Finite Group Databasebeta

← #35298790

Subgroups — #35298790

Order 21248.

396 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 119 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#24910649C211yesmaximal, characteristic
2#24908433C211yesmaximal, characteristic
3#24907597C212yesmaximal
4#24905345C212yesmaximal
5C2×#330×C83C211yesmaximal, characteristic
6C2×#14581703C2211yescharacteristic
7#19127600—22no—
8#19127587—22no—
9#19127587—22no—
10#19127571C2211yescharacteristic
11#19127567C2212yes—
12#19127464—22no—
13#19127464—22no—
14#19127463—22no—
15C2×#125×C83C2211yescharacteristic
16C2×#97×C83C2212yes—
17#330×C83C414yescomplemented
18C22×C83⋊C8—21no—
19C2×C4⋊Dic83—22no—
20C2×C4⋊Dic83—21no—
21C2×C4⋊Dic83—22no—
22C2×C4⋊Dic83D412yes—
23C2×C4⋊Dic83D412yes—
24C2×C4×Dic83—22no—
25#14581703—22no—
26C83⋊#95—22no—
27C83⋊#95—22no—
28C83⋊#95—22no—
29C83⋊#95—22no—
30C22×C8×C83—22no—
31#125×C83—22no—
32#125×C83C2×C412yes—
33#125×C83Q812yes—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83D412yes—
36C2×C42×C83C2311yescharacteristic
37#97×C83—22no—
38#97×C83C2×C414yes—
39C22×Dic83—22no—
40C22×Dic83—22no—
41C22×Dic83—41no—
42C2×C83⋊C8—22no—
43C2×C83⋊C8—41no—
44C2×C83⋊C8—41no—
45C4⋊Dic83—42no—
46C4⋊Dic83—42no—
47C4⋊Dic83—44no—
48C4⋊Dic83—42no—
49C4⋊Dic83—82no—
50C4×Dic83—82no—
51C22×C4×C83—22no—
52C22×C4×C83C2×D412yes—
53C22×C4×C83C2×D411yescharacteristic
54C2×C8×C83—24no—
55C2×C8×C83—42no—
56C2×C8×C83—42no—
57C4⋊C4×C83—24no—
58C4⋊C4×C83—24no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83C4⋊C414yes—
61C42×C83C22×C411yescharacteristic
62C42×C83C2×Q811yescharacteristic
63C42×C83#4112yes—
64C23×C83#12011yescharacteristic
65C2×C4×C83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—21no—
69C2×C4×C83—24no—
70C2×C4×C83—42no—
71C2×C4×C83—42no—
72C2×C4×C83C2×C4⋊C411yescharacteristic
73C2×C4×C83C4×D412yes—
74C2×C4×C83C22⋊Q812yes—
75C2×C4×C83#12311yescharacteristic
76C2×Dic83—42no—
77C2×Dic83—42no—
78C2×Dic83—42no—
79C2×Dic83—42no—
80C2×Dic83—42no—
81C2×Dic83—42no—
82C2×Dic83—82no—
83C2×Dic83—81no—
84C8×C83—44no—
85C83⋊C8—42no—
86C22×C83#38511yescharacteristic
87C22×C83#45412yes—
88C22×C83#45411yescharacteristic
89C22×C83#45512yes—
90C22×C83#45611yescharacteristic
91C4×C83—21no—
92C4×C83—21no—
93C4×C83—22no—
94C4×C83—22no—
95C4×C83—44no—
96Dic83—82no—
97Dic83—82no—
98Dic83—161no—
99#8915—831nomaximal, Sylow
100C2×C83#165811yescharacteristic
101C2×C83#166211yescharacteristic
102C2×C83#166412yes—
103C2×C83#185711yescharacteristic
104C2×C83#186012yes—
105#2382—831no—
106#1992—831no—
107#1836—832no—
108C2×#330D8311yescharacteristic
109#1295—832no—
110C83#891511yescharacteristic, complemented, Sylow
111C2×#125C2×D8311yescharacteristic
112C2×#97—831no—
113C2×#97C2×D8312yes—
114#389—1662no—
115#387—1662no—
116#387—1662no—
117#384—831no—
118#382—832no—
119#340—1662no—
120#340—1662no—
121#340—1662no—
122#330Dic8314yescomplemented
123C22×C8—22no—
124C22×C8—1661no—
125#125—22no—
126#125Dic16612yes—
127#125C2×Dic8312yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—1662no—
132C2×C4⋊C4—1661no—
133C2×C4⋊C4—1662no—
134C2×C4⋊C4D33212yes—
135C2×C42—1662no—
136C2×C42C22×D8311yescharacteristic
137#97—22no—
138#97—1662no—
139#97C2×Dic8314yes—
140#95—1662no—
141#95—1662no—
142#95—1662no—
143#95—1662no—
144C22×C4—22no—
145C22×C4—3321no—
146C22×C4—1662no—
147C22×C4—1662no—
148C22×C4D4×D8312yes—
149C22×C4C2×D33211yescharacteristic
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—3321no—
154C2×C8—3321no—
155C2×C8—1662no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—24no—
159C4⋊C4—3322no—
160C4⋊C4—3322no—
161C4⋊C4—3324no—
162C4⋊C4—3322no—
163C4⋊C4—6642no—
164C4⋊C4C4⋊Dic8314yes—
165C42—6642no—
166C42C22×Dic8311yescharacteristic
167C42C2×Dic16611yescharacteristic
168C42#1305058912yes—
169C23#1458176211yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—3322no—
178C2×C4—3322no—
179C2×C4—3322no—
180C2×C4—3322no—
181C2×C4—3322no—
182C2×C4—3322no—
183C2×C4—6642no—
184C2×C4—6641no—
185C2×C4C2×C4⋊Dic8311yescharacteristic
186C2×C4D4×Dic8312yes—
187C2×C4#1458176912yes—
188C2×C4#1458177611yescharacteristic
189C8—44no—
190C8—3322no—
191C22#1912757611yescharacteristic
192C22#1912787512yes—
193C22#1912787611yescharacteristic
194C22#1912788012yes—
195C22#1912788611yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—13281no—
202C4—6642no—
203C4—6642no—
204C2#2490676611yescharacteristic
205C2#2490678711yescharacteristic
206C2#2490679912yes—
207C2#2490768011yescharacteristic
208C2#2490769812yes—
subgroupG/H
centerC23#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#330×C83C2
socle, prime radicalC23×C83#120
cosocleC2×C42×C83C23
hypercenterC2×#330D83
hypocenterC83#8915
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
17. #330×C83C4semidirect
110. C83#8915semidirect
122. #330Dic83semidirect

Maximal subgroups

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

subgroupindex
#249106492
#249084332
#249075972
#249053452
C2×#330×C832
#891583