Finite Group Databasebeta

← #35329699

Subgroups — #35329699

Order 21248.

2062 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
1C2×#19127612C211yesmaximal, characteristic
2C2×#19127572C212yesmaximal
3C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
4#24905734C211yesmaximal, characteristic, complemented
5#24905726C211yesmaximal, characteristic, complemented
6#24905705C211yesmaximal, characteristic
7#24905701C212yesmaximal
8#24905696C212yesmaximal, complemented
9#24905695C212yesmaximal, complemented
10#24905690C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C83C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×Dic83⋊C4C2212yes—
14C22×(C2×D83)⋊C4C2212yes—
15C22×(C2×D83)⋊C4C2212yes—
16C22×(C2×D83)⋊C4C2211yescharacteristic
17C22×(C2×D83)⋊C4C2211yescharacteristic
18C22×C4×Dic83C2211yescharacteristic
19#19127612C418yescomplemented
20#19127572—28no—
21C4×(C2×D83)⋊C4—24no—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2211yescharacteristic
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
33C23×Q8×C83C2211yescharacteristic
34C2×C4×Q8×C83—28no—
35C2×C4×Q8×C83—24no—
36C22×C4⋊C4×C83C2211yescharacteristic
37C22×C4⋊C4×C83C2212yes—
38C22×C42×C83C2212yes—
39C22×C42×C83C2211yescharacteristic
40C24×D83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46C22×C4×D83—22no—
47C22×C4×D83—22no—
48C22×C4×D83—22no—
49C22×C4×D83—24no—
50C22×C4×D83—42no—
51C2×Dic83⋊C4—24no—
52C2×Dic83⋊C4—24no—
53C2×Dic83⋊C4—42no—
54C2×(C2×D83)⋊C4—24no—
55C2×(C2×D83)⋊C4—24no—
56C2×(C2×D83)⋊C4—22no—
57C2×(C2×D83)⋊C4—22no—
58C2×(C2×D83)⋊C4—24no—
59C2×(C2×D83)⋊C4—24no—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—22no—
62C2×(C2×D83)⋊C4—42no—
63C2×(C2×D83)⋊C4C2×C418yes—
64C2×(C2×D83)⋊C4C2×C414yes—
65C2×(C2×D83)⋊C4Q814yescomplemented
66C2×C4×Dic83—22no—
67C2×C4×Dic83—42no—
68C83⋊#95—24no—
69C83⋊#95—24no—
70C83⋊#95—22no—
71C83⋊#95—22no—
72C83⋊#95—24no—
73C83⋊#95—22no—
74C83⋊#95—24no—
75C83⋊#95—42no—
76C83⋊#95—42no—
77C83⋊#95C2×C414yescomplemented
78C83⋊#95C2×C418yescomplemented
79C22×Q8×C83—24no—
80C22×Q8×C83—22no—
81C22×Q8×C83—22no—
82C22×Q8×C83—24no—
83C22×Q8×C83C2×C414yescomplemented
84C23×C4×C83C2312yes—
85C23×C4×C83C2312yes—
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C4×Q8×C83—232no—
90C2×C4⋊C4×C83—24no—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—24no—
93C2×C4⋊C4×C83—24no—
94C2×C4⋊C4×C83D418yes—
95C2×C42×C83—24no—
96C2×C42×C83—24no—
97C2×C42×C83—24no—
98C2×C42×C83—24no—
99C2×C42×C83D414yes—
100C23×D83—21no—
101C23×D83—22no—
102C23×D83—21no—
103C23×D83—21no—
104C23×D83—22no—
105C23×D83—21no—
106C23×D83—22no—
107C23×D83—22no—
108C23×D83C22×C412yes—
109C23×D83C2×Q812yes—
110C23×D83#4112yes—
111C22×Dic83—21no—
112C22×Dic83—22no—
113C22×Dic83—21no—
114C22×Dic83—21no—
115C22×Dic83—22no—
116C22×Dic83—21no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—21no—
120C22×Dic83—22no—
121C22×Dic83—21no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—22no—
133C22×Dic83—22no—
134C22×Dic83—22no—
135C22×Dic83—22no—
136C22×Dic83C22×C414yes—
137C22×Dic83C22×C412yes—
138C22×Dic83C2×Q812yes—
139C22×Dic83#4112yes—
140C2×C4×D83—28no—
141C2×C4×D83—48no—
142C2×C4×D83—48no—
143C2×C4×D83—44no—
144C2×C4×D83—44no—
145Dic83⋊C4—48no—
146(C2×D83)⋊C4—216no—
147(C2×D83)⋊C4—28no—
148(C2×D83)⋊C4—28no—
149(C2×D83)⋊C4—48no—
150C4×Dic83—44no—
151C24×C83C2411yescharacteristic
152C2×Q8×C83—28no—
153C2×Q8×C83—28no—
154C2×Q8×C83—216no—
155C2×Q8×C83—28no—
156C2×Q8×C83—216no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
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×C83C22×C414yes—
175C22×C4×C83C22×C412yes—
176C22×C4×C83C2×D414yes—
177C22×C4×C83C2×D414yes—
178C22×C4×C83C2×D412yes—
179C22×C4×C83C2×D414yes—
180C22×C4×C83C2×D412yes—
181C22×C4×C83C2×D412yes—
182C22×C4×C83C2×Q812yes—
183C22×C4×C83#4112yes—
184C22×C4×C83#4114yes—
185C22×C4×C83#4112yes—
186C4⋊C4×C83—28no—
187C4⋊C4×C83—216no—
188C42×C83—216no—
189C42×C83—28no—
190C23×C83C23×C411yescharacteristic
191C23×C83C22×D411yescharacteristic
192C23×C83C22×D412yes—
193C23×C83C22×Q811yescharacteristic
194C23×C83C2×#4111yescharacteristic
195C23×C83C2×#4111yescharacteristic
196C23×C83C2×#4111yescharacteristic
197C23×C83C2×#4112yes—
198C23×C83C2×#4111yescharacteristic
199C23×C83H(5,𝔽2)12yes—
200C23×C832−1+412yes—
201C22×D83—24no—
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—24no—
205C22×D83—22no—
206C22×D83—22no—
207C22×D83—22no—
208C22×D83—22no—
209C22×D83—24no—
210C22×D83—24no—
211C22×D83—24no—
212C22×D83—24no—
213C22×D83—41no—
214C22×D83—41no—
215C22×D83—42no—
216C22×D83—41no—
217C22×D83—41no—
218C22×D83—42no—
219C22×D83—42no—
220C22×D83—42no—
221C22×D83—42no—
222C22×D83—42no—
223C22×D83C4×Q814yescomplemented
224Q8×C83—232no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—22no—
229C2×C4×C83—24no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—24no—
233C2×C4×C83—22no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83—42no—
250C2×C4×C83C4×D414yes—
251C2×C4×C83C4×D418yes—
252C2×C4×C83#12014yes—
253C2×C4×C83C22⋊Q814yes—
254C2×Dic83—22no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—24no—
262C2×Dic83—24no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—42no—
270C2×Dic83—42no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—42no—
275C2×Dic83—42no—
276C2×Dic83—42no—
277C2×Dic83—42no—
278C2×Dic83—42no—
279C2×Dic83—42no—
280C2×Dic83—42no—
281C2×Dic83—42no—
282C2×Dic83—42no—
283C4×D83—416no—
284C22×C83C2×C4×D411yescharacteristic
285C22×C83C2×C4×D412yes—
286C22×C83C2×C4×Q811yescharacteristic
287C22×C83C4×#4111yescharacteristic
288C22×C83#52012yes—
289C22×C83C2×#12011yescharacteristic
290C22×C83C2×C22⋊Q811yescharacteristic
291C22×C83#52511yescharacteristic
292C22×C83#52511yescharacteristic
293C22×C83#52512yes—
294C22×C83#52911yescharacteristic
295C22×C83#53311yescharacteristic
296C22×C83#53511yescharacteristic
297C22×C83#53611yescharacteristic
298C22×C83#54212yes—
299C22×C83#54612yes—
300C22×C83#54612yes—
301C22×C83#54812yes—
302C22×C83D4×Q812yes—
303C22×C83#55212yes—
304C22×C83#55412yes—
305C22×C83#55612yes—
306C22×C83#55912yes—
307C2×D83—28no—
308C2×D83—24no—
309C2×D83—42no—
310C2×D83—42no—
311C2×D83—42no—
312C2×D83—44no—
313C2×D83—42no—
314C2×D83—42no—
315C2×D83—44no—
316C2×D83—44no—
317C2×D83—42no—
318C4×C83—24no—
319C4×C83—24no—
320C4×C83—28no—
321C4×C83—24no—
322C4×C83—44no—
323Dic83—42no—
324Dic83—42no—
325Dic83—44no—
326Dic83—42no—
327Dic83—82no—
328#14172—831nomaximal, Sylow
329C2×C83C4×#12011yescharacteristic
330C2×C83C4×C22⋊Q811yescharacteristic
331C2×C83#232611yescharacteristic
332C2×C83#236012yes—
333C2×C83#236712yes—
334C2×C83#240411yescharacteristic
335C2×C83#240911yescharacteristic
336C2×C83#245812yes—
337C2×C83#246512yes—
338C2×C83#258011yescharacteristic
339C2×C83#259911yescharacteristic
340D83—44no—
341C22×C4×Q8D8311yescharacteristic, complemented
342C2×#393—831no—
343C2×#385—832no—
344C2×C4×C22⋊C4—831no—
345#1446—831no—
346#1444—831no—
347#1439—831no—
348#1437—832no—
349#1436—832no—
350#1436—832no—
351#1434—831no—
352C83#1417211yescharacteristic, complemented, Sylow
353C23×Q8C2×D8311yescharacteristic
354C24×C4—831no—
355C2×C4×Q8—28no—
356C2×C4×Q8—24no—
357C22×C4⋊C4—832no—
358C22×C4⋊C4C2×D8311yescharacteristic
359C22×C4⋊C4C2×D8312yes—
360C22×C22⋊C4—832no—
361C22×C22⋊C4—832no—
362C22×C22⋊C4—831no—
363C22×C22⋊C4—831no—
364C22×C42—831no—
365C22×C42C2×D8312yes—
366C22×C42C2×D8311yescharacteristic
367#393—838no—
368#385—1668no—
369C4×C22⋊C4—1664no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95—831no—
373C2×#95—832no—
374C2×#95—832no—
375C2×#95—832no—
376C2×#95—831no—
377C2×#95—831no—
378C2×#95—832no—
379C2×#95—832no—
380C2×#95—831no—
381C25—831no—
382C22×Q8—24no—
383C22×Q8—22no—
384C22×Q8—22no—
385C22×Q8—24no—
386C22×Q8C4×D8314yescomplemented
387C23×C4—3322no—
388C23×C4—832no—
389C23×C4—831no—
390C23×C4—831no—
391C23×C4—831no—
392C23×C4—832no—
393C23×C4—1662no—
394C23×C4—1662no—
395C23×C4—1664no—
396C23×C4—1662no—
397C23×C4C22×D8312yes—
398C23×C4C22×D8312yes—
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401C23×C4C22×D8311yescharacteristic
402C4×Q8—232no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—24no—
407C2×C4⋊C4—3322no—
408C2×C4⋊C4—1664no—
409C2×C4⋊C4—1664no—
410C2×C4⋊C4C22⋊D8318yes—
411C2×C22⋊C4—3322no—
412C2×C22⋊C4—838no—
413C2×C22⋊C4—834no—
414C2×C22⋊C4—834no—
415C2×C22⋊C4—1664no—
416C2×C22⋊C4—1664no—
417C2×C22⋊C4—1662no—
418C2×C22⋊C4—1662no—
419C2×C22⋊C4—1664no—
420C2×C22⋊C4—1664no—
421C2×C22⋊C4—1662no—
422C2×C22⋊C4—1662no—
423C2×C42—24no—
424C2×C42—24no—
425C2×C42—24no—
426C2×C42—24no—
427C2×C42—3322no—
428C2×C42—1662no—
429C2×C42C22⋊D8314yes—
430#95—3322no—
431#95—3322no—
432#95—834no—
433#95—838no—
434#95—1662no—
435#95—1662no—
436#95—1664no—
437#95—1664no—
438#95—1662no—
439#95—1664no—
440#95—1664no—
441C24—832no—
442C24—832no—
443C24—832no—
444C24—1662no—
445C24—1661no—
446C24—1661no—
447C24—1661no—
448C24—1662no—
449C24—1661no—
450C24—1662no—
451C24—1662no—
452C24C23×D8311yescharacteristic
453C2×Q8—28no—
454C2×Q8—216no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
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—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—3324no—
476C22×C4—3328no—
477C22×C4—3328no—
478C22×C4—3324no—
479C22×C4—834no—
480C22×C4—832no—
481C22×C4—832no—
482C22×C4—832no—
483C22×C4—1668no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1661no—
492C22×C4—1661no—
493C22×C4—1661no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1661no—
498C22×C4—1662no—
499C22×C4—1661no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1661no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4C2×C4×D8314yes—
510C22×C4C2×C4×D8312yes—
511C22×C4C2×C22⋊D8314yes—
512C22×C4C2×C22⋊D8314yes—
513C22×C4C2×C22⋊D8312yes—
514C22×C4C2×C22⋊D8312yes—
515C22×C4C2×C22⋊D8314yes—
516C22×C4C2×C22⋊D8312yes—
517C22×C4Q8×D8312yes—
518C22×C4#1305059012yes—
519C22×C4#1305059014yes—
520C22×C4#1305059112yes—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—3328no—
524C22⋊C4—3328no—
525C22⋊C4—1668no—
526C22⋊C4—1668no—
527C22⋊C4—16616no—
528C42—216no—
529C42—28no—
530C42—3324no—
531C23—3322no—
532C23—3322no—
533C23—3322no—
534C23—3322no—
535C23—3322no—
536C23—3321no—
537C23—3322no—
538C23—3321no—
539C23—3321no—
540C23—3321no—
541C23—834no—
542C23—1664no—
543C23—1664no—
544C23—1664no—
545C23—1664no—
546C23—1662no—
547C23—1664no—
548C23—1662no—
549C23—1664no—
550C23—1662no—
551C23—1662no—
552C23—1662no—
553C23—1662no—
554C23C22×C4×D8311yescharacteristic
555C23C22×C22⋊D8312yes—
556C23C22×C22⋊D8311yescharacteristic
557C23C2×Q8×D8311yescharacteristic
558C23#41×D8311yescharacteristic
559C23#41×D8311yescharacteristic
560C23C2×#1305059011yescharacteristic
561C23C2×#1305059012yes—
562C23C2×#1305059111yescharacteristic
563C23#1458183812yes—
564C23#1458184012yes—
565Q8—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—33216no—
592C2×C4—3322no—
593C2×C4—3322no—
594C2×C4—3322no—
595C2×C4—3322no—
596C2×C4—3322no—
597C2×C4—3322no—
598C2×C4—3322no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4—3321no—
604C2×C4—3321no—
605C2×C4—3322no—
606C2×C4—3322no—
607C2×C4—3322no—
608C2×C4—3322no—
609C2×C4—3322no—
610C2×C4—3322no—
611C2×C4—3322no—
612C2×C4—1664no—
613C2×C4—1664no—
614C2×C4—1662no—
615C2×C4—1664no—
616C2×C4—1664no—
617C2×C4—1664no—
618C2×C4—1664no—
619C2×C4—1662no—
620C2×C4—1662no—
621C2×C4C4×C22⋊D8314yes—
622C2×C4C4×C22⋊D8318yes—
623C2×C4#1458176114yes—
624C2×C4#1458176814yes—
625C22—3324no—
626C22—3322no—
627C22—3324no—
628C22—3322no—
629C22—3322no—
630C22—3322no—
631C22—3322no—
632C22—3324no—
633C22—3322no—
634C22—1668no—
635C22—1664no—
636C22C2×C4×C22⋊D8311yescharacteristic
637C22C2×C4×C22⋊D8312yes—
638C22C4×Q8×D8311yescharacteristic
639C22C4×#1305059111yescharacteristic
640C22#1912818312yes—
641C22C2×#1458176111yescharacteristic
642C22C2×#1458176811yescharacteristic
643C22#1912820811yescharacteristic
644C22#1912821212yes—
645C22#1912821211yescharacteristic
646C22#1912823811yescharacteristic
647C22#1912825511yescharacteristic
648C22#1912826511yescharacteristic
649C22#1912827511yescharacteristic
650C22#1912832312yes—
651C22#1912833912yes—
652C22#1912833912yes—
653C22#1912836212yes—
654C22Q8×C22⋊D8312yes—
655C22#1912839212yes—
656C22#1912841112yes—
657C22#1912842312yes—
658C22#1912843612yes—
659C4—28no—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—3322no—
665C4—3322no—
666C4—3322no—
667C4—3324no—
668C4—6642no—
669C2—3324no—
670C2C4×#1458176111yescharacteristic
671C2C4×#1458176811yescharacteristic
672C2#2491017011yescharacteristic
673C2#2491046412yes—
674C2#2491053012yes—
675C2#2491077811yescharacteristic
676C2#2491082211yescharacteristic
677C2#2491137912yes—
678C2#2491147812yes—
679C2#2491282111yescharacteristic
680C2#2491303411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×C4×Q8×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC22×C4×Q8D83
hypocenterC83#14172
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #24905734C2semidirect
5. #24905726C2semidirect
8. #24905696C2semidirect
9. #24905695C2semidirect
10. #24905690C2semidirect
11. C22×C4×Q8×C83C2semidirect
19. #19127612C4semidirect
65. C2×(C2×D83)⋊C4Q8semidirect
77. C83⋊#95C2×C4semidirect
78. C83⋊#95C2×C4semidirect
83. C22×Q8×C83C2×C4semidirect
223. C22×D83C4×Q8semidirect
341. C22×C4×Q8D83semidirect
352. C83#14172semidirect
386. C22×Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276122
C2×#191275722
C2×C4×(C2×D83)⋊C42
#249057342
#249057262
#249057052
#249057012
#249056962
#249056952
#249056902
C22×C4×Q8×C832
#1417283