Finite Group Databasebeta

← #35414973

Subgroups — #35414973

Order 21248.

830 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127756C211yesmaximal, characteristic
2#24909869C211yesmaximal, characteristic
3#24909868C211yesmaximal, characteristic
4#24908223C212yesmaximal, complemented
5#24908218C212yesmaximal, complemented
6#24907661C212yesmaximal
7#24907659C212yesmaximal
8#24907629C211yesmaximal, characteristic, complemented
9#24907597C211yesmaximal, characteristic, complemented
10#24907596C211yesmaximal, characteristic, complemented
11C83×#1847C211yesmaximal, characteristic, complemented
12C2×#14581750—21no—
13C2×#14581750C2211yescharacteristic
14C2×#14581750C2211yescharacteristic
15C2×C42×D83C2211yescharacteristic
16#19127756C418yescomplemented
17C2×#14581725C2212yes—
18C2×#14581723C2212yes—
19C2×Dic83⋊C8C2212yes—
20C2×(C2×D83)⋊C8C2212yes—
21C2×#14581703C2211yescharacteristic
22#19127571C2211yescharacteristic
23#19127570C2211yescharacteristic
24#19127570C2211yescharacteristic
25#19127567C2211yescharacteristic
26#19127566—21no—
27#19127560—21no—
28#19127560—21no—
29#19127558—21no—
30#19127558C2212yes—
31C4⋊C4×Dic83—21no—
32C4⋊C4×Dic83C2212yes—
33#19127552—21no—
34#19127552—21no—
35#19127464C2212yes—
36#19127464C2212yes—
37#19127463C2212yes—
38#19127463C2212yes—
39C2×#107×C83C2212yes—
40C2×#106×C83C2212yes—
41C2×#97×C83C2211yescharacteristic
42#384×C83C2211yescharacteristic
43#382×C83C2211yescharacteristic
44C22×C4×D83D411yescharacteristic
45C22×C4×D83D411yescharacteristic
46C22×C4×D83C2311yescharacteristic
47C22×C83⋊C8C2312yes—
48#14581750—42no—
49#14581750—42no—
50#14581750—44no—
51C2×C4⋊Dic83—21no—
52C2×C4⋊Dic83—21no—
53C2×C4⋊Dic83C2312yes—
54C2×C4⋊Dic83C2311yescharacteristic
55C2×C4⋊Dic83C2311yescharacteristic
56C2×Dic83⋊C4—21no—
57C2×Dic83⋊C4—21no—
58C2×Dic83⋊C4—21no—
59C2×Dic83⋊C4—21no—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—21no—
62C2×(C2×D83)⋊C4—21no—
63C2×C4×Dic83—21no—
64C2×C4×Dic83—21no—
65C2×C4×Dic83—22no—
66C2×C4×Dic83D411yescharacteristic
67C2×C4×Dic83D411yescharacteristic
68C2×C4×Dic83C2311yescharacteristic
69C42×D83C2×C414yes—
70C42×D83Q814yes—
71#14581725—42no—
72#14581723—42no—
73Dic83⋊C8C2×C418yes—
74(C2×D83)⋊C8C2×C418yes—
75#14581703C2×C414yes—
76C83⋊#95—22no—
77C83⋊#95—22no—
78C83⋊#95—21no—
79C83⋊#95—21no—
80C22×C8×C83C2312yes—
81C2×C4⋊C4×C83—21no—
82C2×C4⋊C4×C83—21no—
83C2×C4⋊C4×C83C2312yes—
84C2×C4⋊C4×C83C2311yescharacteristic
85C2×C4⋊C4×C83C2311yescharacteristic
86C2×C42×C83C2311yescharacteristic
87#107×C83—42no—
88#106×C83—42no—
89#97×C83C2×C414yes—
90#95×C83—21no—
91#95×C83—21no—
92C23×D83C2×D411yescharacteristic
93C22×Dic83—21no—
94C22×Dic83—21no—
95C22×Dic83—22no—
96C22×Dic83C2×D411yescharacteristic
97C22×Dic83C2×D411yescharacteristic
98C22×Dic83C2×D411yescharacteristic
99C2×C4×D83—22no—
100C2×C4×D83—22no—
101C2×C4×D83—22no—
102C2×C4×D83—22no—
103C2×C4×D83—41no—
104C2×C4×D83—41no—
105C2×C4×D83C4⋊C414yes—
106C2×C4×D83C4⋊C414yes—
107C2×C4×D83C22×C412yes—
108C2×C4×D83C2×Q812yes—
109C2×C83⋊C8—22no—
110C2×C83⋊C8—22no—
111C2×C83⋊C8C22×C414yes—
112C4⋊Dic83—22no—
113C4⋊Dic83—22no—
114C4⋊Dic83—24no—
115C4⋊Dic83—42no—
116C4⋊Dic83—42no—
117Dic83⋊C4—42no—
118Dic83⋊C4—42no—
119Dic83⋊C4—42no—
120Dic83⋊C4—42no—
121(C2×D83)⋊C4—42no—
122(C2×D83)⋊C4—42no—
123(C2×D83)⋊C4—44no—
124C4×Dic83—42no—
125C4×Dic83—42no—
126C4×Dic83—44no—
127C4×Dic83C4⋊C414yes—
128C4×Dic83C4⋊C414yes—
129C4×Dic83C22×C412yes—
130C4×Dic83C2×Q812yes—
131C22×C4×C83—21no—
132C22×C4×C83—21no—
133C22×C4×C83—22no—
134C22×C4×C83C2×D411yescharacteristic
135C22×C4×C83C2×D411yescharacteristic
136C22×C4×C83C2411yescharacteristic
137C2×C8×C83—22no—
138C2×C8×C83—22no—
139C2×C8×C83C22×C414yes—
140C4⋊C4×C83—22no—
141C4⋊C4×C83—22no—
142C4⋊C4×C83—22no—
143C4⋊C4×C83—22no—
144C4⋊C4×C83—42no—
145C4⋊C4×C83—42no—
146C42×C83C22×C412yes—
147C42×C83C2×Q812yes—
148C23×C83C22×D411yescharacteristic
149C22×D83—21no—
150C22×D83—21no—
151C22×D83—21no—
152C22×D83—21no—
153C22×D83—21no—
154C22×D83—21no—
155C22×D83C2×C4⋊C412yes—
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—21no—
160C2×C4×C83—42no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—42no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—42no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83C2×C4⋊C412yes—
170C2×C4×C83C2×C4⋊C412yes—
171C2×C4×C83C23×C411yescharacteristic
172C2×C4×C83C22×Q811yescharacteristic
173C2×C4×C83H(5,𝔽2)12yes—
174C2×C4×C832−1+412yes—
175C2×Dic83—21no—
176C2×Dic83—21no—
177C2×Dic83—21no—
178C2×Dic83—21no—
179C2×Dic83—41no—
180C2×Dic83—41no—
181C2×Dic83—42no—
182C2×Dic83—42no—
183C2×Dic83—41no—
184C2×Dic83—41no—
185C2×Dic83—41no—
186C2×Dic83—41no—
187C2×Dic83—42no—
188C2×Dic83C2×C4⋊C412yes—
189C2×Dic83C2×C4⋊C412yes—
190C2×Dic83C2×C4⋊C412yes—
191C2×Dic83AGL(1,ℤ/8)12yes—
192C2×Dic83#13712yes—
193C4×D83—24no—
194C4×D83—24no—
195C4×D83—28no—
196C4×D83—44no—
197C8×C83—24no—
198C83⋊C8—24no—
199C22×C83C22×C4⋊C411yescharacteristic
200C22×C83#53511yescharacteristic
201C22×C83#53611yescharacteristic
202C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
203C22×C83C2×#13711yescharacteristic
204C22×C83#57511yescharacteristic
205C22×C83#57511yescharacteristic
206C2×D83—22no—
207C2×D83—22no—
208C2×D83—22no—
209C2×D83—41no—
210C2×D83—41no—
211C2×D83—41no—
212C2×D83—41no—
213C4×C83—22no—
214C4×C83—22no—
215C4×C83—82no—
216C4×C83—81no—
217C4×C83—81no—
218C4×C83#52012yes—
219C4×C83#54412yes—
220Dic83—22no—
221Dic83—22no—
222Dic83—81no—
223Dic83—81no—
224Dic83—82no—
225Dic83#42814yes—
226#21988—831nomaximal, Sylow
227C2×C83#231711yescharacteristic
228C2×C83C2×#42811yescharacteristic
229C2×C83#291111yescharacteristic
230C2×C83#321411yescharacteristic
231C2×C83#321611yescharacteristic
232C2×C83#323511yescharacteristic
233C2×C83#323711yescharacteristic
234D83—42no—
235C2×#432—831no—
236#2285—831no—
237#2285—831no—
238#1948—832no—
239#1947—832no—
240#1854—832no—
241#1854—832no—
242#1847—831no—
243#1847D8311yescharacteristic, complemented
244#1836—831no—
245#1836—831no—
246C83#2198811yescharacteristic, complemented, Sylow
247C2×#117—831no—
248C2×#117—831no—
249C2×#117—1661no—
250C22×C42—831no—
251#432—838no—
252C2×#107—832no—
253C2×#107C2×D8312yes—
254C2×#106—832no—
255C2×#106C2×D8312yes—
256C2×C4⋊C8—832no—
257C2×C22⋊C8—832no—
258C2×#97—831no—
259C2×#97C2×D8311yescharacteristic
260#384—831no—
261#384—831no—
262#384—831no—
263#384C2×D8311yescharacteristic
264#382—831no—
265#382—1661no—
266#382C2×D8311yescharacteristic
267#380—832no—
268#380—1661no—
269#380—1661no—
270#380—1661no—
271C4×C4⋊C4—832no—
272C4×C4⋊C4—1661no—
273C4×C4⋊C4—1661no—
274C4×C4⋊C4—1661no—
275#340—832no—
276#340—832no—
277#340—832no—
278#340—832no—
279C23×C4—831no—
280C23×C4—831no—
281C23×C4—831no—
282C22×C8—832no—
283C22×C8C22×D8312yes—
284#117—3322no—
285#117—3322no—
286#117—3324no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—831no—
290C2×C4⋊C4—831no—
291C2×C4⋊C4—832no—
292C2×C4⋊C4—1661no—
293C2×C4⋊C4—1661no—
294C2×C4⋊C4—1661no—
295C2×C4⋊C4—1661no—
296C2×C4⋊C4—1661no—
297C2×C4⋊C4—1661no—
298C2×C4⋊C4C22×D8311yescharacteristic
299C2×C4⋊C4C22×D8311yescharacteristic
300C2×C4⋊C4C22×D8312yes—
301C2×C22⋊C4—1661no—
302C2×C22⋊C4—1661no—
303C2×C22⋊C4—1662no—
304C2×C42—831no—
305C2×C42—831no—
306C2×C42—831no—
307C2×C42—834no—
308C2×C42—834no—
309C2×C42—1661no—
310C2×C42—1661no—
311C2×C42—1662no—
312C2×C42C22×D8311yescharacteristic
313#107—42no—
314#107—3322no—
315#106—42no—
316#106—3322no—
317C4⋊C8—838no—
318C22⋊C8—838no—
319#97—834no—
320#97C4×D8314yes—
321#95—21no—
322#95—21no—
323#95—1662no—
324#95—1662no—
325#95—1661no—
326#95—1661no—
327C24—831no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—3321no—
332C22×C4—3321no—
333C22×C4—834no—
334C22×C4—831no—
335C22×C4—831no—
336C22×C4—831no—
337C22×C4—832no—
338C22×C4—832no—
339C22×C4—834no—
340C22×C4—1662no—
341C22×C4—1662no—
342C22×C4—1662no—
343C22×C4—1661no—
344C22×C4—1661no—
345C22×C4—1662no—
346C22×C4—1662no—
347C22×C4D4×D8311yescharacteristic
348C22×C4D4×D8311yescharacteristic
349C22×C4C23×D8311yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—834no—
353C2×C8—1662no—
354C2×C8—1662no—
355C2×C8C2×C4×D8314yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—3322no—
363C4⋊C4—3322no—
364C4⋊C4—3322no—
365C4⋊C4—3322no—
366C4⋊C4—3322no—
367C4⋊C4—3322no—
368C4⋊C4—1662no—
369C4⋊C4—1662no—
370C4⋊C4—1664no—
371C22⋊C4—3324no—
372C22⋊C4—3322no—
373C22⋊C4—3322no—
374C42—3324no—
375C42—3322no—
376C42—3322no—
377C42—832no—
378C42—832no—
379C42—834no—
380C42—834no—
381C42C2×C4×D8312yes—
382C42Q8×D8312yes—
383C23—832no—
384C23—1661no—
385C23—1661no—
386C23—1661no—
387C23—1661no—
388C23—1661no—
389C23—1661no—
390C23C2×D4×D8311yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—3324no—
405C2×C4—3322no—
406C2×C4—3321no—
407C2×C4—3321no—
408C2×C4—3321no—
409C2×C4—3322no—
410C2×C4—3321no—
411C2×C4—3321no—
412C2×C4—3321no—
413C2×C4—3322no—
414C2×C4—832no—
415C2×C4—832no—
416C2×C4—832no—
417C2×C4—832no—
418C2×C4—832no—
419C2×C4—1668no—
420C2×C4—1661no—
421C2×C4—1661no—
422C2×C4—1661no—
423C2×C4—1661no—
424C2×C4—1664no—
425C2×C4—1664no—
426C2×C4C4⋊C4×D8312yes—
427C2×C4C4⋊C4×D8312yes—
428C2×C4C22×C4×D8311yescharacteristic
429C2×C4C2×Q8×D8311yescharacteristic
430C2×C4#1458183712yes—
431C2×C4#1458184012yes—
432C8—24no—
433C8—1664no—
434C22—3321no—
435C22—3321no—
436C22—3321no—
437C22—3321no—
438C22—1662no—
439C22—1662no—
440C22—1662no—
441C22C2×C4⋊C4×D8311yescharacteristic
442C22#1912826411yescharacteristic
443C22#1912827611yescharacteristic
444C22AGL(1,ℤ/8)×D8311yescharacteristic
445C22#137×D8311yescharacteristic
446C22#1912849711yescharacteristic
447C22#1912850011yescharacteristic
448C4—22no—
449C4—22no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—834no—
454C4—6642no—
455C4—6641no—
456C4—6641no—
457C4—1662no—
458C4—1662no—
459C4#1912818212yes—
460C4#1912832812yes—
461C2—3322no—
462C2#2491008411yescharacteristic
463C2#428×D8311yescharacteristic
464C2#2491605011yescharacteristic
465C2#2491845311yescharacteristic
466C2#2491846911yescharacteristic
467C2#2491866211yescharacteristic
468C2#2491867811yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1847C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1847D83
hypocenterC83#21988
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #24908223C2semidirect
5. #24908218C2semidirect
8. #24907629C2semidirect
9. #24907597C2semidirect
10. #24907596C2semidirect
11. C83×#1847C2semidirect
16. #19127756C4semidirect
243. #1847D83semidirect
246. C83#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#191277562
#249098692
#249098682
#249082232
#249082182
#249076612
#249076592
#249076292
#249075972
#249075962
C83×#18472
#2198883