Finite Group Databasebeta

← #35414915

Subgroups — #35414915

Order 21248.

966 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127751C211yesmaximal, characteristic
2#24909865C211yesmaximal, characteristic
3C2×#19127570C211yesmaximal, characteristic
4#24908225C212yesmaximal, complemented
5#24908223C212yesmaximal, complemented
6#24907658C211yesmaximal, characteristic
7#24907656C211yesmaximal, characteristic
8#24907655C211yesmaximal, characteristic
9#24907618C212yesmaximal, complemented
10#24907589C212yesmaximal, complemented
11C83×#1853C211yesmaximal, characteristic
12C2×#14581748—21no—
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4—21no—
15C22×Dic83⋊C4—21no—
16C22×C4×Dic83C2211yescharacteristic
17#19127751C418yescomplemented
18C2×#14581724C2212yes—
19C2×#14581723C2212yes—
20C2×#14581723C2212yes—
21C2×Dic83⋊C8C2212yes—
22C2×C22⋊(C83⋊C8)C2211yescharacteristic
23C2×C8×Dic83C2212yes—
24#19127570—21no—
25#19127570—21no—
26#19127570C2212yes—
27#19127570C2212yes—
28#19127570C2211yescharacteristic
29#19127570C2211yescharacteristic
30#19127565—21no—
31#19127561C2211yescharacteristic
32#19127559—21no—
33C4⋊C4×Dic83C2211yescharacteristic
34C4⋊C4×Dic83C2211yescharacteristic
35#19127552—21no—
36#19127464C2212yes—
37#19127463C2212yes—
38#19127463C2212yes—
39C22×C4⋊C4×C83C2211yescharacteristic
40C2×#106×C83C2212yes—
41C2×C22⋊C8×C83C2211yescharacteristic
42#380×C83C2211yescharacteristic
43#340×C83C2212yes—
44C23×Dic83—21no—
45C23×Dic83D411yescharacteristic
46C23×Dic83D411yescharacteristic
47C22×C83⋊C8C2312yes—
48#14581748—44no—
49C2×C4⋊Dic83—21no—
50C2×C4⋊Dic83—42no—
51C2×C4⋊Dic83C2311yescharacteristic
52C2×C4⋊Dic83C2311yescharacteristic
53C2×C4⋊Dic83C2312yes—
54C2×Dic83⋊C4—21no—
55C2×Dic83⋊C4—21no—
56C2×Dic83⋊C4—21no—
57C2×Dic83⋊C4—21no—
58C2×Dic83⋊C4—22no—
59C2×Dic83⋊C4—22no—
60C2×Dic83⋊C4—22no—
61C2×Dic83⋊C4—22no—
62C2×Dic83⋊C4—44no—
63C2×Dic83⋊C4—44no—
64C2×C22⋊Dic83—21no—
65C2×C4×Dic83—21no—
66C2×C4×Dic83—21no—
67C2×C4×Dic83C2×C414yes—
68C2×C4×Dic83D411yescharacteristic
69C2×C4×Dic83D411yescharacteristic
70C2×C4×Dic83Q814yes—
71C2×C4×Dic83C2311yescharacteristic
72C2×C4×Dic83C2311yescharacteristic
73#14581724—42no—
74#14581723—42no—
75#14581723—42no—
76Dic83⋊C8C2×C418yescomplemented
77C22⋊(C83⋊C8)C2×C414yes—
78C8×Dic83C2×C418yescomplemented
79C83⋊#95—21no—
80C83⋊#95—21no—
81C83⋊#95—21no—
82C23×C4×C83—21no—
83C23×C4×C83C2311yescharacteristic
84C22×C8×C83C2312yes—
85C2×C4⋊C4×C83—21no—
86C2×C4⋊C4×C83—42no—
87C2×C4⋊C4×C83C2311yescharacteristic
88C2×C4⋊C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83C2312yes—
90C2×C22⋊C4×C83—21no—
91#106×C83—42no—
92C22⋊C8×C83C2×C414yes—
93#95×C83—21no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—21no—
98C22×Dic83—21no—
99C22×Dic83—42no—
100C22×Dic83—42no—
101C22×Dic83—42no—
102C22×Dic83C4⋊C414yes—
103C22×Dic83C4⋊C414yes—
104C22×Dic83SD1614yes—
105C22×Dic83SD1614yes—
106C22×Dic83C2×D411yescharacteristic
107C22×Dic83C2×D411yescharacteristic
108C22×Dic83C2×D411yescharacteristic
109C22×Dic83C2×D411yescharacteristic
110C2×C83⋊C8—22no—
111C2×C83⋊C8—22no—
112C2×C83⋊C8C22×C414yes—
113C4⋊Dic83—22no—
114C4⋊Dic83—22no—
115C4⋊Dic83—22no—
116C4⋊Dic83—22no—
117C4⋊Dic83—42no—
118Dic83⋊C4—44no—
119Dic83⋊C4—44no—
120Dic83⋊C4—44no—
121Dic83⋊C4—42no—
122Dic83⋊C4—42no—
123Dic83⋊C4—42no—
124Dic83⋊C4—44no—
125Dic83⋊C4—42no—
126C22⋊Dic83—42no—
127C4×Dic83—42no—
128C4×Dic83—42no—
129C4×Dic83C4⋊C414yes—
130C4×Dic83C4⋊C414yes—
131C4×Dic83C22×C412yes—
132C4×Dic83C22×C412yes—
133C4×Dic83C2×Q812yes—
134C4×Dic83C2×Q812yes—
135C24×C83C2×D411yescharacteristic
136C22×C4×C83—22no—
137C22×C4×C83—22no—
138C22×C4×C83—21no—
139C22×C4×C83—21no—
140C22×C4×C83—22no—
141C22×C4×C83—42no—
142C22×C4×C83—42no—
143C22×C4×C83—42no—
144C22×C4×C83C22×C412yes—
145C22×C4×C83C2×D411yescharacteristic
146C22×C4×C83C2×Q812yes—
147C22×C4×C83C2411yescharacteristic
148C2×C8×C83—22no—
149C2×C8×C83—22no—
150C2×C8×C83C22×C414yes—
151C4⋊C4×C83—22no—
152C4⋊C4×C83—22no—
153C4⋊C4×C83—22no—
154C4⋊C4×C83—22no—
155C4⋊C4×C83—42no—
156C22⋊C4×C83—42no—
157C23×C83—21no—
158C23×C83—21no—
159C23×C83—21no—
160C23×C83—21no—
161C23×C83C2×C4⋊C412yes—
162C23×C83C2×SD1612yes—
163C23×C83C2×SD1612yes—
164C23×C83C22×D411yescharacteristic
165C2×C4×C83—21no—
166C2×C4×C83—21no—
167C2×C4×C83—24no—
168C2×C4×C83—24no—
169C2×C4×C83—42no—
170C2×C4×C83—41no—
171C2×C4×C83—41no—
172C2×C4×C83—41no—
173C2×C4×C83—41no—
174C2×C4×C83—42no—
175C2×C4×C83—42no—
176C2×C4×C83—41no—
177C2×C4×C83—41no—
178C2×C4×C83—84no—
179C2×C4×C83C2×C4⋊C412yes—
180C2×C4×C83C23×C411yescharacteristic
181C2×C4×C83C22×Q811yescharacteristic
182C2×C4×C83H(5,𝔽2)12yes—
183C2×C4×C832−1+412yes—
184C2×Dic83—21no—
185C2×Dic83—21no—
186C2×Dic83—28no—
187C2×Dic83—42no—
188C2×Dic83—41no—
189C2×Dic83—41no—
190C2×Dic83—41no—
191C2×Dic83—41no—
192C2×Dic83—42no—
193C2×Dic83—42no—
194C2×Dic83—41no—
195C2×Dic83—41no—
196C2×Dic83—84no—
197C2×Dic83#106116yescomplemented
198C2×Dic83C2×C4⋊C412yes—
199C2×Dic83C2×C4⋊C412yes—
200C2×Dic83C2×C4⋊C412yes—
201C2×Dic83C2×C4⋊C412yes—
202C2×Dic83C2×SD1612yes—
203C2×Dic83C2×SD1612yes—
204C2×Dic83C2×SD1612yes—
205C2×Dic83C2×SD1612yes—
206C2×Dic83AGL(1,ℤ/8)12yes—
207C2×Dic83#13712yes—
208C8×C83—24no—
209C83⋊C8—24no—
210C22×C83—22no—
211C22×C83—22no—
212C22×C83—22no—
213C22×C83—22no—
214C22×C83—22no—
215C22×C83—22no—
216C22×C83C2×#10614yes—
217C22×C83C22×C4⋊C411yescharacteristic
218C22×C83#53511yescharacteristic
219C22×C83#53611yescharacteristic
220C22×C83C22×SD1611yescharacteristic
221C22×C83C22×SD1611yescharacteristic
222C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C83C2×#13711yescharacteristic
224C4×C83—22no—
225C4×C83—81no—
226C4×C83—82no—
227C4×C83—81no—
228C4×C83#52012yes—
229C4×C83#54412yes—
230Dic83—22no—
231Dic83—81no—
232Dic83—82no—
233Dic83—81no—
234Dic83C2×#10614yes—
235Dic83C2×#10614yes—
236Dic83#42814yes—
237#21981—831nomaximal, Sylow
238C2×C83—24no—
239C2×C83#231711yescharacteristic
240C2×C83C22×#10611yescharacteristic
241C2×C83C2×#42811yescharacteristic
242C2×C83#320611yescharacteristic
243C2×C83#320811yescharacteristic
244C2×C83#322711yescharacteristic
245C2×C83#322911yescharacteristic
246C2×#431—831no—
247#2285—831no—
248C2×#384—831no—
249#1948—832no—
250#1948—832no—
251#1853—831no—
252#1853—831no—
253#1853—831no—
254#1853D8311yescharacteristic
255#1844—832no—
256#1834—832no—
257C83#2198111yescharacteristic, complemented, Sylow
258C2×#117—1661no—
259C22×C4⋊C4—831no—
260C22×C4⋊C4—1661no—
261C22×C4⋊C4—1661no—
262C22×C4⋊C4C2×D8311yescharacteristic
263C22×C42—831no—
264#431—838no—
265C2×#106—832no—
266C2×#106—832no—
267C2×#106—832no—
268C2×#106C2×D8312yes—
269C2×C4⋊C8—832no—
270C2×C22⋊C8—831no—
271C2×C22⋊C8C2×D8311yescharacteristic
272C2×C4×C8—832no—
273#384—831no—
274#384—832no—
275#384—831no—
276#384—832no—
277#384—1661no—
278#384—1661no—
279#382—1661no—
280#380—831no—
281#380—1661no—
282#380C2×D8311yescharacteristic
283C4×C4⋊C4—831no—
284C4×C4⋊C4—831no—
285C4×C4⋊C4—1661no—
286#340—832no—
287#340—832no—
288#340—832no—
289#340C2×D8312yes—
290C23×C4—21no—
291C23×C4—831no—
292C23×C4—831no—
293C23×C4—1661no—
294C23×C4C22×D8311yescharacteristic
295C22×C8—832no—
296C22×C8C22×D8312yes—
297#117—3324no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—3324no—
301C2×C4⋊C4—3324no—
302C2×C4⋊C4—3322no—
303C2×C4⋊C4—832no—
304C2×C4⋊C4—831no—
305C2×C4⋊C4—831no—
306C2×C4⋊C4—1662no—
307C2×C4⋊C4—1662no—
308C2×C4⋊C4—1662no—
309C2×C4⋊C4—1662no—
310C2×C4⋊C4—1661no—
311C2×C4⋊C4—1661no—
312C2×C4⋊C4—1661no—
313C2×C4⋊C4—1661no—
314C2×C4⋊C4—1661no—
315C2×C4⋊C4C22×D8312yes—
316C2×C4⋊C4C22×D8311yescharacteristic
317C2×C4⋊C4C22×D8311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—1661no—
320C2×C42—834no—
321C2×C42—831no—
322C2×C42—831no—
323C2×C42—834no—
324C2×C42—831no—
325C2×C42—831no—
326C2×C42—1661no—
327C2×C42—1661no—
328#106—42no—
329#106—3322no—
330#106—3322no—
331#106—3322no—
332C4⋊C8—838no—
333C22⋊C8—834no—
334C22⋊C8C4×D8314yes—
335C4×C8—838no—
336#95—21no—
337#95—1661no—
338#95—1661no—
339#95—1661no—
340C24D4×D8311yescharacteristic
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—3322no—
350C22×C4—3322no—
351C22×C4—3322no—
352C22×C4—834no—
353C22×C4—834no—
354C22×C4—831no—
355C22×C4—831no—
356C22×C4—831no—
357C22×C4—834no—
358C22×C4—831no—
359C22×C4—834no—
360C22×C4—1661no—
361C22×C4—1661no—
362C22×C4—1662no—
363C22×C4—1662no—
364C22×C4—1662no—
365C22×C4C2×C4×D8312yes—
366C22×C4D4×D8311yescharacteristic
367C22×C4Q8×D8312yes—
368C22×C4C23×D8311yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—834no—
372C2×C8—1662no—
373C2×C8—1662no—
374C2×C8C2×C4×D8314yes—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—3322no—
381C4⋊C4—3324no—
382C4⋊C4—3324no—
383C4⋊C4—3324no—
384C4⋊C4—3322no—
385C4⋊C4—3322no—
386C4⋊C4—3322no—
387C4⋊C4—3322no—
388C4⋊C4—3324no—
389C4⋊C4—1662no—
390C4⋊C4—1662no—
391C4⋊C4—1662no—
392C4⋊C4—1662no—
393C22⋊C4—42no—
394C22⋊C4—3322no—
395C42—3322no—
396C42—3322no—
397C42—832no—
398C42—832no—
399C42—832no—
400C42—832no—
401C42—834no—
402C42—834no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23C4⋊C4×D8312yes—
408C23SD16×D8312yes—
409C23SD16×D8312yes—
410C23C2×D4×D8311yescharacteristic
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—84no—
425C2×C4—3321no—
426C2×C4—3321no—
427C2×C4—3322no—
428C2×C4—3322no—
429C2×C4—3321no—
430C2×C4—3321no—
431C2×C4—3322no—
432C2×C4—3321no—
433C2×C4—3321no—
434C2×C4—832no—
435C2×C4—832no—
436C2×C4—832no—
437C2×C4—832no—
438C2×C4—832no—
439C2×C4—832no—
440C2×C4—832no—
441C2×C4—832no—
442C2×C4—832no—
443C2×C4—8316no—
444C2×C4—832no—
445C2×C4—6644no—
446C2×C4—1668no—
447C2×C4—1661no—
448C2×C4—1661no—
449C2×C4C4⋊C4×D8312yes—
450C2×C4C22×C4×D8311yescharacteristic
451C2×C4C2×Q8×D8311yescharacteristic
452C2×C4#1458183712yes—
453C2×C4#1458183912yes—
454C8—24no—
455C8—1664no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22#106×D8314yes—
463C22C2×C4⋊C4×D8311yescharacteristic
464C22#1912826611yescharacteristic
465C22#1912827311yescharacteristic
466C22C2×SD16×D8311yescharacteristic
467C22C2×SD16×D8311yescharacteristic
468C22AGL(1,ℤ/8)×D8311yescharacteristic
469C22#137×D8311yescharacteristic
470C4—22no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4—834no—
475C4—834no—
476C4—834no—
477C4—6641no—
478C4—6641no—
479C4—6642no—
480C4—1662no—
481C4#1912818012yes—
482C4#1912833012yes—
483C2—24no—
484C2#2491008211yescharacteristic
485C2C2×#106×D8311yescharacteristic
486C2#428×D8311yescharacteristic
487C2#2491838611yescharacteristic
488C2#2491840211yescharacteristic
489C2#2491858911yescharacteristic
490C2#2491860511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1853C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1853D83
hypocenterC83#21981
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #24908225C2semidirect
5. #24908223C2semidirect
9. #24907618C2semidirect
10. #24907589C2semidirect
17. #19127751C4semidirect
76. Dic83⋊C8C2×C4semidirect
78. C8×Dic83C2×C4semidirect
197. C2×Dic83#106semidirect
257. C83#21981semidirect

Maximal subgroups

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

subgroupindex
C2×#191277512
#249098652
C2×#191275702
#249082252
#249082232
#249076582
#249076562
#249076552
#249076182
#249075892
C83×#18532
#2198183