Finite Group Databasebeta

← #35422706

Subgroups — #35422706

Order 21248.

974 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×#19127802C211yesmaximal, characteristic
2C2×SD16×Dic83C211yesmaximal, characteristic
3#24910381C211yesmaximal, characteristic, complemented
4#24910368C211yesmaximal, characteristic
5#24908206C211yesmaximal, characteristic, complemented
6#24908204C211yesmaximal, characteristic, complemented
7#24907731C211yesmaximal, characteristic, complemented
8#24907729C211yesmaximal, characteristic, complemented
9#24907724C211yesmaximal, characteristic
10#24907722C211yesmaximal, characteristic
11#24907626C211yesmaximal, characteristic, complemented
12#24907476C211yesmaximal, characteristic, complemented
13#24907471C211yesmaximal, characteristic
14#24907331C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×Q8×Dic83C2211yescharacteristic
17C2×Q8×Dic83C2211yescharacteristic
18C2×D4×Dic83C2211yescharacteristic
19C2×D4×Dic83C2211yescharacteristic
20#19127802—24no—
21SD16×Dic83C418yescomplemented
22C2×#14581726C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
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#19127566—21no—
42#19127566C2211yescharacteristic
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⋊C4D411yescharacteristic
85C2×Dic83⋊C4D411yescharacteristic
86C2×C22⋊Dic83—21no—
87C2×C22⋊Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83C2311yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92#14581726—22no—
93#14581724C2×C414yescomplemented
94#14581723—41no—
95#14581723—41no—
96Dic83⋊C8—22no—
97#14581716—22no—
98#14581716C2×C414yescomplemented
99#14581713—22no—
100#14581713C2×C414yes—
101#14581702—22no—
102C8×Dic83C2×C414yescomplemented
103C83⋊#95—21no—
104C83⋊#95—21no—
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—41no—
166Dic83⋊C4—41no—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169Dic83⋊C4—81no—
170Dic83⋊C4—81no—
171C22⋊Dic83—24no—
172C22⋊Dic83—42no—
173C4×Dic83—21no—
174C4×Dic83—21no—
175C4×Dic83—24no—
176C4×Dic83—42no—
177C4×Dic83C22×C412yes—
178C4×Dic83#4112yes—
179C24×C83—21no—
180C2×Q8×C83—21no—
181C2×Q8×C83—21no—
182C2×Q8×C83—21no—
183C2×Q8×C83C22×C412yes—
184C2×Q8×C83C2×D412yes—
185C2×Q8×C83C2×Q812yes—
186C2×D4×C83—21no—
187C2×D4×C83—21no—
188C2×D4×C83—21no—
189C2×D4×C83C22×C412yes—
190C2×D4×C83C2×D412yes—
191C2×D4×C83C2×Q812yes—
192C22×C4×C83—21no—
193C22×C4×C83—21no—
194C22×C4×C83—21no—
195C22×C4×C83—42no—
196C22×C4×C83—42no—
197C22×C4×C83—81no—
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83C2×D411yescharacteristic
200C22×C4×C83C2411yescharacteristic
201SD16×C83C4⋊C4116yescomplemented
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83C22×C412yes—
206C2×C8×C83C2×D412yes—
207C2×C8×C83C2×Q812yes—
208C4⋊C4×C83—21no—
209C4⋊C4×C83—21no—
210C4⋊C4×C83—21no—
211C4⋊C4×C83—21no—
212C4⋊C4×C83—42no—
213C22⋊C4×C83—42no—
214C42×C83—21no—
215C42×C83—21no—
216C42×C83—42no—
217C23×C83—22no—
218C23×C83—22no—
219C23×C83—22no—
220C23×C83—41no—
221C23×C83—41no—
222C23×C83—41no—
223C23×C83—41no—
224C23×C83C22×D411yescharacteristic
225Q8×C83—22no—
226Q8×C83—22no—
227Q8×C83—22no—
228Q8×C83C2×C4⋊C414yes—
229D4×C83—22no—
230D4×C83—22no—
231D4×C83—22no—
232D4×C83C2×C4⋊C414yes—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—84no—
249C2×C4×C83C23×C411yescharacteristic
250C2×C4×C83C22×D411yescharacteristic
251C2×C4×C83C22×Q811yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83H(5,𝔽2)11yescharacteristic
254C2×C4×C832−1+411yescharacteristic
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—21no—
262C2×Dic83—21no—
263C2×Dic83—21no—
264C2×Dic83—21no—
265C2×Dic83—21no—
266C2×Dic83—21no—
267C2×Dic83—41no—
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—84no—
278C2×Dic83—84no—
279C2×Dic83C4×D412yes—
280C2×Dic83C4×D412yes—
281C8×C83—41no—
282C8×C83C2×C4⋊C414yes—
283C83⋊C8—22no—
284C83⋊C8—41no—
285C22×C83—24no—
286C22×C83—42no—
287C22×C83—42no—
288C22×C83—42no—
289C22×C83—42no—
290C22×C83—42no—
291C22×C83—42no—
292C22×C83C2×C4×D411yescharacteristic
293C22×C83D4211yescharacteristic
294C22×C83D4×Q811yescharacteristic
295C22×C83#57511yescharacteristic
296C22×C83#57511yescharacteristic
297C22×C83#57711yescharacteristic
298C22×C83#57711yescharacteristic
299C4×C83—24no—
300C4×C83—41no—
301C4×C83—41no—
302C4×C83—81no—
303C4×C83—81no—
304C4×C83C22×C4⋊C411yescharacteristic
305C4×C83#52011yescharacteristic
306C4×C83#54711yescharacteristic
307C4×C83#55411yescharacteristic
308Dic83—22no—
309Dic83—22no—
310Dic83—42no—
311Dic83—42no—
312Dic83—41no—
313Dic83—41no—
314Dic83—81no—
315Dic83—81no—
316#22673—831nomaximal, Sylow
317C2×C83—44no—
318C2×C83D4×C4⋊C411yescharacteristic
319C2×C83#294311yescharacteristic
320C2×C83#294911yescharacteristic
321C2×C83#327511yescharacteristic
322C2×C83#327511yescharacteristic
323C2×C83#338811yescharacteristic
324C2×C83#338811yescharacteristic
325C2×#440—831no—
326C2×#440D8311yescharacteristic
327C2×C4×SD16—831no—
328#2351—831no—
329#2349—831no—
330#1944—831no—
331#1944—831no—
332#1865—831no—
333#1865—831no—
334#1864—831no—
335#1864—831no—
336#1846—831no—
337#1805—831no—
338#1804—831no—
339#1770—831no—
340C83#2267311yescharacteristic, complemented, Sylow
341C22×SD16C2×D8311yescharacteristic
342C2×C4×Q8—831no—
343C2×C4×Q8—831no—
344C2×C4×Q8C2×D8311yescharacteristic
345C2×C4×D4—831no—
346C2×C4×D4—831no—
347C2×C4×D4C2×D8311yescharacteristic
348#440—24no—
349#440—1664no—
350C4×SD16—838no—
351C2×#107—831no—
352C2×#107C2×D8311yescharacteristic
353C2×#106—831no—
354C2×#106—831no—
355C2×#106—831no—
356C2×C4⋊C8—831no—
357C2×#103—831no—
358C2×#103—831no—
359C2×#103C2×D8311yescharacteristic
360C2×#102—831no—
361C2×#102—831no—
362C2×#102C2×D8311yescharacteristic
363C2×#97—831no—
364C2×#97C2×D8311yescharacteristic
365C2×C4×C8—831no—
366#389—831no—
367#389—831no—
368#389—831no—
369#389—831no—
370#389—1661no—
371#389—1661no—
372#385—1661no—
373#385—1661no—
374#382—831no—
375#382—831no—
376#382—1661no—
377#380—1661no—
378C4×C4⋊C4—831no—
379C4×C4⋊C4—1661no—
380C2×#95—1661no—
381#340—831no—
382#340—831no—
383#340—831no—
384#340—831no—
385#336—831no—
386#336—831no—
387C22×Q8C22×D8311yescharacteristic
388C22×D4C22×D8311yescharacteristic
389C23×C4—21no—
390C23×C4—1661no—
391C23×C4—1661no—
392C2×SD16Dic16614yes—
393C2×SD16C4×D8314yes—
394C2×SD16C22⋊D8314yes—
395C22×C8—831no—
396C22×C8—831no—
397C22×C8C22×D8311yescharacteristic
398C22×C8C22×D8311yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—3321no—
402C4×Q8—834no—
403C4×Q8—1662no—
404C4×Q8—1662no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—3321no—
408C4×D4—834no—
409C4×D4—1662no—
410C4×D4—1662no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—831no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1661no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4C22×D8311yescharacteristic
425C2×C4⋊C4C22×D8311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1661no—
428C2×C22⋊C4—1661no—
429C2×C42—21no—
430C2×C42—831no—
431C2×C42—831no—
432C2×C42—1661no—
433C2×C42—1661no—
434C2×C42C22×D8311yescharacteristic
435#107—22no—
436#107—1662no—
437#106—3321no—
438#106—3321no—
439#106—834no—
440C4⋊C8—1662no—
441#103—22no—
442#103—834no—
443#103—1662no—
444#102—22no—
445#102—834no—
446#102—1662no—
447#97—22no—
448#97—1662no—
449C4×C8—834no—
450#95—831no—
451#95—831no—
452#95—1661no—
453#95—1661no—
454#95—1661no—
455#95—1661no—
456#95—1661no—
457#95—1661no—
458#95—1661no—
459#95—1661no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D8312yes—
465C2×Q8C2×C22⋊D8312yes—
466C2×Q8C2×Dic16612yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C4×D8312yes—
471C2×D4C2×C22⋊D8312yes—
472C2×D4C2×Dic16612yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—3322no—
480C22×C4—3322no—
481C22×C4—831no—
482C22×C4—831no—
483C22×C4—831no—
484C22×C4—831no—
485C22×C4—6641no—
486C22×C4—6641no—
487C22×C4—6641no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4—1661no—
491C22×C4—1661no—
492C22×C4—1664no—
493C22×C4D4×D8311yescharacteristic
494C22×C4D4×D8311yescharacteristic
495C22×C4C23×D8311yescharacteristic
496SD16Dic83⋊C4116yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—832no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C2×C8C2×C4×D8312yes—
507C2×C8C2×C22⋊D8312yes—
508C2×C8C2×Dic16612yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—3322no—
515C4⋊C4—3322no—
516C4⋊C4—3321no—
517C4⋊C4—3321no—
518C4⋊C4—3322no—
519C4⋊C4—832no—
520C4⋊C4—832no—
521C4⋊C4—6641no—
522C4⋊C4—6641no—
523C4⋊C4—1664no—
524C4⋊C4—1661no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C22⋊C4—42no—
531C22⋊C4—3322no—
532C22⋊C4—1664no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—3322no—
537C42—832no—
538C42—832no—
539C42—1664no—
540C42—1661no—
541C42—1661no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D8311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×Dic83⋊C414yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×Dic83⋊C414yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—84no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—832no—
585C2×C4—832no—
586C2×C4—6644no—
587C2×C4—6644no—
588C2×C4—1661no—
589C2×C4—1661no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4—1661no—
593C2×C4—1661no—
594C2×C4—1661no—
595C2×C4—1661no—
596C2×C4—1661no—
597C2×C4—1661no—
598C2×C4—1661no—
599C2×C4—1661no—
600C2×C4C22×C4×D8311yescharacteristic
601C2×C4C22×C22⋊D8311yescharacteristic
602C2×C4C22×Dic16611yescharacteristic
603C2×C4#41×D8311yescharacteristic
604C2×C4#1458183711yescharacteristic
605C2×C4#1458183911yescharacteristic
606C8—41no—
607C8—3321no—
608C8—1662no—
609C8C2×Dic83⋊C414yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D8311yescharacteristic
618C22D4×C22⋊D8311yescharacteristic
619C22D4×Dic16611yescharacteristic
620C22#1912849711yescharacteristic
621C22#1912850011yescharacteristic
622C22#1912851011yescharacteristic
623C22#1912851111yescharacteristic
624C4—24no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—3322no—
630C4—3321no—
631C4—3321no—
632C4—3322no—
633C4—6641no—
634C4—6641no—
635C4—1662no—
636C4—1662no—
637C4C22×Dic83⋊C411yescharacteristic
638C4#1912818011yescharacteristic
639C4#1912835011yescharacteristic
640C4#1912841011yescharacteristic
641C2—44no—
642C2D4×Dic83⋊C411yescharacteristic
643C2#2491625811yescharacteristic
644C2#2491631911yescharacteristic
645C2#2491908811yescharacteristic
646C2#2491909411yescharacteristic
647C2#2492040911yescharacteristic
648C2#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#22673
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. #24910381C2semidirect
5. #24908206C2semidirect
6. #24908204C2semidirect
7. #24907731C2semidirect
8. #24907729C2semidirect
11. #24907626C2semidirect
12. #24907476C2semidirect
14. #24907331C2semidirect
21. SD16×Dic83C4semidirect
69. Q8×Dic83C2×C4semidirect
93. #14581724C2×C4semidirect
98. #14581716C2×C4semidirect
102. C8×Dic83C2×C4semidirect
201. SD16×C83C4⋊C4semidirect
340. C83#22673semidirect
496. 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
#249103812
#249103682
#249082062
#249082042
#249077312
#249077292
#249077242
#249077222
#249076262
#249074762
#249074712
#249073312
C2×#440×C832
#2267383