Finite Group Databasebeta

← #35467154

Subgroups — #35467154

Order 21248.

848 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127996C211yesmaximal, characteristic, complemented
2C2×#19127744C211yesmaximal, characteristic
3#24911165C211yesmaximal, characteristic
4#24909006C211yesmaximal, characteristic, complemented
5#24908964C211yesmaximal, characteristic, complemented
6#24908844C211yesmaximal, characteristic, complemented
7#24908796C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908034C211yesmaximal, characteristic, complemented
10#24907897C211yesmaximal, characteristic
11#24907771C211yesmaximal, characteristic
12#24907717C211yesmaximal, characteristic, complemented
13#24907662C211yesmaximal, characteristic
14#24907648C211yesmaximal, characteristic
15C83×#2362C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581769—21no—
19C22×C4⋊Dic83C2211yescharacteristic
20#19127996—24no—
21#19127744—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×#14581716C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127633—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127576—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C22⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×#117×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#388×C83—21no—
56#388×C83—21no—
57#387×C83—21no—
58C4×C4⋊C4×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574—22no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581792—22no—
71#14581792—41no—
72#14581769—44no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—41no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725D414yes—
89#14581723—22no—
90#14581723—22no—
91#14581723—41no—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581716—22no—
95#14581716—22no—
96#14581716—22no—
97#14581716—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×Q8×C83C2311yescharacteristic
103C23×C4×C83C2311yescharacteristic
104C22⋊Q8×C83—22no—
105C22⋊Q8×C83Q814yescomplemented
106C4×Q8×C83—22no—
107C4×Q8×C83—41no—
108#117×C83—22no—
109#117×C83—41no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83—21no—
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—42no—
139C22×Dic83—42no—
140C22×Dic83—81no—
141#13050574—28no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8C2×D412yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—42no—
164C4⋊Dic83—81no—
165C4⋊Dic83C2×D412yes—
166Dic83⋊C4—42no—
167Dic83⋊C4—42no—
168C22⋊Dic83—42no—
169C24×C83C2×D411yescharacteristic
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83—21no—
173C2×Q8×C83—21no—
174C2×Q8×C83—21no—
175C2×Q8×C83C2×Q812yes—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×Q812yes—
190C22×C4×C83#4112yes—
191C22×C4×C83C2411yescharacteristic
192C4⋊C4×C83—24no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83—42no—
199C4⋊C4×C83C2×D412yes—
200C4⋊C4×C83C2×Q812yes—
201C22⋊C4×C83—24no—
202C22⋊C4×C83—22no—
203C22⋊C4×C83—22no—
204C22⋊C4×C83—42no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—21no—
209C42×C83—42no—
210C42×C83—42no—
211C42×C83—42no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83C22⋊Q812yes—
219C23×C83C22×D411yescharacteristic
220Q8×C83—22no—
221Q8×C83—22no—
222Q8×C83—41no—
223Q8×C83—41no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—21no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83C22⋊Q812yes—
253C2×C4×C83AGL(1,ℤ/8)12yes—
254C2×C4×C83#13712yes—
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C22×Q811yescharacteristic
257C2×C4×C83C2×#4111yescharacteristic
258C2×C4×C83C2×#4111yescharacteristic
259C2×C4×C83H(5,𝔽2)11yescharacteristic
260C2×C4×C832−1+411yescharacteristic
261C2×Dic83—22no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—84no—
274Dic166—84no—
275C83⋊C8—22no—
276C83⋊C8—22no—
277C83⋊C8—41no—
278C83⋊C8—41no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83C2×C22⋊Q811yescharacteristic
287C22×C83D4211yescharacteristic
288C22×C83#54711yescharacteristic
289C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C83C2×#13711yescharacteristic
291C22×C83#57711yescharacteristic
292C22×C83#57811yescharacteristic
293C4×C83—22no—
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—42no—
297C4×C83—41no—
298C4×C83—42no—
299C4×C83—41no—
300C4×C83—42no—
301C4×C83—81no—
302C4×C83#52511yescharacteristic
303C4×C83#53911yescharacteristic
304C4×C83D4×Q811yescharacteristic
305C4×C83#55411yescharacteristic
306Dic83—42no—
307Dic83—81no—
308Dic83—81no—
309Dic83—81no—
310#26355—831nomaximal, Sylow
311C2×C83—42no—
312C2×C83#244911yescharacteristic
313C2×C83#307911yescharacteristic
314C2×C83#308211yescharacteristic
315C2×C83#327011yescharacteristic
316C2×C83#327211yescharacteristic
317C2×C83#334311yescharacteristic
318C2×C83#335611yescharacteristic
319C2×#477—831no—
320C2×#428—831no—
321#2440—831no—
322#2362D8311yescharacteristic
323#2096—831no—
324#2088—831no—
325#2066—831no—
326#2058—831no—
327#1947—831no—
328#1918—831no—
329#1894—831no—
330#1872—831no—
331#1863—831no—
332#1854—831no—
333#1851—831no—
334C83#2635511yescharacteristic, complemented, Sylow
335C22×M16—831no—
336C2×C4⋊Q8—831no—
337C2×C22⋊Q8—1661no—
338C2×C22⋊Q8C2×D8311yescharacteristic
339C2×C4×Q8C2×D8311yescharacteristic
340C2×#117C2×D8311yescharacteristic
341C22×C4⋊C4—831no—
342#477—1664no—
343#428—1664no—
344C2×#107—831no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#106—831no—
348C2×#106—831no—
349C2×#106—831no—
350C2×C4⋊C8—831no—
351C2×C4⋊C8—831no—
352C2×#103—831no—
353C2×#103—831no—
354C2×#103—831no—
355C2×#103—831no—
356C2×C22⋊C8—831no—
357#398—1661no—
358#395—1661no—
359#391C2×D8311yescharacteristic
360#389—831no—
361#389—831no—
362#389C2×D8311yescharacteristic
363#388—21no—
364#388—21no—
365#387—21no—
366#385—1661no—
367#384—831no—
368#380—831no—
369C4×C4⋊C4—21no—
370C4×C4⋊C4C2×D8311yescharacteristic
371C4×C22⋊C4C2×D8311yescharacteristic
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#340—831no—
378#336—831no—
379#336—831no—
380C22×Q8—1661no—
381C22×Q8C22×D8311yescharacteristic
382C23×C4—1661no—
383C23×C4C22×D8311yescharacteristic
384C2×M16—1662no—
385C2×M16—1662no—
386C2×M16—1662no—
387C22×C8—831no—
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391C4⋊Q8—3321no—
392C4⋊Q8—1662no—
393C22⋊Q8—22no—
394C22⋊Q8—3324no—
395C22⋊Q8Dic16614yescomplemented
396C4×Q8—22no—
397C4×Q8—41no—
398#117—22no—
399#117—41no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—3321no—
404C2×C4⋊C4—831no—
405C2×C4⋊C4—831no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—831no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1662no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4C22×D8311yescharacteristic
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1661no—
421C2×C22⋊C4C22⋊D8311yescharacteristic
422C2×C22⋊C4C22⋊D8311yescharacteristic
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D8311yescharacteristic
427C2×C42C22×D8311yescharacteristic
428#107—834no—
429#107—1662no—
430#107—1662no—
431#106—3321no—
432#106—1662no—
433#106—1662no—
434C4⋊C8—1662no—
435C4⋊C8—1662no—
436#103—3321no—
437#103—1662no—
438#103—1662no—
439#103—1662no—
440C22⋊C8—1662no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—1661no—
446#95—1661no—
447#95—1661no—
448#95C22⋊D8311yescharacteristic
449#95C22⋊D8311yescharacteristic
450C24C2×C22⋊D8311yescharacteristic
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—3322no—
457C2×Q8—3322no—
458C2×Q8—6641no—
459C2×Q8C2×Dic16612yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—3322no—
469C22×C4—3322no—
470C22×C4—6641no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×C22⋊D8311yescharacteristic
477C22×C4C2×C22⋊D8311yescharacteristic
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C2×Dic16612yes—
481C22×C4#1305059012yes—
482C22×C4C23×D8311yescharacteristic
483M16—1668no—
484C2×C8—832no—
485C2×C8—832no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—832no—
509C4⋊C4—6641no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4D4×D8312yes—
518C4⋊C4C2×Dic16612yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—42no—
523C22⋊C4—3322no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C42—42no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1458177112yes—
538C23C22×C22⋊D8311yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—6644no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—6644no—
584C2×C4—1662no—
585C2×C4#1458177112yes—
586C2×C4#1458181712yes—
587C2×C4#1458182212yes—
588C2×C4C2×D4×D8311yescharacteristic
589C2×C4C22×Dic16611yescharacteristic
590C2×C4#41×D8311yescharacteristic
591C2×C4C2×#1305059011yescharacteristic
592C2×C4#1458183711yescharacteristic
593C2×C4#1458183911yescharacteristic
594C8—3321no—
595C8—3321no—
596C8—1662no—
597C8—1662no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22C2×#1458177111yescharacteristic
606C22D4×C22⋊D8311yescharacteristic
607C22#1912835011yescharacteristic
608C22C2×#1458181711yescharacteristic
609C22C2×#1458182211yescharacteristic
610C22#1912850911yescharacteristic
611C22#1912851611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—81no—
621C4—3322no—
622C4—6641no—
623C4—6641no—
624C4—6641no—
625C4#1912821011yescharacteristic
626C4#1912829311yescharacteristic
627C4D4×Dic16611yescharacteristic
628C4#1912841011yescharacteristic
629C2—42no—
630C2#2491126611yescharacteristic
631C2#2491727811yescharacteristic
632C2#2491730511yescharacteristic
633C2#2491902911yescharacteristic
634C2#2491906311yescharacteristic
635C2#2491990311yescharacteristic
636C2#2492007211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2362C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2362D83
hypocenterC83#26355
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#19127996C2semidirect
4. #24909006C2semidirect
5. #24908964C2semidirect
6. #24908844C2semidirect
7. #24908796C2semidirect
8. #24908221C2semidirect
9. #24908034C2semidirect
12. #24907717C2semidirect
105. C22⋊Q8×C83Q8semidirect
334. C83#26355semidirect
395. C22⋊Q8Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191279962
C2×#191277442
#249111652
#249090062
#249089642
#249088442
#249087962
#249082212
#249080342
#249078972
#249077712
#249077172
#249076622
#249076482
C83×#23622
#2635583