Finite Group Databasebeta

← #24909489

Subgroups — #24909489

Order 10624.

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

normal subgroups only

This group has 236 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#19127883C211yesmaximal, characteristic, complemented
2#19127873C212yesmaximal, complemented
3#19127500C211yesmaximal, characteristic, complemented
4#19127498C212yesmaximal, complemented
5#555×C83C211yesmaximal, characteristic, complemented
6#14581840C2211yescharacteristic
7#14581823—21no—
8#14581818—22no—
9#14581720—22no—
10#14581720—21no—
11#14581711C2212yes—
12#14581711C2211yescharacteristic
13C4⋊Q8×C83C2211yescharacteristic
14#124×C83—21no—
15#124×C83C2212yes—
16#121×C83—22no—
17C4×D4×C83—21no—
18#117×C83—22no—
19#13050591—41no—
20#13050590—22no—
21#13050590—21no—
22Q8×D83—41no—
23#13050580—41no—
24#13050578—42no—
25#13050577—41no—
26#13050575—42no—
27#13050574—22no—
28#13050574—21no—
29C2×Q8×C83D411yescharacteristic
30C2×Q8×C83C2311yescharacteristic
31C2×D4×C83—21no—
32C2×D4×C83—21no—
33C2×D4×C83D412yes—
34C22×C4×C83—22no—
35C4⋊C4×C83—21no—
36C4⋊C4×C83—22no—
37C22⋊C4×C83—22no—
38C22⋊C4×C83—22no—
39C22⋊C4×C83—22no—
40C42×C83—21no—
41C42×C83D412yes—
42C42×C83D411yescharacteristic
43C23×C83—22no—
44Q8×C83—41no—
45Q8×C83—41no—
46D4×C83—42no—
47D4×C83—42no—
48D4×C83—81no—
49C2×C4×C83—21no—
50C2×C4×C83—22no—
51C2×C4×C83—21no—
52C2×C4×C83—44no—
53C2×C4×C83C2×D411yescharacteristic
54C2×C4×C83C2×D412yes—
55C22⋊D83—41no—
56D332—22no—
57D332—21no—
58C4×D83—42no—
59C4×D83—41no—
60Dic166—22no—
61Dic166—21no—
62C83⋊C8—42no—
63C83⋊C8—41no—
64C22×C83—42no—
65C22×C83—42no—
66C22×C83—42no—
67C22×C83#12011yescharacteristic
68C2×D83—41no—
69C4×C83—22no—
70C4×C83—22no—
71C4×C83—21no—
72C4×C83—42no—
73C4×C83—41no—
74Dic83—41no—
75C2×C83—21no—
76C2×C83—82no—
77C2×C83B(4,𝔽2)11yescharacteristic
78D83—81no—
79#2204—831nomaximal, Sylow
80C83#220411yescharacteristic, complemented, Sylow
81#555D8311yescharacteristic, complemented
82#456—831no—
83#454—832no—
84#356—831no—
85#355—832no—
862−1+4—831no—
87#137—1661no—
88AGL(1,ℤ/8)—1662no—
89C4⋊Q8C2×D8311yescharacteristic
90#124—21no—
91#124C2×D8312yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—1662no—
96#104—1661no—
97#101—831no—
98#101—832no—
99#41—3321no—
100#41—1662no—
101#41—1661no—
102C2×Q8—3321no—
103C2×Q8C22⋊D8311yescharacteristic
104C2×Q8C22×D8311yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D8312yes—
108C22×C4—22no—
109Q16—3321no—
110SD16—3322no—
111SD16—3321no—
112D8—3322no—
113M16—1662no—
114M16—1661no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D8312yes—
122C42C22⋊D8311yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—1662no—
127Q8—1661no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—3321no—
132D4—1662no—
133D4—1661no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—3322no—
139C2×C4—3321no—
140C2×C4C2×C22⋊D8311yescharacteristic
141C2×C4C2×C22⋊D8312yes—
142C8—3322no—
143C8—3321no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—3321no—
148C22#1458176111yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—3321no—
155C2—21no—
156C2—82no—
157C2—6641no—
158C2#1912788811yescharacteristic
subgroupG/H
centerC2#19127888
commutator, cosocleC2×Q8×C83C23
FrattiniC2×Q8C22×D83
Fitting#555×C83C2
socleC2×C83B(4,𝔽2)
hypercenter#555D83
hypocenterC83#2204
LCSG ▷ C2×Q8×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #19127883C2semidirect
2. #19127873C2semidirect
3. #19127500C2semidirect
4. #19127498C2semidirect
5. #555×C83C2semidirect
80. C83#2204semidirect
81. #555D83semidirect

Maximal subgroups

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

subgroupindex
#191278832
#191278732
#191275002
#191274982
#555×C832
#220483