Finite Group Databasebeta

← #24906607

Subgroups — #24906607

Order 10624.

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

normal subgroups only

This group has 218 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#19128416C211yesmaximal, characteristic
2#19128080C211yesmaximal, characteristic, complemented
3#19127988C211yesmaximal, characteristic, complemented
4#19127449C211yesmaximal, characteristic, complemented
5D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#19127428C211yesmaximal, characteristic, complemented
7#476×C83C211yesmaximal, characteristic, complemented
8C2×#13050580—22no—
9#14581792C2211yescharacteristic
10#14581778—22no—
11#14581766—21no—
12#14581763—21no—
13C4×D332C2211yescharacteristic
14#14581751—21no—
15#14581750—21no—
16#14581723—21no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18#14581712—21no—
19C4×C83⋊C8C2211yescharacteristic
20C4⋊Q8×C83C2211yescharacteristic
21C4×D4×C83C2211yescharacteristic
22#106×C83—21no—
23C4⋊C8×C83C2211yescharacteristic
24#102×C83—21no—
25C2×Dic166D412yes—
26C2×D332D411yescharacteristic
27C2×C22⋊D83—22no—
28C2×C4×D83—21no—
29#13050580—44no—
30C2×C83⋊C8—21no—
31C2×C83⋊C8—21no—
32C4⋊Dic83—21no—
33C4⋊Dic83—21no—
34C4⋊Dic83D411yescharacteristic
35C22⋊Dic83—22no—
36(C2×D83)⋊C4—22no—
37(C2×D83)⋊C4—21no—
38C4×Dic83—22no—
39C2×Q8×C83—21no—
40C2×D4×C83D411yescharacteristic
41C22×C4×C83—21no—
42C2×C8×C83—21no—
43C4⋊C4×C83—21no—
44C4⋊C4×C83—21no—
45C4⋊C4×C83D411yescharacteristic
46C22⋊C4×C83—21no—
47C42×C83C2311yescharacteristic
48C23×C83—21no—
49C22×D83—21no—
50Q8×C83—42no—
51D4×C83—21no—
52D4×C83—21no—
53C2×C4×C83—21no—
54C2×C4×C83—41no—
55C2×C4×C83—41no—
56C2×C4×C83—41no—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C22⋊D83—82no—
61C2×Dic83—21no—
62C2×Dic83—22no—
63D332—21no—
64D332—21no—
65C4×D83—41no—
66C4×D83—41no—
67Dic166—42no—
68C8×C83—41no—
69C83⋊C8—22no—
70C83⋊C8—41no—
71C22×C83—41no—
72C22×C83—41no—
73C22×C83—41no—
74C22×C83#12011yescharacteristic
75C2×D83—41no—
76C2×D83—41no—
77C2×D83—41no—
78C4×C83—21no—
79C4×C83—41no—
80C4×C83—81no—
81C4×C83#13511yescharacteristic
82C4×C83#13511yescharacteristic
83C4×C83#13711yescharacteristic
84C4×C83#13711yescharacteristic
85Dic83—41no—
86Dic83—42no—
87C2×C83—81no—
88C2×C83#45211yescharacteristic
89C2×C83#45211yescharacteristic
90C2×C83#45311yescharacteristic
91D83—81no—
92#1634—831nomaximal, Sylow
93C83#163411yescharacteristic, complemented, Sylow
94#555—831no—
95#494—831no—
96#476—831no—
97#476D8311yescharacteristic, complemented
98#333—831no—
99D4⋊C8—831no—
100D4⋊C8—831no—
101C2×Q16—1662no—
102C4⋊Q8—831no—
103C4⋊Q8C2×D8311yescharacteristic
104#124—1662no—
105#121—1661no—
106#121—1661no—
107C4×D4—831no—
108C4×D4C2×D8311yescharacteristic
109#117—1661no—
110#117—1661no—
111#106—21no—
112#106—1661no—
113C4⋊C8—831no—
114C4⋊C8C2×D8311yescharacteristic
115#102—21no—
116#102—1661no—
117C4×C8—831no—
118C2×Q8—21no—
119C2×Q8—832no—
120C2×D4—831no—
121C2×D4—1662no—
122C2×D4C22⋊D8311yescharacteristic
123C22×C4—21no—
124C22×C4—1661no—
125Q16—3324no—
126C2×C8—21no—
127C2×C8—1661no—
128C2×C8—1661no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—831no—
132C4⋊C4—1661no—
133C4⋊C4—1661no—
134C4⋊C4C22⋊D8311yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1662no—
137C22⋊C4—1662no—
138C22⋊C4—1661no—
139C42—1662no—
140C42C22×D8311yescharacteristic
141C23—21no—
142C23—1661no—
143Q8—42no—
144Q8—3322no—
145D4—21no—
146D4—21no—
147D4—6642no—
148D4—1661no—
149D4—1661no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—3321no—
155C2×C4—3321no—
156C2×C4—1661no—
157C2×C4—1662no—
158C2×C4D4×D8311yescharacteristic
159C2×C4D4×D8311yescharacteristic
160C2×C4C2×C22⋊D8311yescharacteristic
161C8—41no—
162C8—3321no—
163C8—1662no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—3321no—
168C22—3321no—
169C22—3321no—
170C22#1458176011yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—3321no—
175C4—3322no—
176C4#1458181211yescharacteristic
177C4#1458181311yescharacteristic
178C4#1458182111yescharacteristic
179C4#1458182311yescharacteristic
180C2—81no—
181C2—6641no—
182C2#1912786011yescharacteristic
183C2#1912786111yescharacteristic
184C2#1912786711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#476×C83C2
socleC22×C83#120
prime radical#19128416C2
hypercenter#476D83
hypocenterC83#1634
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #19128080C2semidirect
3. #19127988C2semidirect
4. #19127449C2semidirect
5. D4⋊(C83⋊C8)C2semidirect
6. #19127428C2semidirect
7. #476×C83C2semidirect
93. C83#1634semidirect
97. #476D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191280802
#191279882
#191274492
D4⋊(C83⋊C8)2
#191274282
#476×C832
#163483