Finite Group Databasebeta

← #24906661

Subgroups — #24906661

Order 10624.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#19128372C211yesmaximal, characteristic
2#19128049C211yesmaximal, characteristic, complemented
3#19127983C211yesmaximal, characteristic, complemented
4#19127444C211yesmaximal, characteristic, complemented
5#19127431C211yesmaximal, characteristic, complemented
6D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
7#461×C83C211yesmaximal, characteristic, complemented
8C2×#13050590—21no—
9#14581789C2211yescharacteristic
10#14581788—21no—
11#14581766—21no—
12#14581765—22no—
13C4×Dic166C2211yescharacteristic
14#14581754—21no—
15#14581723—21no—
16C4⋊(C83⋊C8)C2211yescharacteristic
17#14581715—21no—
18#14581712—22no—
19C4×C83⋊C8C2211yescharacteristic
20C2×SD16×C83—21no—
21C4⋊Q8×C83C2211yescharacteristic
22C4×D4×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic
24#103×C83—21no—
25#13050590—42no—
26#13050590—42no—
27C2×Dic166D411yescharacteristic
28C2×D332—21no—
29C2×D332—21no—
30C2×D332D412yes—
31C2×C22⋊D83—22no—
32C2×C22⋊D83—22no—
33C2×C4×D83—22no—
34C2×C83⋊C8—21no—
35C2×C83⋊C8—21no—
36C4⋊Dic83D411yescharacteristic
37Dic83⋊C4—21no—
38(C2×D83)⋊C4—22no—
39C4×Dic83—21no—
40C2×Q8×C83—21no—
41C2×D4×C83D411yescharacteristic
42C22×C4×C83—21no—
43SD16×C83—42no—
44C2×C8×C83—21no—
45C4⋊C4×C83—21no—
46C4⋊C4×C83—21no—
47C4⋊C4×C83D411yescharacteristic
48C22⋊C4×C83—21no—
49C42×C83C2311yescharacteristic
50C23×C83—21no—
51C22×D83—22no—
52Q8×C83—41no—
53Q8×C83—41no—
54D4×C83—21no—
55D4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—41no—
58C2×C4×C83—41no—
59C2×C4×C83—41no—
60C2×C4×C83C2×D411yescharacteristic
61C2×C4×C83C2×D411yescharacteristic
62C2×C4×C83C2×D411yescharacteristic
63C22⋊D83—44no—
64C22⋊D83—82no—
65C2×Dic83—21no—
66C2×Dic83—21no—
67D332—42no—
68D332—41no—
69D332—41no—
70D332—41no—
71D332—41no—
72C4×D83—44no—
73Dic166—21no—
74Dic166SD1612yescomplemented
75C8×C83—41no—
76C83⋊C8—41no—
77C83⋊C8—41no—
78C22×C83—41no—
79C22×C83—41no—
80C22×C83—41no—
81C22×C83#12011yescharacteristic
82C2×D83—42no—
83C2×D83—42no—
84C2×D83—42no—
85C4×C83—21no—
86C4×C83—41no—
87C4×C83—81no—
88C4×C83C2×SD1611yescharacteristic
89C4×C83#13511yescharacteristic
90C4×C83AGL(1,ℤ/8)11yescharacteristic
91C4×C83AGL(1,ℤ/8)11yescharacteristic
92Dic83—22no—
93Dic83—41no—
94C2×C83—81no—
95C2×C83#44911yescharacteristic
96C2×C83#45011yescharacteristic
97C2×C83#45511yescharacteristic
98D83—82no—
99#1642—831nomaximal, Sylow
100C83#164211yescharacteristic, complemented, Sylow
101#550—831no—
102#486—831no—
103#475—831no—
104#461D8311yescharacteristic, complemented
105#331—831no—
106Q8⋊C8—831no—
107D4⋊C8—831no—
108C2×#41—1661no—
109C2×SD16—21no—
110C4⋊Q8C2×D8311yescharacteristic
111#127—831no—
112#127—1661no—
113#121—1662no—
114#121—1661no—
115C4×Q8—831no—
116C4×D4—1661no—
117C4×D4C2×D8311yescharacteristic
118#106—1661no—
119C4⋊C8—831no—
120C4⋊C8C2×D8311yescharacteristic
121#103—21no—
122#103—1661no—
123#102—1662no—
124C4×C8—831no—
125#41—3322no—
126#41—3322no—
127C2×Q8—21no—
128C2×Q8—831no—
129C2×D4—832no—
130C2×D4—1662no—
131C2×D4—1662no—
132C2×D4—1661no—
133C2×D4—1661no—
134C2×D4C22⋊D8311yescharacteristic
135C22×C4—21no—
136C22×C4—1662no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—1661no—
140C2×C8—1661no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—831no—
144C4⋊C4—1661no—
145C4⋊C4C22⋊D8311yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—1662no—
148C42—1661no—
149C42C22×D8311yescharacteristic
150C23—21no—
151C23—1662no—
152Q8—41no—
153Q8—41no—
154Q8—832no—
155Q8—1661no—
156D4—21no—
157D4—21no—
158D4—3322no—
159D4—3324no—
160D4—3321no—
161D4—3321no—
162D4—3321no—
163D4—3321no—
164D4—6642no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3324no—
170C2×C4—1661no—
171C2×C4—1661no—
172C2×C4D4×D8311yescharacteristic
173C2×C4D4×D8311yescharacteristic
174C2×C4C2×C22⋊D8311yescharacteristic
175C8—41no—
176C8—3321no—
177C8—3321no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—3322no—
182C22—3322no—
183C22—3322no—
184C22#1458176011yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—3321no—
189C4—1662no—
190C4SD16×D8311yescharacteristic
191C4#1458181211yescharacteristic
192C4#1458181811yescharacteristic
193C4#1458181911yescharacteristic
194C2—81no—
195C2—6642no—
196C2#1912784411yescharacteristic
197C2#1912785011yescharacteristic
198C2#1912787711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#461×C83C2
socleC22×C83#120
prime radical#19128372C2
hypercenter#461D83
hypocenterC83#1642
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #19128049C2semidirect
3. #19127983C2semidirect
4. #19127444C2semidirect
5. #19127431C2semidirect
6. D4⋊(C83⋊C8)C2semidirect
7. #461×C83C2semidirect
74. Dic166SD16semidirect
100. C83#1642semidirect
104. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191283722
#191280492
#191279832
#191274442
#191274312
D4⋊(C83⋊C8)2
#461×C832
#164283