Finite Group Databasebeta

← #35463261

Subgroups — #35463261

Order 21248.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128110C211yesmaximal, characteristic, complemented
2C2×#19127743C211yesmaximal, characteristic
3#24911176C211yesmaximal, characteristic
4#24911165C211yesmaximal, characteristic
5#24908941C211yesmaximal, characteristic, complemented
6#24908936C211yesmaximal, characteristic, complemented
7#24908802C211yesmaximal, characteristic, complemented
8#24908797C211yesmaximal, characteristic, complemented
9#24908201C211yesmaximal, characteristic, complemented
10#24908196C211yesmaximal, characteristic, complemented
11#24907664C211yesmaximal, characteristic
12#24907658C211yesmaximal, characteristic
13#24907620C211yesmaximal, characteristic, complemented
14#24907496C211yesmaximal, characteristic
15C83×#1753C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581769—21no—
19C2×#14581751C2211yescharacteristic
20C22×C4⋊Dic83C2211yescharacteristic
21#19128110—24no—
22#19127743—24no—
23C2×#14581726C2211yescharacteristic
24C2×#14581726C2211yescharacteristic
25C2×#14581726C2211yescharacteristic
26C2×#14581726C2211yescharacteristic
27C2×#14581724C2211yescharacteristic
28C2×#14581724C2211yescharacteristic
29C2×#14581724C2211yescharacteristic
30C2×#14581724C2211yescharacteristic
31C2×#14581724C2211yescharacteristic
32C2×#14581724C2211yescharacteristic
33#19127637—21no—
34#19127633—21no—
35#19127624—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127599C2211yescharacteristic
41#19127598—21no—
42#19127584—21no—
43#19127576—21no—
44#19127571C2211yescharacteristic
45#19127571C2211yescharacteristic
46#19127561C2211yescharacteristic
47#19127561C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53#19127464C2211yescharacteristic
54C22×M16×C83C2211yescharacteristic
55C2×C22⋊C8×C83C2211yescharacteristic
56C2×#97×C83C2211yescharacteristic
57C2×C4×C8×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60#336×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63#14581792—22no—
64#14581792—41no—
65#14581784—22no—
66#14581784—41no—
67#14581769—44no—
68#14581751—22no—
69#14581751—41no—
70C2×C4⋊Dic83—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—41no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×C4⋊Dic83C2311yescharacteristic
84C2×C4⋊Dic83C2311yescharacteristic
85C2×C4⋊Dic83C2311yescharacteristic
86C2×Dic83⋊C4—21no—
87C2×Dic83⋊C4—21no—
88C2×Dic83⋊C4—21no—
89C2×Dic83⋊C4—21no—
90C2×Dic83⋊C4—21no—
91C2×Dic83⋊C4—21no—
92C2×C22⋊Dic83—21no—
93C2×C22⋊Dic83—21no—
94C2×C22⋊Dic83—21no—
95C2×C4×Dic83—21no—
96#14581726—22no—
97#14581726—22no—
98#14581726—22no—
99#14581726D414yes—
100#14581724—22no—
101#14581724—22no—
102#14581724—22no—
103#14581724—22no—
104#14581724—41no—
105#14581724—41no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95—21no—
112C23×C4×C83C2311yescharacteristic
113C2×M16×C83—22no—
114C2×M16×C83—22no—
115C2×M16×C83Q814yescomplemented
116C22×C8×C83C2311yescharacteristic
117C22×C8×C83C2311yescharacteristic
118C22×C8×C83C2311yescharacteristic
119C22×C8×C83C2311yescharacteristic
120C2×C22⋊C4×C83D411yescharacteristic
121C2×C22⋊C4×C83D411yescharacteristic
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124C22⋊C8×C83Q814yescomplemented
125#97×C83—22no—
126C4×C8×C83—22no—
127#95×C83D411yescharacteristic
128#95×C83D411yescharacteristic
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—81no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—42no—
141C22×Dic83—42no—
142C22×Dic83—81no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83—42no—
159C4⋊Dic83—42no—
160C4⋊Dic83—42no—
161C4⋊Dic83—81no—
162C4⋊Dic83—81no—
163C4⋊Dic83C2×D412yes—
164C4⋊Dic83C2×D412yes—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169Dic83⋊C4—42no—
170Dic83⋊C4—81no—
171C22⋊Dic83—42no—
172C22⋊Dic83—42no—
173C22⋊Dic83—81no—
174C4×Dic83—42no—
175C24×C83C2×D411yescharacteristic
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×Q812yes—
186C22×C4×C83C2×Q812yes—
187C22×C4×C83C2411yescharacteristic
188M16×C83—28no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83C2×D412yes—
196C2×C8×C83C2×Q812yes—
197C2×C8×C83C2×Q812yes—
198C2×C8×C83C2×Q812yes—
199C2×C8×C83C2×Q812yes—
200C2×C8×C83#4112yes—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—22no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C4⋊Q812yes—
214C23×C83C22×D411yescharacteristic
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83C4⋊Q812yes—
232C2×C4×C83AGL(1,ℤ/8)12yes—
233C2×C4×C83#13712yes—
234C2×C4×C83C22×D411yescharacteristic
235C2×C4×C83C22×Q811yescharacteristic
236C2×C4×C83C22×Q811yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83H(5,𝔽2)11yescharacteristic
239C2×C4×C832−1+411yescharacteristic
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—84no—
265Dic166—84no—
266C8×C83—22no—
267C8×C83—22no—
268C8×C83—22no—
269C8×C83C22⋊Q814yescomplemented
270C22×C83—22no—
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83C2×C4⋊Q811yescharacteristic
278C22×C83D4211yescharacteristic
279C22×C83#54711yescharacteristic
280C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C83C2×#13711yescharacteristic
282C22×C83#57711yescharacteristic
283C22×C83#57711yescharacteristic
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83C2×C22⋊Q811yescharacteristic
290C4×C83#54411yescharacteristic
291C4×C83D4×Q811yescharacteristic
292C4×C83#55411yescharacteristic
293Dic83—42no—
294Dic83—42no—
295Dic83—81no—
296Dic83—81no—
297Dic83—81no—
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301#26078—831nomaximal, Sylow
302C2×C83—42no—
303C2×C83#244011yescharacteristic
304C2×C83#316311yescharacteristic
305C2×C83#316511yescharacteristic
306C2×C83#327211yescharacteristic
307C2×C83#327811yescharacteristic
308C2×C83#334411yescharacteristic
309C2×C83#334511yescharacteristic
310C2×#501—831no—
311C2×#428—831no—
312#2441—831no—
313#2440—831no—
314#2084—831no—
315#2083—831no—
316#2059—831no—
317#2058—831no—
318#1943—831no—
319#1942—831no—
320#1854—831no—
321#1853—831no—
322#1844—831no—
323#1809—831no—
324#1753D8311yescharacteristic
325C83#2607811yescharacteristic, complemented, Sylow
326C22×M16C2×D8311yescharacteristic
327C2×C4⋊Q8—831no—
328C2×#125—831no—
329C2×C22⋊Q8—1661no—
330C2×#117—831no—
331C22×C4⋊C4—831no—
332#501—1664no—
333#428—1664no—
334C2×#107—831no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#106—831no—
339C2×#106—831no—
340C2×#106—831no—
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×C22⋊C8C2×D8311yescharacteristic
345C2×#97C2×D8311yescharacteristic
346C2×C4×C8C2×D8311yescharacteristic
347#399—1661no—
348#398—1661no—
349#396—1661no—
350#395—1661no—
351#389—831no—
352#389—831no—
353#389—831no—
354#389—831no—
355#389—1661no—
356#387—1661no—
357#385—1661no—
358#384—831no—
359#384—831no—
360#380—831no—
361#380—831no—
362C4×C22⋊C4C2×D8311yescharacteristic
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#336C2×D8311yescharacteristic
370#336C2×D8311yescharacteristic
371C22×Q8—1661no—
372C23×C4—1661no—
373C23×C4C22×D8311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic16614yescomplemented
377C22×C8C22×D8311yescharacteristic
378C22×C8C22×D8311yescharacteristic
379C22×C8C22×D8311yescharacteristic
380C22×C8C22×D8311yescharacteristic
381C4⋊Q8—3321no—
382C4⋊Q8—1662no—
383#125—3321no—
384#125—1662no—
385C22⋊Q8—3324no—
386#117—3321no—
387#117—1662no—
388C2×C4⋊C4—3321no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1662no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4D33211yescharacteristic
414C2×C22⋊C4D33211yescharacteristic
415C2×C42—1661no—
416C2×C42C22×D8311yescharacteristic
417C2×C42C22×D8311yescharacteristic
418#107—834no—
419#107—1662no—
420#107—1662no—
421#107—1662no—
422#106—3321no—
423#106—3321no—
424#106—1662no—
425#106—1662no—
426#106—1662no—
427#106—1662no—
428C22⋊C8Dic16614yescomplemented
429#97—22no—
430C4×C8—22no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95D33211yescharacteristic
438#95D33211yescharacteristic
439C24C2×D33211yescharacteristic
440C2×Q8—3322no—
441C2×Q8—3322no—
442C2×Q8—6641no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—3322no—
448C22×C4—3322no—
449C22×C4—6641no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4C2×D33211yescharacteristic
459C22×C4C2×D33211yescharacteristic
460C22×C4C2×D33211yescharacteristic
461C22×C4C2×D33211yescharacteristic
462C22×C4C2×D33211yescharacteristic
463C22×C4C2×Dic16612yes—
464C22×C4C2×Dic16612yes—
465C22×C4C23×D8311yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8D4×D8312yes—
474C2×C8C2×Dic16612yes—
475C2×C8C2×Dic16612yes—
476C2×C8C2×Dic16612yes—
477C2×C8C2×Dic16612yes—
478C2×C8#1305058912yes—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—3322no—
487C4⋊C4—3322no—
488C4⋊C4—832no—
489C4⋊C4—832no—
490C4⋊C4—6641no—
491C4⋊C4—6641no—
492C4⋊C4—6641no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—3322no—
510C22⋊C4—3322no—
511C22⋊C4—6641no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—3322no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1458179212yes—
524C23C22×D33211yescharacteristic
525Q8—6644no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—6644no—
565C2×C4—1662no—
566C2×C4—1662no—
567C2×C4#1458179212yes—
568C2×C4#1458182012yes—
569C2×C4#1458182512yes—
570C2×C4C2×D4×D8311yescharacteristic
571C2×C4C22×Dic16611yescharacteristic
572C2×C4C22×Dic16611yescharacteristic
573C2×C4C2×#1305058911yescharacteristic
574C2×C4#1458183811yescharacteristic
575C2×C4#1458183911yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1458176914yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1458179211yescharacteristic
588C22D4×D33211yescharacteristic
589C22#1912835511yescharacteristic
590C22C2×#1458182011yescharacteristic
591C22C2×#1458182511yescharacteristic
592C22#1912851311yescharacteristic
593C22#1912851311yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—3322no—
600C4—3322no—
601C4—6641no—
602C4—6641no—
603C4—6641no—
604C4—6641no—
605C4—6641no—
606C4—6641no—
607C4C2×#1458176911yescharacteristic
608C4#1912832911yescharacteristic
609C4D4×Dic16611yescharacteristic
610C4#1912840811yescharacteristic
611C2—42no—
612C2#2491116511yescharacteristic
613C2#2491799411yescharacteristic
614C2#2491800911yescharacteristic
615C2#2491906511yescharacteristic
616C2#2491913111yescharacteristic
617C2#2491992011yescharacteristic
618C2#2491993511yescharacteristic
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1753C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1753D83
hypocenterC83#26078
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#19128110C2semidirect
5. #24908941C2semidirect
6. #24908936C2semidirect
7. #24908802C2semidirect
8. #24908797C2semidirect
9. #24908201C2semidirect
10. #24908196C2semidirect
13. #24907620C2semidirect
115. C2×M16×C83Q8semidirect
124. C22⋊C8×C83Q8semidirect
269. C8×C83C22⋊Q8semidirect
325. C83#26078semidirect
376. C2×M16Dic166semidirect
428. C22⋊C8Dic166semidirect
579. C8#14581769semidirect

Maximal subgroups

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

subgroupindex
C2×#191281102
C2×#191277432
#249111762
#249111652
#249089412
#249089362
#249088022
#249087972
#249082012
#249081962
#249076642
#249076582
#249076202
#249074962
C83×#17532
#2607883