Finite Group Databasebeta

← #35313563

Subgroups — #35313563

Order 21248.

454 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 93 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#24910088C211yesmaximal, characteristic
2#324×D83C212yesmaximal
3#24905408C212yesmaximal, complemented
4#24905407C211yesmaximal, characteristic, complemented
5C83×#1324C211yesmaximal, characteristic, complemented
6C2×#14581750C411yescharacteristic
7C2×#14581749C411yescharacteristic
8C2×C4⋊C4×D83C2211yescharacteristic
9C22⋊C8×D83—22no—
10#19127598—22no—
11#19127575—22no—
12#19127571C411yescharacteristic
13#19127570C2211yescharacteristic
14#19127566C411yescharacteristic
15#19127427C2212yes—
16#19127426—22no—
17#382×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×C4×D83C2×C411yescharacteristic
20C22×C4×D83D412yes—
21C2×C8×D83—42no—
22#14581750—42no—
23#14581749—22no—
24#14581749C814yescomplemented
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—22no—
30C2×Dic83⋊C4D412yes—
31C4⋊C4×D83—42no—
32C2×(C2×D83)⋊C4—21no—
33C2×(C2×D83)⋊C4—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83C2×C411yescharacteristic
36C22⋊(C83⋊C8)—22no—
37(C2×D83)⋊C8—42no—
38C83⋊#95—22no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2311yescharacteristic
41C2×C42×C83C2×C411yescharacteristic
42C22⋊C8×C83—22no—
43#95×C83—22no—
44C23×D83C22⋊C411yescharacteristic
45C22×Dic83—21no—
46C22×Dic83—21no—
47C22×Dic83C22⋊C411yescharacteristic
48C22×Dic83C22⋊C412yes—
49C2×C4×D83—22no—
50C2×C4×D83—22no—
51C2×C4×D83—22no—
52C2×C4×D83—22no—
53C2×C4×D83—82no—
54C2×C4×D83C2×C812yes—
55C2×C4×D83M1612yes—
56C2×C83⋊C8—42no—
57C8×D83—84no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60C4⋊Dic83—82no—
61Dic83⋊C4—24no—
62Dic83⋊C4—42no—
63Dic83⋊C4—82no—
64(C2×D83)⋊C4—24no—
65(C2×D83)⋊C4—42no—
66C4×Dic83—21no—
67C4×Dic83—42no—
68C4×Dic83C2×C812yescomplemented
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83C22×C411yescharacteristic
72C22×C4×C83C2×D412yes—
73C2×C8×C83—42no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—41no—
76C42×C83—21no—
77C42×C83C2×C812yescomplemented
78C23×C83C2×C22⋊C411yescharacteristic
79C22×D83—41no—
80C22×D83—41no—
81C22×D83C22⋊C812yes—
82C22×D83#9912yes—
83C22×D83#10112yes—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—42no—
91C2×C4×C83—42no—
92C2×C4×C83—41no—
93C2×C4×C83C22×C811yescharacteristic
94C2×C4×C83C2×M1611yescharacteristic
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—41no—
100C2×Dic83—41no—
101C2×Dic83—42no—
102C2×Dic83—42no—
103C2×Dic83—41no—
104C2×Dic83C22⋊C812yes—
105C4×D83—24no—
106C4×D83—24no—
107C4×D83—44no—
108C4×D83—82no—
109C8×C83—82no—
110C83⋊C8—82no—
111C22×C83—21no—
112C22×C83—21no—
113C22×C83C2×C22⋊C811yescharacteristic
114C22×C83C2×#9911yescharacteristic
115C22×C83C2×#10111yescharacteristic
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83#32414yes—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—42no—
124C4×C83—42no—
125C4×C83—81no—
126Dic83—22no—
127Dic83—42no—
128Dic83—42no—
129Dic83—81no—
130#11448—831nomaximal, Sylow
131C2×C83—21no—
132C2×C83—21no—
133C2×C83C2×#32411yescharacteristic
134C2×C83#212711yescharacteristic
135C2×C83#213311yescharacteristic
136D83—24no—
137#2317—831no—
138C2×#324—832no—
139#1325—832no—
140#1324—831no—
141#1324D8311yescharacteristic, complemented
142C83#1144811yescharacteristic, complemented, Sylow
143C2×#117—831no—
144C2×#117—831no—
145C22×C4⋊C4—831no—
146C2×C22⋊C8—1662no—
147#389—1662no—
148#385—1662no—
149#384—831no—
150#384—831no—
151#382—831no—
152#382C2×D8311yescharacteristic
153#324—832no—
154#324—1662no—
155#324C2×D8312yes—
156C23×C4—831no—
157C23×C4—832no—
158C22×C8—3322no—
159#117—3322no—
160#117—834no—
161#117—1662no—
162C2×C4⋊C4—3322no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4—832no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—1662no—
167C2×C4⋊C4—1662no—
168C2×C4⋊C4—1661no—
169C2×C4⋊C4C4×D8311yescharacteristic
170C2×C4⋊C4C22×D8311yescharacteristic
171C2×C22⋊C4—1661no—
172C2×C22⋊C4—1661no—
173C2×C42—831no—
174C2×C42—1661no—
175C2×C42C4×D8311yescharacteristic
176C22⋊C8—22no—
177C22⋊C8—3322no—
178C22⋊C8—1662no—
179#95—22no—
180#95—1662no—
181C24—831no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—832no—
185C22×C4—831no—
186C22×C4—832no—
187C22×C4—832no—
188C22×C4—6642no—
189C22×C4—1662no—
190C22×C4—1662no—
191C22×C4—1662no—
192C22×C4—1662no—
193C22×C4—1661no—
194C22×C4—1661no—
195C22×C4C2×C4×D8311yescharacteristic
196C22×C4D4×D8312yes—
197C2×C8—42no—
198C2×C8—3322no—
199C2×C8—6644no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—3321no—
203C4⋊C4—3322no—
204C4⋊C4—3321no—
205C4⋊C4—6642no—
206C4⋊C4—6642no—
207C4⋊C4—1664no—
208C22⋊C4—3322no—
209C22⋊C4—1664no—
210C42—21no—
211C42—3322no—
212C42—832no—
213C42—1661no—
214C42C8×D8312yescomplemented
215C23—3321no—
216C23—3321no—
217C23—832no—
218C23—832no—
219C23—832no—
220C23C22⋊C4×D8311yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—3324no—
231C2×C4—3321no—
232C2×C4—3321no—
233C2×C4—3322no—
234C2×C4—3322no—
235C2×C4—3321no—
236C2×C4—832no—
237C2×C4—6642no—
238C2×C4—1664no—
239C2×C4—1664no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4C2×C8×D8311yescharacteristic
245C2×C4M16×D8311yescharacteristic
246C8—82no—
247C8—6642no—
248C22—21no—
249C22—21no—
250C22—3322no—
251C22—3322no—
252C22—834no—
253C22—1662no—
254C22—1662no—
255C22C22⋊C8×D8311yescharacteristic
256C22#99×D8311yescharacteristic
257C22#101×D8311yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—3322no—
264C4—3322no—
265C4—6641no—
266C4—1662no—
267C2—21no—
268C2—21no—
269C2—1664no—
270C2#324×D8311yescharacteristic
271C2#2490913111yescharacteristic
272C2#2490914911yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1324C2
socleC22×C83C2×#99
prime radicalC23×D83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1324D83
hypocenterC83#11448
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. #24905408C2semidirect
4. #24905407C2semidirect
5. C83×#1324C2semidirect
24. #14581749C8semidirect
68. C4×Dic83C2×C8semidirect
77. C42×C83C2×C8semidirect
141. #1324D83semidirect
142. C83#11448semidirect
214. C42C8×D83semidirect

Maximal subgroups

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

subgroupindex
#249100882
#324×D832
#249054082
#249054072
C83×#13242
#1144883