Finite Group Databasebeta

← #24906646

Subgroups — #24906646

Order 10624.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#19128437C211yesmaximal, characteristic
2#19127983C211yesmaximal, characteristic, complemented
3#19127908C211yesmaximal, characteristic, complemented
4#19127445C211yesmaximal, characteristic
5#19127433C211yesmaximal, characteristic, complemented
6#19127433C211yesmaximal, characteristic, complemented
7#486×C83C211yesmaximal, characteristic
8C2×#13050580—21no—
9#14581779—21no—
10#14581779—21no—
11#14581773—22no—
12C4×Dic166C2211yescharacteristic
13C4×Dic166C2211yescharacteristic
14D4×Dic83—21no—
15D4×Dic83—21no—
16#14581723—21no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18C4⋊(C83⋊C8)C2211yescharacteristic
19#14581716—21no—
20#14581715—21no—
21C4⋊Q8×C83C2211yescharacteristic
22#127×C83C2211yescharacteristic
23#102×C83—22no—
24C4×C8×C83C2211yescharacteristic
25C2×Dic166D411yescharacteristic
26C2×Dic166D411yescharacteristic
27C22×Dic83—22no—
28#13050580—42no—
29C2×C83⋊C8—21no—
30C2×C83⋊C8—21no—
31C4⋊Dic83D411yescharacteristic
32C4⋊Dic83D411yescharacteristic
33Dic83⋊C4—21no—
34Dic83⋊C4—21no—
35C22⋊Dic83—22no—
36C22⋊Dic83—22no—
37C22⋊Dic83—22no—
38C4×Dic83—21no—
39C4×Dic83—21no—
40C2×Q8×C83—21no—
41C2×D4×C83—21no—
42C2×D4×C83—21no—
43C2×D4×C83D412yes—
44C2×C8×C83—21no—
45C4⋊C4×C83—21no—
46C4⋊C4×C83—21no—
47C42×C83C2311yescharacteristic
48C23×C83—22no—
49Q8×C83—41no—
50Q8×C83—41no—
51D4×C83—42no—
52D4×C83—41no—
53D4×C83—41no—
54D4×C83—41no—
55D4×C83—41no—
56C2×C4×C83—41no—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C2×Dic83—21no—
61C2×Dic83—21no—
62C2×Dic83—21no—
63C2×Dic83—21no—
64C2×Dic83—82no—
65Dic166—21no—
66Dic166—21no—
67Dic166—21no—
68Dic166—21no—
69C8×C83—41no—
70C83⋊C8—41no—
71C83⋊C8—41no—
72C22×C83—42no—
73C22×C83—42no—
74C22×C83—42no—
75C22×C83#12011yescharacteristic
76C4×C83—21no—
77C4×C83—81no—
78C4×C83#13511yescharacteristic
79C4×C83#13511yescharacteristic
80C4×C83AGL(1,ℤ/8)11yescharacteristic
81C4×C83#13711yescharacteristic
82Dic83—41no—
83Dic83—41no—
84Dic83—41no—
85Dic83—41no—
86C2×C83—82no—
87C2×C83#44911yescharacteristic
88C2×C83#45211yescharacteristic
89C2×C83#45511yescharacteristic
90#1640—831nomaximal, Sylow
91C83#164011yescharacteristic, complemented, Sylow
92#559—831no—
93#486D8311yescharacteristic
94#475—831no—
95#462—831no—
96#331—831no—
97Q8⋊C8—831no—
98Q8⋊C8—831no—
99C2×Q16—1661no—
100C4⋊Q8C2×D8311yescharacteristic
101#127C2×D8311yescharacteristic
102#124—1661no—
103#124—1661no—
104#123—1662no—
105C4×Q8—831no—
106C4×Q8—831no—
107C4×D4—1661no—
108C4×D4—1661no—
109#106—1661no—
110C4⋊C8—831no—
111C4⋊C8—831no—
112#103—1661no—
113#103—1661no—
114#102—22no—
115C4×C8C2×D8311yescharacteristic
116C2×Q8—21no—
117C2×Q8—831no—
118C2×Q8—831no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D8312yes—
122C22×C4—1662no—
123Q16—3322no—
124C2×C8—21no—
125C2×C8—1661no—
126C2×C8—1661no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—831no—
130C4⋊C4—831no—
131C4⋊C4—1661no—
132C4⋊C4—1661no—
133C22⋊C4—1662no—
134C22⋊C4—1662no—
135C22⋊C4—1662no—
136C42—1661no—
137C42—1661no—
138C42C22×D8311yescharacteristic
139C23—22no—
140Q8—41no—
141Q8—41no—
142Q8—1661no—
143Q8—1661no—
144Q8—1661no—
145Q8—1661no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—41no—
152C2×C4—6642no—
153C2×C4—1661no—
154C2×C4—1661no—
155C2×C4—1661no—
156C2×C4—1661no—
157C2×C4D4×D8311yescharacteristic
158C2×C4D4×D8311yescharacteristic
159C2×C4C2×C22⋊D8311yescharacteristic
160C8—41no—
161C8—3321no—
162C8—3321no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1458176011yescharacteristic
167C4—21no—
168C4—81no—
169C4—3321no—
170C4—3321no—
171C4—3321no—
172C4—3321no—
173C4#1458181111yescharacteristic
174C4#1458181311yescharacteristic
175C4#1458181611yescharacteristic
176C4#1458182111yescharacteristic
177C2—82no—
178C2#1912784311yescharacteristic
179C2#1912786011yescharacteristic
180C2#1912787811yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#486×C83C2
socleC22×C83#120
prime radical#127×C83C22
hypercenter#486D83
hypocenterC83#1640
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #19127983C2semidirect
3. #19127908C2semidirect
5. #19127433C2semidirect
6. #19127433C2semidirect
91. C83#1640semidirect

Maximal subgroups

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

subgroupindex
#191284372
#191279832
#191279082
#191274452
#191274332
#486×C832
#164083