Finite Group Databasebeta

← #35326205

Subgroups — #35326205

Order 21248.

2158 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581757C211yesmaximal, characteristic, complemented
2C2×#19127613C211yesmaximal, characteristic
3C2×C22⋊C4×Dic83C212yesmaximal
4C2×C22⋊C4×Dic83C211yesmaximal, characteristic
5#24905733C211yesmaximal, characteristic, complemented
6#24905722C211yesmaximal, characteristic, complemented
7#24905718C212yesmaximal, complemented
8#24905706C212yesmaximal
9#95×Dic83C211yesmaximal, characteristic, complemented
10#95×Dic83C212yesmaximal, complemented
11C83×#1439C211yesmaximal, characteristic
12C23×Dic166C2211yescharacteristic
13C24×Dic83C2211yescharacteristic
14C2×#14581757—28no—
15C2×#14581757—24no—
16C22×Dic83⋊C4C2212yes—
17C22×C22⋊Dic83C2211yescharacteristic
18C22×C22⋊Dic83C2212yes—
19C22×C4×Dic83C2212yes—
20C22×C4×Dic83C2211yescharacteristic
21C22×C4×Dic83C2211yescharacteristic
22C22×C4×Dic83C2212yes—
23#19127613C418yescomplemented
24C22⋊C4×Dic83—28no—
25C22⋊C4×Dic83—24no—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2211yescharacteristic
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2211yescharacteristic
35C22×C4⋊C4×C83C2211yescharacteristic
36C22×C22⋊C4×C83C2212yes—
37C22×C22⋊C4×C83C2211yescharacteristic
38C2×#95×C83C2211yescharacteristic
39C2×#95×C83C2212yes—
40C22×Dic166—24no—
41C22×Dic166—24no—
42C22×Dic166—22no—
43C22×Dic166—22no—
44C22×Dic166C2×C414yescomplemented
45C23×Dic83—22no—
46C23×Dic83—24no—
47C23×Dic83—24no—
48C23×Dic83—22no—
49C23×Dic83Q814yescomplemented
50C23×Dic83C2312yes—
51C23×Dic83C2311yescharacteristic
52C23×Dic83C2311yescharacteristic
53C23×Dic83C2311yescharacteristic
54C23×Dic83C2312yes—
55C23×Dic83C2311yescharacteristic
56#14581757—232no—
57C2×Dic83⋊C4—24no—
58C2×Dic83⋊C4—24no—
59C2×Dic83⋊C4—24no—
60C2×C22⋊Dic83—22no—
61C2×C22⋊Dic83—24no—
62C2×C22⋊Dic83—22no—
63C2×C22⋊Dic83—24no—
64C2×C22⋊Dic83—22no—
65C2×C22⋊Dic83C2×C418yes—
66C2×C4×Dic83—24no—
67C2×C4×Dic83—24no—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—22no—
70C2×C4×Dic83—24no—
71C2×C4×Dic83—24no—
72C2×C4×Dic83—24no—
73C2×C4×Dic83—24no—
74C2×C4×Dic83D418yes—
75C2×C4×Dic83D414yes—
76C83⋊#95—22no—
77C83⋊#95—24no—
78C83⋊#95—24no—
79C83⋊#95—24no—
80C83⋊#95—24no—
81C83⋊#95—22no—
82C83⋊#95—24no—
83C83⋊#95C2×C418yescomplemented
84C83⋊#95C2×C414yescomplemented
85C25×C83C2311yescharacteristic
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2312yes—
88C23×C4×C83C2312yes—
89C23×C4×C83C2311yescharacteristic
90C2×C4⋊C4×C83—24no—
91C2×C4⋊C4×C83—22no—
92C2×C22⋊C4×C83—24no—
93C2×C22⋊C4×C83—24no—
94C2×C22⋊C4×C83—22no—
95C2×C22⋊C4×C83—24no—
96C2×C22⋊C4×C83—22no—
97C2×C22⋊C4×C83C2×C414yes—
98#95×C83—24no—
99#95×C83—22no—
100C2×Dic166—216no—
101C2×Dic166—28no—
102C2×Dic166—28no—
103C2×Dic166—28no—
104C2×Dic166—216no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—21no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—28no—
128C22×Dic83—216no—
129C22×Dic83—44no—
130C22×Dic83—44no—
131C22×Dic83—48no—
132C22×Dic83C22×C414yes—
133C22×Dic83C22×C414yes—
134C22×Dic83C2×D412yes—
135C22×Dic83C2×D414yes—
136C22×Dic83C2×D414yes—
137C22×Dic83C2×D412yes—
138C22×Dic83C2×Q812yes—
139C22×Dic83C2×Q812yes—
140C22×Dic83#4112yes—
141C22×Dic83#4114yes—
142C22×Dic83#4114yes—
143C22×Dic83#4112yes—
144Dic83⋊C4—216no—
145C22⋊Dic83—28no—
146C22⋊Dic83—216no—
147C4×Dic83—216no—
148C4×Dic83—28no—
149C4×Dic83—44no—
150C4×Dic83—48no—
151C24×C83—21no—
152C24×C83—22no—
153C24×C83—22no—
154C24×C83—21no—
155C24×C83—21no—
156C24×C83—22no—
157C24×C83—22no—
158C24×C83—21no—
159C24×C83C22×C412yes—
160C24×C83C2×Q812yes—
161C24×C83#4112yes—
162C24×C83C2411yescharacteristic
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—21no—
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—21no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83—22no—
180C22×C4×C83—22no—
181C22×C4×C83C22×C412yes—
182C22×C4×C83C22×C412yes—
183C22×C4×C83C2×D414yes—
184C22×C4×C83C2×D412yes—
185C22×C4×C83#4114yes—
186C22×C4×C83#4112yes—
187C4⋊C4×C83—28no—
188C22⋊C4×C83—216no—
189C22⋊C4×C83—28no—
190C23×C83—22no—
191C23×C83—24no—
192C23×C83—24no—
193C23×C83—22no—
194C23×C83—22no—
195C23×C83—24no—
196C23×C83—24no—
197C23×C83—22no—
198C23×C83—24no—
199C23×C83—22no—
200C23×C83—22no—
201C23×C83—24no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—42no—
205C23×C83—41no—
206C23×C83—42no—
207C23×C83—42no—
208C23×C83—42no—
209C23×C83—42no—
210C23×C83—42no—
211C23×C83—41no—
212C23×C83C4×Q814yescomplemented
213C23×C83C23×C411yescharacteristic
214C23×C83C22×D412yes—
215C23×C83C22×D411yescharacteristic
216C23×C83C22×Q811yescharacteristic
217C23×C83C2×#4112yes—
218C23×C83C2×#4112yes—
219C23×C83C2×#4111yescharacteristic
220C23×C83C2×#4111yescharacteristic
221C23×C83C2×#4111yescharacteristic
222C23×C83C2×#4111yescharacteristic
223C23×C83C2×#4112yes—
224C2×C4×C83—24no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—24no—
228C2×C4×C83—22no—
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—24no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—42no—
239C2×C4×C83—42no—
240C2×C4×C83—41no—
241C2×C4×C83—42no—
242C2×C4×C83—42no—
243C2×C4×C83—42no—
244C2×C4×C83—42no—
245C2×C4×C83—42no—
246C2×C4×C83—41no—
247C2×C4×C83C4×D414yes—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—24no—
260C2×Dic83—22no—
261C2×Dic83—22no—
262C2×Dic83—24no—
263C2×Dic83—22no—
264C2×Dic83—41no—
265C2×Dic83—42no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—42no—
269C2×Dic83—41no—
270C2×Dic83—42no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—42no—
275C2×Dic83—42no—
276C2×Dic83—416no—
277C2×Dic83C4×D418yes—
278C2×Dic83#12014yes—
279C2×Dic83C22⋊Q814yes—
280C2×Dic83C22⋊Q818yes—
281C2×Dic83#12414yes—
282C2×Dic83#12418yes—
283Dic166—232no—
284C22×C83—24no—
285C22×C83—28no—
286C22×C83—42no—
287C22×C83—42no—
288C22×C83—42no—
289C22×C83—44no—
290C22×C83—42no—
291C22×C83—44no—
292C22×C83—42no—
293C22×C83—44no—
294C22×C83—42no—
295C22×C83C2×C4×D412yes—
296C22×C83C2×C4×D411yescharacteristic
297C22×C83C2×C4×Q811yescharacteristic
298C22×C83C4×#4112yes—
299C22×C83C4×#4111yescharacteristic
300C22×C83C2×#12011yescharacteristic
301C22×C83C2×C22⋊Q811yescharacteristic
302C22×C83C2×C22⋊Q812yes—
303C22×C83#52512yes—
304C22×C83#52511yescharacteristic
305C22×C83#52511yescharacteristic
306C22×C83#52512yes—
307C22×C83C2×#12411yescharacteristic
308C22×C83C2×#12412yes—
309C22×C83#52912yes—
310C22×C83#52912yes—
311C22×C83#52912yes—
312C22×C83#52911yescharacteristic
313C22×C83#52912yes—
314C22×C83#53212yes—
315C22×C83#53211yescharacteristic
316C22×C83#53312yes—
317C22×C83#53311yescharacteristic
318C4×C83—24no—
319C4×C83—44no—
320C4×C83—44no—
321C4×C83—42no—
322Dic83—28no—
323Dic83—42no—
324Dic83—42no—
325Dic83—42no—
326Dic83—44no—
327Dic83#39318yescomplemented
328C4×#393—831nomaximal, Sylow
329C2×C83—44no—
330C2×C83C4×#12011yescharacteristic
331C2×C83C4×C22⋊Q811yescharacteristic
332C2×C83C4×C22⋊Q812yes—
333C2×C83C4×#12412yes—
334C2×C83C4×#12411yescharacteristic
335C2×C83C2×#39311yescharacteristic
336C2×C83#238811yescharacteristic
337C2×C83#239611yescharacteristic
338C2×C83#239612yes—
339C2×C83#239811yescharacteristic
340C2×C83#239812yes—
341C22×C4×Q8—831no—
342C2×#393—831no—
343C2×C4×C22⋊C4—832no—
344C2×C4×C22⋊C4—831no—
345#1446—831no—
346#1442—832no—
347#1442—831no—
348#1439—832no—
349#1439D8311yescharacteristic
350C4×#95—831no—
351C4×#95—832no—
352C83C4×#39311yescharacteristic, complemented, Sylow
353C23×Q8—831no—
354C24×C4—831no—
355C2×C4×Q8—1664no—
356C2×C4×Q8—1668no—
357C22×C4⋊C4—832no—
358C22×C4⋊C4C2×D8311yescharacteristic
359C22×C22⋊C4—832no—
360C22×C22⋊C4—831no—
361C22×C22⋊C4C2×D8312yes—
362C22×C22⋊C4C2×D8311yescharacteristic
363C22×C42—832no—
364C22×C42—831no—
365C22×C42—831no—
366C22×C42—832no—
367#393—838no—
368C4×C22⋊C4—1668no—
369C4×C22⋊C4—1664no—
370C2×#95—832no—
371C2×#95—831no—
372C2×#95—832no—
373C2×#95—832no—
374C2×#95—832no—
375C2×#95—831no—
376C2×#95—832no—
377C2×#95—832no—
378C2×#95—831no—
379C2×#95C2×D8312yes—
380C2×#95C2×D8311yescharacteristic
381C25C22×D8311yescharacteristic
382C22×Q8—834no—
383C22×Q8—1664no—
384C22×Q8—1664no—
385C22×Q8—1662no—
386C22×Q8—1662no—
387C23×C4—831no—
388C23×C4—832no—
389C23×C4—834no—
390C23×C4—831no—
391C23×C4—832no—
392C23×C4—831no—
393C23×C4—831no—
394C23×C4—1664no—
395C23×C4—1662no—
396C23×C4—1664no—
397C23×C4—1662no—
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8312yes—
400C23×C4C22×D8312yes—
401C23×C4C22×D8311yescharacteristic
402C4×Q8—16632no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—1664no—
406C2×C4⋊C4—1664no—
407C2×C4⋊C4—1664no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—838no—
414C2×C22⋊C4—1664no—
415C2×C22⋊C4—1662no—
416C2×C22⋊C4—1664no—
417C2×C22⋊C4—1662no—
418C2×C22⋊C4—1662no—
419C2×C22⋊C4C4×D8314yes—
420C2×C42—838no—
421C2×C42—834no—
422C2×C42—1662no—
423C2×C42—1664no—
424C2×C42—1664no—
425C2×C42—1664no—
426C2×C42—1664no—
427C2×C42—1664no—
428C2×C42—1664no—
429C2×C42—1664no—
430#95—24no—
431#95—22no—
432#95—838no—
433#95—834no—
434#95—1664no—
435#95—1662no—
436#95—1664no—
437#95—1664no—
438#95—1664no—
439#95—1662no—
440#95—1664no—
441C24—22no—
442C24—21no—
443C24—22no—
444C24—21no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24—22no—
449C24C2×C4×D8312yes—
450C24Q8×D8312yes—
451C24#1305058912yes—
452C24C23×D8311yescharacteristic
453C2×Q8—16616no—
454C2×Q8—1668no—
455C2×Q8—1668no—
456C2×Q8—16616no—
457C2×Q8—1668no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—3324no—
477C22×C4—3324no—
478C22×C4—3328no—
479C22×C4—834no—
480C22×C4—832no—
481C22×C4—832no—
482C22×C4—834no—
483C22×C4—832no—
484C22×C4—834no—
485C22×C4—834no—
486C22×C4—834no—
487C22×C4—834no—
488C22×C4—832no—
489C22×C4—832no—
490C22×C4—832no—
491C22×C4—1668no—
492C22×C4—16616no—
493C22×C4—1662no—
494C22×C4—1661no—
495C22×C4—1661no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1661no—
505C22×C4—1662no—
506C22×C4—1661no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4—1661no—
510C22×C4—1661no—
511C22×C4—1662no—
512C22×C4—1662no—
513C22×C4—1662no—
514C22×C4—1662no—
515C22×C4C2×C4×D8312yes—
516C22×C4C2×C4×D8312yes—
517C22×C4D4×D8314yes—
518C22×C4D4×D8312yes—
519C22×C4#1305058914yes—
520C22×C4#1305058912yes—
521C4⋊C4—28no—
522C4⋊C4—16616no—
523C22⋊C4—216no—
524C22⋊C4—28no—
525C22⋊C4—16616no—
526C22⋊C4—1668no—
527C42—3328no—
528C42—3324no—
529C42—1668no—
530C42—16616no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23#1458175714yescomplemented
554C23C22×C4×D8311yescharacteristic
555C23C2×D4×D8312yes—
556C23C2×D4×D8311yescharacteristic
557C23C2×Q8×D8311yescharacteristic
558C23#41×D8312yes—
559C23#41×D8312yes—
560C23#41×D8311yescharacteristic
561C23#41×D8311yescharacteristic
562C23C2×#1305058911yescharacteristic
563C23C2×#1305058911yescharacteristic
564C23C2×#1305058912yes—
565Q8—16632no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—3321no—
590C2×C4—3322no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3322no—
594C2×C4—3321no—
595C2×C4—3322no—
596C2×C4—3322no—
597C2×C4—3322no—
598C2×C4—3322no—
599C2×C4—3322no—
600C2×C4—3322no—
601C2×C4—33216no—
602C2×C4—834no—
603C2×C4—834no—
604C2×C4—838no—
605C2×C4—834no—
606C2×C4—838no—
607C2×C4—838no—
608C2×C4—1664no—
609C2×C4—1664no—
610C2×C4—1664no—
611C2×C4—1662no—
612C2×C4—1662no—
613C2×C4—1664no—
614C2×C4—1662no—
615C2×C4—1662no—
616C2×C4—1664no—
617C2×C4—1662no—
618C2×C4—1664no—
619C2×C4—1664no—
620C2×C4—1664no—
621C2×C4—1664no—
622C2×C4—1664no—
623C2×C4—1664no—
624C2×C4(C22⋊D83)⋊C414yes—
625C22—24no—
626C22—28no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22C4×D4×D8312yes—
637C22C2×(C22⋊D83)⋊C411yescharacteristic
638C22C2×#1458175711yescharacteristic
639C22C4×#1305058912yes—
640C22#1912816911yescharacteristic
641C22#120×D8311yescharacteristic
642C22C22⋊Q8×D8311yescharacteristic
643C22C22⋊Q8×D8312yes—
644C22#1912820712yes—
645C22#1912820911yescharacteristic
646C22#1912821111yescharacteristic
647C22#1912821112yes—
648C22#124×D8311yescharacteristic
649C22#124×D8312yes—
650C22#1912822812yes—
651C22#1912822812yes—
652C22#1912823212yes—
653C22#1912823211yescharacteristic
654C22#1912823412yes—
655C22#1912824612yes—
656C22#1912824911yescharacteristic
657C22#1912825312yes—
658C22#1912825311yescharacteristic
659C4—24no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—3322no—
664C4—3322no—
665C4—3322no—
666C4—3324no—
667C4—838no—
668C4—1668no—
669C2—44no—
670C2#2490996811yescharacteristic
671C2#2490999111yescharacteristic
672C2#2490999512yes—
673C2#2491000412yes—
674C2#2491001011yescharacteristic
675C2#393×D8311yescharacteristic
676C2#2491067511yescharacteristic
677C2#2491073211yescharacteristic
678C2#2491073812yes—
679C2#2491074811yescharacteristic
680C2#2491075112yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1439C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenter#1439D83
hypocenterC83C4×#393
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#14581757C2semidirect
5. #24905733C2semidirect
6. #24905722C2semidirect
7. #24905718C2semidirect
9. #95×Dic83C2semidirect
10. #95×Dic83C2semidirect
23. #19127613C4semidirect
44. C22×Dic166C2×C4semidirect
49. C23×Dic83Q8semidirect
83. C83⋊#95C2×C4semidirect
84. C83⋊#95C2×C4semidirect
212. C23×C83C4×Q8semidirect
327. Dic83#393semidirect
352. C83C4×#393semidirect
553. C23#14581757semidirect

Maximal subgroups

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

subgroupindex
C22×#145817572
C2×#191276132
C2×C22⋊C4×Dic832
#249057332
#249057222
#249057182
#249057062
#95×Dic832
C83×#14392
C4×#39383