Finite Group Databasebeta

← #35335693

Subgroups — #35335693

Order 21248.

2734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581768C211yesmaximal, characteristic
2C22×C42×D83C211yesmaximal, characteristic
3C2×#19127608C211yesmaximal, characteristic, complemented
4C2×#19127558C211yesmaximal, characteristic
5#24905694C214yesmaximal, complemented
6#24905693C211yesmaximal, characteristic, complemented
7#24905692C211yesmaximal, characteristic, complemented
8(C2×D83)⋊#95C214yesmaximal
9C2×#391×C83C211yesmaximal, characteristic, complemented
10C23×C4×D83C2212yes—
11C23×C4×D83C2211yescharacteristic
12C2×#14581768—24no—
13C2×#14581768—24no—
14C2×#14581768C418yescomplemented
15C22×C4⋊Dic83C2211yescharacteristic
16C22×C4⋊Dic83C2211yescharacteristic
17C22×Dic83⋊C4C2212yes—
18C22×(C2×D83)⋊C4C2212yes—
19C22×(C2×D83)⋊C4C2212yes—
20C22×C4×Dic83C2212yes—
21C22×C4×Dic83C2211yescharacteristic
22C2×C42×D83—24no—
23C2×C42×D83—24no—
24C2×C42×D83—24no—
25#19127608—24no—
26#19127558—24no—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2214yes—
29C2×C83⋊#95C2214yes—
30C2×C83⋊#95C2214yes—
31C23×Q8×C83C2211yescharacteristic
32C22×C4⋊C4×C83C2211yescharacteristic
33C22×C42×C83C2211yescharacteristic
34#391×C83—24no—
35C2×#95×C83C2214yes—
36C24×D83C2311yescharacteristic
37C23×Dic83C2312yes—
38C23×Dic83C2312yes—
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2312yes—
41C22×C4×D83—24no—
42C22×C4×D83—24no—
43C22×C4×D83—24no—
44C22×C4×D83—24no—
45C22×C4×D83—22no—
46C22×C4×D83—24no—
47C22×C4×D83—22no—
48C22×C4×D83—22no—
49C22×C4×D83—22no—
50C22×C4×D83—24no—
51C22×C4×D83C2×C414yes—
52C22×C4×D83D414yes—
53C22×C4×D83D414yes—
54C22×C4×D83Q818yescomplemented
55#14581768—232no—
56C2×C4⋊Dic83—22no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83—22no—
59C2×C4⋊Dic83—22no—
60C2×C4⋊Dic83—22no—
61C2×C4⋊Dic83C2×C414yescomplemented
62C2×Dic83⋊C4—24no—
63C2×Dic83⋊C4—24no—
64C2×Dic83⋊C4C2×C418yescomplemented
65C2×(C2×D83)⋊C4—24no—
66C2×(C2×D83)⋊C4—24no—
67C2×(C2×D83)⋊C4—24no—
68C2×(C2×D83)⋊C4—28no—
69C2×(C2×D83)⋊C4C2×C418yes—
70C2×C4×Dic83—24no—
71C2×C4×Dic83—24no—
72C2×C4×Dic83—22no—
73C2×C4×Dic83—24no—
74C2×C4×Dic83—22no—
75C2×C4×Dic83D414yes—
76C42×D83—232no—
77C83⋊#95—24no—
78C83⋊#95—28no—
79C83⋊#95—28no—
80C83⋊#95—28no—
81C22×Q8×C83—22no—
82C22×Q8×C83—22no—
83C22×Q8×C83—22no—
84C22×Q8×C83—22no—
85C22×Q8×C83—22no—
86C22×Q8×C83—22no—
87C22×Q8×C83C2×C414yescomplemented
88C23×C4×C83C2312yes—
89C23×C4×C83C2312yes—
90C23×C4×C83C2312yes—
91C23×C4×C83C2311yescharacteristic
92C2×C4⋊C4×C83—22no—
93C2×C4⋊C4×C83—22no—
94C2×C4⋊C4×C83—22no—
95C2×C42×C83—22no—
96C2×C42×C83—22no—
97C2×C42×C83D414yes—
98#95×C83—28no—
99C23×D83—21no—
100C23×D83—21no—
101C23×D83—22no—
102C23×D83—21no—
103C23×D83—21no—
104C23×D83—22no—
105C23×D83C22×C412yes—
106C23×D83C2×D412yes—
107C23×D83C2×D412yes—
108C23×D83C2×Q814yes—
109C23×D83#4114yes—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—21no—
116C22×Dic83—21no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—24no—
121C22×Dic83—24no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—24no—
125C22×Dic83—24no—
126C22×Dic83—22no—
127C22×Dic83C22×C414yes—
128C22×Dic83C22×C412yes—
129C22×Dic83C2×D412yes—
130C22×Dic83C2×D412yes—
131C22×Dic83C2×D414yes—
132C22×Dic83C2×Q814yes—
133C22×Dic83#4114yes—
134C2×C4×D83—216no—
135C2×C4×D83—216no—
136C2×C4×D83—216no—
137C2×C4×D83—216no—
138C2×C4×D83—216no—
139C2×C4×D83—28no—
140C2×C4×D83—28no—
141C2×C4×D83—28no—
142C2×C4×D83—28no—
143C2×C4×D83—28no—
144C2×C4×D83—216no—
145C2×C4×D83—28no—
146C2×C4×D83—216no—
147C2×C4×D83C22⋊C4116yes—
148C4⋊Dic83—28no—
149C4⋊Dic83—44no—
150Dic83⋊C4—216no—
151(C2×D83)⋊C4—216no—
152(C2×D83)⋊C4—48no—
153C4×Dic83—216no—
154C4×Dic83—28no—
155C24×C83C2411yescharacteristic
156C2×Q8×C83—28no—
157C2×Q8×C83—28no—
158C2×Q8×C83—28no—
159C2×Q8×C83—44no—
160C2×Q8×C83—44no—
161C2×Q8×C83—44no—
162C2×Q8×C83—44no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83C22×C414yes—
179C22×C4×C83C22×C412yes—
180C22×C4×C83C2×D412yes—
181C22×C4×C83C2×D412yes—
182C22×C4×C83C2×D412yes—
183C22×C4×C83C2×D414yes—
184C22×C4×C83C2×D412yes—
185C22×C4×C83C2×Q814yes—
186C22×C4×C83#4112yes—
187C22×C4×C83#4112yes—
188C22×C4×C83#4114yes—
189C4⋊C4×C83—44no—
190C42×C83—28no—
191C23×C83C23×C411yescharacteristic
192C23×C83C22×D411yescharacteristic
193C23×C83C22×D411yescharacteristic
194C23×C83C22×D411yescharacteristic
195C23×C83C22×D411yescharacteristic
196C23×C83C22×Q812yes—
197C23×C83C2×#4112yes—
198C23×C83C2×#4112yes—
199C23×C83C2×#4111yescharacteristic
200C23×C83C2×#4111yescharacteristic
201C23×C83C2×#4112yes—
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—24no—
205C22×D83—22no—
206C22×D83—22no—
207C22×D83—22no—
208C22×D83—24no—
209C22×D83—24no—
210C22×D83—22no—
211C22×D83—24no—
212C22×D83—24no—
213C22×D83—24no—
214C22×D83—24no—
215C22×D83—24no—
216C22×D83—22no—
217C22×D83—22no—
218C22×D83—24no—
219C22×D83—24no—
220C22×D83C2×C22⋊C414yes—
221C22×D83C4×Q818yes—
222C22×D83C22⋊Q818yes—
223C22×D83#12414yes—
224C22×D83C4⋊Q814yes—
225Q8×C83—416no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—42no—
239C2×C4×C83—42no—
240C2×C4×C83—44no—
241C2×C4×C83—42no—
242C2×C4×C83—44no—
243C2×C4×C83—42no—
244C2×C4×C83C2×C22⋊C414yes—
245C2×C4×C83C4×D414yes—
246C2×C4×C83C4×D414yes—
247C2×C4×C83#12014yes—
248C2×C4×C83#12114yes—
249C2×C4×C83#12114yes—
250C2×C4×C83C22⋊Q818yescomplemented
251C2×C4×C83#12314yes—
252C2×Dic83—22no—
253C2×Dic83—22no—
254C2×Dic83—24no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—22no—
261C2×Dic83—24no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—24no—
265C2×Dic83—28no—
266C2×Dic83—44no—
267C2×Dic83—42no—
268C2×Dic83—42no—
269C2×Dic83—42no—
270C2×Dic83—44no—
271C2×Dic83—42no—
272C2×Dic83—44no—
273C2×Dic83—42no—
274C2×Dic83C2×C22⋊C414yes—
275C2×Dic83C22⋊Q818yescomplemented
276C4×D83—264no—
277C4×D83—232no—
278C22×C83C22×C22⋊C411yescharacteristic
279C22×C83C2×C4×D411yescharacteristic
280C22×C83C2×C4×D411yescharacteristic
281C22×C83C2×C4×Q812yes—
282C22×C83C4×#4112yes—
283C22×C83C2×#12011yescharacteristic
284C22×C83C2×#12111yescharacteristic
285C22×C83C2×#12111yescharacteristic
286C22×C83C2×C22⋊Q812yes—
287C22×C83C2×C22⋊Q812yes—
288C22×C83C2×C22⋊Q812yes—
289C22×C83C2×#12311yescharacteristic
290C22×C83#52512yes—
291C22×C83#52512yes—
292C22×C83#52512yes—
293C22×C83C2×#12411yescharacteristic
294C22×C83#52912yes—
295C22×C83#52912yes—
296C22×C83C2×C4⋊Q811yescharacteristic
297C22×C83#53211yescharacteristic
298C22×C83#53211yescharacteristic
299C22×C83#53312yes—
300C22×C83#53312yes—
301C2×D83—28no—
302C2×D83—24no—
303C2×D83—24no—
304C2×D83—28no—
305C2×D83—24no—
306C2×D83—24no—
307C2×D83—24no—
308C2×D83—24no—
309C2×D83—28no—
310C2×D83—28no—
311C2×D83#39118yescomplemented
312C4×C83—28no—
313C4×C83—44no—
314C4×C83—82no—
315C4×C83#38618yes—
316Dic83—28no—
317Dic83—24no—
318Dic83—44no—
319Dic83—82no—
320#14702—831nomaximal, Sylow
321C2×C83C2×#38611yescharacteristic
322C2×C83C2×#39111yescharacteristic
323C2×C83#229711yescharacteristic
324C2×C83C4×C22⋊Q812yes—
325C2×C83C4×C22⋊Q812yes—
326C2×C83#284812yes—
327C2×C83#285011yescharacteristic
328C2×C83#285011yescharacteristic
329C2×C83#285512yes—
330C2×C83#285711yescharacteristic
331C2×C83#285711yescharacteristic
332D83—28no—
333C22×C22⋊Q8—831no—
334C23×C42—831no—
335C2×#391—831no—
336C2×#391D8311yescharacteristic, complemented
337C2×#380—831no—
338#1436—834no—
339C4⋊#95—831no—
340C4⋊#95—831no—
341C22⋊#95—834no—
342C83#1470211yescharacteristic, complemented, Sylow
343C23×Q8C2×D8311yescharacteristic
344C24×C4—831no—
345C24×C4—832no—
346C2×C22⋊Q8—838no—
347C2×C22⋊Q8—1664no—
348C2×C22⋊Q8—1664no—
349C22×C4⋊C4—832no—
350C22×C4⋊C4—831no—
351C22×C4⋊C4—831no—
352C22×C4⋊C4C2×D8311yescharacteristic
353C22×C22⋊C4—832no—
354C22×C22⋊C4—832no—
355C22×C42—831no—
356C22×C42—832no—
357C22×C42—1664no—
358C22×C42—1664no—
359C22×C42—1664no—
360C22×C42C2×D8311yescharacteristic
361#391—24no—
362#391—1664no—
363#380—1664no—
364C2×#95—834no—
365C2×#95—834no—
366C2×#95—834no—
367C2×#95—832no—
368C2×#95C2×D8314yes—
369C25—831no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8C4×D8314yescomplemented
377C23×C4—831no—
378C23×C4—832no—
379C23×C4—834no—
380C23×C4—834no—
381C23×C4—834no—
382C23×C4—832no—
383C23×C4—838no—
384C23×C4—832no—
385C23×C4—1664no—
386C23×C4—1664no—
387C23×C4—1664no—
388C23×C4—1664no—
389C23×C4—1662no—
390C23×C4—1664no—
391C23×C4—1662no—
392C23×C4—1662no—
393C23×C4—1662no—
394C23×C4—1664no—
395C23×C4C22×D8312yes—
396C23×C4C22×D8312yes—
397C23×C4C22×D8312yes—
398C23×C4C22×D8311yescharacteristic
399C22⋊Q8—16632no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—838no—
404C2×C4⋊C4—834no—
405C2×C4⋊C4—1662no—
406C2×C4⋊C4—1662no—
407C2×C4⋊C4—1662no—
408C2×C4⋊C4—1664no—
409C2×C4⋊C4—1664no—
410C2×C4⋊C4—1662no—
411C2×C4⋊C4—1662no—
412C2×C22⋊C4—838no—
413C2×C22⋊C4—1664no—
414C2×C22⋊C4—1668no—
415C2×C22⋊C4—1664no—
416C2×C22⋊C4—1664no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—834no—
420C2×C42—16632no—
421C2×C42—1664no—
422C2×C42—1664no—
423C2×C42—1662no—
424C2×C42—1664no—
425C2×C42—1662no—
426C2×C42C22⋊D8314yes—
427#95—28no—
428#95—1664no—
429#95—1668no—
430#95—1668no—
431#95—1668no—
432C24—832no—
433C24—832no—
434C24—832no—
435C24—834no—
436C24—834no—
437C24—1661no—
438C24—1661no—
439C24—1662no—
440C24—1661no—
441C24—1661no—
442C24—1662no—
443C24C23×D8311yescharacteristic
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—832no—
467C22×C4—834no—
468C22×C4—832no—
469C22×C4—832no—
470C22×C4—834no—
471C22×C4—834no—
472C22×C4—8316no—
473C22×C4—834no—
474C22×C4—16616no—
475C22×C4—16616no—
476C22×C4—16616no—
477C22×C4—16616no—
478C22×C4—16616no—
479C22×C4—16616no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1664no—
483C22×C4—1662no—
484C22×C4—1664no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1664no—
488C22×C4—1664no—
489C22×C4—1661no—
490C22×C4—1662no—
491C22×C4—1661no—
492C22×C4—1662no—
493C22×C4—1662no—
494C22×C4—1661no—
495C22×C4—1661no—
496C22×C4—1662no—
497C22×C4—1668no—
498C22×C4—1668no—
499C22×C4—1668no—
500C22×C4—16616no—
501C22×C4—1668no—
502C22×C4—1668no—
503C22×C4—1668no—
504C22×C4C2×C4×D8314yes—
505C22×C4C2×C4×D8312yes—
506C22×C4D4×D8312yes—
507C22×C4D4×D8312yes—
508C22×C4D4×D8312yes—
509C22×C4C2×C22⋊D8314yes—
510C22×C4C2×C22⋊D8312yes—
511C22×C4Q8×D8314yes—
512C22×C4#1305058912yes—
513C22×C4#1305059012yes—
514C22×C4#1305059114yes—
515C4⋊C4—44no—
516C4⋊C4—3324no—
517C4⋊C4—16616no—
518C4⋊C4—1668no—
519C22⋊C4—3328no—
520C22⋊C4—16616no—
521C42—28no—
522C42—16616no—
523C42—1668no—
524C23—834no—
525C23—838no—
526C23—838no—
527C23—834no—
528C23—834no—
529C23—1662no—
530C23—1662no—
531C23—1664no—
532C23—1662no—
533C23—1662no—
534C23—1662no—
535C23—1664no—
536C23—1664no—
537C23—1662no—
538C23—1664no—
539C23—1664no—
540C23—1664no—
541C23—1664no—
542C23—1664no—
543C23—1662no—
544C23—1662no—
545C23—1664no—
546C23—1664no—
547C23C22×C4×D8311yescharacteristic
548C23C2×D4×D8311yescharacteristic
549C23C2×D4×D8311yescharacteristic
550C23C2×D4×D8311yescharacteristic
551C23C22×C22⋊D8311yescharacteristic
552C23C2×Q8×D8312yes—
553C23#41×D8312yes—
554C23#41×D8312yes—
555C23C2×#1305058911yescharacteristic
556C23C2×#1305059011yescharacteristic
557C23C2×#1305059112yes—
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—3324no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3324no—
582C2×C4—3322no—
583C2×C4—3324no—
584C2×C4—3322no—
585C2×C4—834no—
586C2×C4—838no—
587C2×C4—16664no—
588C2×C4—1664no—
589C2×C4—1668no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4—1664no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1664no—
597C2×C4—1664no—
598C2×C4—1662no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—16632no—
603C2×C4C22⋊C4×D8314yes—
604C2×C4(C22⋊D83)⋊C414yes—
605C2×C4C4×C22⋊D8314yes—
606C2×C4#1458176014yes—
607C2×C4#1458176314yes—
608C2×C4#1458176514yes—
609C2×C4#1458176818yescomplemented
610C2×C4#1458177414yes—
611C22—838no—
612C22—1668no—
613C22—1664no—
614C22—1664no—
615C22—1664no—
616C22—1668no—
617C22—1664no—
618C22—1664no—
619C22—1664no—
620C22—1668no—
621C22—1668no—
622C22C2×C22⋊C4×D8311yescharacteristic
623C22C2×(C22⋊D83)⋊C411yescharacteristic
624C22C2×C4×C22⋊D8311yescharacteristic
625C22C4×Q8×D8312yes—
626C22C4×#1305059112yes—
627C22C2×#1458176011yescharacteristic
628C22C2×#1458176311yescharacteristic
629C22C2×#1458176511yescharacteristic
630C22C22⋊Q8×D8312yes—
631C22C22⋊Q8×D8312yes—
632C22C2×#1458176812yes—
633C22C2×#1458177411yescharacteristic
634C22#1912820812yes—
635C22#1912821412yes—
636C22#1912821412yes—
637C22#124×D8311yescharacteristic
638C22#1912823012yes—
639C22#1912823812yes—
640C22C4⋊Q8×D8311yescharacteristic
641C22#1912825011yescharacteristic
642C22#1912825211yescharacteristic
643C22#1912825512yes—
644C22#1912825612yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—3324no—
649C4—6642no—
650C4—1668no—
651C4—1664no—
652C4#1912757718yes—
653C2—1668no—
654C2C2×#1912757711yescharacteristic
655C2#391×D8311yescharacteristic
656C2#2490995211yescharacteristic
657C2C4×#1458176812yes—
658C2#2491000112yes—
659C2#2491574712yes—
660C2#2491575911yescharacteristic
661C2#2491576711yescharacteristic
662C2#2491580412yes—
663C2#2491581011yescharacteristic
664C2#2491581211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#391×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×#391D83
hypocenterC83#14702
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#19127608C2semidirect
5. #24905694C2semidirect
6. #24905693C2semidirect
7. #24905692C2semidirect
9. C2×#391×C83C2semidirect
14. C2×#14581768C4semidirect
54. C22×C4×D83Q8semidirect
61. C2×C4⋊Dic83C2×C4semidirect
64. C2×Dic83⋊C4C2×C4semidirect
87. C22×Q8×C83C2×C4semidirect
250. C2×C4×C83C22⋊Q8semidirect
275. C2×Dic83C22⋊Q8semidirect
311. C2×D83#391semidirect
336. C2×#391D83semidirect
342. C83#14702semidirect
376. C22×Q8C4×D83semidirect
609. C2×C4#14581768semidirect

Maximal subgroups

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

subgroupindex
C22×#145817682
C22×C42×D832
C2×#191276082
C2×#191275582
#249056942
#249056932
#249056922
(C2×D83)⋊#952
C2×#391×C832
#1470283