Finite Group Databasebeta

← #35413737

Subgroups — #35413737

Order 21248.

1374 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128251C211yesmaximal, characteristic
2#24909861C211yesmaximal, characteristic, complemented
3#24908389C212yesmaximal, complemented
4#24908385C212yesmaximal, complemented
5#24908274C214yesmaximal, complemented
6#24907404C214yesmaximal, complemented
7C2×#431×C83C211yesmaximal, characteristic, complemented
8C22×#13050590C2212yes—
9#19128251—24no—
10#19128251C418yescomplemented
11C2×#14581792C2211yescharacteristic
12C2×#14581789C2211yescharacteristic
13C2×#14581780C2212yes—
14C2×#14581766—21no—
15C2×#14581766—21no—
16C2×C4×D332—21no—
17C2×C4×D332—21no—
18C2×#14581717C2218yes—
19C2×#14581714C2218yes—
20#19127569—21no—
21#19127567C2212yes—
22#19127564—21no—
23#19127561C2212yes—
24C4×C4⋊Dic83C2212yes—
25C4×C22⋊Dic83—21no—
26#19127547—21no—
27C22×C42×C83C2211yescharacteristic
28#431×C83—24no—
29C2×C4⋊C8×C83C2212yes—
30C2×C22⋊C8×C83C2212yes—
31C2×C4×C8×C83C2212yes—
32C2×#13050590—24no—
33C2×#13050590—48no—
34C2×#13050590C2×C418yes—
35C22×Dic166C2312yes—
36C22×D332—21no—
37C22×D332—21no—
38C22×D332C2312yes—
39C22×C22⋊D83—22no—
40C22×C4×D83—22no—
41#14581792—22no—
42#14581792C2×C414yes—
43#14581789—22no—
44#14581789C2×C414yes—
45#14581780—24no—
46#14581780C2×C418yes—
47#14581766—28no—
48#14581766—28no—
49C4×D332—28no—
50C4×D332—28no—
51C2×C4⋊Dic83—21no—
52C2×C4⋊Dic83—21no—
53C2×C4⋊Dic83C2314yes—
54C2×C22⋊Dic83—22no—
55C2×(C2×D83)⋊C4—22no—
56C2×(C2×D83)⋊C4—22no—
57C2×C4×Dic83—22no—
58#14581717—48no—
59#14581714—48no—
60C83⋊#95—22no—
61C83⋊#95—22no—
62C83⋊#95—22no—
63C83⋊#95—22no—
64C23×C4×C83D411yescharacteristic
65C23×C4×C83D411yescharacteristic
66C23×C4×C83C2311yescharacteristic
67C22×C8×C83C2314yes—
68C2×C42×C83C2×C414yes—
69C2×C42×C83D411yescharacteristic
70C2×C42×C83D411yescharacteristic
71C2×C42×C83D414yes—
72C2×C42×C83C2311yescharacteristic
73C2×C42×C83C2311yescharacteristic
74C4⋊C8×C83—24no—
75C22⋊C8×C83—24no—
76C4×C8×C83—24no—
77C23×D83—22no—
78#13050590—416no—
79#13050590—416no—
80C2×Dic166—28no—
81C2×Dic166—22no—
82C2×Dic166C22×C414yes—
83C2×D332—28no—
84C2×D332—24no—
85C2×D332—24no—
86C2×D332—22no—
87C2×D332—44no—
88C2×D332—44no—
89C2×D332C22×C414yes—
90C2×C22⋊D83—28no—
91C2×C22⋊D83—48no—
92C22×Dic83—22no—
93C22×Dic83—24no—
94C2×C4×D83—28no—
95C2×C4×D83—48no—
96C4⋊Dic83—24no—
97C4⋊Dic83—24no—
98C4⋊Dic83—24no—
99C4⋊Dic83—24no—
100C22⋊Dic83—44no—
101(C2×D83)⋊C4—28no—
102(C2×D83)⋊C4—28no—
103C4×Dic83—44no—
104C24×C83C2×D411yescharacteristic
105C22×C4×C83—24no—
106C22×C4×C83—24no—
107C22×C4×C83—42no—
108C22×C4×C83C22⋊C414yes—
109C22×C4×C83C22⋊C414yes—
110C22×C4×C83C22×C412yes—
111C22×C4×C83C2×D412yes—
112C22×C4×C83C2×D411yescharacteristic
113C22×C4×C83C2×D411yescharacteristic
114C22×C4×C83C2×D411yescharacteristic
115C22×C4×C83C2×D411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83C2411yescharacteristic
118C2×C8×C83—24no—
119C2×C8×C83—28no—
120C42×C83C22⋊C414yes—
121C42×C83C22⋊C414yes—
122C42×C83C22×C412yes—
123C42×C83C22×C412yes—
124C42×C83C2×D412yes—
125C42×C83C2×D412yes—
126C23×C83—22no—
127C23×C83—22no—
128C23×C83—22no—
129C23×C83C2×C22⋊C412yes—
130C23×C83C22×D411yescharacteristic
131C22×D83—24no—
132C22×D83—44no—
133C22×D83—44no—
134C22×D83—44no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—24no—
140C2×C4×C83—24no—
141C2×C4×C83—28no—
142C2×C4×C83—44no—
143C2×C4×C83C2×C22⋊C412yes—
144C2×C4×C83C2×C22⋊C412yes—
145C2×C4×C83C2×C22⋊C412yes—
146C2×C4×C83C2×C22⋊C412yes—
147C2×C4×C83C2×C22⋊C412yes—
148C2×C4×C83#13514yes—
149C2×C4×C83C23×C411yescharacteristic
150C2×C4×C83C22×D411yescharacteristic
151C2×C4×C83H(5,𝔽2)14yes—
152C22⋊D83—416no—
153C2×Dic83—24no—
154C2×Dic83—44no—
155C2×Dic83—44no—
156C2×Dic83—44no—
157C2×Dic83—44no—
158D332—28no—
159D332—48no—
160D332—44no—
161D332—48no—
162C4×D83—416no—
163Dic166—28no—
164Dic166—44no—
165C8×C83—44no—
166C22×C83—22no—
167C22×C83—22no—
168C22×C83—22no—
169C22×C83—44no—
170C22×C83C22×C22⋊C411yescharacteristic
171C22×C83#53512yes—
172C22×C83C2×#13512yes—
173C22×C83#57512yes—
174C2×D83—44no—
175C2×D83—44no—
176C2×D83—44no—
177C2×D83—88no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—22no—
181C4×C83—22no—
182C4×C83#41614yes—
183C4×C83#51812yes—
184C4×C83#53512yes—
185Dic83—44no—
186Dic83—84no—
187#21807—831nomaximal, Sylow
188C2×C83—42no—
189C2×C83#230911yescharacteristic
190C2×C83C2×#41611yescharacteristic
191C2×C83#289611yescharacteristic
192C2×C83#321214yes—
193D83—84no—
194C2×#532—831no—
195C2×#431D8311yescharacteristic, complemented
196#2283—831no—
197#1982—832no—
198#1981—832no—
199#1958—834no—
200#1788—834no—
201C83#2180711yescharacteristic, complemented, Sylow
202C22×#41—832no—
203#532—838no—
204#532—1664no—
205C2×C4⋊Q8—831no—
206C2×#127—831no—
207C2×#124—832no—
208C2×#121—1661no—
209C2×#121—1661no—
210C2×C4×D4—1661no—
211C2×C4×D4—1661no—
212C22×C42C2×D8311yescharacteristic
213#431—24no—
214C2×C4⋊C8C2×D8312yes—
215C2×#103—838no—
216C2×#102—838no—
217C2×C22⋊C8C2×D8312yes—
218C2×C4×C8C2×D8312yes—
219#383—1661no—
220#382—832no—
221#381—1661no—
222#380—832no—
223C4×C4⋊C4—832no—
224C4×C22⋊C4—1661no—
225#376—1661no—
226C2×#41—3328no—
227C2×#41—838no—
228C2×#41—1664no—
229C22×Q8—832no—
230C22×D4—832no—
231C22×D4—1662no—
232C22×D4—1661no—
233C22×D4—1661no—
234C23×C4—1662no—
235C23×C4D33211yescharacteristic
236C23×C4D33211yescharacteristic
237C23×C4C22×D8311yescharacteristic
238C22×C8C22×D8314yes—
239C4⋊Q8—834no—
240C4⋊Q8—1662no—
241#127—834no—
242#127—1662no—
243#124—838no—
244#124—1664no—
245#121—1668no—
246#121—1668no—
247C4×D4—1668no—
248C4×D4—1668no—
249C2×C4⋊C4—834no—
250C2×C4⋊C4—1661no—
251C2×C4⋊C4—1661no—
252C2×C22⋊C4—1662no—
253C2×C22⋊C4—1662no—
254C2×C22⋊C4—1662no—
255C2×C42—1662no—
256C2×C42C4×D8314yes—
257C2×C42D33211yescharacteristic
258C2×C42D33211yescharacteristic
259C2×C42C22⋊D8314yes—
260C2×C42C22×D8311yescharacteristic
261C2×C42C22×D8311yescharacteristic
262C4⋊C8—24no—
263#103—3328no—
264#102—3328no—
265C22⋊C8—24no—
266C4×C8—24no—
267#95—1662no—
268#95—1662no—
269#95—1662no—
270#95—1662no—
271C24—1662no—
272C24C2×D33211yescharacteristic
273#41—33216no—
274#41—33216no—
275C2×Q8—834no—
276C2×Q8—1668no—
277C2×Q8—1662no—
278C2×D4—3324no—
279C2×D4—3328no—
280C2×D4—3324no—
281C2×D4—834no—
282C2×D4—1668no—
283C2×D4—1662no—
284C2×D4—1668no—
285C2×D4—1664no—
286C2×D4—1664no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—3328no—
291C22×C4—1664no—
292C22×C4—1662no—
293C22×C4—1668no—
294C22×C4(C2×D83)⋊C414yes—
295C22×C4(C2×D83)⋊C414yes—
296C22×C4C2×C4×D8312yes—
297C22×C4C2×C22⋊D8312yes—
298C22×C4C2×D33211yescharacteristic
299C22×C4C2×D33211yescharacteristic
300C22×C4C2×D33211yescharacteristic
301C22×C4C2×D33211yescharacteristic
302C22×C4C2×D33211yescharacteristic
303C22×C4C23×D8311yescharacteristic
304C2×C8—24no—
305C2×C8—28no—
306C4⋊C4—1664no—
307C4⋊C4—1664no—
308C4⋊C4—1664no—
309C4⋊C4—1664no—
310C22⋊C4—3324no—
311C22⋊C4—1668no—
312C22⋊C4—1668no—
313C42—3324no—
314C42(C2×D83)⋊C414yes—
315C42(C2×D83)⋊C414yes—
316C42C2×C4×D8312yes—
317C42C2×C4×D8312yes—
318C42C2×C22⋊D8312yes—
319C42C2×C22⋊D8312yes—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—3324no—
324C23—3324no—
325C23—3324no—
326C23—1664no—
327C23C2×(C2×D83)⋊C412yes—
328C23C22×D33211yescharacteristic
329Q8—3324no—
330Q8—1668no—
331D4—3328no—
332D4—33216no—
333D4—3328no—
334D4—3324no—
335D4—1668no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—33216no—
345C2×C4—3324no—
346C2×C4—3324no—
347C2×C4—3324no—
348C2×C4—3324no—
349C2×C4—1664no—
350C2×C4C2×(C2×D83)⋊C412yes—
351C2×C4C2×(C2×D83)⋊C412yes—
352C2×C4C2×(C2×D83)⋊C412yes—
353C2×C4C2×(C2×D83)⋊C412yes—
354C2×C4C2×(C2×D83)⋊C412yes—
355C2×C4#1458181414yes—
356C2×C4C22×C4×D8311yescharacteristic
357C2×C4C22×C22⋊D8311yescharacteristic
358C2×C4#1458183814yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3324no—
365C22—3324no—
366C22—3324no—
367C22—6648no—
368C22C22×(C2×D83)⋊C411yescharacteristic
369C22#1912827012yes—
370C22C2×#1458181412yes—
371C22#1912850112yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—3324no—
377C4—6644no—
378C4#1912769814yes—
379C4#1912817512yes—
380C4#1912826512yes—
381C2—42no—
382C2—6644no—
383C2#2491004311yescharacteristic
384C2C2×#1912769811yescharacteristic
385C2#2491598311yescharacteristic
386C2#2491844114yes—
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#431×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128251C2
cosocleC22×C4×C83C24
hypercenterC2×#431D83
hypocenterC83#21807
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
2. #24909861C2semidirect
3. #24908389C2semidirect
4. #24908385C2semidirect
5. #24908274C2semidirect
6. #24907404C2semidirect
7. C2×#431×C83C2semidirect
10. #19128251C4semidirect
195. C2×#431D83semidirect
201. C83#21807semidirect

Maximal subgroups

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

subgroupindex
C2×#191282512
#249098612
#249083892
#249083852
#249082742
#249074042
C2×#431×C832
#2180783