Finite Group Databasebeta

← #16818659

Subgroups — #16818659

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#14271514C211yesmaximal, characteristic, complemented
3#14271458C211yesmaximal, characteristic, complemented
4#14271454C211yesmaximal, characteristic
5#14270908C211yesmaximal, characteristic, complemented
6#14270892C211yesmaximal, characteristic
7C29×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D29—21no—
9C4×Q8×C29C2211yescharacteristic
10#11745259—21no—
11#11745259—22no—
12C4×D116—21no—
13C4×D116C2211yescharacteristic
14#11745231—21no—
15#11745231C2211yescharacteristic
16#11745231C2211yescharacteristic
17C29×C4⋊C8C2211yescharacteristic
18C4×C8×C29C2211yescharacteristic
19#11745183—21no—
20#11745169—21no—
21#11745165—21no—
22#11745164—22no—
23#11745155—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⋊Dic29D412yes—
47C4⋊Dic29D411yescharacteristic
48C2×#6597—21no—
49C22×D29—21no—
50Q8×C29—21no—
51Q8×C29SD1612yes—
52C2×C4×C29—21no—
53C2×C4×C29—21no—
54C2×C4×C29C2×D411yescharacteristic
55C2×C4×C29C2×D411yescharacteristic
56C2×C4×C29C2×D411yescharacteristic
57C2×Dic29—21no—
58C2×Dic29—22no—
59C2×Dic29—41no—
60D116—21no—
61D116Q1612yes—
62C4×D29—44no—
63C4×D29—41no—
64C4×D29—41no—
65Dic58—44no—
66Dic58—41no—
67Dic58—41no—
68Dic58—42no—
69C8×C29—41no—
70C8×C29—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×C29C2×SD1611yescharacteristic
81C4×C29C2×Q1611yescharacteristic
82C4×C29AGL(1,ℤ/8)11yescharacteristic
83C4×C29#13711yescharacteristic
84Dic29—41no—
85Dic29—42no—
86Dic29—81no—
87D4×Q8—291no—
88#487—291no—
89#477—291no—
90#476—291no—
91#332—291no—
92Q8⋊C8D2911yescharacteristic, complemented
93D4⋊C8—291no—
94C2×C29#45011yescharacteristic
95C2×C29#45111yescharacteristic
96C2×C29#45411yescharacteristic
97D29—42no—
98C22×Q8—581no—
99C4⋊Q8—291no—
100C4⋊Q8—291no—
101C4⋊Q8—581no—
102C22⋊Q8—582no—
103C22⋊Q8—581no—
104C4×Q8C2×D2911yescharacteristic
105C4×D4—291no—
106C4×D4—581no—
107#107—581no—
108#106—581no—
109C4⋊C8—291no—
110C4⋊C8C2×D2911yescharacteristic
111#103—582no—
112#103—581no—
113#102—581no—
114C4×C8C2×D2911yescharacteristic
115C29#163711yescharacteristic, complemented, Sylow
116C2×Q8—582no—
117C2×Q8—581no—
118C2×Q8—581no—
119C2×Q8—1162no—
120C2×Q8—1162no—
121C2×Q8D11611yescharacteristic
122C2×D4—291no—
123C22×C4—582no—
124C22×C4—581no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—581no—
128C4⋊C4—21no—
129C4⋊C4—292no—
130C4⋊C4—291no—
131C4⋊C4—582no—
132C4⋊C4—581no—
133C4⋊C4—581no—
134C4⋊C4—581no—
135C4⋊C4D11611yescharacteristic
136C22⋊C4—582no—
137C22⋊C4—581no—
138C42—21no—
139C42C22×D2911yescharacteristic
140C23—581no—
141Q8—21no—
142Q8—1164no—
143Q8—1162no—
144Q8—1161no—
145Q8—1161no—
146Q8#9077312yes—
147D4—292no—
148D4—581no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—582no—
152C2×C4—581no—
153C2×C4—1164no—
154C2×C4—1161no—
155C2×C4—1161no—
156C2×C4—1161no—
157C2×C4C2×D11611yescharacteristic
158C2×C4D4×D2911yescharacteristic
159C2×C4D4×D2911yescharacteristic
160C8—41no—
161C8—41no—
162C8—1161no—
163C22—582no—
164C22—1161no—
165C22—1161no—
166C22#1174524411yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—1162no—
171C4—1161no—
172C4—2321no—
173C4C2×#9077311yescharacteristic
174C4#1174527011yescharacteristic
175C4#1174527711yescharacteristic
176C4Q16×D2911yescharacteristic
177C2—1162no—
178C2#1427131611yescharacteristic
179C2#1427132011yescharacteristic
180C2#1427133711yescharacteristic
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×Q8⋊C8C2
socleC22×C29#120
prime radicalC2×D116D4
hypercenterQ8⋊C8D29
hypocenterC29#1637
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. #14271514C2semidirect
3. #14271458C2semidirect
5. #14270908C2semidirect
7. C29×Q8⋊C8C2semidirect
92. Q8⋊C8D29semidirect
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
#142715142
#142714582
#142714542
#142709082
#142708922
C29×Q8⋊C82
#163729