Finite Group Databasebeta

← #35406242

Subgroups — #35406242

Order 21248.

1866 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128265C211yesmaximal, characteristic
2C22×#14581708C212yesmaximal, complemented
3#24915908C211yesmaximal, characteristic, complemented
4#24907926C218yesmaximal, complemented
5#24907201C212yesmaximal, complemented
6C2×#568×C83C211yesmaximal, characteristic, complemented
7C23×D332C2211yescharacteristic
8C22×#13050574C2212yes—
9#19128265—24no—
10#19128265—24no—
11C2×#14581788C412yes—
12C2×#14581781C412yes—
13C2×#14581766C2214yes—
14C2×#14581760C414yes—
15C2×#14581751C2211yescharacteristic
16C2×#14581708—28no—
17C2×#14581708C2218yes—
18#19127668—24no—
19#19127665C2214yescomplemented
20#19127469C2218yescomplemented
21C22×#41×C83C2211yescharacteristic
22#568×C83—28no—
23C2×#131×C83—24no—
24C22×M16×C83C2212yes—
25C22×M16×C83C2214yescomplemented
26C24×D83C2×C412yes—
27C22×D332—24no—
28C22×D332—22no—
29C22×D332—22no—
30C22×D332—24no—
31C22×D332C2×C412yes—
32C22×D332C2312yes—
33C22×C22⋊D83C2×C418yes—
34C2×#13050574—22no—
35C2×#13050574—24no—
36C2×#13050574—24no—
37C2×#13050574C2314yes—
38C22×C83⋊C8—22no—
39#14581788—28no—
40#14581781—28no—
41#14581766—28no—
42#14581766—28no—
43#14581760—216no—
44#14581751—22no—
45#14581751—22no—
46C2×C4⋊Dic83C2312yes—
47C2×C22⋊Dic83C2×C412yes—
48C2×(C2×D83)⋊C4C2×C418yes—
49C2×C4×Dic83C2×C412yes—
50#14581708—216no—
51C22⋊(C83⋊C8)—28no—
52C2×#41×C83—24no—
53C2×#41×C83—24no—
54C2×#41×C83D414yes—
55C2×#41×C83D414yes—
56C22×Q8×C83C2×C411yescharacteristic
57C22×D4×C83C2×C412yes—
58C22×D4×C83C2×C411yescharacteristic
59C23×C4×C83C2312yes—
60C23×C4×C83C2311yescharacteristic
61#131×C83—232no—
62C2×M16×C83—22no—
63C2×M16×C83—24no—
64C2×M16×C83—24no—
65C2×M16×C83—216no—
66C2×M16×C83D418yescomplemented
67C2×M16×C83C2314yes—
68C22×C8×C83—22no—
69C22×C8×C83—24no—
70C23×D83—24no—
71C23×D83—28no—
72C23×D83—84no—
73C23×D83C22×C414yes—
74C2×D332—24no—
75C2×D332—28no—
76C2×D332—22no—
77C2×D332—22no—
78C2×D332—24no—
79C2×D332—24no—
80C2×D332—44no—
81C2×D332—44no—
82C2×D332—44no—
83C2×D332—44no—
84C2×C22⋊D83—216no—
85C2×C22⋊D83—88no—
86C22×Dic83C22×C414yes—
87#13050574—24no—
88#13050574—24no—
89#13050574—44no—
90C2×C83⋊C8—24no—
91C2×C83⋊C8—44no—
92C4⋊Dic83—22no—
93C4⋊Dic83—22no—
94C22⋊Dic83—24no—
95(C2×D83)⋊C4—216no—
96C4×Dic83—24no—
97C24×C83C22×C412yes—
98C24×C83C22×C411yescharacteristic
99#41×C83—28no—
100#41×C83—28no—
101#41×C83—28no—
102#41×C83—28no—
103C2×Q8×C83—24no—
104C2×Q8×C83C22⋊C414yes—
105C2×D4×C83—28no—
106C2×D4×C83—24no—
107C2×D4×C83C22⋊C418yes—
108C2×D4×C83C22⋊C414yes—
109C22×C4×C83—24no—
110C22×C4×C83—24no—
111C22×C4×C83C22×C412yes—
112C22×C4×C83C22×C411yescharacteristic
113C22×C4×C83C2×D414yes—
114C22×C4×C83C2×D412yes—
115C22×C4×C83C2×D414yes—
116C22×C4×C83C2×D414yes—
117C22×C4×C83C2×D412yes—
118C22×C4×C83#4114yes—
119C22×C4×C83C2411yescharacteristic
120M16×C83—216no—
121M16×C83—28no—
122M16×C83—24no—
123M16×C83C2×D418yes—
124C2×C8×C83—216no—
125C2×C8×C83—28no—
126C2×C8×C83—24no—
127C23×C83—24no—
128C23×C83—42no—
129C23×C83—44no—
130C23×C83C2×C22⋊C414yes—
131C23×C83C2×C22⋊C412yes—
132C23×C83C4×D414yes—
133C23×C83C23×C411yescharacteristic
134C22×D83—24no—
135C22×D83—28no—
136C22×D83—48no—
137C22×D83—48no—
138C22×D83—88no—
139C22×D83—816no—
140Q8×C83—28no—
141D4×C83—216no—
142D4×C83—28no—
143C2×C4×C83—24no—
144C2×C4×C83—24no—
145C2×C4×C83—28no—
146C2×C4×C83—24no—
147C2×C4×C83C2×C22⋊C414yes—
148C2×C4×C83C2×C22⋊C412yes—
149C2×C4×C83C4×D414yes—
150C2×C4×C83#12014yes—
151C2×C4×C83#12114yes—
152C2×C4×C83#12114yes—
153C2×C4×C83#12314yes—
154C2×C4×C83C22×D412yes—
155C2×C4×C83C22×D411yescharacteristic
156C2×C4×C83C22×D411yescharacteristic
157C2×C4×C83C2×#4112yes—
158C22⋊D83—816no—
159C2×Dic83—24no—
160C2×Dic83—44no—
161D332—42no—
162D332—44no—
163D332—42no—
164D332—48no—
165C8×C83—28no—
166C83⋊C8—44no—
167C22×C83—22no—
168C22×C83—24no—
169C22×C83—28no—
170C22×C83—24no—
171C22×C83—44no—
172C22×C83—84no—
173C22×C83#38614yes—
174C22×C83C22×C22⋊C411yescharacteristic
175C22×C83C2×C4×D412yes—
176C2×D83—48no—
177C2×D83—44no—
178C2×D83—816no—
179C2×D83—88no—
180C4×C83—24no—
181C4×C83#38614yes—
182C4×C83C2×#12011yescharacteristic
183C4×C83C2×#12112yes—
184C4×C83C2×#12311yescharacteristic
185Dic83—44no—
186#20990—831nomaximal, Sylow
187C2×C83—22no—
188C2×C83—24no—
189C2×C83—44no—
190C2×C83C2×#38611yescharacteristic
191C2×C83#288812yes—
192D83—84no—
193C2×#568D8311yescharacteristic, complemented
194C2×#535—831no—
195C22×#100—832no—
196#2879—831no—
197#1900—838no—
198#1742—832no—
199C83#2099011yescharacteristic, complemented, Sylow
200C22×#41C2×D8311yescharacteristic
201C23×D4—831no—
202#568—28no—
203C2×#131—24no—
204C22×M16—832no—
205C22×M16C2×D8312yes—
206C22×M16C2×D8314yescomplemented
207#535—1664no—
208#535—1664no—
209C2×#127—832no—
210C2×#124—832no—
211C2×#121—834no—
212C2×#120—834no—
213C2×#117—831no—
214C2×#100—838no—
215C2×#100—1668no—
216#408—1664no—
217#407—834no—
218#343—838no—
219C25—832no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41D33214yes—
223C2×#41C22⋊D8314yes—
224C22×Q8C4×D8311yescharacteristic
225C22×D4—838no—
226C22×D4—832no—
227C22×D4—832no—
228C22×D4—1662no—
229C22×D4—1664no—
230C22×D4—1662no—
231C22×D4—1664no—
232C22×D4C4×D8312yes—
233C22×D4C4×D8311yescharacteristic
234C23×C4C22×D8312yes—
235C23×C4C22×D8311yescharacteristic
236#131—232no—
237C2×M16—22no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—216no—
241C2×M16—834no—
242C2×M16—1664no—
243C2×M16—1664no—
244C2×M16—1662no—
245C2×M16C22⋊D8318yescomplemented
246C2×M16C22×D8314yes—
247C22×C8—22no—
248C22×C8—24no—
249C22×C8—1662no—
250#127—1668no—
251#124—1668no—
252#121—1668no—
253#121—1668no—
254#120—16616no—
255#117—1662no—
256#117—1662no—
257C2×C4⋊C4—832no—
258C2×C22⋊C4—832no—
259C2×C22⋊C4—838no—
260C2×C42—832no—
261#100—16616no—
262C22⋊C8—1668no—
263C24—834no—
264C24—6644no—
265C24—1664no—
266C24—1668no—
267C24C2×C4×D8312yes—
268C24C2×C4×D8311yescharacteristic
269#41—28no—
270#41—28no—
271#41—28no—
272#41—28no—
273C2×Q8—24no—
274C2×Q8(C2×D83)⋊C414yes—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—3324no—
278C2×D4—3324no—
279C2×D4—3324no—
280C2×D4—3324no—
281C2×D4—6648no—
282C2×D4—16616no—
283C2×D4—1664no—
284C2×D4—1662no—
285C2×D4—1662no—
286C2×D4—1664no—
287C2×D4—1664no—
288C2×D4—1668no—
289C2×D4(C2×D83)⋊C418yes—
290C2×D4(C2×D83)⋊C414yes—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—834no—
294C22×C4C2×C4×D8312yes—
295C22×C4C2×C4×D8311yescharacteristic
296C22×C4C2×C22⋊D8314yes—
297C22×C4C2×C22⋊D8314yes—
298C22×C4C2×C22⋊D8312yes—
299C22×C4C2×D33214yes—
300C22×C4C2×D33212yes—
301C22×C4#1305059014yes—
302C22×C4C23×D8311yescharacteristic
303M16—216no—
304M16—28no—
305M16—24no—
306M16—3324no—
307M16—1664no—
308M16—1664no—
309M16C2×C22⋊D8318yes—
310C2×C8—216no—
311C2×C8—28no—
312C2×C8—24no—
313C2×C8—3324no—
314C2×C8—1664no—
315C4⋊C4—1662no—
316C4⋊C4—1662no—
317C22⋊C4—1664no—
318C22⋊C4—16616no—
319C42—1664no—
320C23—24no—
321C23—42no—
322C23—44no—
323C23—3328no—
324C23—3328no—
325C23—6648no—
326C23—66416no—
327C23—1664no—
328C23—1668no—
329C23C2×(C2×D83)⋊C414yes—
330C23C2×(C2×D83)⋊C412yes—
331C23C4×C22⋊D8314yes—
332C23C22×C4×D8311yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336D4—3328no—
337D4—3322no—
338D4—3324no—
339D4—3322no—
340D4—66416no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—3324no—
346C2×C4—1664no—
347C2×C4C2×(C2×D83)⋊C414yes—
348C2×C4C2×(C2×D83)⋊C412yes—
349C2×C4C4×C22⋊D8314yes—
350C2×C4#1458176114yes—
351C2×C4#1458176614yes—
352C2×C4#1458176614yes—
353C2×C4#1458177514yes—
354C2×C4C22×C22⋊D8312yes—
355C2×C4C22×C22⋊D8311yescharacteristic
356C2×C4C22×D33211yescharacteristic
357C2×C4C2×#1305059012yes—
358C8—28no—
359C8—3324no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—44no—
365C22—84no—
366C22—3324no—
367C22—3328no—
368C22—6648no—
369C22—66416no—
370C22#1912758014yes—
371C22C22×(C2×D83)⋊C411yescharacteristic
372C22C2×C4×C22⋊D8312yes—
373C4—24no—
374C4—3324no—
375C4#1912758014yes—
376C4C2×#1458176111yescharacteristic
377C4C2×#1458176612yes—
378C4C2×#1458177511yescharacteristic
379C2—22no—
380C2—24no—
381C2—44no—
382C2—6644no—
383C2C2×#1912758011yescharacteristic
384C2#2491594112yes—
subgroupG/H
centerC22C22×(C2×D83)⋊C4
commutatorC23×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#568×C83C2
socleC22×C83C22×C22⋊C4
prime radicalC2×#19128265C2
cosocleC22×C4×C83C24
hypercenterC2×#568D83
hypocenterC83#20990
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C22×#14581708C2semidirect
3. #24915908C2semidirect
4. #24907926C2semidirect
5. #24907201C2semidirect
6. C2×#568×C83C2semidirect
19. #19127665C22semidirect
20. #19127469C22semidirect
25. C22×M16×C83C22semidirect
66. C2×M16×C83D4semidirect
193. C2×#568D83semidirect
199. C83#20990semidirect
206. C22×M16C2×D83semidirect
245. C2×M16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282652
C22×#145817082
#249159082
#249079262
#249072012
C2×#568×C832
#2099083