Finite Group Databasebeta

← #16818655

Subgroups — #16818655

Order 3712.

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

normal subgroups only

This group has 228 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#14271513C211yesmaximal, characteristic, complemented
3#14271450C211yesmaximal, characteristic
4#14270910C211yesmaximal, characteristic, complemented
5Q8⋊#6597C211yesmaximal, characteristic, complemented
6#14270890C211yesmaximal, characteristic
7C29×#477C211yesmaximal, characteristic, complemented
8C2×Q8×D29—21no—
9C29×C4⋊Q8C2211yescharacteristic
10C4×Q8×C29C2211yescharacteristic
11#11745259—21no—
12#11745259—22no—
13C4×D116—21no—
14C4×D116C2211yescharacteristic
15#11745231—21no—
16#11745231C2211yescharacteristic
17C29×#107—21no—
18C29×C4⋊C8C2211yescharacteristic
19C29×#103—21no—
20#11745183—22no—
21#11745157—21no—
22#11745156—21no—
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#90781—22no—
42#90781—21no—
43C4⋊Dic29—22no—
44C4⋊Dic29—21no—
45C4⋊Dic29D411yescharacteristic
46C4⋊Dic29D412yes—
47C2×#6597—21no—
48C2×#6597—21no—
49C22×D29—21no—
50Q8×C29—21no—
51Q8×C29—42no—
52Q8×C29SD1612yes—
53C2×C4×C29—21no—
54C2×C4×C29—21no—
55C2×C4×C29—41no—
56C2×C4×C29C2×D411yescharacteristic
57C2×C4×C29C2×D411yescharacteristic
58C2×C4×C29C2×D411yescharacteristic
59C2×Dic29—21no—
60C2×Dic29—22no—
61D116—21no—
62D116Q1612yes—
63C4×D29—44no—
64C4×D29—41no—
65C4×D29—41no—
66Dic58—44no—
67Dic58—41no—
68Dic58—41no—
69C8×C29—41no—
70#6597—41no—
71#6597—41no—
72#1637—291nomaximal, Sylow
73C22×C29#12011yescharacteristic
74C2×D29—22no—
75C2×D29—41no—
76C2×D29—41no—
77C4×C29—22no—
78C4×C29—21no—
79C4×C29—41no—
80C4×C29—81no—
81C4×C29C2×SD1611yescharacteristic
82C4×C29C2×Q1611yescharacteristic
83C4×C29AGL(1,ℤ/8)11yescharacteristic
84C4×C29#13711yescharacteristic
85Dic29—41no—
86Dic29—42no—
87D4×Q8—291no—
88#487—291no—
89#477D2911yescharacteristic, complemented
90#476—291no—
91#332—291no—
92Q8⋊C8—291no—
93D4⋊C8—291no—
94C2×C29#45011yescharacteristic
95C2×C29#45111yescharacteristic
96C2×C29#45411yescharacteristic
97D29—42no—
98C22×Q8—581no—
99C4⋊Q8—291no—
100C4⋊Q8—581no—
101C4⋊Q8C2×D2911yescharacteristic
102C22⋊Q8—582no—
103C22⋊Q8—581no—
104C4×Q8C2×D2911yescharacteristic
105C4×D4—291no—
106C4×D4—581no—
107#107—21no—
108#106—581no—
109C4⋊C8—291no—
110C4⋊C8C2×D2911yescharacteristic
111#103—21no—
112#103—582no—
113#102—581no—
114C4×C8—291no—
115C29#163711yescharacteristic, complemented, Sylow
116C2×Q8—21no—
117C2×Q8—582no—
118C2×Q8—581no—
119C2×Q8—1162no—
120C2×Q8—1162no—
121C2×Q8C22⋊D2911yescharacteristic
122C2×D4—291no—
123C22×C4—582no—
124C22×C4—581no—
125C2×C8—21no—
126C2×C8—581no—
127C2×C8—581no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—292no—
132C4⋊C4—291no—
133C4⋊C4—582no—
134C4⋊C4—581no—
135C4⋊C4C22⋊D2911yescharacteristic
136C22⋊C4—582no—
137C22⋊C4—581no—
138C42—21no—
139C42C22×D2911yescharacteristic
140C23—581no—
141Q8—21no—
142Q8—42no—
143Q8—1164no—
144Q8—1161no—
145Q8—1161no—
146Q8#9078412yes—
147D4—292no—
148D4—581no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—582no—
153C2×C4—581no—
154C2×C4—1164no—
155C2×C4—1161no—
156C2×C4—1161no—
157C2×C4D4×D2911yescharacteristic
158C2×C4D4×D2911yescharacteristic
159C2×C4C2×C22⋊D2911yescharacteristic
160C8—41no—
161C8—1161no—
162C8—1161no—
163C22—582no—
164C22—1161no—
165C22—1161no—
166C22#1174529911yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—81no—
171C4—1162no—
172C4—1161no—
173C4Q16×D2911yescharacteristic
174C4#1174528011yescharacteristic
175C4C2×#9078411yescharacteristic
176C4#1174531111yescharacteristic
177C2—1162no—
178C2#1427131311yescharacteristic
179C2#1427131711yescharacteristic
180C2#1427133411yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#477C2
socleC22×C29#120
prime radicalC2×D116D4
hypercenter#477D29
hypocenterC29#1637
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #14271513C2semidirect
4. #14270910C2semidirect
5. Q8⋊#6597C2semidirect
7. C29×#477C2semidirect
89. #477D29semidirect
115. C29#1637semidirect

Maximal subgroups

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

subgroupindex
Q8×D1162
#142715132
#142714502
#142709102
Q8⋊#65972
#142708902
C29×#4772
#163729