Finite Group Databasebeta

← #25115013

Subgroups — #25115013

Order 11008.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic43C211yesmaximal, characteristic
2C2×SD16×Dic43C211yesmaximal, characteristic
3#19137513C211yesmaximal, characteristic, complemented
4#19137500C211yesmaximal, characteristic
5#19135323C211yesmaximal, characteristic, complemented
6#19135321C211yesmaximal, characteristic, complemented
7#19134849C211yesmaximal, characteristic, complemented
8#19134847C211yesmaximal, characteristic, complemented
9#19134842C211yesmaximal, characteristic
10#19134840C211yesmaximal, characteristic
11#19134756C211yesmaximal, characteristic, complemented
12#19134608C211yesmaximal, characteristic, complemented
13#19134603C211yesmaximal, characteristic
14C8×Dic43⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C43C211yesmaximal, characteristic
16C2×Q8×Dic43C2211yescharacteristic
17C2×Q8×Dic43C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20SD16×Dic43—24no—
21SD16×Dic43C418yescomplemented
22C2×#13063258C2211yescharacteristic
23C2×#13063258C2211yescharacteristic
24C2×#13063257C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063249C2211yescharacteristic
29C2×#13063246C2211yescharacteristic
30C2×#13063246C2211yescharacteristic
31C2×C8×Dic43C2211yescharacteristic
32C2×C8×Dic43C2211yescharacteristic
33#14582797—21no—
34#14582797C2211yescharacteristic
35#14582797C2211yescharacteristic
36#14582795—21no—
37#14582795C2211yescharacteristic
38#14582795C2211yescharacteristic
39#14582772—21no—
40#14582771—21no—
41#14582764—21no—
42#14582764C2211yescharacteristic
43#14582764C2211yescharacteristic
44#14582757—21no—
45C4×Dic43⋊C4—21no—
46C4×Dic43⋊C4C2211yescharacteristic
47C2×C43⋊#95—21no—
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53#14582653C2211yescharacteristic
54C22×SD16×C43C2211yescharacteristic
55C2×C4×Q8×C43C2211yescharacteristic
56C2×C4×D4×C43C2211yescharacteristic
57C4×SD16×C43—24no—
58C2×#106×C43C2211yescharacteristic
59C2×#103×C43C2211yescharacteristic
60C2×#102×C43C2211yescharacteristic
61C2×C4×C8×C43C2211yescharacteristic
62C23×Dic43—21no—
63C23×Dic43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66Q8×Dic43—22no—
67Q8×Dic43—22no—
68Q8×Dic43—41no—
69Q8×Dic43C2×C414yescomplemented
70D4×Dic43—22no—
71D4×Dic43—22no—
72D4×Dic43—41no—
73D4×Dic43C2×C414yes—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4D411yescharacteristic
85C2×Dic43⋊C4D411yescharacteristic
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43C2311yescharacteristic
91C2×C4×Dic43C2311yescharacteristic
92#13063258—41no—
93#13063258—41no—
94#13063257—22no—
95#13063257C2×C414yescomplemented
96Dic43⋊C8—22no—
97#13063249—22no—
98#13063249C2×C414yescomplemented
99#13063246—22no—
100#13063246C2×C414yes—
101C8×Dic43—22no—
102C8×Dic43C2×C414yescomplemented
103C43⋊#95—21no—
104C43⋊#95—21no—
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×C43C2×C414yes—
117C2×SD16×C43D414yes—
118C2×SD16×C43Q814yes—
119C22×C8×C43C2311yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C4×Q8×C43—22no—
122C4×Q8×C43—41no—
123C4×D4×C43—22no—
124C4×D4×C43—41no—
125C2×C4⋊C4×C43—21no—
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C22⋊C4×C43—21no—
129C2×C42×C43—21no—
130C2×C42×C43C2311yescharacteristic
131#106×C43—22no—
132#103×C43—22no—
133#102×C43—22no—
134C4×C8×C43—22no—
135C24×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43C22×C412yes—
140C2×Q8×C43C2×D412yes—
141C2×Q8×C43C2×Q812yes—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43C22×C412yes—
146C2×D4×C43C2×D412yes—
147C2×D4×C43C2×Q812yes—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—42no—
152C22×C4×C43—42no—
153C22×C4×C43—81no—
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2411yescharacteristic
157C22×Dic43—24no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—42no—
163C22×Dic43—42no—
164C22×Dic43—81no—
165C22×Dic43—81no—
166C22×Dic43—81no—
167C22×Dic43C2×D411yescharacteristic
168C22×Dic43C2×D411yescharacteristic
169C22×Dic43C2×D411yescharacteristic
170C22×Dic43C2×D411yescharacteristic
171SD16×C43C4⋊C4116yescomplemented
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
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—42no—
183C22⋊C4×C43—42no—
184C42×C43—21no—
185C42×C43—21no—
186C42×C43—42no—
187C22⋊Dic43—24no—
188C22⋊Dic43—42no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—24no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—42no—
197C4⋊Dic43C22×C412yes—
198C4⋊Dic43C22×C412yes—
199Dic43⋊C4—22no—
200Dic43⋊C4—22no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203Dic43⋊C4—81no—
204Dic43⋊C4—81no—
205C4×Dic43—21no—
206C4×Dic43—21no—
207C4×Dic43—24no—
208C4×Dic43—42no—
209C4×Dic43C22×C412yes—
210C4×Dic43#4112yes—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8C22×C412yes—
217C23×C43—22no—
218C23×C43—22no—
219C23×C43—22no—
220C23×C43—41no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43—41no—
224C23×C43C22×D411yescharacteristic
225Q8×C43—22no—
226Q8×C43—22no—
227Q8×C43—22no—
228Q8×C43C2×C4⋊C414yes—
229D4×C43—22no—
230D4×C43—22no—
231D4×C43—22no—
232D4×C43C2×C4⋊C414yes—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43—21no—
236C2×C4×C43—22no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—84no—
248C2×C4×C43#13512yes—
249C2×C4×C43C23×C411yescharacteristic
250C2×C4×C43C22×D411yescharacteristic
251C2×C4×C43C22×Q811yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43H(5,𝔽2)11yescharacteristic
254C2×C4×C432−1+411yescharacteristic
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—22no—
262C2×Dic43—22no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
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—41no—
275C2×Dic43—41no—
276C2×Dic43—84no—
277C2×Dic43—84no—
278C2×Dic43C4×D412yes—
279C2×Dic43C4×D412yes—
280C2×Dic43#13512yes—
281C8×C43—41no—
282C8×C43C2×C4⋊C414yes—
283C43⋊C8—22no—
284C43⋊C8—41no—
285#22683—431nomaximal, Sylow
286C22×C43—24no—
287C22×C43—42no—
288C22×C43—42no—
289C22×C43—42no—
290C22×C43—42no—
291C22×C43—42no—
292C22×C43—42no—
293C22×C43C2×C4×D411yescharacteristic
294C22×C43D4211yescharacteristic
295C22×C43D4×Q811yescharacteristic
296C22×C43C2×#13511yescharacteristic
297C22×C43C2×#13511yescharacteristic
298C22×C43#57711yescharacteristic
299C22×C43#57711yescharacteristic
300C4×C43—24no—
301C4×C43—22no—
302C4×C43—41no—
303C4×C43—81no—
304C4×C43—81no—
305C4×C43C22×C4⋊C411yescharacteristic
306C4×C43#52011yescharacteristic
307C4×C43#54711yescharacteristic
308C4×C43#55411yescharacteristic
309Dic43—22no—
310Dic43—22no—
311Dic43—22no—
312Dic43—42no—
313Dic43—42no—
314Dic43—41no—
315Dic43—81no—
316Dic43—81no—
317C2×C4×SD16—431no—
318C2×C4×SD16—431no—
319C2×C4×SD16D4311yescharacteristic
320#2351—431no—
321#2349—431no—
322#1941—431no—
323#1941—431no—
324#1863—431no—
325#1863—431no—
326#1862—431no—
327#1862—431no—
328#1846—431no—
329#1805—431no—
330#1804—431no—
331C8×C4⋊C4—431no—
332C2×C43—44no—
333C2×C43D4×C4⋊C411yescharacteristic
334C2×C43C4×#13511yescharacteristic
335C2×C43#294911yescharacteristic
336C2×C43#328211yescharacteristic
337C2×C43#328211yescharacteristic
338C2×C43#338111yescharacteristic
339C2×C43#338111yescharacteristic
340C22×SD16C2×D4311yescharacteristic
341C2×C4×Q8—431no—
342C2×C4×Q8—431no—
343C2×C4×Q8C2×D4311yescharacteristic
344C2×C4×D4—431no—
345C2×C4×D4—431no—
346C2×C4×D4C2×D4311yescharacteristic
347C4×SD16—24no—
348C4×SD16—438no—
349C4×SD16—864no—
350C2×#107—431no—
351C2×#107—431no—
352C2×#106—431no—
353C2×#106—431no—
354C2×#106C2×D4311yescharacteristic
355C2×C4⋊C8—431no—
356C2×#103—431no—
357C2×#103—431no—
358C2×#103C2×D4311yescharacteristic
359C2×#102—431no—
360C2×#102—431no—
361C2×#102C2×D4311yescharacteristic
362C2×C4×C8—431no—
363C2×C4×C8—431no—
364C2×C4×C8C2×D4311yescharacteristic
365#389—431no—
366#389—431no—
367#389—431no—
368#389—431no—
369#389—861no—
370#389—861no—
371#385—861no—
372#385—861no—
373#382—431no—
374#382—431no—
375#382—861no—
376#380—861no—
377C4×C4⋊C4—431no—
378C4×C4⋊C4—861no—
379C2×#95—861no—
380#340—431no—
381#340—431no—
382#340—431no—
383#340—431no—
384#336—431no—
385#336—431no—
386C43#2268311yescharacteristic, complemented, Sylow
387C22×Q8C22×D4311yescharacteristic
388C22×D4C22×D4311yescharacteristic
389C23×C4—21no—
390C23×C4—861no—
391C23×C4—861no—
392C2×SD16Dic8614yes—
393C2×SD16C4×D4314yes—
394C2×SD16C22⋊D4314yes—
395C22×C8—431no—
396C22×C8—431no—
397C22×C8C22×D4311yescharacteristic
398C22×C8C22×D4311yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—434no—
402C4×Q8—862no—
403C4×Q8—862no—
404C4×Q8—1721no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—434no—
408C4×D4—862no—
409C4×D4—862no—
410C4×D4—1721no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4—431no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4C22×D4311yescharacteristic
425C2×C4⋊C4C22×D4311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—861no—
428C2×C22⋊C4—861no—
429C2×C42—21no—
430C2×C42—431no—
431C2×C42—431no—
432C2×C42—861no—
433C2×C42—861no—
434C2×C42C22×D4311yescharacteristic
435#107—1721no—
436#107—1721no—
437#106—22no—
438#106—434no—
439#106—862no—
440C4⋊C8—862no—
441#103—22no—
442#103—434no—
443#103—862no—
444#102—22no—
445#102—434no—
446#102—862no—
447C4×C8—22no—
448C4×C8—434no—
449C4×C8—862no—
450#95—431no—
451#95—431no—
452#95—861no—
453#95—861no—
454#95—861no—
455#95—861no—
456#95—861no—
457#95—861no—
458#95—861no—
459#95—861no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×Dic8612yes—
465C2×Q8C2×C4×D4312yes—
466C2×Q8C2×C22⋊D4312yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×Dic8612yes—
471C2×D4C2×C4×D4312yes—
472C2×D4C2×C22⋊D4312yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—431no—
480C22×C4—431no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—864no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—3441no—
489C22×C4—3441no—
490C22×C4—3441no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4D4×D4311yescharacteristic
494C22×C4D4×D4311yescharacteristic
495C22×C4C23×D4311yescharacteristic
496SD16Dic43⋊C4116yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—432no—
501C2×C8—861no—
502C2×C8—861no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8C2×Dic8612yes—
507C2×C8C2×C4×D4312yes—
508C2×C8C2×C22⋊D4312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—432no—
515C4⋊C4—432no—
516C4⋊C4—862no—
517C4⋊C4—864no—
518C4⋊C4—862no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4—861no—
522C4⋊C4—861no—
523C4⋊C4—861no—
524C4⋊C4—861no—
525C4⋊C4—3441no—
526C4⋊C4—3441no—
527C4⋊C4—1722no—
528C4⋊C4—1722no—
529C4⋊C4—1722no—
530C22⋊C4—42no—
531C22⋊C4—864no—
532C22⋊C4—1722no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—432no—
537C42—432no—
538C42—864no—
539C42—861no—
540C42—861no—
541C42—1722no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D4311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×Dic43⋊C414yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×Dic43⋊C414yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—432no—
576C2×C4—861no—
577C2×C4—861no—
578C2×C4—861no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—862no—
583C2×C4—862no—
584C2×C4—861no—
585C2×C4—861no—
586C2×C4—861no—
587C2×C4—3444no—
588C2×C4—3444no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4#1306334612yes—
600C2×C4C22×C4×D4311yescharacteristic
601C2×C4C22×C22⋊D4311yescharacteristic
602C2×C4C22×Dic8611yescharacteristic
603C2×C4#41×D4311yescharacteristic
604C2×C4#1306337011yescharacteristic
605C2×C4#1306337211yescharacteristic
606C8—41no—
607C8—862no—
608C8—1721no—
609C8C2×Dic43⋊C414yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D4311yescharacteristic
618C22D4×C22⋊D4311yescharacteristic
619C22D4×Dic8611yescharacteristic
620C22#135×D4311yescharacteristic
621C22C2×#1306334611yescharacteristic
622C22#1458370511yescharacteristic
623C22#1458370611yescharacteristic
624C4—22no—
625C4—24no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—862no—
630C4—862no—
631C4—862no—
632C4—3441no—
633C4—3441no—
634C4—1722no—
635C4—1721no—
636C4—1722no—
637C4C22×Dic43⋊C411yescharacteristic
638C4#1458337811yescharacteristic
639C4#1458354811yescharacteristic
640C4#1458360811yescharacteristic
641C2—44no—
642C2D4×Dic43⋊C411yescharacteristic
643C2C4×#1306334611yescharacteristic
644C2#1914344911yescharacteristic
645C2#1914629611yescharacteristic
646C2#1914630211yescharacteristic
647C2#1914746811yescharacteristic
648C2#1914747111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C4×SD16×C43C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenterC2×C4×SD16D43
hypocenterC43#22683
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #19137513C2semidirect
5. #19135323C2semidirect
6. #19135321C2semidirect
7. #19134849C2semidirect
8. #19134847C2semidirect
11. #19134756C2semidirect
12. #19134608C2semidirect
14. C8×Dic43⋊C4C2semidirect
21. SD16×Dic43C4semidirect
69. Q8×Dic43C2×C4semidirect
95. #13063257C2×C4semidirect
98. #13063249C2×C4semidirect
102. C8×Dic43C2×C4semidirect
171. SD16×C43C4⋊C4semidirect
386. C43#22683semidirect
496. SD16Dic43⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic432
#191375132
#191375002
#191353232
#191353212
#191348492
#191348472
#191348422
#191348402
#191347562
#191346082
#191346032
C8×Dic43⋊C42
C2×C4×SD16×C432
#2268343