Finite Group Databasebeta

← #16818621

Subgroups — #16818621

Order 3712.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#14271835C211yesmaximal, characteristic
2#14271536C211yesmaximal, characteristic, complemented
3#14271458C211yesmaximal, characteristic, complemented
4#14271454C211yesmaximal, characteristic, complemented
5#14270906C211yesmaximal, characteristic, complemented
6#14270892C211yesmaximal, characteristic, complemented
7C29×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#90809—21no—
9C4×Q8×C29C2211yescharacteristic
10C2×#90773—22no—
11#11745257—22no—
12#11745257—21no—
13#11745236—21no—
14#11745236C2211yescharacteristic
15C4×D116—21no—
16C4×D116C2211yescharacteristic
17#11745231C2211yescharacteristic
18C29×C4⋊C8C2211yescharacteristic
19C4×C8×C29C2211yescharacteristic
20#11745183—21no—
21#11745169—21no—
22#11745165—21no—
23#11745155—21no—
24C4⋊#6597C2211yescharacteristic
25C2×Q8×C29D411yescharacteristic
26#90809—42no—
27#90809—42no—
28C2×D116—22no—
29C2×D116—21no—
30C2×D116—21no—
31C2×D116—22no—
32C2×D116D411yescharacteristic
33C2×D116D412yes—
34C2×C4×D29—22no—
35C2×C4×D29—21no—
36C2×Dic58—21no—
37C2×C8×C29—21no—
38C2×C8×C29—21no—
39C4⋊C4×C29—21no—
40C4⋊C4×C29D411yescharacteristic
41C42×C29—21no—
42C42×C29C2311yescharacteristic
43#90781—22no—
44#90781—21no—
45C4⋊Dic29—21no—
46C4⋊Dic29—21no—
47C4⋊Dic29D411yescharacteristic
48C2×#6597—21no—
49#90773—44no—
50C22×D29—21no—
51C22×D29—22no—
52Q8×C29—21no—
53Q8×C29SD1612yescomplemented
54C2×C4×C29—21no—
55C2×C4×C29—21no—
56C2×C4×C29C2×D411yescharacteristic
57C2×C4×C29C2×D411yescharacteristic
58C2×C4×C29C2×D411yescharacteristic
59C2×Dic29—21no—
60C2×Dic29—41no—
61D116—21no—
62D116—21no—
63D116—42no—
64D116—44no—
65D116—41no—
66D116—41no—
67D116—41no—
68D116—41no—
69D116—82no—
70C4×D29—44no—
71C4×D29—41no—
72C4×D29—41no—
73Dic58—42no—
74C8×C29—22no—
75C8×C29—41no—
76#6597—41no—
77#1631—291nomaximal, Sylow
78C22×C29#12011yescharacteristic
79C2×D29—42no—
80C2×D29—41no—
81C2×D29—41no—
82C2×D29—42no—
83C2×D29—42no—
84C2×D29—41no—
85C4×C29—22no—
86C4×C29—21no—
87C4×C29—41no—
88C4×C29C2×SD1611yescharacteristic
89C4×C29#13511yescharacteristic
90C4×C29AGL(1,ℤ/8)11yescharacteristic
91C4×C29#13711yescharacteristic
92Dic29—41no—
93Dic29—81no—
94#550—291no—
95#492—291no—
96#477—291no—
97#476—291no—
98#331—291no—
99Q8⋊C8D2911yescharacteristic, complemented
100D4⋊C8—291no—
101C2×C29#45011yescharacteristic
102C2×C29#45211yescharacteristic
103C2×C29#45311yescharacteristic
104D29—81no—
105D29—82no—
106C2×#41—581no—
107C2×SD16—582no—
108C4⋊Q8—291no—
109#127—291no—
110#127—581no—
111#121—582no—
112#121—581no—
113C4×Q8C2×D2911yescharacteristic
114C4×D4—291no—
115C4×D4—581no—
116#107—581no—
117#106—581no—
118C4⋊C8—291no—
119C4⋊C8C2×D2911yescharacteristic
120#103—581no—
121#102—581no—
122C4×C8C2×D2911yescharacteristic
123C29#163111yescharacteristic, complemented, Sylow
124#41—1162no—
125#41—1162no—
126C2×Q8—581no—
127C2×Q8D11611yescharacteristic
128C2×D4—291no—
129C2×D4—292no—
130C2×D4—582no—
131C2×D4—582no—
132C2×D4—581no—
133C2×D4—581no—
134C22×C4—582no—
135C22×C4—581no—
136SD16—1164no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—581no—
140C4⋊C4—21no—
141C4⋊C4—291no—
142C4⋊C4—581no—
143C4⋊C4—581no—
144C4⋊C4D11611yescharacteristic
145C22⋊C4—582no—
146C22⋊C4—581no—
147C42—21no—
148C42C22×D2911yescharacteristic
149C23—581no—
150C23—582no—
151Q8—21no—
152Q8—1162no—
153Q8#9077312yescomplemented
154D4—581no—
155D4—581no—
156D4—1162no—
157D4—1164no—
158D4—1161no—
159D4—1161no—
160D4—1161no—
161D4—1161no—
162D4—2322no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—581no—
166C2×C4—1164no—
167C2×C4—1161no—
168C2×C4—1161no—
169C2×C4—1161no—
170C2×C4C2×D11611yescharacteristic
171C2×C4D4×D2911yescharacteristic
172C2×C4D4×D2911yescharacteristic
173C8—22no—
174C8—41no—
175C8—1161no—
176C22—1162no—
177C22—1161no—
178C22—1161no—
179C22—1162no—
180C22—1162no—
181C22—1161no—
182C22#1174524411yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—1161no—
187C4—2321no—
188C4C2×#9077311yescharacteristic
189C4#1174527011yescharacteristic
190C4#1174527811yescharacteristic
191C4#1174528011yescharacteristic
192C2—2321no—
193C2—2322no—
194C2#1427131611yescharacteristic
195C2#1427132811yescharacteristic
196C2#1427133211yescharacteristic
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×Q8⋊C8C2
socleC22×C29#120
prime radical#14271835C2
hypercenterQ8⋊C8D29
hypocenterC29#1631
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. #14271536C2semidirect
3. #14271458C2semidirect
4. #14271454C2semidirect
5. #14270906C2semidirect
6. #14270892C2semidirect
7. C29×Q8⋊C8C2semidirect
53. Q8×C29SD16semidirect
99. Q8⋊C8D29semidirect
123. C29#1631semidirect
153. Q8#90773semidirect

Maximal subgroups

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

subgroupindex
#142718352
#142715362
#142714582
#142714542
#142709062
#142708922
C29×Q8⋊C82
#163129