Finite Group Databasebeta

← #16818754

Subgroups — #16818754

Order 3712.

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×D116C211yesmaximal, characteristic
2#14271559C211yesmaximal, characteristic, complemented
3#14271365C211yesmaximal, characteristic
4#14270910C211yesmaximal, characteristic, complemented
5Q8⋊#6597C211yesmaximal, characteristic, complemented
6#14270890C211yesmaximal, characteristic
7C29×#475C211yesmaximal, characteristic, complemented
8C2×Q8×D29—21no—
9C29×C4⋊Q8C2211yescharacteristic
10C4×Q8×C29C2211yescharacteristic
11C2×#90784—21no—
12#11745259—21no—
13#11745259—22no—
14C4×D116—21no—
15C4×D116C2211yescharacteristic
16#11745231—21no—
17#11745231C2211yescharacteristic
18C29×#106—21no—
19C29×C4⋊C8C2211yescharacteristic
20C29×#103—21no—
21#11745183—21no—
22#11745156—22no—
23C4⋊#6597C2211yescharacteristic
24C4×#6597C2211yescharacteristic
25C2×Q8×C29—21no—
26C2×Q8×C29D411yescharacteristic
27Q8×D29—42no—
28Q8×D29—42no—
29C2×D116D411yescharacteristic
30C2×C4×D29—22no—
31C2×C4×D29—21no—
32C2×Dic58—22no—
33C2×Dic58—21no—
34C2×C8×C29—21no—
35C4⋊C4×C29—21no—
36C4⋊C4×C29—21no—
37C4⋊C4×C29—21no—
38C4⋊C4×C29D411yescharacteristic
39C42×C29—21no—
40C42×C29C2311yescharacteristic
41#90784—42no—
42#90781—22no—
43#90781—21no—
44C4⋊Dic29—22no—
45C4⋊Dic29—21no—
46C4⋊Dic29D412yes—
47C4⋊Dic29D411yescharacteristic
48C2×#6597—21no—
49C2×#6597—21no—
50C22×D29—21no—
51Q8×C29—21no—
52Q8×C29—41no—
53Q8×C29—41no—
54Q8×C29SD1612yescomplemented
55C2×C4×C29—21no—
56C2×C4×C29—21no—
57C2×C4×C29—41no—
58C2×C4×C29C2×D411yescharacteristic
59C2×C4×C29C2×D411yescharacteristic
60C2×C4×C29C2×D411yescharacteristic
61C2×Dic29—21no—
62C2×Dic29—22no—
63D116—21no—
64D116SD1612yes—
65C4×D29—44no—
66C4×D29—41no—
67C4×D29—41no—
68Dic58—44no—
69Dic58—41no—
70Dic58—41no—
71C8×C29—41no—
72#6597—22no—
73#6597—41no—
74#1651—291nomaximal, Sylow
75C22×C29#12011yescharacteristic
76C2×D29—22no—
77C2×D29—41no—
78C2×D29—41no—
79C4×C29—22no—
80C4×C29—21no—
81C4×C29—41no—
82C4×C29—81no—
83C4×C29C2×SD1611yescharacteristic
84C4×C29C2×SD1611yescharacteristic
85C4×C29AGL(1,ℤ/8)11yescharacteristic
86C4×C29#13711yescharacteristic
87Dic29—41no—
88Dic29—42no—
89D4×Q8—291no—
90#498—291no—
91#475D2911yescharacteristic, complemented
92#461—291no—
93#332—291no—
94Q8⋊C8—291no—
95D4⋊C8—291no—
96C2×C29#44811yescharacteristic
97C2×C29#45011yescharacteristic
98C2×C29#45611yescharacteristic
99D29—42no—
100C22×Q8—581no—
101C2×SD16—581no—
102C4⋊Q8—291no—
103C4⋊Q8—581no—
104C4⋊Q8C2×D2911yescharacteristic
105C22⋊Q8—582no—
106C22⋊Q8—581no—
107C4×Q8C2×D2911yescharacteristic
108C4×D4—291no—
109C4×D4—581no—
110#106—21no—
111#106—582no—
112C4⋊C8—291no—
113C4⋊C8C2×D2911yescharacteristic
114#103—21no—
115#103—581no—
116C4×C8—291no—
117C29#165111yescharacteristic, complemented, Sylow
118C2×Q8—21no—
119C2×Q8—582no—
120C2×Q8—581no—
121C2×Q8—1162no—
122C2×Q8—1162no—
123C2×Q8C22⋊D2911yescharacteristic
124C2×D4—291no—
125C22×C4—582no—
126C22×C4—581no—
127SD16—1162no—
128C2×C8—21no—
129C2×C8—581no—
130C2×C8—581no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—291no—
135C4⋊C4—292no—
136C4⋊C4—582no—
137C4⋊C4—581no—
138C4⋊C4C22⋊D2911yescharacteristic
139C22⋊C4—582no—
140C22⋊C4—581no—
141C42—21no—
142C42C22×D2911yescharacteristic
143C23—581no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—1164no—
148Q8—1161no—
149Q8—1161no—
150Q8#9078412yescomplemented
151D4—292no—
152D4—581no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—582no—
157C2×C4—581no—
158C2×C4—1164no—
159C2×C4—1161no—
160C2×C4—1161no—
161C2×C4D4×D2911yescharacteristic
162C2×C4D4×D2911yescharacteristic
163C2×C4C2×C22⋊D2911yescharacteristic
164C8—41no—
165C8—582no—
166C8—1161no—
167C22—582no—
168C22—1161no—
169C22—1161no—
170C22#1174529911yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—1162no—
176C4—1161no—
177C4SD16×D2911yescharacteristic
178C4#1174527711yescharacteristic
179C4C2×#9078411yescharacteristic
180C4#1174531111yescharacteristic
181C2—1162no—
182C2#1427130011yescharacteristic
183C2#1427131311yescharacteristic
184C2#1427134411yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#475C2
socleC22×C29#120
prime radicalC2×D116D4
hypercenter#475D29
hypocenterC29#1651
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #14271559C2semidirect
4. #14270910C2semidirect
5. Q8⋊#6597C2semidirect
7. C29×#475C2semidirect
54. Q8×C29SD16semidirect
91. #475D29semidirect
117. C29#1651semidirect
150. Q8#90784semidirect

Maximal subgroups

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

subgroupindex
Q8×D1162
#142715592
#142713652
#142709102
Q8⋊#65972
#142708902
C29×#4752
#165129