Finite Group Databasebeta

← #24906648

Subgroups — #24906648

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#19128436C211yesmaximal, characteristic
2#19128049C211yesmaximal, characteristic
3#19127908C211yesmaximal, characteristic, complemented
4#19127444C211yesmaximal, characteristic
5Q8⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#19127431C211yesmaximal, characteristic, complemented
7#475×C83C211yesmaximal, characteristic, complemented
8C2×#13050580—21no—
9#14581789C2211yescharacteristic
10#14581781—21no—
11#14581780—21no—
12#14581777—22no—
13C4×Dic166C2211yescharacteristic
14C4×D332—21no—
15#14581754—21no—
16C4⋊(C83⋊C8)C2211yescharacteristic
17#14581716—21no—
18#14581712—22no—
19C4×C83⋊C8C2211yescharacteristic
20C4⋊Q8×C83C2211yescharacteristic
21C4×Q8×C83C2211yescharacteristic
22#106×C83—21no—
23C4⋊C8×C83C2211yescharacteristic
24#103×C83—21no—
25C2×Dic166D411yescharacteristic
26C2×D332—21no—
27C2×D332—21no—
28C2×D332D412yes—
29C2×C4×D83—22no—
30#13050580—42no—
31C2×C83⋊C8—21no—
32C2×C83⋊C8—21no—
33C4⋊Dic83D411yescharacteristic
34Dic83⋊C4—21no—
35(C2×D83)⋊C4—22no—
36(C2×D83)⋊C4—22no—
37(C2×D83)⋊C4—22no—
38C4×Dic83—21no—
39C2×Q8×C83—21no—
40C2×Q8×C83D411yescharacteristic
41C2×C8×C83—21no—
42C4⋊C4×C83—21no—
43C4⋊C4×C83—21no—
44C4⋊C4×C83—21no—
45C4⋊C4×C83D411yescharacteristic
46C42×C83—21no—
47C42×C83C2311yescharacteristic
48C22×D83—22no—
49Q8×C83—21no—
50Q8×C83—21no—
51Q8×C83—41no—
52Q8×C83—41no—
53C2×C4×C83—21no—
54C2×C4×C83—21no—
55C2×C4×C83—41no—
56C2×C4×C83C2×D411yescharacteristic
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×Dic83—21no—
60C2×Dic83—21no—
61D332—42no—
62D332—41no—
63D332—41no—
64D332—41no—
65D332—41no—
66C4×D83—82no—
67Dic166—21no—
68Dic166—21no—
69C8×C83—41no—
70C83⋊C8—41no—
71C83⋊C8—41no—
72C22×C83#12011yescharacteristic
73C2×D83—42no—
74C2×D83—42no—
75C2×D83—42no—
76C4×C83—21no—
77C4×C83—41no—
78C4×C83—41no—
79C4×C83—81no—
80C4×C83#13511yescharacteristic
81C4×C83#13511yescharacteristic
82C4×C83AGL(1,ℤ/8)11yescharacteristic
83C4×C83#13711yescharacteristic
84Dic83—41no—
85Dic83—41no—
86C2×C83#44911yescharacteristic
87C2×C83#45211yescharacteristic
88C2×C83#45511yescharacteristic
89D83—82no—
90#1640—831nomaximal, Sylow
91C83#164011yescharacteristic, complemented, Sylow
92#559—831no—
93#486—831no—
94#475D8311yescharacteristic, complemented
95#462—831no—
96#331—831no—
97Q8⋊C8—831no—
98Q8⋊C8—831no—
99C2×Q16—1661no—
100C4⋊Q8C2×D8311yescharacteristic
101#127—831no—
102#124—1661no—
103#124—1661no—
104#123—1662no—
105C4×Q8—831no—
106C4×Q8C2×D8311yescharacteristic
107C4×D4—1661no—
108C4×D4—1661no—
109#106—21no—
110C4⋊C8—831no—
111C4⋊C8C2×D8311yescharacteristic
112#103—21no—
113#103—1661no—
114#102—1662no—
115C4×C8—831no—
116C2×Q8—21no—
117C2×Q8—831no—
118C2×Q8C22⋊D8311yescharacteristic
119C2×D4—832no—
120C2×D4—1661no—
121C2×D4—1661no—
122C22×C4—1662no—
123Q16—3322no—
124C2×C8—21no—
125C2×C8—1661no—
126C2×C8—1661no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—831no—
131C4⋊C4—1661no—
132C4⋊C4C22⋊D8311yescharacteristic
133C22⋊C4—1662no—
134C22⋊C4—1662no—
135C22⋊C4—1662no—
136C42—21no—
137C42—1661no—
138C42C22×D8311yescharacteristic
139C23—1662no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—1661no—
145Q8—1661no—
146D4—3322no—
147D4—3321no—
148D4—3321no—
149D4—3321no—
150D4—3321no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—6642no—
155C2×C4—1661no—
156C2×C4—1661no—
157C2×C4D4×D8311yescharacteristic
158C2×C4D4×D8311yescharacteristic
159C2×C4C2×C22⋊D8311yescharacteristic
160C8—41no—
161C8—3321no—
162C8—3321no—
163C22—3322no—
164C22—3322no—
165C22—3322no—
166C22#1458176011yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—3321no—
172C4—3321no—
173C4#1458181211yescharacteristic
174C4#1458181311yescharacteristic
175C4#1458181911yescharacteristic
176C4#1458182311yescharacteristic
177C2—6642no—
178C2#1912784511yescharacteristic
179C2#1912786111yescharacteristic
180C2#1912787711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#475×C83C2
socleC22×C83#120
prime radical#14581789C22
hypercenter#475D83
hypocenterC83#1640
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #19127908C2semidirect
5. Q8⋊(C83⋊C8)C2semidirect
6. #19127431C2semidirect
7. #475×C83C2semidirect
91. C83#1640semidirect
94. #475D83semidirect

Maximal subgroups

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

subgroupindex
#191284362
#191280492
#191279082
#191274442
Q8⋊(C83⋊C8)2
#191274312
#475×C832
#164083