Finite Group Databasebeta

← #1824×D83 #35404187

Subgroups — #1824×D83 #35404187

Order 21248.

1878 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D83C211yesmaximal, characteristic
2C2×C22⋊C8×D83C211yesmaximal, characteristic
3#380×D83C211yesmaximal, characteristic
4#24907554C211yesmaximal, characteristic, complemented
5#24907553C211yesmaximal, characteristic, complemented
6#24907552C214yesmaximal, complemented
7#24907551C211yesmaximal, characteristic, complemented
8#340×D83C214yesmaximal, complemented
9C83×#1824C211yesmaximal, characteristic, complemented
10C23×C4×D83—21no—
11C23×C4×D83C2211yescharacteristic
12C22×C8×D83C2212yes—
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4—21no—
15C2×C4⋊C4×D83—24no—
16C2×C4⋊C4×D83—21no—
17C2×C4⋊C4×D83—21no—
18C2×C4⋊C4×D83C418yescomplemented
19C2×C4⋊C4×D83C2212yes—
20C2×C4⋊C4×D83C2212yes—
21C2×C4⋊C4×D83C2212yes—
22C2×C22⋊C4×D83—21no—
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24C2×(C2×D83)⋊C8C2212yes—
25C22⋊C8×D83—24no—
26#19127561C2211yescharacteristic
27#19127560—21no—
28#19127559—21no—
29#19127558C2212yes—
30#95×D83—21no—
31#19127464C2214yescomplemented
32#19127463C2214yescomplemented
33#19127463C2214yescomplemented
34C22×C4⋊C4×C83C2211yescharacteristic
35C2×C22⋊C8×C83C2211yescharacteristic
36#380×C83C2211yescharacteristic
37#340×C83C2214yescomplemented
38C24×D83D411yescharacteristic
39C23×Dic83—21no—
40C23×Dic83D411yescharacteristic
41C22×C4×D83—28no—
42C22×C4×D83—28no—
43C22×C4×D83—22no—
44C22×C4×D83—22no—
45C22×C4×D83—22no—
46C22×C4×D83—48no—
47C22×C4×D83C2×C414yes—
48C22×C4×D83D411yescharacteristic
49C22×C4×D83D411yescharacteristic
50C22×C4×D83D414yes—
51C22×C4×D83C2311yescharacteristic
52C22×C4×D83C2311yescharacteristic
53C22×C83⋊C8C2312yes—
54C2×C8×D83—24no—
55C2×C8×D83—28no—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83C2×C414yescomplemented
59C2×C4⋊Dic83C2312yes—
60C2×C4⋊Dic83C2312yes—
61C2×Dic83⋊C4—28no—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4—22no—
64C2×Dic83⋊C4—22no—
65C4⋊C4×D83—28no—
66C4⋊C4×D83—28no—
67C4⋊C4×D83—24no—
68C4⋊C4×D83—24no—
69C4⋊C4×D83—28no—
70C4⋊C4×D83C2×C418yes—
71C4⋊C4×D83C2×C418yescomplemented
72C2×C22⋊Dic83—21no—
73C2×(C2×D83)⋊C4—22no—
74C22⋊C4×D83—44no—
75C22⋊(C83⋊C8)—22no—
76(C2×D83)⋊C8—24no—
77C83⋊#95—22no—
78C83⋊#95—21no—
79C23×C4×C83—21no—
80C23×C4×C83C2311yescharacteristic
81C22×C8×C83C2312yes—
82C2×C4⋊C4×C83—21no—
83C2×C4⋊C4×C83—22no—
84C2×C4⋊C4×C83C2×C414yescomplemented
85C2×C4⋊C4×C83C2312yes—
86C2×C4⋊C4×C83C2312yes—
87C2×C22⋊C4×C83—21no—
88C22⋊C8×C83—22no—
89#95×C83—21no—
90C23×D83—24no—
91C23×D83—24no—
92C23×D83C22⋊C414yes—
93C23×D83SD1614yes—
94C23×D83Q1614yes—
95C23×D83C2×D411yescharacteristic
96C23×D83C2×D411yescharacteristic
97C22×Dic83—24no—
98C22×Dic83—24no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—44no—
102C22×Dic83C22⋊C414yes—
103C22×Dic83C2×D411yescharacteristic
104C22×Dic83C2×D411yescharacteristic
105C2×C4×D83—24no—
106C2×C4×D83—24no—
107C2×C4×D83—24no—
108C2×C4×D83—216no—
109C2×C4×D83—216no—
110C2×C4×D83—28no—
111C2×C4×D83—28no—
112C2×C4×D83—48no—
113C2×C4×D83—432no—
114C2×C4×D83—48no—
115C2×C4×D83—44no—
116C2×C4×D83—44no—
117C2×C4×D83—44no—
118C2×C4×D83C22⋊C414yes—
119C2×C4×D83C22⋊C414yes—
120C2×C4×D83C22×C412yes—
121C2×C4×D83C22×C412yes—
122C2×C4×D83C2×D412yes—
123C2×C4×D83C2×D412yes—
124C2×C4×D83#4118yes—
125C2×C83⋊C8—22no—
126C2×C83⋊C8—24no—
127C8×D83—48no—
128C4⋊Dic83—24no—
129C4⋊Dic83—22no—
130C4⋊Dic83—24no—
131C4⋊Dic83C22×C414yes—
132Dic83⋊C4—28no—
133Dic83⋊C4—28no—
134Dic83⋊C4—44no—
135C22⋊Dic83—42no—
136(C2×D83)⋊C4—44no—
137C24×C83C2×D411yescharacteristic
138C22×C4×C83—24no—
139C22×C4×C83—24no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—44no—
143C22×C4×C83C22×C412yes—
144C22×C4×C83C2×D411yescharacteristic
145C22×C4×C83C2×D412yes—
146C22×C4×C83C2411yescharacteristic
147C2×C8×C83—22no—
148C2×C8×C83—24no—
149C4⋊C4×C83—24no—
150C4⋊C4×C83—22no—
151C4⋊C4×C83—24no—
152C4⋊C4×C83C22×C414yes—
153C22⋊C4×C83—42no—
154C23×C83—22no—
155C23×C83—22no—
156C23×C83C2×C22⋊C412yes—
157C23×C83C2×SD1612yes—
158C23×C83C2×Q1612yes—
159C23×C83C22×D411yescharacteristic
160C22×D83—22no—
161C22×D83—22no—
162C22×D83—28no—
163C22×D83—28no—
164C22×D83—216no—
165C22×D83—216no—
166C22×D83#103116yes—
167C22×D83C2×C22⋊C412yes—
168C22×D83C2×C22⋊C412yes—
169C22×D83#12314yes—
170C22×D83C2×SD1612yes—
171C22×D83C2×SD1612yes—
172C22×D83C2×Q1612yes—
173C22×D83C2×Q1612yes—
174C22×D83AGL(1,ℤ/8)14yes—
175C2×C4×C83—22no—
176C2×C4×C83—24no—
177C2×C4×C83—24no—
178C2×C4×C83—24no—
179C2×C4×C83—48no—
180C2×C4×C83—44no—
181C2×C4×C83—42no—
182C2×C4×C83—42no—
183C2×C4×C83—42no—
184C2×C4×C83C2×C22⋊C412yes—
185C2×C4×C83C23×C411yescharacteristic
186C2×C4×C83C22×D411yescharacteristic
187C2×C4×C83C2×#4114yes—
188C2×Dic83—22no—
189C2×Dic83—28no—
190C2×Dic83—24no—
191C2×Dic83—48no—
192C2×Dic83—44no—
193C2×Dic83—42no—
194C2×Dic83—42no—
195C2×Dic83—42no—
196C2×Dic83C2×C22⋊C412yes—
197C2×Dic83C2×C22⋊C412yes—
198C2×Dic83#12314yes—
199C4×D83—28no—
200C4×D83—24no—
201C4×D83—24no—
202C4×D83—28no—
203C4×D83—416no—
204C4×D83—416no—
205C4×D83—88no—
206C4×D83#11718yes—
207C4×D83#12318yes—
208C8×C83—42no—
209C83⋊C8—42no—
210C22×C83—24no—
211C22×C83—24no—
212C22×C83—22no—
213C22×C83—22no—
214C22×C83C2×#10314yes—
215C22×C83C22×C22⋊C411yescharacteristic
216C22×C83C2×#12312yes—
217C22×C83C22×SD1611yescharacteristic
218C22×C83C22×Q1611yescharacteristic
219C22×C83C2×AGL(1,ℤ/8)12yes—
220C2×D83—24no—
221C2×D83—24no—
222C2×D83—22no—
223C2×D83—22no—
224C2×D83—232no—
225C2×D83#38114yes—
226C2×D83C2×#10314yes—
227C2×D83C2×#10314yes—
228C2×D83#41814yes—
229C2×D83#48118yes—
230C2×D83#48418yes—
231C4×C83—22no—
232C4×C83—44no—
233C4×C83—82no—
234C4×C83C2×#11712yes—
235C4×C83C2×#12312yes—
236Dic83—22no—
237Dic83—44no—
238Dic83—82no—
239Dic83#38114yes—
240C2×#1824—831nomaximal, Sylow
241C2×C83—24no—
242C2×C83C2×#38111yescharacteristic
243C2×C83C22×#10311yescharacteristic
244C2×C83C2×#41811yescharacteristic
245C2×C83C2×#48112yes—
246C2×C83C2×#48412yes—
247D83—24no—
248D83#182418yesdirect product factor
249C23×C4⋊C4—831no—
250C22×C22⋊C8—831no—
251C2×#380—831no—
252#1824—831no—
253#1824—831no—
254#1824—834no—
255#1824—831no—
256#1824D8311yescharacteristic, direct product factor
257C2×#340—834no—
258C83C2×#182411yescharacteristic, complemented, Sylow
259C24×C4—831no—
260C24×C4—1661no—
261C23×C8—832no—
262C22×C4⋊C4—831no—
263C22×C4⋊C4—838no—
264C22×C4⋊C4—832no—
265C22×C4⋊C4—832no—
266C22×C4⋊C4—832no—
267C22×C4⋊C4—1661no—
268C22×C4⋊C4—1661no—
269C22×C4⋊C4—1664no—
270C22×C4⋊C4—1661no—
271C22×C4⋊C4C2×D8311yescharacteristic
272C22×C22⋊C4—1661no—
273C2×C22⋊C8—831no—
274C2×C22⋊C8—832no—
275C2×C22⋊C8—1664no—
276C2×C22⋊C8C2×D8311yescharacteristic
277#380—831no—
278#380—832no—
279#380—1661no—
280#380—1661no—
281#380C2×D8311yescharacteristic
282C2×#95—1661no—
283#340—834no—
284#340—834no—
285#340—834no—
286#340C2×D8314yescomplemented
287C25—831no—
288C23×C4—21no—
289C23×C4—3328no—
290C23×C4—831no—
291C23×C4—831no—
292C23×C4—834no—
293C23×C4—831no—
294C23×C4—831no—
295C23×C4—834no—
296C23×C4—831no—
297C23×C4—1662no—
298C23×C4—1662no—
299C23×C4—1662no—
300C23×C4—1668no—
301C23×C4—1668no—
302C23×C4—1661no—
303C23×C4C22×D8311yescharacteristic
304C22×C8—832no—
305C22×C8—1668no—
306C22×C8—1664no—
307C22×C8C22×D8312yes—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—834no—
311C2×C4⋊C4—832no—
312C2×C4⋊C4—838no—
313C2×C4⋊C4—838no—
314C2×C4⋊C4—832no—
315C2×C4⋊C4—1668no—
316C2×C4⋊C4—1664no—
317C2×C4⋊C4—1664no—
318C2×C4⋊C4—1668no—
319C2×C4⋊C4—1668no—
320C2×C4⋊C4—1662no—
321C2×C4⋊C4—1662no—
322C2×C4⋊C4—1662no—
323C2×C4⋊C4—1668no—
324C2×C4⋊C4—1662no—
325C2×C4⋊C4—1661no—
326C2×C4⋊C4C4×D8314yescomplemented
327C2×C4⋊C4C22×D8312yes—
328C2×C4⋊C4C22×D8312yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—3324no—
331C2×C22⋊C4—1662no—
332C2×C22⋊C4—1661no—
333C22⋊C8—22no—
334C22⋊C8—1664no—
335C22⋊C8—1662no—
336#95—21no—
337#95—1662no—
338#95—1661no—
339C24—831no—
340C24—834no—
341C24—834no—
342C24—831no—
343C24—834no—
344C24—1664no—
345C24—1664no—
346C24D4×D8311yescharacteristic
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—44no—
352C22×C4—3328no—
353C22×C4—33232no—
354C22×C4—3328no—
355C22×C4—3324no—
356C22×C4—3324no—
357C22×C4—3324no—
358C22×C4—3324no—
359C22×C4—831no—
360C22×C4—832no—
361C22×C4—832no—
362C22×C4—838no—
363C22×C4—831no—
364C22×C4—834no—
365C22×C4—832no—
366C22×C4—832no—
367C22×C4—834no—
368C22×C4—834no—
369C22×C4—1668no—
370C22×C4—1668no—
371C22×C4—1664no—
372C22×C4—16616no—
373C22×C4—16616no—
374C22×C4—1664no—
375C22×C4—1662no—
376C22×C4—1662no—
377C22×C4—1664no—
378C22×C4—1664no—
379C22×C4—1664no—
380C22×C4C2×C4×D8312yes—
381C22×C4D4×D8312yes—
382C22×C4D4×D8311yescharacteristic
383C22×C4C23×D8311yescharacteristic
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—3328no—
387C2×C8—1662no—
388C2×C8—1664no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—3324no—
393C4⋊C4—834no—
394C4⋊C4—1668no—
395C4⋊C4—1668no—
396C4⋊C4—1664no—
397C4⋊C4—1662no—
398C4⋊C4—1664no—
399C4⋊C4C2×C4×D8314yes—
400C22⋊C4—42no—
401C22⋊C4—3324no—
402C22⋊C4—3322no—
403C23—22no—
404C23—22no—
405C23—832no—
406C23—834no—
407C23—832no—
408C23—832no—
409C23—834no—
410C23—832no—
411C23—8316no—
412C23—832no—
413C23—832no—
414C23—1668no—
415C23—1668no—
416C23—16616no—
417C23—16616no—
418C23—1662no—
419C23—1662no—
420C23C22⋊C4×D8312yes—
421C23SD16×D8312yes—
422C23Q16×D8312yes—
423C23C2×D4×D8311yescharacteristic
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—33216no—
434C2×C4—33216no—
435C2×C4—3328no—
436C2×C4—3324no—
437C2×C4—3322no—
438C2×C4—3322no—
439C2×C4—3322no—
440C2×C4—832no—
441C2×C4—834no—
442C2×C4—838no—
443C2×C4—838no—
444C2×C4—832no—
445C2×C4—6648no—
446C2×C4—1668no—
447C2×C4—1668no—
448C2×C4—1664no—
449C2×C4—1668no—
450C2×C4—1662no—
451C2×C4—1664no—
452C2×C4—1664no—
453C2×C4C22⋊C4×D8312yes—
454C2×C4C22×C4×D8311yescharacteristic
455C2×C4C2×D4×D8311yescharacteristic
456C2×C4#41×D8314yes—
457C8—42no—
458C8—3322no—
459C22—22no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—834no—
464C22—838no—
465C22—838no—
466C22—834no—
467C22—834no—
468C22—834no—
469C22—16632no—
470C22—1662no—
471C22—1664no—
472C22—1664no—
473C22—1662no—
474C22#103×D8314yes—
475C22C2×C22⋊C4×D8311yescharacteristic
476C22#123×D8312yes—
477C22C2×SD16×D8311yescharacteristic
478C22C2×Q16×D8311yescharacteristic
479C22AGL(1,ℤ/8)×D8312yes—
480C4—22no—
481C4—44no—
482C4—82no—
483C4—3324no—
484C4—834no—
485C4—6642no—
486C4—1662no—
487C4#117×D8312yes—
488C4#123×D8312yes—
489C2—24no—
490C2—838no—
491C2—1664no—
492C2#381×D8311yescharacteristic
493C2C2×#103×D8311yescharacteristic
494C2#418×D8311yescharacteristic
495C2#481×D8312yes—
496C2#484×D8312yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1824C2
socleC23×C83C22×D4
prime radicalC24×D83D4
cosocleC22×C4×C83C24
hypercenter#1824D83
hypocenterC83C2×#1824
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #24907554C2semidirect
5. #24907553C2semidirect
6. #24907552C2semidirect
7. #24907551C2semidirect
8. #340×D83C2semidirect
9. C83×#1824C2semidirect
18. C2×C4⋊C4×D83C4semidirect
31. #19127464C22semidirect
32. #19127463C22semidirect
33. #19127463C22semidirect
37. #340×C83C22semidirect
58. C2×C4⋊Dic83C2×C4semidirect
71. C4⋊C4×D83C2×C4semidirect
84. C2×C4⋊C4×C83C2×C4semidirect
248. D83#1824direct (N × H)
256. #1824D83direct (N × H)
258. C83C2×#1824semidirect
286. #340C2×D83semidirect
326. C2×C4⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D832
C2×C22⋊C8×D832
#380×D832
#249075542
#249075532
#249075522
#249075512
#340×D832
C83×#18242
C2×#182483