Finite Group Databasebeta

← #35504456

Subgroups — #35504456

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
1#24915815C211yesmaximal, characteristic, complemented
2#24909021C211yesmaximal, characteristic, complemented
3#24908991C211yesmaximal, characteristic, complemented
4#24908796C211yesmaximal, characteristic, complemented
5#24908599C211yesmaximal, characteristic, complemented
6#24908055C212yesmaximal, complemented
7#24907902C212yesmaximal
8#24907724C212yesmaximal, complemented
9#24907494C212yesmaximal, complemented
10#24907157C211yesmaximal, characteristic, complemented
11C4×#121×C83C211yesmaximal, characteristic
12C22×#13050574C2212yes—
13C2×#14581792C2211yescharacteristic
14C2×#14581771—21no—
15C2×#14581725C2212yes—
16C2×#14581723C2212yes—
17C2×#14581713C2212yes—
18C2×#14581713C2212yes—
19#19127607—21no—
20#19127599—21no—
21#19127599C2212yes—
22#19127571C2211yescharacteristic
23#19127561—21no—
24#19127561C2212yes—
25#19127463C2212yes—
26#19127463C2212yes—
27#19127463C2212yes—
28#19127463C2212yes—
29#19127456C2212yes—
30#19127456C2211yescharacteristic
31#19127456C2211yescharacteristic
32C2×#121×C83C2211yescharacteristic
33C2×C4×D4×C83—21no—
34C2×C4×D4×C83C2211yescharacteristic
35C22×C42×C83C2211yescharacteristic
36#390×C83C2211yescharacteristic
37#389×C83C2211yescharacteristic
38#388×C83—21no—
39#386×C83—21no—
40#380×C83C2211yescharacteristic
41C4×C4⋊C4×C83C2211yescharacteristic
42C4×C22⋊C4×C83—21no—
43C22×Dic166—21no—
44C2×#13050574—24no—
45C2×#13050574—24no—
46C2×#13050574—42no—
47C22×C83⋊C8C2312yes—
48C22×C83⋊C8C2312yes—
49#14581792—42no—
50#14581771—44no—
51C2×C4⋊Dic83—21no—
52C2×C4⋊Dic83—21no—
53C2×C4⋊Dic83—21no—
54C2×C4⋊Dic83C2312yes—
55C2×C4⋊Dic83C2312yes—
56C2×Dic83⋊C4—22no—
57C2×C22⋊Dic83—22no—
58#14581725—24no—
59#14581723—42no—
60#14581713—24no—
61#14581713—42no—
62C83⋊#95—22no—
63C83⋊#95—22no—
64C22×D4×C83—21no—
65C22×D4×C83C2311yescharacteristic
66C23×C4×C83—21no—
67C23×C4×C83D411yescharacteristic
68C23×C4×C83D411yescharacteristic
69C23×C4×C83C2311yescharacteristic
70#121×C83—22no—
71#121×C83Q814yescomplemented
72C4×D4×C83—42no—
73C4×D4×C83—44no—
74C2×C4⋊C4×C83—21no—
75C2×C4⋊C4×C83C2311yescharacteristic
76C2×C4⋊C4×C83C2311yescharacteristic
77C2×C4⋊C4×C83C2311yescharacteristic
78C2×C22⋊C4×C83—21no—
79C2×C22⋊C4×C83—21no—
80C2×C22⋊C4×C83—21no—
81C2×C22⋊C4×C83—21no—
82C2×C22⋊C4×C83—21no—
83C2×C42×C83—21no—
84C2×C42×C83—42no—
85C2×C42×C83D411yescharacteristic
86C2×C42×C83D411yescharacteristic
87C2×C42×C83C2311yescharacteristic
88C2×C42×C83C2311yescharacteristic
89#95×C83—21no—
90#95×C83—21no—
91#95×C83—21no—
92C2×Dic166—42no—
93C2×Dic166—42no—
94C2×Dic166—42no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97#13050574—48no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8—22no—
100C2×C83⋊C8—22no—
101C2×C83⋊C8—22no—
102C2×C83⋊C8—22no—
103C2×C83⋊C8#4114yes—
104C4⋊Dic83—22no—
105C4⋊Dic83—22no—
106C4⋊Dic83—22no—
107C4⋊Dic83—22no—
108C4⋊Dic83—42no—
109C4⋊Dic83—42no—
110C4⋊Dic83—42no—
111Dic83⋊C4—44no—
112C22⋊Dic83—44no—
113C24×C83—21no—
114C24×C83C2×D411yescharacteristic
115C2×D4×C83—24no—
116C2×D4×C83—22no—
117C2×D4×C83—21no—
118C2×D4×C83—22no—
119C2×D4×C83—42no—
120C2×D4×C83—42no—
121C2×D4×C83C2×Q812yes—
122C22×C4×C83—22no—
123C22×C4×C83—22no—
124C22×C4×C83—21no—
125C22×C4×C83—22no—
126C22×C4×C83—22no—
127C22×C4×C83—22no—
128C22×C4×C83—21no—
129C22×C4×C83—21no—
130C22×C4×C83—41no—
131C22×C4×C83—41no—
132C22×C4×C83—41no—
133C22×C4×C83—42no—
134C22×C4×C83—42no—
135C22×C4×C83—42no—
136C22×C4×C83C2×D411yescharacteristic
137C22×C4×C83C2×D411yescharacteristic
138C22×C4×C83C2×D411yescharacteristic
139C22×C4×C83C2×D411yescharacteristic
140C22×C4×C83C2×D411yescharacteristic
141C22×C4×C83C2×Q812yes—
142C22×C4×C83#4112yes—
143C22×C4×C83C2411yescharacteristic
144C4⋊C4×C83—21no—
145C4⋊C4×C83—22no—
146C4⋊C4×C83—22no—
147C4⋊C4×C83—42no—
148C4⋊C4×C83C2×Q812yes—
149C22⋊C4×C83—24no—
150C22⋊C4×C83—42no—
151C22⋊C4×C83—42no—
152C22⋊C4×C83—42no—
153C22⋊C4×C83—42no—
154C42×C83—22no—
155C42×C83—22no—
156C42×C83—41no—
157C42×C83—41no—
158C42×C83—42no—
159C23×C83—21no—
160C23×C83—21no—
161C23×C83—21no—
162C23×C83—21no—
163C23×C83—21no—
164C23×C83—21no—
165C23×C83—22no—
166C23×C83—41no—
167C23×C83—41no—
168C23×C83—41no—
169C23×C83—41no—
170C23×C83—41no—
171C23×C83—41no—
172C23×C83C22⋊Q812yes—
173C23×C83C22×D411yescharacteristic
174D4×C83—24no—
175D4×C83—48no—
176D4×C83—42no—
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—21no—
184C2×C4×C83—21no—
185C2×C4×C83—24no—
186C2×C4×C83—24no—
187C2×C4×C83—22no—
188C2×C4×C83—21no—
189C2×C4×C83—21no—
190C2×C4×C83—21no—
191C2×C4×C83—44no—
192C2×C4×C83—44no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—41no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83—41no—
200C2×C4×C83—41no—
201C2×C4×C83—84no—
202C2×C4×C83C22⋊Q812yes—
203C2×C4×C83AGL(1,ℤ/8)12yes—
204C2×C4×C83AGL(1,ℤ/8)12yes—
205C2×C4×C83#13712yes—
206C2×C4×C83C22×Q811yescharacteristic
207C2×C4×C83C2×#4112yes—
208C2×C4×C83C2×#4111yescharacteristic
209C2×C4×C83H(5,𝔽2)12yes—
210C2×Dic83—42no—
211C2×Dic83—42no—
212C2×Dic83—42no—
213C2×Dic83—42no—
214C2×Dic83—42no—
215C2×Dic83—42no—
216Dic166—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×C83—42no—
227C22×C83—42no—
228C22×C83—42no—
229C22×C83—81no—
230C22×C83—81no—
231C22×C83—81no—
232C22×C83—81no—
233C22×C83C2×C22⋊Q811yescharacteristic
234C22×C83#53211yescharacteristic
235C22×C83#53411yescharacteristic
236C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C83C2×#13711yescharacteristic
239C22×C83#57511yescharacteristic
240C4×C83—22no—
241C4×C83—22no—
242C4×C83—22no—
243C4×C83—22no—
244C4×C83—42no—
245C4×C83—41no—
246C4×C83—81no—
247C4×C83—81no—
248C4×C83#55112yes—
249C4×C83#55412yes—
250Dic83—82no—
251Dic83—82no—
252#29163—831nomaximal, Sylow
253C2×C83—42no—
254C2×C83—82no—
255C2×C83#256411yescharacteristic
256C2×C83#307711yescharacteristic
257C2×C83#307911yescharacteristic
258C2×C83#317211yescharacteristic
259C2×C83#318411yescharacteristic
260C2×C83#319311yescharacteristic
261C2×C83#319411yescharacteristic
262#2857—831no—
263C4×#121D8311yescharacteristic
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⋊Q8—831no—
276C2×C22⋊Q8—1661no—
277C2×#121C2×D8311yescharacteristic
278C2×C4×D4—21no—
279C2×C4×D4C2×D8311yescharacteristic
280C22×C42C2×D8311yescharacteristic
281C2×#107—832no—
282C2×#106—832no—
283C2×#102—832no—
284C2×#102—832no—
285#391—1661no—
286#390C2×D8311yescharacteristic
287#389—832no—
288#389—1661no—
289#389C2×D8311yescharacteristic
290#388—21no—
291#386—21no—
292#384—831no—
293#380—832no—
294#380—1661no—
295#380C2×D8311yescharacteristic
296C4×C4⋊C4C2×D8311yescharacteristic
297C4×C22⋊C4—21no—
298#340—832no—
299#340—832no—
300#340—832no—
301#340—832no—
302#336—831no—
303#336—831no—
304#336—832no—
305C22×Q8—1661no—
306C22×D4—21no—
307C22×D4C22×D8311yescharacteristic
308C23×C4—21no—
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—3322no—
318C22⋊Q8—3324no—
319#121—22no—
320#121Dic16614yescomplemented
321C4×D4—42no—
322C4×D4—44no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—832no—
325C2×C4⋊C4—832no—
326C2×C4⋊C4—1661no—
327C2×C4⋊C4—1662no—
328C2×C4⋊C4—1661no—
329C2×C4⋊C4—1661no—
330C2×C4⋊C4C22×D8311yescharacteristic
331C2×C4⋊C4C22×D8311yescharacteristic
332C2×C4⋊C4C22×D8311yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1662no—
339C2×C42—21no—
340C2×C42—42no—
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—21no—
350#95—21no—
351#95—21no—
352#95—1662no—
353#95—1662no—
354C24—21no—
355C24C2×C22⋊D8311yescharacteristic
356C2×Q8—3322no—
357C2×Q8—3322no—
358C2×Q8—3322no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4C2×Dic16612yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—1662no—
381C22×C4—1662no—
382C22×C4C2×C22⋊D8311yescharacteristic
383C22×C4C2×C22⋊D8311yescharacteristic
384C22×C4C2×C22⋊D8311yescharacteristic
385C22×C4C2×C22⋊D8311yescharacteristic
386C22×C4C2×C22⋊D8311yescharacteristic
387C22×C4C2×Dic16612yes—
388C22×C4#1305059012yes—
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—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—3324no—
402C4⋊C4—3322no—
403C4⋊C4—3322no—
404C4⋊C4—3322no—
405C4⋊C4—1662no—
406C4⋊C4—1662no—
407C4⋊C4—1662no—
408C4⋊C4—1662no—
409C4⋊C4C2×Dic16612yes—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—3324no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—42no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23#1458177112yes—
435C23C22×C22⋊D8311yescharacteristic
436Q8—6644no—
437D4—24no—
438D4—48no—
439D4—42no—
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—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—84no—
465C2×C4—3322no—
466C2×C4—3322no—
467C2×C4—3322no—
468C2×C4—3322no—
469C2×C4—3322no—
470C2×C4—3322no—
471C2×C4#1458177112yes—
472C2×C4#1458181712yes—
473C2×C4#1458181812yes—
474C2×C4#1458182212yes—
475C2×C4C22×Dic16611yescharacteristic
476C2×C4#41×D8312yes—
477C2×C4C2×#1305059011yescharacteristic
478C2×C4#1458183712yes—
479C8—3322no—
480C8—1664no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22C2×#1458177111yescharacteristic
496C22#1912824711yescharacteristic
497C22#1912826011yescharacteristic
498C22C2×#1458181711yescharacteristic
499C22C2×#1458181811yescharacteristic
500C22C2×#1458182211yescharacteristic
501C22#1912849911yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—41no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—6642no—
511C4—6642no—
512C4#1912838012yes—
513C4#1912841012yes—
514C2—42no—
515C2—82no—
516C2#2491263911yescharacteristic
517C2#2491726011yescharacteristic
518C2#2491727811yescharacteristic
519C2#2491807011yescharacteristic
520C2#2491819711yescharacteristic
521C2#2491826911yescharacteristic
522C2#2491827311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC4×#121×C83C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenterC4×#121D83
hypocenterC83#29163
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #24915815C2semidirect
2. #24909021C2semidirect
3. #24908991C2semidirect
4. #24908796C2semidirect
5. #24908599C2semidirect
6. #24908055C2semidirect
8. #24907724C2semidirect
9. #24907494C2semidirect
10. #24907157C2semidirect
71. #121×C83Q8semidirect
273. C83#29163semidirect
320. #121Dic166semidirect

Maximal subgroups

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

subgroupindex
#249158152
#249090212
#249089912
#249087962
#249085992
#249080552
#249079022
#249077242
#249074942
#249071572
C4×#121×C832
#2916383