Finite Group Databasebeta

← #35302256

Subgroups — #35302256

Order 21248.

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

normal subgroups only

This group has 542 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#24916544C211yesmaximal, characteristic, complemented
2#24912368C211yesmaximal, characteristic
3#24905887C212yesmaximal
4#24905323C212yesmaximal, complemented
5C83×#1801C211yesmaximal, characteristic, complemented
6C2×#14581768—21no—
7C2×C4×D332C2211yescharacteristic
8Q8×C83⋊C8—24no—
9#19127754—22no—
10#19127754C2212yes—
11#19127631C412yes—
12#19127618C412yes—
13#19127592—22no—
14#19127567C2211yescharacteristic
15#19127558—21no—
16#19127428—24no—
17C2×C4×Q8×C83C2211yescharacteristic
18#423×C83C2212yes—
19C2×#101×C83—21no—
20#343×C83—21no—
21C22×D332C2×C411yescharacteristic
22C22×C4×D83—21no—
23#14581768—42no—
24#14581768—42no—
25C4×D332—22no—
26C4×D332—22no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×Dic83⋊C4—22no—
30C2×(C2×D83)⋊C4—22no—
31C2×(C2×D83)⋊C4—22no—
32C2×(C2×D83)⋊C4—21no—
33C4⋊(C83⋊C8)—24no—
34C4⋊(C83⋊C8)—22no—
35C22⋊(C83⋊C8)—22no—
36C4×C83⋊C8—24no—
37C4×C83⋊C8D414yes—
38C83⋊#95—22no—
39C83⋊#95—21no—
40C83⋊#95C2×C412yes—
41C22×Q8×C83D411yescharacteristic
42C2×M16×C83—22no—
43C4×Q8×C83—22no—
44C4×Q8×C83—22no—
45C2×C4⋊C4×C83C2×C412yes—
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C42×C83D412yes—
48C2×C42×C83C2311yescharacteristic
49C4⋊C8×C83—22no—
50C4⋊C8×C83—22no—
51#101×C83—42no—
52C23×D83—21no—
53C2×D332—22no—
54C2×D332—42no—
55C22×Dic83—21no—
56C22×Dic83—22no—
57C2×C4×D83—42no—
58C2×C4×D83—41no—
59C2×C4×D83—41no—
60C2×C4×D83—42no—
61C2×C83⋊C8—24no—
62C4⋊Dic83—22no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65Dic83⋊C4—44no—
66(C2×D83)⋊C4—44no—
67(C2×D83)⋊C4—44no—
68(C2×D83)⋊C4—41no—
69(C2×D83)⋊C4—41no—
70C2×Q8×C83—22no—
71C2×Q8×C83—22no—
72C2×Q8×C83D812yes—
73C2×Q8×C83SD1612yes—
74C22×C4×C83C22⋊C412yes—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79M16×C83—42no—
80M16×C83—42no—
81C2×C8×C83—42no—
82C4⋊C4×C83—24no—
83C4⋊C4×C83—22no—
84C42×C83—22no—
85C42×C83—22no—
86C42×C83C2×D412yes—
87C42×C83#4112yes—
88C23×C83C2×C22⋊C411yescharacteristic
89C22×D83—22no—
90C22×D83—41no—
91C22×D83—41no—
92C22×D83—41no—
93C22×D83—41no—
94C22×D83—41no—
95C22×D83—41no—
96Q8×C83—24no—
97Q8×C83—24no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—42no—
104C2×C4×C83#10214yes—
105C2×C4×C83C4×D412yes—
106C2×C4×C83#12011yescharacteristic
107C2×C4×C83#12112yes—
108C2×C4×C83#12311yescharacteristic
109C2×C4×C83C2×D811yescharacteristic
110C2×C4×C83C2×SD1611yescharacteristic
111C2×C4×C83#13512yes—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—41no—
115C2×Dic83—42no—
116C2×Dic83—41no—
117C2×Dic83—41no—
118D332—24no—
119D332—42no—
120C4×D83—82no—
121C4×D83—82no—
122C8×C83—82no—
123C83⋊C8—44no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83#38611yescharacteristic
127C22×C83C2×#10211yescharacteristic
128C22×C83#41611yescharacteristic
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—81no—
133C2×D83—81no—
134C2×D83—81no—
135C2×D83—81no—
136C4×C83—22no—
137C4×C83—24no—
138C4×C83—21no—
139C4×C83—21no—
140C4×C83—42no—
141C4×C83#46512yes—
142C4×C83#46612yes—
143Dic83—82no—
144Dic83—81no—
145#9486—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83#189111yescharacteristic
149C2×C83#189311yescharacteristic
150C2×C83#191211yescharacteristic
151D83—82no—
152#2982—831no—
153#2536—831no—
154#1801D8311yescharacteristic, complemented
155#1486—832no—
156#1286—832no—
157C83#948611yescharacteristic, complemented, Sylow
158C2×C22⋊Q8—1661no—
159C2×C4×Q8C2×D8311yescharacteristic
160C2×C4×D4—831no—
161C8×Q8—1664no—
162#431—832no—
163#431—1662no—
164#423C2×D8312yes—
165C2×#101—21no—
166#398—832no—
167#394—832no—
168#388—1662no—
169#382—831no—
170#380—1661no—
171#343—21no—
172D4⋊C8—1664no—
173C22×Q8C22⋊D8311yescharacteristic
174C22×D4—831no—
175C23×C4—1661no—
176C2×M16—22no—
177C22⋊Q8—3322no—
178C22⋊Q8—3322no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1662no—
182C4×D4—1662no—
183C2×C4⋊C4—831no—
184C2×C4⋊C4—1662no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4C4×D8312yes—
187C2×C4⋊C4C22⋊D8311yescharacteristic
188C2×C22⋊C4—1662no—
189C2×C22⋊C4—1662no—
190C2×C22⋊C4—1661no—
191C2×C42D33212yes—
192C2×C42C22×D8311yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—1664no—
196C4⋊C8—1662no—
197#101—42no—
198C22⋊C8—1662no—
199C4×C8—834no—
200C4×C8—1664no—
201#95—832no—
202#95—1662no—
203#95—1661no—
204C24—1661no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1305057512yes—
208C2×Q8#1305057712yes—
209C2×D4—3322no—
210C2×D4—1662no—
211C22×C4—3322no—
212C22×C4—3321no—
213C22×C4—3321no—
214C22×C4—3322no—
215C22×C4—1662no—
216C22×C4—1661no—
217C22×C4(C2×D83)⋊C412yes—
218C22×C4(C2×D83)⋊C412yes—
219C22×C4C2×C4×D8311yescharacteristic
220C22×C4C2×C22⋊D8311yescharacteristic
221C22×C4C2×D33211yescharacteristic
222M16—42no—
223M16—42no—
224C2×C8—42no—
225C2×C8—1664no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—3321no—
229C4⋊C4—3321no—
230C4⋊C4—3324no—
231C4⋊C4—1662no—
232C22⋊C4—3324no—
233C22⋊C4—3321no—
234C22⋊C4—3321no—
235C22⋊C4—3324no—
236C42—22no—
237C42—22no—
238C42D4×D8312yes—
239C42#1305058912yes—
240C23—3321no—
241C23—3321no—
242C23—3321no—
243C23—3321no—
244C23—3321no—
245C23—3321no—
246C23—1662no—
247C23C2×(C2×D83)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—3322no—
251D4—1664no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3322no—
259C2×C4—3322no—
260C2×C4—3321no—
261C2×C4—3322no—
262C2×C4—3321no—
263C2×C4—3321no—
264C2×C4—6642no—
265C2×C4—6642no—
266C2×C4#1458171214yes—
267C2×C4(C22⋊D83)⋊C412yes—
268C2×C4#1458176211yescharacteristic
269C2×C4#1458176412yes—
270C2×C4#1458177611yescharacteristic
271C2×C4C2×#1305057511yescharacteristic
272C2×C4C2×#1305057711yescharacteristic
273C2×C4#1458181212yes—
274C8—82no—
275C8—3324no—
276C22—21no—
277C22—21no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—6641no—
282C22—6641no—
283C22—6641no—
284C22—6641no—
285C22#1912757811yescharacteristic
286C22C2×#1458171211yescharacteristic
287C22#1912769511yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—6642no—
294C4—6641no—
295C4#1912793212yes—
296C4#1912793512yes—
297C2—21no—
298C2—21no—
299C2—6642no—
300C2#2490787911yescharacteristic
301C2#2490788911yescharacteristic
302C2#2490799811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1801C2
socleC22×C83#386
prime radicalC22×D332C2×C4
cosocleC2×C42×C83C23
hypercenter#1801D83
hypocenterC83#9486
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #24916544C2semidirect
4. #24905323C2semidirect
5. C83×#1801C2semidirect
154. #1801D83semidirect
157. C83#9486semidirect

Maximal subgroups

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

subgroupindex
#249165442
#249123682
#249058872
#249053232
C83×#18012
#948683