Finite Group Databasebeta

← #25114944

Subgroups — #25114944

Order 11008.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583001C211yesmaximal, characteristic
2C2×#14582998C211yesmaximal, characteristic
3C2×#14582986C211yesmaximal, characteristic
4#19137512C211yesmaximal, characteristic, complemented
5#19137495C211yesmaximal, characteristic
6#19135331C211yesmaximal, characteristic, complemented
7#19135326C211yesmaximal, characteristic, complemented
8#19134860C211yesmaximal, characteristic, complemented
9#19134853C211yesmaximal, characteristic
10#19134846C211yesmaximal, characteristic, complemented
11#19134839C211yesmaximal, characteristic
12#19134602C211yesmaximal, characteristic
13#19134597C211yesmaximal, characteristic, complemented
14#19134467C211yesmaximal, characteristic, complemented
15C43×#1847C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C2×C4×Dic86C2211yescharacteristic
18C2×#13063290C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22C2×#13063287C2211yescharacteristic
23C2×C4⋊C4×D43—21no—
24#14583001C418yescomplemented
25#14582998—24no—
26#14582986—24no—
27C2×#13063259C2211yescharacteristic
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×#13063248C2211yescharacteristic
32C2×#13063247C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×#13063245C2211yescharacteristic
35C2×#13063235C2211yescharacteristic
36C2×C8×Dic43C2211yescharacteristic
37#14582796—21no—
38#14582796—21no—
39#14582795C2211yescharacteristic
40#14582795C2211yescharacteristic
41#14582795C2211yescharacteristic
42#14582795C2211yescharacteristic
43#14582770—21no—
44#14582770—21no—
45#14582763—21no—
46#14582756—21no—
47C4×C4⋊Dic43C2211yescharacteristic
48#14582750—21no—
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582653C2211yescharacteristic
54#14582653C2211yescharacteristic
55C2×#107×C43C2211yescharacteristic
56C2×#107×C43C2211yescharacteristic
57C2×#106×C43C2211yescharacteristic
58C2×#106×C43C2211yescharacteristic
59C2×#97×C43C2211yescharacteristic
60C43×#384C2211yescharacteristic
61C43×#382C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C2×#10621873C2×C414yes—
68C2×#10621873D414yes—
69C2×#10621873Q814yes—
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72C4×Dic86—22no—
73C4×Dic86C2×C414yescomplemented
74#13063290—22no—
75#13063290—22no—
76#13063290—41no—
77#13063290—41no—
78C4×D172—22no—
79C4×D172C2×C414yes—
80#13063287—22no—
81#13063287—22no—
82#13063287—41no—
83#13063287—41no—
84C2×C4⋊Dic43D411yescharacteristic
85C2×C4⋊Dic43D411yescharacteristic
86C2×C4⋊Dic43C2311yescharacteristic
87C2×C4⋊Dic43C2311yescharacteristic
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C4⋊C4×D43—44no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4—21no—
98C2×(C2×D43)⋊C4—21no—
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43—21no—
102C2×C4×Dic43C2311yescharacteristic
103C2×C4×Dic43C2311yescharacteristic
104#13063259C2×C414yescomplemented
105C4⋊(C43⋊C8)—22no—
106#13063250C2×C414yescomplemented
107#13063248—22no—
108#13063248—22no—
109#13063247C2×C414yes—
110#13063245—22no—
111#13063245—22no—
112#13063235—22no—
113C8×Dic43—22no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95D411yescharacteristic
119C43⋊#95D411yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C22×C8×C43C2311yescharacteristic
122C2×C4⋊C4×C43—21no—
123C2×C4⋊C4×C43—21no—
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C42×C43C2311yescharacteristic
129#107×C43—22no—
130#107×C43—41no—
131#106×C43—22no—
132#106×C43—41no—
133#97×C43C2×C414yescomplemented
134#95×C43—21no—
135#95×C43—21no—
136C23×D43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2411yescharacteristic
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150C22×Dic43C2×D411yescharacteristic
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172C22×C412yes—
155C2×D172C2×D412yes—
156C2×D172C2×Q812yes—
157C2×C4×D43—24no—
158C2×C4×D43—42no—
159C2×C4×D43—42no—
160C2×C4×D43—42no—
161C2×C4×D43—42no—
162C2×C4×D43—42no—
163C2×C4×D43—42no—
164C2×C4×D43—81no—
165C2×C4×D43—81no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86C22×C412yes—
170C2×Dic86C2×D412yes—
171C2×Dic86C2×Q812yes—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43C22×C412yes—
175C2×C8×C43C22×C412yes—
176C2×C8×C43C2×D412yes—
177C2×C8×C43C2×Q812yes—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—42no—
187C4⋊C4×C43—81no—
188C42×C43C22×C412yes—
189C42×C43#4112yes—
190(C2×D43)⋊C4—24no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—42no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—22no—
196C4⋊Dic43—41no—
197C4⋊Dic43C22×C412yes—
198C4⋊Dic43C22×C412yes—
199Dic43⋊C4—24no—
200Dic43⋊C4—42no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203Dic43⋊C4—42no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206C4×Dic43—24no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—42no—
212C4×Dic43—42no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219#10621873C4⋊C4116yescomplemented
220C23×C43C22×D411yescharacteristic
221C22×D43—22no—
222C22×D43—22no—
223C22×D43—22no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43C4×D412yes—
245C2×C4×C43C4×D412yes—
246C2×C4×C43C23×C411yescharacteristic
247C2×C4×C43C22×D411yescharacteristic
248C2×C4×C43C22×Q811yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43H(5,𝔽2)11yescharacteristic
251C2×C4×C432−1+411yescharacteristic
252C2×Dic43—22no—
253C2×Dic43—21no—
254C2×Dic43—22no—
255C2×Dic43—22no—
256C2×Dic43—21no—
257C2×Dic43—22no—
258C2×Dic43—22no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43AGL(1,ℤ/8)12yes—
273D172—22no—
274D172—22no—
275D172—22no—
276D172C2×C4⋊C414yes—
277C4×D43—48no—
278C4×D43—84no—
279C4×D43—84no—
280Dic86—22no—
281Dic86—22no—
282Dic86—22no—
283Dic86C2×C4⋊C414yescomplemented
284C8×C43—22no—
285C8×C43C2×C4⋊C414yescomplemented
286C43⋊C8—41no—
287C43⋊C8—41no—
288#22677—431nomaximal, Sylow
289C22×C43C2×C4×D411yescharacteristic
290C22×C43D4211yescharacteristic
291C22×C43D4×Q811yescharacteristic
292C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C43#57511yescharacteristic
294C22×C43#57711yescharacteristic
295C22×C43#57811yescharacteristic
296C2×D43—24no—
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—42no—
300C2×D43—42no—
301C2×D43—42no—
302C2×D43—42no—
303C4×C43—22no—
304C4×C43—22no—
305C4×C43—81no—
306C4×C43—81no—
307C4×C43—81no—
308C4×C43—81no—
309C4×C43C22×C4⋊C411yescharacteristic
310C4×C43#52011yescharacteristic
311C4×C43#54711yescharacteristic
312C4×C43#55411yescharacteristic
313Dic43—24no—
314Dic43—22no—
315Dic43—42no—
316Dic43—42no—
317Dic43—41no—
318Dic43—41no—
319Dic43—41no—
320C2×#440—431no—
321C2×#440—431no—
322C2×C4×SD16—431no—
323#2351—431no—
324D4×C4⋊C4—431no—
325#1943—431no—
326#1942—431no—
327#1865—431no—
328#1864—431no—
329#1863—431no—
330#1862—431no—
331#1847D4311yescharacteristic, complemented
332#1804—431no—
333#1803—431no—
334#1771—431no—
335C2×C43D4×C4⋊C411yescharacteristic
336C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
337C2×C43#294711yescharacteristic
338C2×C43#326511yescharacteristic
339C2×C43#326911yescharacteristic
340C2×C43#338611yescharacteristic
341C2×C43#338811yescharacteristic
342D43—44no—
343C22×SD16—431no—
344C2×C4×Q8—431no—
345C2×C4×Q8—431no—
346C2×C4×Q8—431no—
347C2×C4×D4—431no—
348C2×C4×D4—431no—
349C2×C4×D4—431no—
350C22×C4⋊C4—861no—
351#440—438no—
352#440—864no—
353C4×SD16—864no—
354C2×#107—431no—
355C2×#107C2×D4311yescharacteristic
356C2×#107C2×D4311yescharacteristic
357C2×#106C2×D4311yescharacteristic
358C2×#106C2×D4311yescharacteristic
359C2×C4⋊C8—431no—
360C2×#103—431no—
361C2×#103—431no—
362C2×#103—431no—
363C2×#102—431no—
364C2×#102—431no—
365C2×#102—431no—
366C2×#97—431no—
367C2×#97C2×D4311yescharacteristic
368C2×C4×C8—431no—
369#389—431no—
370#389—431no—
371#389—431no—
372#389—431no—
373#389—861no—
374#389—861no—
375#385—861no—
376#385—861no—
377#384C2×D4311yescharacteristic
378#382—861no—
379#382C2×D4311yescharacteristic
380#380—861no—
381C4×C4⋊C4—431no—
382C4×C4⋊C4—861no—
383#340—431no—
384#340—431no—
385#340—431no—
386#340—431no—
387#336—431no—
388#336—431no—
389C43#2267711yescharacteristic, complemented, Sylow
390C22×Q8—431no—
391C22×D4—431no—
392C23×C4—861no—
393C23×C4—861no—
394C23×C4—861no—
395C2×SD16—434no—
396C2×SD16—434no—
397C2×SD16—434no—
398C22×C8—431no—
399C22×C8—431no—
400C22×C8C22×D4311yescharacteristic
401C22×C8C22×D4311yescharacteristic
402C4×Q8—434no—
403C4×Q8—862no—
404C4×Q8—862no—
405C4×Q8—862no—
406C4×Q8—1721no—
407C4×Q8—1721no—
408C4×D4—434no—
409C4×D4—862no—
410C4×D4—862no—
411C4×D4—862no—
412C4×D4—1721no—
413C4×D4—1721no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—431no—
419C2×C4⋊C4—431no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—1724no—
428C2×C4⋊C4C22×D4311yescharacteristic
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C4⋊C4C22×D4311yescharacteristic
431C2×C4⋊C4C22×D4311yescharacteristic
432C2×C22⋊C4—861no—
433C2×C22⋊C4—861no—
434C2×C22⋊C4—861no—
435C2×C42—431no—
436C2×C42—431no—
437C2×C42—861no—
438C2×C42—861no—
439C2×C42—861no—
440C2×C42C22×D4311yescharacteristic
441#107—22no—
442#107—41no—
443#107—434no—
444#106—22no—
445#106—41no—
446C4⋊C8—862no—
447#103—434no—
448#103—862no—
449#103—862no—
450#102—434no—
451#102—862no—
452#102—862no—
453#97—862no—
454#97C4×D4314yescomplemented
455C4×C8—862no—
456#95—21no—
457#95—21no—
458#95—431no—
459#95—431no—
460#95—861no—
461#95—861no—
462#95—861no—
463#95—861no—
464C24—861no—
465C2×Q8—432no—
466C2×Q8—432no—
467C2×Q8—432no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×D4—432no—
472C2×D4—432no—
473C2×D4—432no—
474C2×D4—861no—
475C2×D4—861no—
476C2×D4—861no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—431no—
484C22×C4—431no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—864no—
489C22×C4—3441no—
490C22×C4—3441no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1722no—
494C22×C4—1722no—
495C22×C4—1722no—
496C22×C4—1722no—
497C22×C4D4×D4311yescharacteristic
498C22×C4D4×D4311yescharacteristic
499C22×C4C23×D4311yescharacteristic
500SD16—4316no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8—861no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8C2×C4×D4312yes—
510C2×C8C2×C4×D4312yes—
511C2×C8D4×D4312yes—
512C2×C8Q8×D4312yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—432no—
524C4⋊C4—432no—
525C4⋊C4—862no—
526C4⋊C4—864no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—1721no—
530C4⋊C4—1722no—
531C4⋊C4—1722no—
532C4⋊C4—1722no—
533C4⋊C4—1722no—
534C4⋊C4—1722no—
535C4⋊C4—1722no—
536C22⋊C4—864no—
537C22⋊C4—1722no—
538C22⋊C4—1722no—
539C42—864no—
540C42—861no—
541C42—861no—
542C42—861no—
543C42—861no—
544C42—1722no—
545C42—1722no—
546C42C2×C4×D4312yes—
547C42#1062190212yes—
548C23—862no—
549C23—862no—
550C23—862no—
551C23—1721no—
552C23—1721no—
553C23—1721no—
554C23—1721no—
555C23C2×D4×D4311yescharacteristic
556Q8—434no—
557Q8—862no—
558Q8—862no—
559Q8—862no—
560D4—434no—
561D4—862no—
562D4—862no—
563D4—862no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—432no—
581C2×C4—862no—
582C2×C4—862no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—862no—
586C2×C4—862no—
587C2×C4—862no—
588C2×C4—861no—
589C2×C4—861no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—861no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—3444no—
598C2×C4—3444no—
599C2×C4—1728no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4#1306328712yes—
605C2×C4#1306328712yes—
606C2×C4C22×C4×D4311yescharacteristic
607C2×C4C2×D4×D4311yescharacteristic
608C2×C4C2×Q8×D4311yescharacteristic
609C2×C4C2×#1062190211yescharacteristic
610C2×C4#1306337011yescharacteristic
611C2×C4#1306337311yescharacteristic
612C8—22no—
613C8—1721no—
614C8—1721no—
615C8C4⋊C4×D4314yescomplemented
616C22—864no—
617C22—1722no—
618C22—1722no—
619C22—1722no—
620C22—1722no—
621C22—1722no—
622C22—1722no—
623C22C2×#1306328711yescharacteristic
624C22#1458353411yescharacteristic
625C22#1458356411yescharacteristic
626C22AGL(1,ℤ/8)×D4311yescharacteristic
627C22#1458369811yescharacteristic
628C22#1458370911yescharacteristic
629C22#1458371511yescharacteristic
630C4—22no—
631C4—22no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—862no—
637C4—864no—
638C4—1721no—
639C4—1721no—
640C4—1721no—
641C4—1722no—
642C4—1722no—
643C4C2×C4⋊C4×D4311yescharacteristic
644C4#1458338011yescharacteristic
645C4#1458355211yescharacteristic
646C4#1458361011yescharacteristic
647C2—1724no—
648C2#1913749511yescharacteristic
649C2#1914340211yescharacteristic
650C2#1914343611yescharacteristic
651C2#1914608911yescharacteristic
652C2#1914614911yescharacteristic
653C2#1914751311yescharacteristic
654C2#1914753911yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1847C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#1847D43
hypocenterC43#22677
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #19137512C2semidirect
6. #19135331C2semidirect
7. #19135326C2semidirect
8. #19134860C2semidirect
10. #19134846C2semidirect
13. #19134597C2semidirect
14. #19134467C2semidirect
15. C43×#1847C2semidirect
24. #14583001C4semidirect
73. C4×Dic86C2×C4semidirect
104. #13063259C2×C4semidirect
106. #13063250C2×C4semidirect
133. #97×C43C2×C4semidirect
219. #10621873C4⋊C4semidirect
283. Dic86C2×C4⋊C4semidirect
285. C8×C43C2×C4⋊C4semidirect
331. #1847D43semidirect
389. C43#22677semidirect
454. #97C4×D43semidirect
615. C8C4⋊C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145830012
C2×#145829982
C2×#145829862
#191375122
#191374952
#191353312
#191353262
#191348602
#191348532
#191348462
#191348392
#191346022
#191345972
#191344672
C43×#18472
#2267743