Finite Group Databasebeta

← #35504462

Subgroups — #35504462

Order 21248.

888 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14581766C211yesmaximal, characteristic
2#24909020C211yesmaximal, characteristic, complemented
3#24908992C211yesmaximal, characteristic, complemented
4#24908793C211yesmaximal, characteristic, complemented
5#24908596C211yesmaximal, characteristic, complemented
6#24908052C212yesmaximal, complemented
7#24907899C212yesmaximal
8#24907723C212yesmaximal, complemented
9#24907494C212yesmaximal, complemented
10#24907157C211yesmaximal, characteristic, complemented
11C83×#2857C211yesmaximal, characteristic, complemented
12C22×#13050574C2212yes—
13C2×#14581766C2211yescharacteristic
14C2×C4×D332C2211yescharacteristic
15C2×C4×C22⋊D83—21no—
16C2×#14581725C2212yes—
17C2×#14581723C2212yes—
18C2×#14581712C2212yes—
19C2×#14581712C2212yes—
20#19127603C2211yescharacteristic
21#19127599C2211yescharacteristic
22#19127592—21no—
23#19127580—21no—
24#19127561C2211yescharacteristic
25C4×C4⋊Dic83C2211yescharacteristic
26C4×(C2×D83)⋊C4—21no—
27#19127463C2212yes—
28#19127463C2212yes—
29#19127463C2212yes—
30#19127463C2212yes—
31#19127456C2212yes—
32#19127456C2211yescharacteristic
33#19127456C2211yescharacteristic
34C2×C4⋊Q8×C83C2211yescharacteristic
35C2×C22⋊Q8×C83—21no—
36C22×C42×C83C2211yescharacteristic
37#391×C83—21no—
38#389×C83—21no—
39#389×C83C2212yes—
40#384×C83C2211yescharacteristic
41#380×C83—21no—
42#380×C83C2212yes—
43C22×D332C2311yescharacteristic
44C22×C22⋊D83—21no—
45C22×C4×D83—21no—
46C2×#13050574—24no—
47C2×#13050574—24no—
48C2×#13050574—42no—
49C22×C83⋊C8C2312yes—
50C22×C83⋊C8C2312yes—
51#14581766—22no—
52#14581766Q814yescomplemented
53C4×D332—42no—
54C4×C22⋊D83—44no—
55C2×C4⋊Dic83C2311yescharacteristic
56C2×C4⋊Dic83C2311yescharacteristic
57C2×C4⋊Dic83C2311yescharacteristic
58C2×Dic83⋊C4—21no—
59C2×C22⋊Dic83—21no—
60C2×(C2×D83)⋊C4—21no—
61C2×(C2×D83)⋊C4—21no—
62C2×(C2×D83)⋊C4—21no—
63C2×(C2×D83)⋊C4—21no—
64C2×C4×Dic83—21no—
65#14581725—24no—
66#14581723—42no—
67#14581712—24no—
68#14581712—42no—
69C83⋊#95—21no—
70C83⋊#95—21no—
71C83⋊#95—21no—
72C22×Q8×C83—21no—
73C23×C4×C83D411yescharacteristic
74C23×C4×C83D411yescharacteristic
75C23×C4×C83C2311yescharacteristic
76C4⋊Q8×C83—42no—
77C22⋊Q8×C83—44no—
78C2×C4⋊C4×C83—21no—
79C2×C4⋊C4×C83—22no—
80C2×C4⋊C4×C83—21no—
81C2×C4⋊C4×C83—21no—
82C2×C4⋊C4×C83C2312yes—
83C2×C4⋊C4×C83C2312yes—
84C2×C22⋊C4×C83—22no—
85C2×C42×C83—42no—
86C2×C42×C83D411yescharacteristic
87C2×C42×C83D411yescharacteristic
88C2×C42×C83C2311yescharacteristic
89C2×C42×C83C2311yescharacteristic
90#95×C83—22no—
91#95×C83—22no—
92C23×D83—21no—
93C2×D332—22no—
94C2×D332—22no—
95C2×D332—21no—
96C2×D332C2×Q812yes—
97C2×C22⋊D83—24no—
98C2×C22⋊D83—42no—
99C2×C22⋊D83—42no—
100C22×Dic83—21no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C2×C4×D83—42no—
104C2×C4×D83—42no—
105C2×C4×D83—42no—
106#13050574—48no—
107C2×C83⋊C8—22no—
108C2×C83⋊C8—22no—
109C2×C83⋊C8—22no—
110C2×C83⋊C8—22no—
111C2×C83⋊C8—22no—
112C2×C83⋊C8#4114yes—
113C4⋊Dic83—21no—
114C4⋊Dic83—22no—
115C4⋊Dic83—22no—
116C4⋊Dic83C2×Q812yes—
117Dic83⋊C4—42no—
118C22⋊Dic83—42no—
119(C2×D83)⋊C4—24no—
120(C2×D83)⋊C4—42no—
121(C2×D83)⋊C4—42no—
122(C2×D83)⋊C4—42no—
123C4×Dic83—42no—
124C24×C83C2×D411yescharacteristic
125C2×Q8×C83—42no—
126C2×Q8×C83—42no—
127C2×Q8×C83—42no—
128C22×C4×C83—22no—
129C22×C4×C83—22no—
130C22×C4×C83—22no—
131C22×C4×C83—22no—
132C22×C4×C83—22no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—41no—
136C22×C4×C83—41no—
137C22×C4×C83—41no—
138C22×C4×C83C2×D411yescharacteristic
139C22×C4×C83C2×D411yescharacteristic
140C22×C4×C83C2×D411yescharacteristic
141C22×C4×C83C2×D411yescharacteristic
142C22×C4×C83C2×D411yescharacteristic
143C22×C4×C83C2×Q812yes—
144C22×C4×C83#4112yes—
145C22×C4×C83C2411yescharacteristic
146C4⋊C4×C83—22no—
147C4⋊C4×C83—22no—
148C4⋊C4×C83—22no—
149C4⋊C4×C83—22no—
150C4⋊C4×C83—44no—
151C4⋊C4×C83—42no—
152C4⋊C4×C83—42no—
153C4⋊C4×C83—42no—
154C22⋊C4×C83—44no—
155C42×C83—22no—
156C42×C83—22no—
157C42×C83—41no—
158C42×C83—41no—
159C23×C83—21no—
160C23×C83—21no—
161C23×C83—21no—
162C23×C83—21no—
163C23×C83—21no—
164C23×C83—21no—
165C23×C83C22⋊Q812yes—
166C23×C83C22×D411yescharacteristic
167C22×D83—22no—
168C22×D83—41no—
169C22×D83—41no—
170C22×D83—41no—
171C22×D83—41no—
172C22×D83—41no—
173C22×D83—41no—
174Q8×C83—84no—
175C2×C4×C83—21no—
176C2×C4×C83—21no—
177C2×C4×C83—21no—
178C2×C4×C83—21no—
179C2×C4×C83—21no—
180C2×C4×C83—21no—
181C2×C4×C83—21no—
182C2×C4×C83—21no—
183C2×C4×C83—24no—
184C2×C4×C83—24no—
185C2×C4×C83—21no—
186C2×C4×C83—21no—
187C2×C4×C83—21no—
188C2×C4×C83—44no—
189C2×C4×C83—44no—
190C2×C4×C83—42no—
191C2×C4×C83—42no—
192C2×C4×C83—42no—
193C2×C4×C83—42no—
194C2×C4×C83—42no—
195C2×C4×C83—42no—
196C2×C4×C83C22⋊Q812yes—
197C2×C4×C83AGL(1,ℤ/8)12yes—
198C2×C4×C83AGL(1,ℤ/8)12yes—
199C2×C4×C83#13712yes—
200C2×C4×C83C22×Q811yescharacteristic
201C2×C4×C83C2×#4112yes—
202C2×C4×C83C2×#4111yescharacteristic
203C2×C4×C83H(5,𝔽2)12yes—
204C22⋊D83—48no—
205C2×Dic83—22no—
206C2×Dic83—41no—
207C2×Dic83—41no—
208C2×Dic83—41no—
209C2×Dic83—41no—
210C2×Dic83—41no—
211C2×Dic83—41no—
212C2×Dic83—41no—
213C2×Dic83—41no—
214D332—24no—
215D332—42no—
216C4×D83—84no—
217C83⋊C8—24no—
218C83⋊C8—42no—
219C22×C83—22no—
220C22×C83—22no—
221C22×C83—22no—
222C22×C83—41no—
223C22×C83—41no—
224C22×C83—41no—
225C22×C83—41no—
226C22×C83C2×C22⋊Q811yescharacteristic
227C22×C83#53211yescharacteristic
228C22×C83#53411yescharacteristic
229C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C83C2×#13711yescharacteristic
232C22×C83#57511yescharacteristic
233C2×D83—42no—
234C2×D83—42no—
235C2×D83—42no—
236C2×D83—81no—
237C2×D83—81no—
238C2×D83—81no—
239C2×D83—81no—
240C4×C83—22no—
241C4×C83—22no—
242C4×C83—22no—
243C4×C83—22no—
244C4×C83—41no—
245C4×C83—82no—
246C4×C83—82no—
247C4×C83#55112yes—
248C4×C83#55412yes—
249Dic83—42no—
250Dic83—81no—
251Dic83—81no—
252#29163—831nomaximal, Sylow
253C2×C83—42no—
254C2×C83#256411yescharacteristic
255C2×C83#307711yescharacteristic
256C2×C83#307911yescharacteristic
257C2×C83#317211yescharacteristic
258C2×C83#318411yescharacteristic
259C2×C83#319311yescharacteristic
260C2×C83#319411yescharacteristic
261D83—82no—
262#2857D8311yescharacteristic, complemented
263C4×#121—831no—
264#2099—831no—
265#2093—831no—
266#2058—831no—
267#2023—831no—
268#1921—832no—
269#1895—832no—
270#1864—832no—
271#1809—832no—
272#1728—831no—
273C83#2916311yescharacteristic, complemented, Sylow
274C22×M16—832no—
275C2×C4⋊Q8C2×D8311yescharacteristic
276C2×C22⋊Q8—21no—
277C2×#121—831no—
278C2×C4×D4—831no—
279C2×C4×D4—1661no—
280C22×C42C2×D8311yescharacteristic
281C2×#107—832no—
282C2×#106—832no—
283C2×#102—832no—
284C2×#102—832no—
285#391—21no—
286#390—831no—
287#389—21no—
288#389—831no—
289#389C2×D8312yes—
290#388—1661no—
291#386—1661no—
292#384C2×D8311yescharacteristic
293#380—21no—
294#380—831no—
295#380C2×D8312yes—
296C4×C4⋊C4—831no—
297C4×C22⋊C4—1661no—
298#340—832no—
299#340—832no—
300#340—832no—
301#340—832no—
302#336—831no—
303#336—831no—
304#336—832no—
305C22×Q8—21no—
306C22×D4—831no—
307C22×D4—1661no—
308C23×C4—1661no—
309C23×C4C22⋊D8311yescharacteristic
310C23×C4C22⋊D8311yescharacteristic
311C23×C4C22×D8311yescharacteristic
312C2×M16—3322no—
313C2×M16—1664no—
314C2×M16—1664no—
315C22×C8—832no—
316C22×C8—832no—
317C4⋊Q8—42no—
318C22⋊Q8—44no—
319#121—834no—
320#121—1662no—
321C4×D4—3322no—
322C4×D4—3324no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—831no—
328C2×C4⋊C4—831no—
329C2×C4⋊C4—831no—
330C2×C4⋊C4—1661no—
331C2×C4⋊C4C22×D8312yes—
332C2×C4⋊C4C22×D8312yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—1661no—
335C2×C22⋊C4—1661no—
336C2×C22⋊C4—1661no—
337C2×C22⋊C4—1661no—
338C2×C22⋊C4—1661no—
339C2×C42—42no—
340C2×C42—1661no—
341C2×C42C22⋊D8311yescharacteristic
342C2×C42C22⋊D8311yescharacteristic
343C2×C42C22×D8311yescharacteristic
344C2×C42C22×D8311yescharacteristic
345#107—1664no—
346#106—3322no—
347#102—3322no—
348#102—1664no—
349#95—22no—
350#95—22no—
351#95—1661no—
352#95—1661no—
353#95—1661no—
354C24—1661no—
355C24C2×C22⋊D8311yescharacteristic
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×D4—3322no—
360C2×D4—3322no—
361C2×D4—832no—
362C2×D4—1664no—
363C2×D4—1662no—
364C2×D4—1661no—
365C2×D4—1662no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—3322no—
377C22×C4—3322no—
378C22×C4—3322no—
379C22×C4—1661no—
380C22×C4—1661no—
381C22×C4—1661no—
382C22×C4C2×C22⋊D8311yescharacteristic
383C22×C4C2×C22⋊D8311yescharacteristic
384C22×C4C2×C22⋊D8311yescharacteristic
385C22×C4C2×C22⋊D8311yescharacteristic
386C22×C4C2×C22⋊D8311yescharacteristic
387C22×C4Q8×D8312yes—
388C22×C4#1305059112yes—
389C22×C4C23×D8311yescharacteristic
390M16—3328no—
391C2×C8—834no—
392C2×C8—1662no—
393C2×C8—1662no—
394C2×C8—1662no—
395C2×C8—1662no—
396C2×C8—1662no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—3322no—
406C4⋊C4—832no—
407C4⋊C4—1662no—
408C4⋊C4—1662no—
409C4⋊C4—1661no—
410C22⋊C4—44no—
411C22⋊C4—3322no—
412C22⋊C4—3322no—
413C22⋊C4—3322no—
414C22⋊C4—3322no—
415C22⋊C4—1664no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—3322no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—3321no—
428C23—3321no—
429C23—3321no—
430C23—3321no—
431C23—3321no—
432C23—3321no—
433C23—1662no—
434C23#1458176812yes—
435C23C22×C22⋊D8311yescharacteristic
436Q8—84no—
437D4—3328no—
438D4—3322no—
439D4—1664no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—3321no—
462C2×C4—3321no—
463C2×C4—3321no—
464C2×C4—3321no—
465C2×C4—3321no—
466C2×C4—3321no—
467C2×C4—3321no—
468C2×C4—3321no—
469C2×C4—6644no—
470C2×C4—1662no—
471C2×C4#1458176812yes—
472C2×C4#1458181712yes—
473C2×C4#1458181812yes—
474C2×C4#1458182312yes—
475C2×C4C2×Q8×D8311yescharacteristic
476C2×C4#41×D8312yes—
477C2×C4C2×#1305059111yescharacteristic
478C2×C4#1458183712yes—
479C8—3322no—
480C8—1664no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—3322no—
489C22—3322no—
490C22—3322no—
491C22—6641no—
492C22—6641no—
493C22—6641no—
494C22—6641no—
495C22C2×#1458176811yescharacteristic
496C22#1912824711yescharacteristic
497C22#1912826011yescharacteristic
498C22C2×#1458181711yescharacteristic
499C22C2×#1458181811yescharacteristic
500C22C2×#1458182311yescharacteristic
501C22#1912849911yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—41no—
507C4—82no—
508C4—82no—
509C4—3322no—
510C4—6641no—
511C4—6641no—
512C4#1912837812yes—
513C4#1912840512yes—
514C2—42no—
515C2—6642no—
516C2#2491264511yescharacteristic
517C2#2491725611yescharacteristic
518C2#2491727211yescharacteristic
519C2#2491806311yescharacteristic
520C2#2491819711yescharacteristic
521C2#2491826511yescharacteristic
522C2#2491827711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2857C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
hypercenter#2857D83
hypocenterC83#29163
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. #24909020C2semidirect
3. #24908992C2semidirect
4. #24908793C2semidirect
5. #24908596C2semidirect
6. #24908052C2semidirect
8. #24907723C2semidirect
9. #24907494C2semidirect
10. #24907157C2semidirect
11. C83×#2857C2semidirect
52. #14581766Q8semidirect
262. #2857D83semidirect
273. C83#29163semidirect

Maximal subgroups

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

subgroupindex
C4×#145817662
#249090202
#249089922
#249087932
#249085962
#249080522
#249078992
#249077232
#249074942
#249071572
C83×#28572
#2916383