×C43#53511yescharacteristic
288C4×C43#53611yescharacteristic
289Dic43—22no—
290Dic43—22no—
291Dic43—22no—
292Dic43—42no—
293Dic43—42no—
294Dic43—82no—
295C2×#529—431no—
296C2×#423—431no—
297#2285—431no—
298#1989—432no—
299#1983—432no—
300#1895—432no—
301#1894—432no—
302#1803—431no—
303#1803—431no—
304#1802—431no—
305#1802D4311yescharacteristic, complemented
306C2×C43—82no—
307C2×C43#231111yescharacteristic
308C2×C43C2×#41711yescharacteristic
309C2×C43#289611yescharacteristic
310C2×C43#325111yescharacteristic
311C2×C43#325111yescharacteristic
312C2×C43#325211yescharacteristic
313C2×C43#325211yescharacteristic
314D43—42no—
315C22×M16—432no—
316#529—438no—
317#529—864no—
318C2×#126—861no—
319C2×#125—431no—
320C2×#124—431no—
321C2×#123—861no—
322C2×C22⋊Q8—431no—
323C2×#121—431no—
324C2×C4×Q8—431no—
325C2×C4×D4—861no—
326C2×C4×D4C2×D4311yescharacteristic
327C2×#117—861no—
328C2×#117—861no—
329C22×C42—431no—
330#423—864no—
331C2×C4⋊C8—432no—
332C2×C4⋊C8—431no—
333C2×C4⋊C8C2×D4311yescharacteristic
334C2×#103—432no—
335C2×#103—432no—
336C2×#102—432no—
337C2×#102C2×D4312yes—
338#384—431no—
339#384C2×D4311yescharacteristic
340#382—861no—
341#380—432no—
342#380—861no—
343C4×C4⋊C4—432no—
344C4×C4⋊C4—861no—
345#340—432no—
346#340—432no—
347#340—432no—
348#340C2×D4312yes—
349C43#2189011yescharacteristic, complemented, Sylow
350C22×Q8—431no—
351C22×D4—861no—
352C22×D4C22×D4311yescharacteristic
353C23×C4—21no—
354C23×C4—431no—
355C23×C4—431no—
356C23×C4—431no—
357C2×M16—864no—
358C2×M16—1722no—
359C2×M16—1722no—
360C22×C8—432no—
361C22×C8C22×D4312yes—
362#126—868no—
363#125—434no—
364#125—862no—
365#124—434no—
366#124—862no—
367#123—868no—
368C22⋊Q8—434no—
369C22⋊Q8—862no—
370#121—434no—
371#121—862no—
372C4×Q8—434no—
373C4×Q8—862no—
374C4×D4—22no—
375C4×D4—868no—
376C4×D4C4×D4314yes—
377#117—868no—
378#117—1724no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—431no—
382C2×C4⋊C4—432no—
383C2×C4⋊C4—861no—
384C2×C4⋊C4—861no—
385C2×C4⋊C4—861no—
386C2×C4⋊C4—861no—
387C2×C4⋊C4—861no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4C22×D4312yes—
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—861no—
393C2×C22⋊C4—861no—
394C2×C22⋊C4—861no—
395C2×C22⋊C4—861no—
396C2×C22⋊C4—862no—
397C2×C42—431no—
398C2×C42—431no—
399C2×C42—431no—
400C2×C42—434no—
401C2×C42—434no—
402C2×C42—861no—
403C2×C42—862no—
404C2×C42C22×D4311yescharacteristic
405C4⋊C8—22no—
406C4⋊C8—864no—
407C4⋊C8—862no—
408#103—1722no—
409#103—1722no—
410#102—42no—
411#102—1722no—
412#95—862no—
413#95—862no—
414C24—21no—
415C24—431no—
416C2×Q8—432no—
417C2×Q8—861no—
418C2×Q8—861no—
419C2×Q8—861no—
420C2×Q8—861no—
421C2×Q8—861no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—864no—
428C2×D4—1722no—
429C2×D4—1722no—
430C2×D4C2×C4×D4312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—431no—
437C22×C4—431no—
438C22×C4—431no—
439C22×C4—434no—
440C22×C4—432no—
441C22×C4—432no—
442C22×C4—434no—
443C22×C4—862no—
444C22×C4—862no—
445C22×C4—861no—
446C22×C4—861no—
447C22×C4—861no—
448C22×C4—861no—
449C22×C4—861no—
450C22×C4—861no—
451C22×C4—862no—
452C22×C4—1721no—
453C22×C4—1721no—
454C22×C4D4×D4311yescharacteristic
455C22×C4D4×D4311yescharacteristic
456C22×C4C23×D4311yescharacteristic
457M16—1728no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—862no—
462C2×C8—862no—
463C2×C8—862no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—432no—
470C4⋊C4—864no—
471C4⋊C4—864no—
472C4⋊C4—864no—
473C4⋊C4—864no—
474C4⋊C4—861no—
475C4⋊C4—862no—
476C4⋊C4—862no—
477C4⋊C4—1722no—
478C4⋊C4—1722no—
479C4⋊C4C2×C4×D4312yes—
480C22⋊C4—24no—
481C22⋊C4—864no—
482C22⋊C4—864no—
483C22⋊C4—864no—
484C22⋊C4—864no—
485C22⋊C4—1724no—
486C42—434no—
487C42—434no—
488C42—432no—
489C42—432no—
490C42—864no—
491C42—1724no—
492C42C2×C4×D4312yes—
493C42D4×D4312yes—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—432no—
502C23—861no—
503C23—861no—
504C23—861no—
505C23—861no—
506C23—861no—
507C23—861no—
508C23C2×D4×D4311yescharacteristic
509Q8—862no—
510Q8—862no—
511Q8—1721no—
512Q8—1721no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517D4—1728no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—432no—
528C2×C4—432no—
529C2×C4—432no—
530C2×C4—861no—
531C2×C4—861no—
532C2×C4—861no—
533C2×C4—861no—
534C2×C4—861no—
535C2×C4—861no—
536C2×C4—862no—
537C2×C4—862no—
538C2×C4—868no—
539C2×C4—862no—
540C2×C4—862no—
541C2×C4—862no—
542C2×C4—862no—
543C2×C4—1724no—
544C2×C4—1722no—
545C2×C4—1722no—
546C2×C4—1722no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—1721no—
551C2×C4C22⋊C4×D4312yes—
552C2×C4C22⋊C4×D4312yes—
553C2×C4#1306334912yes—
554C2×C4#1306335412yes—
555C2×C4C22×C4×D4311yescharacteristic
556C2×C4C2×D4×D4311yescharacteristic
557C2×C4#1306337011yescharacteristic
558C2×C4#1306337111yescharacteristic
559C2×C4#1306337211yescharacteristic
560C2×C4#1306337311yescharacteristic
561C8—42no—
562C8—1722no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—862no—
571C22—862no—
572C22—862no—
573C22—1721no—
574C22—1721no—
575C22—1721no—
576C22—1721no—
577C22C2×C22⋊C4×D4311yescharacteristic
578C22#1458347811yescharacteristic
579C22#1458347911yescharacteristic
580C22C2×#1306334911yescharacteristic
581C22C2×#1306335411yescharacteristic
582C22#1458369511yescharacteristic
583C22#1458369811yescharacteristic
584C4—22no—
585C4—42no—
586C4—82no—
587C4—862no—
588C4—862no—
589C4—862no—
590C4—3442no—
591C4—1722no—
592C4—1722no—
593C4#1458289714yes—
594C4#1458337911yescharacteristic
595C4#1458338011yescharacteristic
596C4#1458346511yescharacteristic
597C4#1458347211yescharacteristic
598C2—82no—
599C2—1722no—
600C2#1913718411yescharacteristic
601C2C2×#1458289711yescharacteristic
602C2#1914311111yescharacteristic
603C2#1914595311yescharacteristic
604C2#1914595911yescharacteristic
605C2#1914596411yescharacteristic
606C2#1914597011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1802C2
socleC23×C43C22×D4
prime radicalC2×#13063296C22
cosocleC22×C4×C43C24
hypercenter#1802D43
hypocenterC43#21890
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#14582921C2semidirect
3. #19137000C2semidirect
4. #19135549C2semidirect
5. #19135526C2semidirect
7. #19135027C2semidirect
8. #19134601C2semidirect
9. #19134597C2semidirect
10. #19134595C2semidirect
11. C43×#1802C2semidirect
14. #14583431C4semidirect
305. #1802D43semidirect
349. C43#21890semidirect

Maximal subgroups

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

subgroupindex
C2×#145834312
C2×#145829212
#191370002
#191355492
#191355262
#191350302
#191350272
#191346012
#191345972
#191345952
C43×#18022
#2189043