Finite Group Databasebeta

← #24906690

Subgroups — #24906690

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#19128416C211yesmaximal, characteristic
2#19128053C211yesmaximal, characteristic, complemented
3#19127903C211yesmaximal, characteristic, complemented
4#19127449C211yesmaximal, characteristic, complemented
5D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#19127428C211yesmaximal, characteristic, complemented
7#474×C83C211yesmaximal, characteristic, complemented
8C2×#13050577—21no—
9#14581792C2211yescharacteristic
10#14581778—22no—
11#14581766—21no—
12#14581763—21no—
13C4×D332C2211yescharacteristic
14#14581751—21no—
15#14581750—21no—
16C4⋊(C83⋊C8)C2211yescharacteristic
17#14581716—21no—
18#14581715—22no—
19C4×C83⋊C8C2211yescharacteristic
20C4⋊Q8×C83C2211yescharacteristic
21C4×D4×C83C2211yescharacteristic
22#107×C83—21no—
23C4⋊C8×C83C2211yescharacteristic
24#102×C83—21no—
25C2×Dic166D412yes—
26C2×D332D411yescharacteristic
27C2×C22⋊D83—22no—
28C2×C4×D83—21no—
29#13050577—42no—
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—41no—
51Q8×C83—41no—
52D4×C83—21no—
53D4×C83—21no—
54C2×C4×C83—21no—
55C2×C4×C83—41no—
56C2×C4×C83—41no—
57C2×C4×C83—41no—
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C2×C4×C83C2×D411yescharacteristic
61C22⋊D83—82no—
62C2×Dic83—21no—
63C2×Dic83—22no—
64D332—21no—
65D332—21no—
66C4×D83—41no—
67C4×D83—41no—
68Dic166—42no—
69C8×C83—41no—
70C83⋊C8—41no—
71C83⋊C8—41no—
72C22×C83—41no—
73C22×C83—41no—
74C22×C83—41no—
75C22×C83#12011yescharacteristic
76C2×D83—41no—
77C2×D83—41no—
78C2×D83—41no—
79C4×C83—21no—
80C4×C83—41no—
81C4×C83—81no—
82C4×C83#13511yescharacteristic
83C4×C83#13511yescharacteristic
84C4×C83AGL(1,ℤ/8)11yescharacteristic
85C4×C83#13711yescharacteristic
86Dic83—41no—
87Dic83—42no—
88C2×C83—81no—
89C2×C83#44911yescharacteristic
90C2×C83#45211yescharacteristic
91C2×C83#45511yescharacteristic
92D83—81no—
93#1646—831nomaximal, Sylow
94C83#164611yescharacteristic, complemented, Sylow
95#555—831no—
96#487—831no—
97#474D8311yescharacteristic, complemented
98#461—831no—
99#333—831no—
100D4⋊C8—831no—
101D4⋊C8—831no—
102C2×SD16—1661no—
103C4⋊Q8—831no—
104C4⋊Q8C2×D8311yescharacteristic
105#124—1662no—
106#121—1661no—
107#121—1661no—
108C4×D4—831no—
109C4×D4C2×D8311yescharacteristic
110#117—1661no—
111#117—1661no—
112#107—21no—
113C4⋊C8—831no—
114C4⋊C8C2×D8311yescharacteristic
115#103—1662no—
116#103—1661no—
117#102—21no—
118C4×C8—831no—
119C2×Q8—21no—
120C2×Q8—832no—
121C2×D4—831no—
122C2×D4—1662no—
123C2×D4C22⋊D8311yescharacteristic
124C22×C4—21no—
125C22×C4—1661no—
126SD16—3322no—
127C2×C8—21no—
128C2×C8—1661no—
129C2×C8—1661no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—831no—
133C4⋊C4—1661no—
134C4⋊C4—1661no—
135C4⋊C4C22⋊D8311yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1662no—
138C22⋊C4—1662no—
139C22⋊C4—1661no—
140C42—1662no—
141C42C22×D8311yescharacteristic
142C23—21no—
143C23—1661no—
144Q8—41no—
145Q8—41no—
146Q8—3322no—
147D4—21no—
148D4—21no—
149D4—6642no—
150D4—1661no—
151D4—1661no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3321no—
157C2×C4—3321no—
158C2×C4—1661no—
159C2×C4—1662no—
160C2×C4D4×D8311yescharacteristic
161C2×C4D4×D8311yescharacteristic
162C2×C4C2×C22⋊D8311yescharacteristic
163C8—41no—
164C8—3321no—
165C8—3321no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—3321no—
170C22—3321no—
171C22—3321no—
172C22#1458176011yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—3321no—
177C4—3322no—
178C4#1458181111yescharacteristic
179C4#1458181211yescharacteristic
180C4#1458181611yescharacteristic
181C4#1458182311yescharacteristic
182C2—81no—
183C2—6641no—
184C2#1912784311yescharacteristic
185C2#1912786111yescharacteristic
186C2#1912787711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#474×C83C2
socleC22×C83#120
prime radical#19128416C2
hypercenter#474D83
hypocenterC83#1646
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #19128053C2semidirect
3. #19127903C2semidirect
4. #19127449C2semidirect
5. D4⋊(C83⋊C8)C2semidirect
6. #19127428C2semidirect
7. #474×C83C2semidirect
94. C83#1646semidirect
97. #474D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191280532
#191279032
#191274492
D4⋊(C83⋊C8)2
#191274282
#474×C832
#164683