Finite Group Databasebeta

← #35410707

Subgroups — #35410707

Order 21248.

1004 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
1C2×#19127920C211yesmaximal, characteristic
2C2×#19127806C211yesmaximal, characteristic, complemented
3C2×#19127801C211yesmaximal, characteristic
4C42×Dic166C211yesmaximal, characteristic, complemented
5#24908396C211yesmaximal, characteristic
6#24908229C211yesmaximal, characteristic, complemented
7#24908101C211yesmaximal, characteristic, complemented
8#24908087C211yesmaximal, characteristic
9#24907821C211yesmaximal, characteristic
10#24907807C211yesmaximal, characteristic, complemented
11#24907475C211yesmaximal, characteristic, complemented
12#24907316C211yesmaximal, characteristic, complemented
13#24907293C211yesmaximal, characteristic, complemented
14#24907288C211yesmaximal, characteristic
15C4×#124×C83C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C22×#13050578C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20C2×C4×Dic166C2211yescharacteristic
21#19127920C418yescomplemented
22#19127806—24no—
23#19127801—24no—
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581715C2211yescharacteristic
33C2×#14581713C2211yescharacteristic
34C2×#14581713C2211yescharacteristic
35C2×#14581713C2211yescharacteristic
36C2×#14581703C2211yescharacteristic
37C2×#14581703C2211yescharacteristic
38C4×Dic83⋊C4—21no—
39C4×Dic83⋊C4—21no—
40C4×Dic83⋊C4—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42C4×C4⋊Dic83C2211yescharacteristic
43C4×C4⋊Dic83C2211yescharacteristic
44C42×Dic83—21no—
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127456C2211yescharacteristic
50#19127456C2211yescharacteristic
51C2×#124×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#391×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#388×C83—21no—
57#388×C83—21no—
58#382×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83—21no—
61C43×C83C2211yescharacteristic
62C22×Dic166C2311yescharacteristic
63C2×#13050580—22no—
64C2×#13050580—22no—
65C2×#13050580C2×C414yescomplemented
66C2×#13050578—22no—
67C2×#13050578—22no—
68C2×#13050578C2×C414yes—
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71C22×C83⋊C8C2311yescharacteristic
72C22×C83⋊C8C2311yescharacteristic
73C4×Dic166—22no—
74C4×Dic166—22no—
75C4×Dic166—22no—
76C4×Dic166—22no—
77C4×Dic166C2×C414yescomplemented
78C4×Dic166D414yes—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83—21no—
91#14581725—22no—
92#14581723—22no—
93C4⋊(C83⋊C8)—22no—
94C4⋊(C83⋊C8)C2×C414yescomplemented
95#14581716—22no—
96#14581716—22no—
97#14581716C2×C414yescomplemented
98#14581715—22no—
99#14581715—22no—
100#14581713—22no—
101#14581713—22no—
102#14581713C2×C414yes—
103#14581703—22no—
104#14581703—22no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C22×Q8×C83C2311yescharacteristic
109C22×D4×C83C2311yescharacteristic
110C23×C4×C83—21no—
111#124×C83—22no—
112#124×C83C2×C414yes—
113C4×Q8×C83—22no—
114C4×Q8×C83—22no—
115C4×D4×C83—22no—
116C4×D4×C83—22no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83—21no—
122C2×C22⋊C4×C83—21no—
123C2×C22⋊C4×C83—21no—
124C2×C22⋊C4×C83—21no—
125C2×C42×C83—21no—
126C2×C42×C83D411yescharacteristic
127C2×C42×C83D411yescharacteristic
128C2×C42×C83D411yescharacteristic
129C2×C42×C83D411yescharacteristic
130C2×C42×C83C2311yescharacteristic
131C2×C42×C83C2311yescharacteristic
132C2×C42×C83C2311yescharacteristic
133#95×C83—21no—
134#95×C83—21no—
135#95×C83—21no—
136#95×C83—21no—
137C2×Dic166—21no—
138C2×Dic166—21no—
139C2×Dic166—21no—
140C2×Dic166C22×C412yes—
141C2×Dic166C2×D412yes—
142C2×Dic166#4112yes—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C22×Dic83—21no—
147#13050580—28no—
148#13050578—28no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8C22×C412yes—
160C2×C83⋊C8C22×C412yes—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83C22×C412yes—
165C4⋊Dic83C2×D412yes—
166C4⋊Dic83#4112yes—
167Dic83⋊C4—24no—
168Dic83⋊C4—24no—
169Dic83⋊C4—24no—
170C4×Dic83—24no—
171C4×Dic83—24no—
172C4×Dic83—24no—
173C4×Dic83—42no—
174C4×Dic83—42no—
175C4×Dic83—42no—
176C24×C83—21no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83—21no—
181C2×Q8×C83—21no—
182C2×Q8×C83C22×C412yes—
183C2×D4×C83—21no—
184C2×D4×C83—21no—
185C2×D4×C83—21no—
186C2×D4×C83—21no—
187C2×D4×C83—21no—
188C2×D4×C83C22×C412yes—
189C22×C4×C83—21no—
190C22×C4×C83—21no—
191C22×C4×C83—21no—
192C22×C4×C83—42no—
193C22×C4×C83—42no—
194C22×C4×C83—42no—
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83C2×D411yescharacteristic
200C22×C4×C83C2×D411yescharacteristic
201C22×C4×C83C2411yescharacteristic
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—42no—
207C22⋊C4×C83—24no—
208C22⋊C4×C83—24no—
209C22⋊C4×C83—42no—
210C22⋊C4×C83—42no—
211C22⋊C4×C83—42no—
212C42×C83—22no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—41no—
216C42×C83—41no—
217C42×C83—41no—
218C42×C83—42no—
219C42×C83C22×C412yes—
220C42×C83C2×D412yes—
221C42×C83#4112yes—
222C42×C83#4112yes—
223C23×C83—22no—
224C23×C83—41no—
225C23×C83—41no—
226C23×C83—41no—
227C23×C83—41no—
228C23×C83—41no—
229C23×C83—41no—
230C23×C83C22×D411yescharacteristic
231Q8×C83—22no—
232Q8×C83—22no—
233Q8×C83—41no—
234Q8×C83—41no—
235D4×C83—22no—
236D4×C83—22no—
237D4×C83—41no—
238D4×C83—41no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—22no—
250C2×C4×C83—21no—
251C2×C4×C83—21no—
252C2×C4×C83—21no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—41no—
260C2×C4×C83—41no—
261C2×C4×C83—84no—
262C2×C4×C83C4×D412yes—
263C2×C4×C83C4×D412yes—
264C2×C4×C83#12112yes—
265C2×C4×C83#12112yes—
266C2×C4×C83#13712yes—
267C2×C4×C83C23×C411yescharacteristic
268C2×C4×C83C22×D411yescharacteristic
269C2×C4×C83C2×#4111yescharacteristic
270C2×C4×C83C2×#4111yescharacteristic
271C2×C4×C83C2×#4111yescharacteristic
272C2×C4×C83C2×#4111yescharacteristic
273C2×Dic83—22no—
274C2×Dic83—22no—
275C2×Dic83—22no—
276C2×Dic83—22no—
277C2×Dic83—22no—
278C2×Dic83—22no—
279C2×Dic83—41no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283C2×Dic83—41no—
284C2×Dic83—41no—
285Dic166—22no—
286Dic166—22no—
287Dic166—22no—
288Dic166C4×D414yes—
289C83⋊C8—22no—
290C83⋊C8—22no—
291C83⋊C8—41no—
292C83⋊C8—41no—
293C22×C83—42no—
294C22×C83—42no—
295C22×C83—42no—
296C22×C83—81no—
297C22×C83—81no—
298C22×C83—81no—
299C22×C83—81no—
300C22×C83C2×C4×D411yescharacteristic
301C22×C83C2×#12111yescharacteristic
302C22×C83#52511yescharacteristic
303C22×C83C2×#13711yescharacteristic
304C22×C83#57511yescharacteristic
305C22×C83#57511yescharacteristic
306C22×C83#57511yescharacteristic
307C4×C83—22no—
308C4×C83—22no—
309C4×C83—22no—
310C4×C83—22no—
311C4×C83—22no—
312C4×C83—41no—
313C4×C83—42no—
314C4×C83—81no—
315C4×C83—81no—
316C4×C83C2×C4×D411yescharacteristic
317C4×C83C4×#4111yescharacteristic
318C4×C83#52911yescharacteristic
319C4×C83#53211yescharacteristic
320Dic83—24no—
321Dic83—42no—
322Dic83—42no—
323Dic83—42no—
324#21508—831nomaximal, Sylow
325C2×C83—82no—
326C2×C83C4×#12111yescharacteristic
327C2×C83#294311yescharacteristic
328C2×C83C4×#13711yescharacteristic
329C2×C83#303711yescharacteristic
330C2×C83#304011yescharacteristic
331C2×C83#311111yescharacteristic
332C2×C83#311711yescharacteristic
333C2×#464—831no—
334C2×#441—831no—
335C2×#440—831no—
336C4×#124D8311yescharacteristic
337C42×Q8—831no—
338#1983—831no—
339#1949—831no—
340#1928—831no—
341#1926—831no—
342#1880—831no—
343#1878—831no—
344#1805—831no—
345#1767—831no—
346#1763—831no—
347#1762—831no—
348C83#2150811yescharacteristic, complemented, Sylow
349C22×Q16—831no—
350C22×SD16—831no—
351C2×#124C2×D8311yescharacteristic
352C2×C4×Q8—831no—
353C2×C4×Q8—831no—
354C2×C4×Q8—831no—
355C2×C4×Q8C2×D8311yescharacteristic
356C2×C4×D4C2×D8311yescharacteristic
357#464—838no—
358#441—1664no—
359#440—1664no—
360C2×#107—831no—
361C2×#106—831no—
362C2×C4⋊C8—831no—
363C2×C4⋊C8—831no—
364C2×#103—831no—
365C2×#103—831no—
366C2×#103—831no—
367C2×#103—831no—
368C2×#103—831no—
369C2×#102—831no—
370C2×#102—831no—
371C2×#102—831no—
372C2×#97—831no—
373C2×#97—831no—
374#391C2×D8311yescharacteristic
375#390C2×D8311yescharacteristic
376#388—21no—
377#388—21no—
378#382C2×D8311yescharacteristic
379C4×C4⋊C4—831no—
380C4×C4⋊C4—831no—
381C4×C4⋊C4—831no—
382C4×C4⋊C4—1661no—
383C4×C4⋊C4—1661no—
384C4×C4⋊C4—1661no—
385C4×C22⋊C4—21no—
386C4×C22⋊C4—21no—
387C43—1661no—
388C43C2×D8311yescharacteristic
389#340—831no—
390#340—831no—
391#340—831no—
392#340—831no—
393#336—831no—
394#336—831no—
395C22×Q8—831no—
396C22×Q8C22×D8311yescharacteristic
397C22×D4C22×D8311yescharacteristic
398C23×C4—21no—
399C2×Q16—834no—
400C2×Q16—1662no—
401C2×Q16—1662no—
402C2×SD16—834no—
403C2×SD16—1662no—
404C2×SD16—1662no—
405C22×C8—831no—
406C22×C8—831no—
407C22×C8—831no—
408C22×C8—831no—
409#124—22no—
410#124C4×D8314yes—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—834no—
414C4×Q8—834no—
415C4×Q8—1662no—
416C4×Q8—1662no—
417C4×Q8—1662no—
418C4×Q8—1662no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—831no—
423C2×C4⋊C4—831no—
424C2×C4⋊C4—831no—
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—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C42—21no—
436C2×C42—1661no—
437C2×C42—1661no—
438C2×C42—1661no—
439C2×C42—1661no—
440C2×C42—1661no—
441C2×C42—1661no—
442C2×C42C22⋊D8311yescharacteristic
443C2×C42C22⋊D8311yescharacteristic
444C2×C42C22⋊D8311yescharacteristic
445C2×C42C22⋊D8311yescharacteristic
446C2×C42C22×D8311yescharacteristic
447C2×C42C22×D8311yescharacteristic
448C2×C42C22×D8311yescharacteristic
449#107—1662no—
450#106—1662no—
451C4⋊C8—834no—
452C4⋊C8—1662no—
453#103—834no—
454#103—1662no—
455#103—1662no—
456#103—1662no—
457#103—1662no—
458#102—834no—
459#102—1662no—
460#102—1662no—
461#97—1662no—
462#97—1662no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1661no—
468#95—1661no—
469#95—1661no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—832no—
477C2×Q8—832no—
478C2×Q8—832no—
479C2×Q8—1661no—
480C2×Q8—1661no—
481C2×Q8—1661no—
482C2×Q8C2×C4×D8312yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D8312yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—1661no—
496C22×C4—1661no—
497C22×C4—1661no—
498C22×C4—1661no—
499C22×C4C2×C22⋊D8311yescharacteristic
500C22×C4C2×C22⋊D8311yescharacteristic
501C22×C4C2×C22⋊D8311yescharacteristic
502C22×C4C2×C22⋊D8311yescharacteristic
503C22×C4C2×C22⋊D8311yescharacteristic
504C22×C4C2×C22⋊D8311yescharacteristic
505C22×C4C23×D8311yescharacteristic
506Q16—1668no—
507SD16—1668no—
508C2×C8—832no—
509C2×C8—832no—
510C2×C8—1661no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8—1661no—
519C2×C8—1661no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—832no—
526C4⋊C4—832no—
527C4⋊C4—832no—
528C4⋊C4—1664no—
529C4⋊C4—1664no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1664no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—3322no—
547C42—3322no—
548C42—3322no—
549C42—1664no—
550C42—1664no—
551C42—1664no—
552C42C2×C4×D8312yes—
553C42D4×D8312yes—
554C42#1305058912yes—
555C42#1305059012yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C22×C22⋊D8311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—834no—
569Q8—1662no—
570Q8—1662no—
571Q8—1662no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4—3321no—
604C2×C4—3321no—
605C2×C4—1662no—
606C2×C4—1662no—
607C2×C4—1662no—
608C2×C4—1662no—
609C2×C4—1662no—
610C2×C4—1662no—
611C2×C4C4×C22⋊D8312yes—
612C2×C4C4×C22⋊D8312yes—
613C2×C4#1458176312yes—
614C2×C4#1458176312yes—
615C2×C4#1458182212yes—
616C2×C4C22×C4×D8311yescharacteristic
617C2×C4C2×D4×D8311yescharacteristic
618C2×C4#41×D8311yescharacteristic
619C2×C4#41×D8311yescharacteristic
620C2×C4C2×#1305058911yescharacteristic
621C2×C4C2×#1305059011yescharacteristic
622C8—3321no—
623C8—3321no—
624C8—1662no—
625C8—1662no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×C4×C22⋊D8311yescharacteristic
634C22C2×#1458176311yescharacteristic
635C22#1912820811yescharacteristic
636C22C2×#1458182211yescharacteristic
637C22#1912849911yescharacteristic
638C22#1912849911yescharacteristic
639C22#1912849911yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—3322no—
650C4—3322no—
651C4—3322no—
652C4—1664no—
653C4C4×D4×D8311yescharacteristic
654C4C4×#1305058911yescharacteristic
655C4#1912823311yescharacteristic
656C4#1912824811yescharacteristic
657C2—82no—
658C2C4×#1458176311yescharacteristic
659C2#2491626111yescharacteristic
660C2C4×#1458182211yescharacteristic
661C2#2491693111yescharacteristic
662C2#2491695311yescharacteristic
663C2#2491756611yescharacteristic
664C2#2491763011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×#124×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC4×#124D83
hypocenterC83#21508
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. C2×#19127806C2semidirect
4. C42×Dic166C2semidirect
6. #24908229C2semidirect
7. #24908101C2semidirect
10. #24907807C2semidirect
11. #24907475C2semidirect
12. #24907316C2semidirect
13. #24907293C2semidirect
21. #19127920C4semidirect
65. C2×#13050580C2×C4semidirect
77. C4×Dic166C2×C4semidirect
94. C4⋊(C83⋊C8)C2×C4semidirect
97. #14581716C2×C4semidirect
348. C83#21508semidirect

Maximal subgroups

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

subgroupindex
C2×#191279202
C2×#191278062
C2×#191278012
C42×Dic1662
#249083962
#249082292
#249081012
#249080872
#249078212
#249078072
#249074752
#249073162
#249072932
#249072882
C4×#124×C832
#2150883