Finite Group Databasebeta

← #35422763

Subgroups — #35422763

Order 21248.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127802C211yesmaximal, characteristic
2C2×SD16×Dic83C211yesmaximal, characteristic
3Q8×Dic83⋊C4C211yesmaximal, characteristic, complemented
4#24910368C211yesmaximal, characteristic
5#24908201C211yesmaximal, characteristic, complemented
6#24908194C211yesmaximal, characteristic, complemented
7#24907731C211yesmaximal, characteristic, complemented
8#24907722C211yesmaximal, characteristic
9#24907715C211yesmaximal, characteristic, complemented
10#24907710C211yesmaximal, characteristic
11#24907630C211yesmaximal, characteristic, complemented
12#24907476C211yesmaximal, characteristic, complemented
13#24907461C211yesmaximal, characteristic
14#24907339C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×Q8×Dic83C2211yescharacteristic
17C2×Q8×Dic83C2211yescharacteristic
18C2×D4×Dic83C2211yescharacteristic
19C2×D4×Dic83C2211yescharacteristic
20#19127802C418yescomplemented
21SD16×Dic83—24no—
22C2×#14581726C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581724C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×Dic83⋊C8C2211yescharacteristic
27C2×#14581716C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581702C2211yescharacteristic
32C2×C8×Dic83C2211yescharacteristic
33#19127599—21no—
34#19127599C2211yescharacteristic
35#19127599C2211yescharacteristic
36#19127597—21no—
37#19127597C2211yescharacteristic
38#19127597C2211yescharacteristic
39#19127574—21no—
40#19127573—21no—
41#19127570—21no—
42#19127570C2211yescharacteristic
43#19127566C2211yescharacteristic
44#19127559—21no—
45C4×Dic83⋊C4—21no—
46C4×Dic83⋊C4C2211yescharacteristic
47C2×C83⋊#95—21no—
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127456C2211yescharacteristic
53#19127455C2211yescharacteristic
54C22×SD16×C83C2211yescharacteristic
55C2×C4×Q8×C83C2211yescharacteristic
56C2×C4×D4×C83C2211yescharacteristic
57#440×C83—24no—
58C2×#107×C83C2211yescharacteristic
59C2×#103×C83C2211yescharacteristic
60C2×#102×C83C2211yescharacteristic
61C2×#97×C83C2211yescharacteristic
62C23×Dic83—21no—
63C23×Dic83—21no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66Q8×Dic83—22no—
67Q8×Dic83—22no—
68Q8×Dic83—41no—
69Q8×Dic83C2×C414yescomplemented
70D4×Dic83—22no—
71D4×Dic83—22no—
72D4×Dic83—41no—
73D4×Dic83C2×C414yes—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4D411yescharacteristic
87C2×Dic83⋊C4D411yescharacteristic
88C2×C22⋊Dic83—21no—
89C2×C22⋊Dic83—21no—
90C2×C4×Dic83—21no—
91C2×C4×Dic83—21no—
92C2×C4×Dic83C2311yescharacteristic
93C2×C4×Dic83C2311yescharacteristic
94#14581726C2×C414yescomplemented
95#14581725—41no—
96#14581724—22no—
97#14581723—41no—
98Dic83⋊C8—22no—
99#14581716—22no—
100#14581716C2×C414yescomplemented
101#14581713—22no—
102#14581713C2×C414yes—
103#14581702C2×C414yescomplemented
104C8×Dic83—22no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95D411yescharacteristic
112C83⋊#95D411yescharacteristic
113C22×Q8×C83C2311yescharacteristic
114C22×D4×C83C2311yescharacteristic
115C23×C4×C83—21no—
116C2×SD16×C83C2×C414yes—
117C2×SD16×C83D414yes—
118C2×SD16×C83Q814yes—
119C22×C8×C83C2311yescharacteristic
120C22×C8×C83C2311yescharacteristic
121C4×Q8×C83—22no—
122C4×Q8×C83—41no—
123C4×D4×C83—22no—
124C4×D4×C83—41no—
125C2×C4⋊C4×C83—21no—
126C2×C4⋊C4×C83C2311yescharacteristic
127C2×C4⋊C4×C83C2311yescharacteristic
128C2×C22⋊C4×C83—21no—
129C2×C42×C83—21no—
130C2×C42×C83C2311yescharacteristic
131#107×C83—22no—
132#103×C83—22no—
133#102×C83—22no—
134#97×C83—22no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—24no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—42no—
141C22×Dic83—42no—
142C22×Dic83—81no—
143C22×Dic83—81no—
144C22×Dic83—81no—
145C22×Dic83C2×D411yescharacteristic
146C22×Dic83C2×D411yescharacteristic
147C22×Dic83C2×D411yescharacteristic
148C22×Dic83C2×D411yescharacteristic
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8C22×C412yes—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—24no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—42no—
163C4⋊Dic83C22×C412yes—
164C4⋊Dic83C22×C412yes—
165Dic83⋊C4—22no—
166Dic83⋊C4—41no—
167Dic83⋊C4—42no—
168Dic83⋊C4—81no—
169Dic83⋊C4—81no—
170Dic83⋊C4—81no—
171Dic83⋊C4—81no—
172Dic83⋊C4—81no—
173C22⋊Dic83—24no—
174C22⋊Dic83—42no—
175C4×Dic83—21no—
176C4×Dic83—21no—
177C4×Dic83—24no—
178C4×Dic83—42no—
179C4×Dic83C22×C412yes—
180C4×Dic83#4112yes—
181C24×C83—21no—
182C2×Q8×C83—21no—
183C2×Q8×C83—21no—
184C2×Q8×C83—21no—
185C2×Q8×C83C22×C412yes—
186C2×Q8×C83C2×D412yes—
187C2×Q8×C83C2×Q812yes—
188C2×D4×C83—21no—
189C2×D4×C83—21no—
190C2×D4×C83—21no—
191C2×D4×C83C22×C412yes—
192C2×D4×C83C2×D412yes—
193C2×D4×C83C2×Q812yes—
194C22×C4×C83—21no—
195C22×C4×C83—21no—
196C22×C4×C83—21no—
197C22×C4×C83—42no—
198C22×C4×C83—42no—
199C22×C4×C83—81no—
200C22×C4×C83C2×D411yescharacteristic
201C22×C4×C83C2×D411yescharacteristic
202C22×C4×C83C2411yescharacteristic
203SD16×C83C4⋊C4116yescomplemented
204C2×C8×C83—21no—
205C2×C8×C83—21no—
206C2×C8×C83—21no—
207C2×C8×C83C22×C412yes—
208C2×C8×C83C2×D412yes—
209C2×C8×C83C2×Q812yes—
210C4⋊C4×C83—21no—
211C4⋊C4×C83—21no—
212C4⋊C4×C83—21no—
213C4⋊C4×C83—21no—
214C4⋊C4×C83—42no—
215C22⋊C4×C83—42no—
216C42×C83—21no—
217C42×C83—21no—
218C42×C83—42no—
219C23×C83—22no—
220C23×C83—22no—
221C23×C83—22no—
222C23×C83—41no—
223C23×C83—41no—
224C23×C83—41no—
225C23×C83—41no—
226C23×C83C22×D411yescharacteristic
227Q8×C83—22no—
228Q8×C83—22no—
229Q8×C83—22no—
230Q8×C83C2×C4⋊C414yes—
231D4×C83—22no—
232D4×C83—22no—
233D4×C83—22no—
234D4×C83C2×C4⋊C414yes—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—84no—
251C2×C4×C83C23×C411yescharacteristic
252C2×C4×C83C22×D411yescharacteristic
253C2×C4×C83C22×Q811yescharacteristic
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83H(5,𝔽2)11yescharacteristic
256C2×C4×C832−1+411yescharacteristic
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—21no—
262C2×Dic83—21no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—21no—
266C2×Dic83—21no—
267C2×Dic83—21no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—84no—
279C2×Dic83—84no—
280C2×Dic83C4×D412yes—
281C2×Dic83C4×D412yes—
282C2×Dic83#13712yes—
283C8×C83—41no—
284C8×C83C2×C4⋊C414yes—
285C83⋊C8—22no—
286C83⋊C8—41no—
287C22×C83—24no—
288C22×C83—42no—
289C22×C83—42no—
290C22×C83—42no—
291C22×C83—42no—
292C22×C83—42no—
293C22×C83—42no—
294C22×C83C2×C4×D411yescharacteristic
295C22×C83D4211yescharacteristic
296C22×C83D4×Q811yescharacteristic
297C22×C83C2×#13711yescharacteristic
298C22×C83#57511yescharacteristic
299C22×C83#57611yescharacteristic
300C22×C83#57711yescharacteristic
301C4×C83—24no—
302C4×C83—41no—
303C4×C83—41no—
304C4×C83—81no—
305C4×C83—81no—
306C4×C83C22×C4⋊C411yescharacteristic
307C4×C83#52011yescharacteristic
308C4×C83#54711yescharacteristic
309C4×C83#55411yescharacteristic
310Dic83—22no—
311Dic83—22no—
312Dic83—22no—
313Dic83—42no—
314Dic83—42no—
315Dic83—41no—
316Dic83—81no—
317Dic83—81no—
318#22678—831nomaximal, Sylow
319C2×C83—44no—
320C2×C83D4×C4⋊C411yescharacteristic
321C2×C83C4×#13711yescharacteristic
322C2×C83#294611yescharacteristic
323C2×C83#326611yescharacteristic
324C2×C83#326811yescharacteristic
325C2×C83#338511yescharacteristic
326C2×C83#338811yescharacteristic
327C2×#440—831no—
328C2×#440D8311yescharacteristic
329C2×C4×SD16—831no—
330Q8×C4⋊C4—831no—
331#2349—831no—
332#1943—831no—
333#1942—831no—
334#1865—831no—
335#1864—831no—
336#1863—831no—
337#1862—831no—
338#1847—831no—
339#1805—831no—
340#1802—831no—
341#1771—831no—
342C83#2267811yescharacteristic, complemented, Sylow
343C22×SD16C2×D8311yescharacteristic
344C2×C4×Q8—831no—
345C2×C4×Q8—831no—
346C2×C4×Q8C2×D8311yescharacteristic
347C2×C4×D4—831no—
348C2×C4×D4—831no—
349C2×C4×D4C2×D8311yescharacteristic
350#440—24no—
351#440—838no—
352C4×SD16—1664no—
353C2×#107—831no—
354C2×#107—831no—
355C2×#107C2×D8311yescharacteristic
356C2×#106—831no—
357C2×#106—831no—
358C2×C4⋊C8—831no—
359C2×#103—831no—
360C2×#103—831no—
361C2×#103C2×D8311yescharacteristic
362C2×#102—831no—
363C2×#102—831no—
364C2×#102C2×D8311yescharacteristic
365C2×#97—831no—
366C2×#97C2×D8311yescharacteristic
367C2×C4×C8—831no—
368#389—831no—
369#389—831no—
370#389—831no—
371#389—831no—
372#389—1661no—
373#389—1661no—
374#385—1661no—
375#385—1661no—
376#384—831no—
377#384—1661no—
378#382—831no—
379#380—1661no—
380C4×C4⋊C4—831no—
381C4×C4⋊C4—1661no—
382C2×#95—1661no—
383#340—831no—
384#340—831no—
385#340—831no—
386#340—831no—
387#336—831no—
388#336—831no—
389C22×Q8C22×D8311yescharacteristic
390C22×D4C22×D8311yescharacteristic
391C23×C4—21no—
392C23×C4—1661no—
393C23×C4—1661no—
394C2×SD16Dic16614yes—
395C2×SD16C4×D8314yes—
396C2×SD16C22⋊D8314yes—
397C22×C8—831no—
398C22×C8—831no—
399C22×C8C22×D8311yescharacteristic
400C22×C8C22×D8311yescharacteristic
401C4×Q8—22no—
402C4×Q8—41no—
403C4×Q8—3321no—
404C4×Q8—834no—
405C4×Q8—1662no—
406C4×Q8—1662no—
407C4×D4—22no—
408C4×D4—41no—
409C4×D4—3321no—
410C4×D4—834no—
411C4×D4—1662no—
412C4×D4—1662no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—831no—
419C2×C4⋊C4—831no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1661no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4—1661no—
425C2×C4⋊C4—1661no—
426C2×C4⋊C4—1661no—
427C2×C4⋊C4—1661no—
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1661no—
432C2×C22⋊C4—1661no—
433C2×C42—21no—
434C2×C42—831no—
435C2×C42—831no—
436C2×C42—1661no—
437C2×C42—1661no—
438C2×C42C22×D8311yescharacteristic
439#107—22no—
440#107—3321no—
441#107—834no—
442#106—3321no—
443#106—1662no—
444C4⋊C8—1662no—
445#103—22no—
446#103—834no—
447#103—1662no—
448#102—22no—
449#102—834no—
450#102—1662no—
451#97—22no—
452#97—834no—
453C4×C8—1662no—
454#95—831no—
455#95—831no—
456#95—1661no—
457#95—1661no—
458#95—1661no—
459#95—1661no—
460#95—1661no—
461#95—1661no—
462C24—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D8312yes—
467C2×Q8C2×C22⋊D8312yes—
468C2×Q8C2×Dic16612yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D8312yes—
473C2×D4C2×C22⋊D8312yes—
474C2×D4C2×Dic16612yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—3322no—
482C22×C4—3322no—
483C22×C4—831no—
484C22×C4—831no—
485C22×C4—831no—
486C22×C4—831no—
487C22×C4—6641no—
488C22×C4—6641no—
489C22×C4—6641no—
490C22×C4—1661no—
491C22×C4—1661no—
492C22×C4—1661no—
493C22×C4—1661no—
494C22×C4—1664no—
495C22×C4D4×D8311yescharacteristic
496C22×C4D4×D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498SD16Dic83⋊C4116yescomplemented
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—832no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C2×C8—1661no—
507C2×C8—1661no—
508C2×C8C2×C4×D8312yes—
509C2×C8C2×C22⋊D8312yes—
510C2×C8C2×Dic16612yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—3322no—
517C4⋊C4—3321no—
518C4⋊C4—3322no—
519C4⋊C4—832no—
520C4⋊C4—832no—
521C4⋊C4—6641no—
522C4⋊C4—6641no—
523C4⋊C4—6641no—
524C4⋊C4—6641no—
525C4⋊C4—6641no—
526C4⋊C4—1662no—
527C4⋊C4—1664no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C22⋊C4—42no—
535C22⋊C4—3322no—
536C22⋊C4—1664no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—3322no—
541C42—832no—
542C42—832no—
543C42—1664no—
544C42—1661no—
545C42—1661no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D8311yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C2×Dic83⋊C414yes—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C2×Dic83⋊C414yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4—3321no—
588C2×C4—832no—
589C2×C4—832no—
590C2×C4—832no—
591C2×C4—6644no—
592C2×C4—6644no—
593C2×C4—1661no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1661no—
597C2×C4—1661no—
598C2×C4—1661no—
599C2×C4—1661no—
600C2×C4—1661no—
601C2×C4—1661no—
602C2×C4—1661no—
603C2×C4—1661no—
604C2×C4C22×C4×D8311yescharacteristic
605C2×C4C22×C22⋊D8311yescharacteristic
606C2×C4C22×Dic16611yescharacteristic
607C2×C4#41×D8311yescharacteristic
608C2×C4#1458183711yescharacteristic
609C2×C4#1458183911yescharacteristic
610C8—41no—
611C8—3321no—
612C8—1662no—
613C8C2×Dic83⋊C414yes—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C4×D4×D8311yescharacteristic
622C22D4×C22⋊D8311yescharacteristic
623C22D4×Dic16611yescharacteristic
624C22#137×D8311yescharacteristic
625C22#1912849711yescharacteristic
626C22#1912850511yescharacteristic
627C22#1912851011yescharacteristic
628C4—24no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—3322no—
634C4—3322no—
635C4—3321no—
636C4—6641no—
637C4—6641no—
638C4—1662no—
639C4—1662no—
640C4—1662no—
641C4C22×Dic83⋊C411yescharacteristic
642C4#1912818011yescharacteristic
643C4#1912835011yescharacteristic
644C4#1912841011yescharacteristic
645C2—44no—
646C2D4×Dic83⋊C411yescharacteristic
647C2#2491628511yescharacteristic
648C2#2491629411yescharacteristic
649C2#2491896711yescharacteristic
650C2#2491899711yescharacteristic
651C2#2492037411yescharacteristic
652C2#2492041311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#440×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#440D83
hypocenterC83#22678
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. Q8×Dic83⋊C4C2semidirect
5. #24908201C2semidirect
6. #24908194C2semidirect
7. #24907731C2semidirect
9. #24907715C2semidirect
11. #24907630C2semidirect
12. #24907476C2semidirect
14. #24907339C2semidirect
20. #19127802C4semidirect
69. Q8×Dic83C2×C4semidirect
94. #14581726C2×C4semidirect
100. #14581716C2×C4semidirect
103. #14581702C2×C4semidirect
203. SD16×C83C4⋊C4semidirect
342. C83#22678semidirect
498. SD16Dic83⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#191278022
C2×SD16×Dic832
Q8×Dic83⋊C42
#249103682
#249082012
#249081942
#249077312
#249077222
#249077152
#249077102
#249076302
#249074762
#249074612
#249073392
C2×#440×C832
#2267883