Finite Group Databasebeta

← #25159644

Subgroups — #25159644

Order 11008.

884 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583201C211yesmaximal, characteristic
2C2×SD16×Dic43C211yesmaximal, characteristic
3#19138459C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic43C211yesmaximal, characteristic
5#19136133C211yesmaximal, characteristic, complemented
6#19136073C211yesmaximal, characteristic, complemented
7#19135986C211yesmaximal, characteristic, complemented
8#19135932C211yesmaximal, characteristic, complemented
9#19135327C211yesmaximal, characteristic, complemented
10#19135151C211yesmaximal, characteristic, complemented
11#19134931C211yesmaximal, characteristic
12#19134865C211yesmaximal, characteristic, complemented
13#19134842C211yesmaximal, characteristic
14#19134840C211yesmaximal, characteristic
15C43×#1877C211yesmaximal, characteristic
16C2×#13063316C2211yescharacteristic
17C2×Q8×Dic43C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20C22×C4⋊Dic43—21no—
21#14583201—24no—
22SD16×Dic43—24no—
23C2×#13063258C2211yescharacteristic
24C2×#13063258C2211yescharacteristic
25C2×#13063258C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27C2×#13063257C2211yescharacteristic
28C2×Dic43⋊C8C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063246C2211yescharacteristic
32C2×#13063246C2211yescharacteristic
33C2×C8×Dic43C2211yescharacteristic
34#14582841—21no—
35#14582831—21no—
36#14582806C2211yescharacteristic
37#14582798C2211yescharacteristic
38#14582798C2211yescharacteristic
39#14582797C2211yescharacteristic
40#14582796—21no—
41#14582795C2211yescharacteristic
42#14582786—21no—
43#14582774—21no—
44#14582774—21no—
45#14582769C2211yescharacteristic
46#14582759—21no—
47C4×C4⋊Dic43C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582654C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C22×SD16×C43C2211yescharacteristic
55C2×C4×D4×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C23×Dic43—21no—
62C23×Dic43—21no—
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65#13063316—22no—
66#13063316—41no—
67Q8×Dic43—22no—
68Q8×Dic43—41no—
69D4×Dic43—22no—
70D4×Dic43—22no—
71D4×Dic43—41no—
72D4×Dic43—41no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43—44no—
80C2×C4⋊Dic43D411yescharacteristic
81C2×C4⋊Dic43D411yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×C22⋊Dic43—21no—
90C2×C22⋊Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43C2311yescharacteristic
93C2×C4×Dic43C2311yescharacteristic
94#13063258—22no—
95#13063258—22no—
96#13063258—41no—
97#13063257—22no—
98#13063257—22no—
99Dic43⋊C8—22no—
100#13063249—22no—
101#13063249—41no—
102#13063246—22no—
103#13063246D414yes—
104C8×Dic43—22no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95D411yescharacteristic
112C43⋊#95D411yescharacteristic
113C22×Q8×C43C2311yescharacteristic
114C22×D4×C43C2311yescharacteristic
115C23×C4×C43—21no—
116C2×SD16×C43—22no—
117C2×SD16×C43—22no—
118C2×SD16×C43Q814yes—
119C22×C8×C43C2311yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C4×D4×C43—22no—
122C4×D4×C43—22no—
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C42×C43C2311yescharacteristic
127#103×C43—22no—
128#102×C43—22no—
129#95×C43—21no—
130#95×C43—21no—
131C24×C43—21no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43C2×Q812yes—
138C2×D4×C43—21no—
139C2×D4×C43—21no—
140C2×D4×C43—21no—
141C2×D4×C43C2×D412yes—
142C2×D4×C43C2×Q812yes—
143C2×D4×C43#4112yes—
144C22×C4×C43—24no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43—42no—
149C22×C4×C43—42no—
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2411yescharacteristic
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—42no—
158C22×Dic43—42no—
159C22×Dic43—42no—
160C22×Dic43—42no—
161C22×Dic43—81no—
162C22×Dic43—81no—
163C22×Dic43C2×D411yescharacteristic
164C22×Dic43C2×D411yescharacteristic
165C22×Dic43C2×D411yescharacteristic
166C22×Dic43C2×D411yescharacteristic
167SD16×C43—28no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43C2×Q812yes—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43#4112yes—
178C22⋊C4×C43—24no—
179C42×C43—21no—
180C42×C43—21no—
181C22⋊Dic43—42no—
182C22⋊Dic43—42no—
183C4⋊Dic43—22no—
184C4⋊Dic43—21no—
185C4⋊Dic43—22no—
186C4⋊Dic43—21no—
187C4⋊Dic43—21no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—42no—
193C4⋊Dic43—42no—
194C4⋊Dic43—42no—
195C4⋊Dic43—42no—
196C4⋊Dic43—42no—
197C4⋊Dic43—81no—
198C4⋊Dic43C2×D412yes—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201Dic43⋊C4—42no—
202C4×Dic43—21no—
203C4×Dic43—21no—
204C4×Dic43—21no—
205C4×Dic43—21no—
206C4×Dic43—42no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8C2×D412yes—
213C23×C43—22no—
214C23×C43—22no—
215C23×C43—22no—
216C23×C43—41no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43C22×D411yescharacteristic
221Q8×C43—22no—
222Q8×C43—22no—
223Q8×C43—41no—
224Q8×C43—41no—
225D4×C43—22no—
226D4×C43—22no—
227D4×C43—22no—
228D4×C43C22⋊Q814yes—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—22no—
233C2×C4×C43—22no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—48no—
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×C43#13512yes—
245C2×C4×C43C22×D411yescharacteristic
246C2×C4×C43C22×Q811yescharacteristic
247C2×C4×C43C2×#4111yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C2×C4×C43H(5,𝔽2)11yescharacteristic
250C2×C4×C432−1+411yescharacteristic
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—22no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—84no—
275C2×Dic43—84no—
276C2×Dic43AGL(1,ℤ/8)12yes—
277C8×C43—22no—
278C8×C43—41no—
279C43⋊C8—22no—
280C43⋊C8—41no—
281#26374—431nomaximal, Sylow
282C22×C43—24no—
283C22×C43—42no—
284C22×C43—42no—
285C22×C43—42no—
286C22×C43—42no—
287C22×C43—42no—
288C22×C43—42no—
289C22×C43#52511yescharacteristic
290C22×C43D4211yescharacteristic
291C22×C43D4×Q811yescharacteristic
292C22×C43C2×#13511yescharacteristic
293C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C43#57711yescharacteristic
295C22×C43#57811yescharacteristic
296C4×C43—22no—
297C4×C43—42no—
298C4×C43—41no—
299C4×C43—42no—
300C4×C43—81no—
301C4×C43C2×C22⋊Q811yescharacteristic
302C4×C43#53911yescharacteristic
303C4×C43#54711yescharacteristic
304C4×C43#55411yescharacteristic
305Dic43—22no—
306Dic43—41no—
307Dic43—41no—
308Dic43—41no—
309Dic43—42no—
310Dic43—81no—
311Dic43—81no—
312Dic43—81no—
313C2×#478—431no—
314C2×C4×SD16—431no—
315#2453—431no—
316D4×C4⋊C4—431no—
317#2095—431no—
318#2084—431no—
319#2068—431no—
320#2059—431no—
321#1942—431no—
322#1916—431no—
323#1877—431no—
324#1877D4311yescharacteristic
325#1865—431no—
326#1862—431no—
327#1862—431no—
328C2×C43—44no—
329C2×C43#244911yescharacteristic
330C2×C43#310511yescharacteristic
331C2×C43#312011yescharacteristic
332C2×C43#326511yescharacteristic
333C2×C43#328311yescharacteristic
334C2×C43#338111yescharacteristic
335C2×C43#338611yescharacteristic
336C22×SD16C2×D4311yescharacteristic
337C2×#125—431no—
338C2×C4×Q8—431no—
339C2×C4×D4—431no—
340C2×C4×D4—431no—
341C2×C4×D4C2×D4311yescharacteristic
342C22×C4⋊C4—861no—
343#478—864no—
344C4×SD16—864no—
345C2×#107—431no—
346C2×#107—431no—
347C2×#107—431no—
348C2×#106—431no—
349C2×#106—431no—
350C2×C4⋊C8—431no—
351C2×#103—431no—
352C2×#103—431no—
353C2×#103C2×D4311yescharacteristic
354C2×#102—431no—
355C2×#102—431no—
356C2×#102C2×D4311yescharacteristic
357C2×C4×C8—431no—
358#400—861no—
359#398—861no—
360#391—431no—
361#391C2×D4311yescharacteristic
362#389—431no—
363#389—431no—
364#389—431no—
365#389—431no—
366#389—861no—
367#387—861no—
368#385—861no—
369#385—861no—
370#384—431no—
371#380—861no—
372C4×C4⋊C4—431no—
373#340—431no—
374#340—431no—
375#340—431no—
376#340C2×D4311yescharacteristic
377#336—431no—
378#336—431no—
379#336—431no—
380#336C2×D4311yescharacteristic
381C43#2637411yescharacteristic, complemented, Sylow
382C22×Q8C22×D4311yescharacteristic
383C22×D4C22×D4311yescharacteristic
384C23×C4—21no—
385C23×C4—861no—
386C23×C4—861no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16Dic8614yes—
390C22×C8—431no—
391C22×C8—431no—
392C22×C8C22×D4311yescharacteristic
393C22×C8C22×D4311yescharacteristic
394#125—862no—
395#125—1721no—
396C4×Q8—862no—
397C4×Q8—1721no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—862no—
401C4×D4—862no—
402C4×D4—1721no—
403C4×D4—1721no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4—431no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—861no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—1724no—
420C2×C4⋊C4C22×D4311yescharacteristic
421C2×C4⋊C4C22×D4311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4—861no—
425C2×C42—431no—
426C2×C42—431no—
427C2×C42—861no—
428C2×C42C22×D4311yescharacteristic
429#107—862no—
430#107—862no—
431#107—1721no—
432#106—862no—
433#106—862no—
434C4⋊C8—862no—
435#103—22no—
436#103—862no—
437#103—1721no—
438#102—22no—
439#102—434no—
440#102—862no—
441C4×C8—862no—
442#95—21no—
443#95—21no—
444#95—431no—
445#95—431no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452C24—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8C2×Dic8612yes—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4C2×Dic8612yes—
463C2×D4D4×D4312yes—
464C2×D4#1062190012yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—431no—
472C22×C4—431no—
473C22×C4—431no—
474C22×C4—431no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—861no—
479C22×C4—3441no—
480C22×C4—3441no—
481C22×C4—1722no—
482C22×C4—1722no—
483C22×C4—1722no—
484C22×C4—1722no—
485C22×C4D4×D4311yescharacteristic
486C22×C4D4×D4311yescharacteristic
487C22×C4C23×D4311yescharacteristic
488SD16—28no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—432no—
495C2×C8—861no—
496C2×C8—861no—
497C2×C8—861no—
498C2×C8—861no—
499C2×C8—861no—
500C2×C8C2×Dic8612yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—432no—
505C4⋊C4—862no—
506C4⋊C4—862no—
507C4⋊C4—861no—
508C4⋊C4—861no—
509C4⋊C4—861no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—3441no—
515C4⋊C4—1722no—
516C4⋊C4—1722no—
517C4⋊C4—1722no—
518C4⋊C4—1722no—
519C4⋊C4—1722no—
520C4⋊C4—1722no—
521C4⋊C4—1722no—
522C4⋊C4—1722no—
523C4⋊C4#1062190012yes—
524C22⋊C4—24no—
525C22⋊C4—1722no—
526C22⋊C4—1722no—
527C42—21no—
528C42—21no—
529C42—861no—
530C42—861no—
531C42—861no—
532C42—861no—
533C42—1722no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D4311yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1306330214yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—432no—
566C2×C4—861no—
567C2×C4—861no—
568C2×C4—861no—
569C2×C4—861no—
570C2×C4—861no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—861no—
574C2×C4—861no—
575C2×C4—861no—
576C2×C4—861no—
577C2×C4—862no—
578C2×C4—3444no—
579C2×C4—3444no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4#1306334612yes—
592C2×C4C2×D4×D4311yescharacteristic
593C2×C4C22×Dic8611yescharacteristic
594C2×C4C2×#1062190011yescharacteristic
595C2×C4C2×#1062190011yescharacteristic
596C2×C4#1306337111yescharacteristic
597C2×C4#1306337211yescharacteristic
598C8—22no—
599C8—41no—
600C8—862no—
601C8—1721no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1458340511yescharacteristic
610C22#1458353311yescharacteristic
611C22D4×Dic8611yescharacteristic
612C22C2×#1306334611yescharacteristic
613C22AGL(1,ℤ/8)×D4311yescharacteristic
614C22#1458370611yescharacteristic
615C22#1458371511yescharacteristic
616C4—22no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—81no—
621C4—862no—
622C4—3441no—
623C4—3441no—
624C4—3441no—
625C4—1721no—
626C4—1721no—
627C4—1722no—
628C4—1721no—
629C4C2×#1306330211yescharacteristic
630C4#1458349311yescharacteristic
631C4#1458354911yescharacteristic
632C4#1458360611yescharacteristic
633C2—44no—
634C2#1913841011yescharacteristic
635C2#1914464311yescharacteristic
636C2#1914480211yescharacteristic
637C2#1914609611yescharacteristic
638C2#1914631211yescharacteristic
639C2#1914746811yescharacteristic
640C2#1914751711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1877C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#1877D43
hypocenterC43#26374
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #19138459C2semidirect
5. #19136133C2semidirect
6. #19136073C2semidirect
7. #19135986C2semidirect
8. #19135932C2semidirect
9. #19135327C2semidirect
10. #19135151C2semidirect
12. #19134865C2semidirect
381. C43#26374semidirect

Maximal subgroups

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

subgroupindex
C2×#145832012
C2×SD16×Dic432
#191384592
D4×C4⋊Dic432
#191361332
#191360732
#191359862
#191359322
#191353272
#191351512
#191349312
#191348652
#191348422
#191348402
C43×#18772
#2637443