Finite Group Databasebeta

← #16818758

Subgroups — #16818758

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#14271561C211yesmaximal, characteristic, complemented
3#14271448C211yesmaximal, characteristic, complemented
4#14271369C211yesmaximal, characteristic
5#14270908C211yesmaximal, characteristic, complemented
6#14270892C211yesmaximal, characteristic
7C29×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D29—21no—
9C4×Q8×C29C2211yescharacteristic
10C2×#90773—21no—
11#11745259—21no—
12#11745259—22no—
13C4×D116—21no—
14C4×D116C2211yescharacteristic
15#11745231—21no—
16#11745231C2211yescharacteristic
17#11745231C2211yescharacteristic
18C29×C4⋊C8C2211yescharacteristic
19C4×C8×C29C2211yescharacteristic
20#11745183—21no—
21#11745165—22no—
22#11745164—21no—
23#11745156—21no—
24C4⋊#6597C2211yescharacteristic
25C2×Q8×C29D411yescharacteristic
26Q8×D29—42no—
27Q8×D29—42no—
28C2×D116D411yescharacteristic
29C2×C4×D29—22no—
30C2×C4×D29—21no—
31C2×Dic58—22no—
32C2×Dic58—21no—
33C2×Dic58—21no—
34C2×C8×C29—21no—
35C2×C8×C29—21no—
36C4⋊C4×C29—21no—
37C4⋊C4×C29D411yescharacteristic
38C42×C29—21no—
39C42×C29C2311yescharacteristic
40#90781—22no—
41#90781—21no—
42C4⋊Dic29—22no—
43C4⋊Dic29—21no—
44C4⋊Dic29—21no—
45C4⋊Dic29—21no—
46C4⋊Dic29D411yescharacteristic
47C4⋊Dic29D412yes—
48C2×#6597—21no—
49#90773—42no—
50C22×D29—21no—
51Q8×C29—21no—
52Q8×C29SD1612yescomplemented
53C2×C4×C29—21no—
54C2×C4×C29—21no—
55C2×C4×C29C2×D411yescharacteristic
56C2×C4×C29C2×D411yescharacteristic
57C2×C4×C29C2×D411yescharacteristic
58C2×Dic29—21no—
59C2×Dic29—22no—
60C2×Dic29—41no—
61D116—21no—
62D116SD1612yes—
63C4×D29—44no—
64C4×D29—41no—
65C4×D29—41no—
66Dic58—44no—
67Dic58—41no—
68Dic58—41no—
69Dic58—41no—
70Dic58—41no—
71C8×C29—22no—
72C8×C29—41no—
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×C29C2×SD1611yescharacteristic
83C4×C29C2×SD1611yescharacteristic
84C4×C29AGL(1,ℤ/8)11yescharacteristic
85C4×C29#13711yescharacteristic
86Dic29—41no—
87Dic29—42no—
88Dic29—81no—
89D4×Q8—291no—
90#498—291no—
91#475—291no—
92#461—291no—
93#332—291no—
94Q8⋊C8D2911yescharacteristic, complemented
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—291no—
104C4⋊Q8—581no—
105C22⋊Q8—582no—
106C22⋊Q8—581no—
107C4×Q8C2×D2911yescharacteristic
108C4×D4—291no—
109C4×D4—581no—
110#106—582no—
111#106—581no—
112C4⋊C8—291no—
113C4⋊C8C2×D2911yescharacteristic
114#103—581no—
115#103—581no—
116C4×C8C2×D2911yescharacteristic
117C29#165111yescharacteristic, complemented, Sylow
118C2×Q8—582no—
119C2×Q8—581no—
120C2×Q8—581no—
121C2×Q8—1162no—
122C2×Q8—1162no—
123C2×Q8D11611yescharacteristic
124C2×D4—291no—
125C22×C4—582no—
126C22×C4—581no—
127SD16—1162no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—581no—
131C4⋊C4—21no—
132C4⋊C4—291no—
133C4⋊C4—292no—
134C4⋊C4—582no—
135C4⋊C4—581no—
136C4⋊C4—581no—
137C4⋊C4—581no—
138C4⋊C4D11611yescharacteristic
139C22⋊C4—582no—
140C22⋊C4—581no—
141C42—21no—
142C42C22×D2911yescharacteristic
143C23—581no—
144Q8—21no—
145Q8—1164no—
146Q8—1161no—
147Q8—1161no—
148Q8—1161no—
149Q8—1161no—
150Q8#9077312yescomplemented
151D4—292no—
152D4—581no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—582no—
156C2×C4—581no—
157C2×C4—1164no—
158C2×C4—1161no—
159C2×C4—1161no—
160C2×C4—1161no—
161C2×C4C2×D11611yescharacteristic
162C2×C4D4×D2911yescharacteristic
163C2×C4D4×D2911yescharacteristic
164C8—22no—
165C8—41no—
166C8—1161no—
167C22—582no—
168C22—1161no—
169C22—1161no—
170C22#1174524411yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—1162no—
175C4—1161no—
176C4—2321no—
177C4C2×#9077311yescharacteristic
178C4#1174527011yescharacteristic
179C4SD16×D2911yescharacteristic
180C4#1174528011yescharacteristic
181C2—1162no—
182C2#1427130311yescharacteristic
183C2#1427131611yescharacteristic
184C2#1427134711yescharacteristic
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×Q8⋊C8C2
socleC22×C29#120
prime radicalC2×D116D4
hypercenterQ8⋊C8D29
hypocenterC29#1651
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14271561C2semidirect
3. #14271448C2semidirect
5. #14270908C2semidirect
7. C29×Q8⋊C8C2semidirect
52. Q8×C29SD16semidirect
94. Q8⋊C8D29semidirect
117. C29#1651semidirect
150. Q8#90773semidirect

Maximal subgroups

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

subgroupindex
Q8×D1162
#142715612
#142714482
#142713692
#142709082
#142708922
C29×Q8⋊C82
#165129