Finite Group Databasebeta

← #24906725

Subgroups — #24906725

Order 10624.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D332C211yesmaximal, characteristic
2#19128097C211yesmaximal, characteristic, complemented
3#19127903C211yesmaximal, characteristic
4#19127448C211yesmaximal, characteristic, complemented
5Q8⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#19127428C211yesmaximal, characteristic
7#475×C83C211yesmaximal, characteristic, complemented
8C2×Q8×D83—21no—
9C2×#13050577—21no—
10#14581792—21no—
11#14581792C2211yescharacteristic
12#14581772—22no—
13#14581772—21no—
14C4×D332—21no—
15C4×D332C2211yescharacteristic
16#14581723—22no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18#14581716—21no—
19C4×C83⋊C8C2211yescharacteristic
20C4⋊Q8×C83C2211yescharacteristic
21C4×Q8×C83C2211yescharacteristic
22#106×C83—21no—
23C4⋊C8×C83C2211yescharacteristic
24#103×C83—21no—
25C2×Dic166—22no—
26C2×Dic166—21no—
27Q8×D83—42no—
28Q8×D83—42no—
29C2×D332D411yescharacteristic
30C2×C4×D83—22no—
31C2×C4×D83—21no—
32#13050577—42no—
33C2×C83⋊C8—21no—
34C2×C83⋊C8—21no—
35C4⋊Dic83—22no—
36C4⋊Dic83—21no—
37C4⋊Dic83D412yes—
38C4⋊Dic83D411yescharacteristic
39(C2×D83)⋊C4—22no—
40(C2×D83)⋊C4—21no—
41C2×Q8×C83—21no—
42C2×Q8×C83D411yescharacteristic
43C2×C8×C83—21no—
44C4⋊C4×C83—21no—
45C4⋊C4×C83—21no—
46C4⋊C4×C83—21no—
47C4⋊C4×C83D411yescharacteristic
48C42×C83—21no—
49C42×C83C2311yescharacteristic
50C22×D83—21no—
51Q8×C83—21no—
52Q8×C83—41no—
53Q8×C83—41no—
54Q8×C83SD1612yescomplemented
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—41no—
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C2×C4×C83C2×D411yescharacteristic
61C2×Dic83—22no—
62C2×Dic83—21no—
63D332—21no—
64D332SD1612yes—
65C4×D83—44no—
66C4×D83—41no—
67C4×D83—41no—
68Dic166—44no—
69Dic166—41no—
70Dic166—41no—
71C8×C83—41no—
72C83⋊C8—22no—
73C83⋊C8—41no—
74C22×C83#12011yescharacteristic
75C2×D83—22no—
76C2×D83—41no—
77C2×D83—41no—
78C4×C83—22no—
79C4×C83—21no—
80C4×C83—41no—
81C4×C83—81no—
82C4×C83C2×SD1611yescharacteristic
83C4×C83C2×SD1611yescharacteristic
84C4×C83AGL(1,ℤ/8)11yescharacteristic
85C4×C83#13711yescharacteristic
86Dic83—41no—
87Dic83—42no—
88C2×C83#44811yescharacteristic
89C2×C83#45011yescharacteristic
90C2×C83#45611yescharacteristic
91D83—42no—
92#1651—831nomaximal, Sylow
93C83#165111yescharacteristic, complemented, Sylow
94D4×Q8—831no—
95#498—831no—
96#475D8311yescharacteristic, complemented
97#461—831no—
98#332—831no—
99Q8⋊C8—831no—
100D4⋊C8—831no—
101C22×Q8—1661no—
102C2×SD16—1661no—
103C4⋊Q8—831no—
104C4⋊Q8—1661no—
105C4⋊Q8C2×D8311yescharacteristic
106C22⋊Q8—1662no—
107C22⋊Q8—1661no—
108C4×Q8C2×D8311yescharacteristic
109C4×D4—831no—
110C4×D4—1661no—
111#106—21no—
112#106—1662no—
113C4⋊C8—831no—
114C4⋊C8C2×D8311yescharacteristic
115#103—21no—
116#103—1661no—
117C4×C8—831no—
118C2×Q8—21no—
119C2×Q8—3322no—
120C2×Q8—3322no—
121C2×Q8—1662no—
122C2×Q8—1661no—
123C2×Q8C22⋊D8311yescharacteristic
124C2×D4—831no—
125C22×C4—1662no—
126C22×C4—1661no—
127SD16—3322no—
128C2×C8—21no—
129C2×C8—1661no—
130C2×C8—1661no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—831no—
135C4⋊C4—832no—
136C4⋊C4—1662no—
137C4⋊C4—1661no—
138C4⋊C4C22⋊D8311yescharacteristic
139C22⋊C4—1662no—
140C22⋊C4—1661no—
141C42—21no—
142C42C22×D8311yescharacteristic
143C23—1661no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—3324no—
148Q8—3321no—
149Q8—3321no—
150Q8#1305057712yescomplemented
151D4—832no—
152D4—1661no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—3324no—
157C2×C4—3321no—
158C2×C4—3321no—
159C2×C4—1662no—
160C2×C4—1661no—
161C2×C4D4×D8311yescharacteristic
162C2×C4D4×D8311yescharacteristic
163C2×C4C2×C22⋊D8311yescharacteristic
164C8—41no—
165C8—3321no—
166C8—1662no—
167C22—3321no—
168C22—3321no—
169C22—1662no—
170C22#1458176011yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—3322no—
176C4—3321no—
177C4SD16×D8311yescharacteristic
178C4C2×#1305057711yescharacteristic
179C4#1458181711yescharacteristic
180C4#1458182111yescharacteristic
181C2—3322no—
182C2#1912783811yescharacteristic
183C2#1912785111yescharacteristic
184C2#1912788211yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#475×C83C2
socleC22×C83#120
prime radicalC2×D332D4
hypercenter#475D83
hypocenterC83#1651
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #19128097C2semidirect
4. #19127448C2semidirect
5. Q8⋊(C83⋊C8)C2semidirect
7. #475×C83C2semidirect
54. Q8×C83SD16semidirect
93. C83#1651semidirect
96. #475D83semidirect
150. Q8#13050577semidirect

Maximal subgroups

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

subgroupindex
Q8×D3322
#191280972
#191279032
#191274482
Q8⋊(C83⋊C8)2
#191274282
#475×C832
#165183