Finite Group Databasebeta

← #35307489

Subgroups — #35307489

Order 21248.

454 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 101 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#24908291C211yesmaximal, characteristic
2#24906022C212yesmaximal
3#24905321C212yesmaximal, complemented
4#24905296C211yesmaximal, characteristic, complemented
5C83×#2862C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19127754C2212yes—
8#19127725C2211yescharacteristic
9C2×#14581720—21no—
10C2×#14581720—21no—
11#19127434—24no—
12C2×C4⋊Q8×C83—21no—
13#391×C83—24no—
14#382×C83—21no—
15#382×C83C2212yes—
16C43×C83C2211yescharacteristic
17C2×#13050590—21no—
18C2×#13050574—21no—
19#14581792Q811yescharacteristic, complemented
20#14581789Q811yescharacteristic, complemented
21#14581780—21no—
22#14581766—22no—
23C4×D332—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26#14581720—42no—
27#14581720—42no—
28C22⋊(C83⋊C8)—22no—
29C4×C83⋊C8—22no—
30C22×Q8×C83—22no—
31C4⋊Q8×C83—42no—
32C4⋊Q8×C83—42no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83—22no—
35C2×C42×C83—22no—
36C2×C42×C83D411yescharacteristic
37C2×C42×C83D411yescharacteristic
38C2×C42×C83C2311yescharacteristic
39#95×C83—24no—
40#95×C83—22no—
41#95×C83D414yes—
42#13050590—41no—
43#13050590—41no—
44#13050590—41no—
45#13050590—41no—
46C2×Dic166—21no—
47C2×D332—21no—
48C2×D332—22no—
49C2×C22⋊D83—41no—
50C2×C4×D83—41no—
51#13050574—41no—
52#13050574—41no—
53C2×C83⋊C8—41no—
54C2×C83⋊C8—42no—
55C4⋊Dic83—22no—
56(C2×D83)⋊C4—42no—
57C2×Q8×C83—44no—
58C2×Q8×C83—48no—
59C22×C4×C83—21no—
60C22×C4×C83—21no—
61C22×C4×C83—24no—
62C22×C4×C83C2×D412yes—
63C22×C4×C83C2×D411yescharacteristic
64C4⋊C4×C83—44no—
65C4⋊C4×C83—42no—
66C4⋊C4×C83—42no—
67C42×C83—21no—
68C42×C83—21no—
69C42×C83—44no—
70C42×C83D812yes—
71C42×C83SD1612yes—
72C42×C83C2×Q811yescharacteristic
73C42×C83#4111yescharacteristic
74C42×C83#4112yes—
75C23×C83#12011yescharacteristic
76C22×D83—41no—
77Q8×C83—88no—
78C2×C4×C83—24no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—42no—
82C2×C4×C83—42no—
83C2×C4×C83—48no—
84C2×C4×C83—44no—
85C2×C4×C83C22⋊Q812yes—
86C2×C4×C83C22⋊Q811yescharacteristic
87C2×C4×C83#12412yes—
88C2×C4×C83#12411yescharacteristic
89C2×C4×C83C2×D811yescharacteristic
90C2×C4×C83C2×SD1611yescharacteristic
91C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C83#13711yescharacteristic
93C22⋊D83—81no—
94C22⋊D83—81no—
95C2×Dic83—41no—
96D332—41no—
97D332—41no—
98D332—82no—
99C4×D83—81no—
100C4×D83—81no—
101Dic166—41no—
102Dic166—41no—
103C83⋊C8—82no—
104C83⋊C8—81no—
105C22×C83—22no—
106C22×C83#39311yescharacteristic
107C22×C83#44711yescharacteristic
108C22×C83#44811yescharacteristic
109C2×D83—81no—
110C2×D83—82no—
111C4×C83—24no—
112C4×C83—22no—
113C4×C83—22no—
114C4×C83—42no—
115C4×C83—84no—
116C4×C83#47411yescharacteristic
117C4×C83#47611yescharacteristic
118C4×C83#48611yescharacteristic
119C4×C83#48911yescharacteristic
120Dic83—81no—
121#10362—831nomaximal, Sylow
122C2×C83—21no—
123C2×C83—21no—
124C2×C83#202311yescharacteristic
125C2×C83#203112yes—
126D83—161no—
127#2862D8311yescharacteristic, complemented
128#1962—831no—
129#1517—832no—
130#1286—832no—
131#1276—831no—
132C83#1036211yescharacteristic, complemented, Sylow
133#532—831no—
134C2×C4⋊Q8—21no—
135#431—832no—
136#423—831no—
137C2×#104—1661no—
138C2×#104—1661no—
139#391—24no—
140#382—21no—
141#382C2×D8312yes—
142C43C2×D8311yescharacteristic
143#327—1664no—
144C2×#41—1661no—
145C22×Q8—22no—
146C2×M16—1661no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—831no—
150#127—831no—
151#124—1661no—
152#121—1662no—
153C4×D4—1662no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C42—22no—
157C2×C42C22⋊D8311yescharacteristic
158C2×C42C22⋊D8311yescharacteristic
159C2×C42C22×D8311yescharacteristic
160C4⋊C8—1662no—
161C4⋊C8—1662no—
162#104—3322no—
163#104—3322no—
164C22⋊C8—1662no—
165C4×C8—1662no—
166#95—24no—
167#95—22no—
168#95C22⋊D8314yes—
169#41—3321no—
170#41—3321no—
171#41—3321no—
172#41—3321no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—1661no—
176C2×D4—3321no—
177C2×D4—1661no—
178C2×D4—1662no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—3321no—
183C22×C4C2×C22⋊D8312yes—
184C22×C4C2×C22⋊D8311yescharacteristic
185M16—3321no—
186M16—3321no—
187C2×C8—3322no—
188C2×C8—3321no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—1662no—
193C22⋊C4—3322no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1305057512yes—
198C42#1305057712yes—
199C42Q8×D8311yescharacteristic
200C42#1305059111yescharacteristic
201C42#1305059112yes—
202C23—3321no—
203C23#1458176111yescharacteristic
204Q8—88no—
205Q8—3321no—
206Q8—3321no—
207D4—3321no—
208D4—3321no—
209D4—6642no—
210D4—6641no—
211D4—6641no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—3321no—
220C2×C4—6641no—
221C2×C4—6641no—
222C2×C4#1458176812yes—
223C2×C4#1458176811yescharacteristic
224C2×C4#1458178112yes—
225C2×C4#1458178111yescharacteristic
226C2×C4C2×#1305057511yescharacteristic
227C2×C4C2×#1305057711yescharacteristic
228C2×C4#1458181811yescharacteristic
229C2×C4#1458182311yescharacteristic
230C8—6642no—
231C8—6641no—
232C22—22no—
233C22—6641no—
234C22—6642no—
235C22#1912761211yescharacteristic
236C22#1912783511yescharacteristic
237C22#1912784011yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—84no—
243C4—6641no—
244C4#1912797811yescharacteristic
245C4#1912798811yescharacteristic
246C4#1912804911yescharacteristic
247C4#1912806311yescharacteristic
248C2—21no—
249C2—21no—
250C2—13281no—
251C2#2490859611yescharacteristic
252C2#2490863212yes—
subgroupG/H
centerC22#19127612
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2862C2
socleC22×C83#393
prime radical#19128251C22
hypercenter#2862D83
hypocenterC83#10362
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #24905321C2semidirect
4. #24905296C2semidirect
5. C83×#2862C2semidirect
19. #14581792Q8semidirect
20. #14581789Q8semidirect
127. #2862D83semidirect
132. C83#10362semidirect

Maximal subgroups

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

subgroupindex
#249082912
#249060222
#249053212
#249052962
C83×#28622
#1036283