Finite Group Databasebeta

← #35307920

Subgroups — #35307920

Order 21248.

630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912379C211yesmaximal, characteristic
2#24907460C212yesmaximal, complemented
3C2×#19127430C212yesmaximal, complemented
4#24905346C211yesmaximal, characteristic, complemented
5C83×#1844C211yesmaximal, characteristic, complemented
6C2×#14581767—22no—
7C2×C4×D332C2212yes—
8C2×C4⋊(C83⋊C8)C2212yes—
9C2×#14581712—22no—
10#19127618—22no—
11#19127558—22no—
12#19127463—22no—
13#19127430—24no—
14C2×#106×C83—22no—
15C2×C4×C8×C83C2211yescharacteristic
16#384×C83C2211yescharacteristic
17#384×C83C2211yescharacteristic
18C22×D332—22no—
19C22×D332D412yes—
20C22×C4×D83—22no—
21C22×C83⋊C8—22no—
22#14581767—44no—
23#14581767—44no—
24C4×D332—22no—
25C4×D332—22no—
26C4×D332—22no—
27C4×D332—22no—
28C2×C4⋊Dic83D412yes—
29C2×(C2×D83)⋊C4—24no—
30C2×(C2×D83)⋊C4—22no—
31C4⋊(C83⋊C8)—22no—
32C4⋊(C83⋊C8)—22no—
33#14581712—44no—
34C83⋊#95—22no—
35C22×C8×C83—21no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83D412yes—
40C2×C42×C83C2311yescharacteristic
41#106×C83—44no—
42C4×C8×C83—21no—
43C4×C8×C83D412yescomplemented
44C23×D83—22no—
45C2×D332—22no—
46C2×D332—22no—
47C2×D332—22no—
48C2×D332—22no—
49C2×D332—22no—
50C2×D332—48no—
51C2×D332—82no—
52C2×D332SD1614yes—
53C22×Dic83—22no—
54C2×C4×D83—42no—
55C2×C4×D83—42no—
56C2×C4×D83—42no—
57C2×C4×D83—42no—
58C2×C4×D83—42no—
59C2×C4×D83—42no—
60C2×C83⋊C8—42no—
61C2×C83⋊C8—42no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—22no—
64C4⋊Dic83—22no—
65(C2×D83)⋊C4—48no—
66(C2×D83)⋊C4—42no—
67(C2×D83)⋊C4—42no—
68C22×C4×C83—22no—
69C22×C4×C83—41no—
70C22×C4×C83C2×D412yes—
71C22×C4×C83C2×D411yescharacteristic
72C2×C8×C83—22no—
73C2×C8×C83—22no—
74C2×C8×C83—22no—
75C4⋊C4×C83—42no—
76C4⋊C4×C83—42no—
77C4⋊C4×C83—42no—
78C4⋊C4×C83—42no—
79C4⋊C4×C83—42no—
80C4⋊C4×C83—42no—
81C42×C83C2×D411yescharacteristic
82C42×C83#4111yescharacteristic
83C42×C83#4112yes—
84C23×C83#12011yescharacteristic
85C22×D83—24no—
86C22×D83—42no—
87C22×D83—42no—
88C22×D83—42no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83—42no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—44no—
95C2×C4×C83—42no—
96C2×C4×C83—82no—
97C2×C4×C83—81no—
98C2×C4×C83#12111yescharacteristic
99C2×C4×C83#12112yes—
100C2×C4×C83#12312yes—
101C2×C4×C83#12411yescharacteristic
102C2×C4×C83C2×SD1612yes—
103C2×C4×C83#13512yes—
104C2×C4×C83AGL(1,ℤ/8)12yes—
105C2×C4×C83AGL(1,ℤ/8)12yes—
106C2×Dic83—42no—
107C2×Dic83—42no—
108C2×Dic83—42no—
109D332—24no—
110D332—24no—
111D332—42no—
112D332—42no—
113D332—88no—
114C4×D83—82no—
115C4×D83—82no—
116C4×D83—82no—
117C4×D83—82no—
118C8×C83—24no—
119C83⋊C8—82no—
120C22×C83#39411yescharacteristic
121C22×C83#44912yes—
122C22×C83#45012yes—
123C22×C83#45311yescharacteristic
124C22×C83#45611yescharacteristic
125C2×D83—44no—
126C2×D83—44no—
127C2×D83—44no—
128C2×D83—82no—
129C2×D83—82no—
130C2×D83—82no—
131C2×D83—82no—
132C4×C83—21no—
133C4×C83—21no—
134C4×C83—82no—
135C4×C83—161no—
136C4×C83#46512yes—
137C4×C83#46912yes—
138C4×C83#48112yes—
139C4×C83#48212yes—
140Dic83—82no—
141#10432—831nomaximal, Sylow
142C2×C83#162411yescharacteristic
143C2×C83#162911yescharacteristic
144C2×C83#165412yes—
145C2×C83#203412yes—
146C2×C83#203911yescharacteristic
147D83—84no—
148#2537—831no—
149#1844D8311yescharacteristic, complemented
150#1802—832no—
151C2×D4⋊C8—832no—
152#1296—831no—
153C83#1043211yescharacteristic, complemented, Sylow
154C2×#121—1662no—
155C2×C4×D4—832no—
156C2×#106—22no—
157C2×C4⋊C8—832no—
158C2×#102—1662no—
159C2×C4×C8C2×D8311yescharacteristic
160#394—1662no—
161#384C2×D8311yescharacteristic
162#384C2×D8311yescharacteristic
163#380—1662no—
164#340—1662no—
165D4⋊C8—1664no—
166C22×D4—832no—
167C22×D4—1662no—
168C23×C4—1662no—
169C22×C8—21no—
170C22×C8—1662no—
171#121—3324no—
172#121—3324no—
173C4×D4—1662no—
174C4×D4—1662no—
175C4×D4—1662no—
176C4×D4—1662no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—832no—
181C2×C4⋊C4C22⋊D8312yes—
182C2×C22⋊C4—1664no—
183C2×C22⋊C4—1662no—
184C2×C42C22×D8311yescharacteristic
185#106—44no—
186C4⋊C8—1662no—
187C4⋊C8—1662no—
188#102—3324no—
189C4×C8—21no—
190C4×C8D33212yescomplemented
191#95—1662no—
192C24—1662no—
193C2×D4—3328no—
194C2×D4—834no—
195C2×D4—6642no—
196C2×D4—1662no—
197C2×D4—1662no—
198C2×D4—1662no—
199C2×D4—1662no—
200C2×D4—1662no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—3322no—
204C22×C4—3322no—
205C22×C4—3322no—
206C22×C4—3322no—
207C22×C4—3322no—
208C22×C4—3322no—
209C22×C4—1662no—
210C22×C4D4×D8312yes—
211C22×C4C2×C22⋊D8311yescharacteristic
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—3322no—
216C2×C8—3322no—
217C2×C8—3322no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—1662no—
225C4⋊C4—1662no—
226C22⋊C4—3328no—
227C22⋊C4—3322no—
228C22⋊C4—3322no—
229C42C2×D33211yescharacteristic
230C42#1305059011yescharacteristic
231C42#1305059112yes—
232C23—3322no—
233C23—3322no—
234C23—3322no—
235C23—3322no—
236C23—3322no—
237C23—3322no—
238C23—1664no—
239C23#1458176011yescharacteristic
240D4—3322no—
241D4—3322no—
242D4—6648no—
243D4—1664no—
244D4—1664no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—3322no—
252C2×C4—3322no—
253C2×C4—3322no—
254C2×C4—6642no—
255C2×C4—6642no—
256C2×C4—6642no—
257C2×C4—6642no—
258C2×C4#1458176611yescharacteristic
259C2×C4#1458176712yes—
260C2×C4#1458177712yes—
261C2×C4#1458178111yescharacteristic
262C2×C4SD16×D8312yes—
263C2×C4#1458181312yes—
264C2×C4#1458181912yes—
265C2×C4#1458181912yes—
266C8—24no—
267C8—6642no—
268C22—3324no—
269C22—3324no—
270C22—3324no—
271C22—6642no—
272C22—6642no—
273C22—6642no—
274C22—6642no—
275C22#1912761811yescharacteristic
276C22#1912784512yes—
277C22#1912785012yes—
278C22#1912786811yescharacteristic
279C22#1912788311yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—161no—
284C4—6642no—
285C4#1912793112yes—
286C4#1912795512yes—
287C4#1912802112yes—
288C4#1912802712yes—
289C2—6644no—
290C2#2490654411yescharacteristic
291C2#2490657711yescharacteristic
292C2#2490674412yes—
293C2#2490865412yes—
294C2#2490868711yescharacteristic
subgroupG/H
centerC23#14581760
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1844C2
socleC23×C83#120
prime radical#24912379C2
hypercenter#1844D83
hypocenterC83#10432
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. #24907460C2semidirect
3. C2×#19127430C2semidirect
4. #24905346C2semidirect
5. C83×#1844C2semidirect
43. C4×C8×C83D4semidirect
149. #1844D83semidirect
153. C83#10432semidirect
190. C4×C8D332semidirect

Maximal subgroups

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

subgroupindex
#249123792
#249074602
C2×#191274302
#249053462
C83×#18442
#1043283