Finite Group Databasebeta

← #24909484

Subgroups — #24909484

Order 10624.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#19127883C212yesmaximal, complemented
2#19127873C211yesmaximal, characteristic
3#19127500C212yesmaximal, complemented
4#19127498C211yesmaximal, characteristic, complemented
5#563×C83C211yesmaximal, characteristic, complemented
6#14581840C2211yescharacteristic
7#14581823—22no—
8#14581818—21no—
9#14581720—21no—
10#14581720—22no—
11#14581711C2211yescharacteristic
12#14581711C2212yes—
13C4⋊Q8×C83C2212yes—
14#126×C83—22no—
15#125×C83—21no—
16#124×C83C2211yescharacteristic
17#123×C83—21no—
18C22⋊Q8×C83—22no—
19#13050591—41no—
20#13050590—21no—
21#13050590—22no—
22Q8×D83—41no—
23#13050580—42no—
24#13050578—41no—
25#13050577—42no—
26#13050575—41no—
27#13050574—21no—
28#13050574—22no—
29C2×Q8×C83D412yes—
30C2×Q8×C83C2311yescharacteristic
31C2×D4×C83D411yescharacteristic
32C22×C4×C83—21no—
33C4⋊C4×C83—21no—
34C4⋊C4×C83—22no—
35C4⋊C4×C83—21no—
36C4⋊C4×C83—22no—
37C4⋊C4×C83—22no—
38C22⋊C4×C83—22no—
39C22⋊C4×C83—21no—
40C22⋊C4×C83—22no—
41C42×C83D411yescharacteristic
42C42×C83D412yes—
43C23×C83—21no—
44Q8×C83—41no—
45Q8×C83—42no—
46D4×C83—41no—
47C2×C4×C83—21no—
48C2×C4×C83—22no—
49C2×C4×C83—22no—
50C2×C4×C83—81no—
51C2×C4×C83C2×D412yes—
52C2×C4×C83C2×D411yescharacteristic
53C22⋊D83—41no—
54D332—21no—
55D332—22no—
56C4×D83—41no—
57C4×D83—42no—
58Dic166—21no—
59Dic166—22no—
60C83⋊C8—41no—
61C83⋊C8—42no—
62C22×C83—42no—
63C22×C83—41no—
64C22×C83#12011yescharacteristic
65C2×D83—41no—
66C4×C83—21no—
67C4×C83—22no—
68C4×C83—41no—
69C4×C83—42no—
70C4×C83—42no—
71Dic83—41no—
72C2×C83—21no—
73C2×C83—81no—
74C2×C83B(4,𝔽2)11yescharacteristic
75D83—81no—
76#2203—831nomaximal, Sylow
77C83#220311yescharacteristic, complemented, Sylow
78#563D8311yescharacteristic, complemented
79#456—832no—
80#454—831no—
81#356—832no—
82#355—831no—
832−1+4—831no—
84#137—1662no—
85AGL(1,ℤ/8)—1661no—
86C4⋊Q8C2×D8312yes—
87#126—22no—
88#125—21no—
89#124C2×D8311yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—1661no—
93#104—1662no—
94#101—832no—
95#101—831no—
96#41—3321no—
97#41—1661no—
98#41—1662no—
99C2×Q8—3321no—
100C2×Q8C22⋊D8312yes—
101C2×Q8C22×D8311yescharacteristic
102C2×D4C22⋊D8311yescharacteristic
103C22×C4—21no—
104Q16—3322no—
105SD16—3321no—
106SD16—3322no—
107D8—3321no—
108M16—1661no—
109M16—1662no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D8311yescharacteristic
119C42C22⋊D8312yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—1661no—
124Q8—1662no—
125D4—41no—
126D4—3321no—
127D4—1661no—
128D4—1662no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—3321no—
134C2×C4—3322no—
135C2×C4C2×C22⋊D8312yes—
136C2×C4C2×C22⋊D8311yescharacteristic
137C8—3321no—
138C8—3322no—
139C22—42no—
140C22—41no—
141C22—3321no—
142C22#1458176111yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—3321no—
149C2—21no—
150C2—81no—
151C2—6641no—
152C2#1912788811yescharacteristic
subgroupG/H
centerC2#19127888
commutator, cosocleC2×Q8×C83C23
FrattiniC2×Q8C22×D83
Fitting#563×C83C2
socleC2×C83B(4,𝔽2)
prime radical#19127873C2
hypercenter#563D83
hypocenterC83#2203
LCSG ▷ C2×Q8×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #19127883C2semidirect
3. #19127500C2semidirect
4. #19127498C2semidirect
5. #563×C83C2semidirect
77. C83#2203semidirect
78. #563D83semidirect

Maximal subgroups

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

subgroupindex
#191278832
#191278732
#191275002
#191274982
#563×C832
#220383