Finite Group Databasebeta

← #16818678

Subgroups — #16818678

Order 3712.

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

normal subgroups only

This group has 212 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#14271509C211yesmaximal, characteristic
2#14271448C211yesmaximal, characteristic, complemented
3#14271373C211yesmaximal, characteristic, complemented
4#14270907C211yesmaximal, characteristic
5Q8⋊#6597C211yesmaximal, characteristic, complemented
6Q8⋊#6597C211yesmaximal, characteristic, complemented
7C29×#559C211yesmaximal, characteristic
8C29×#127C2211yescharacteristic
9C29×#124—21no—
10C29×#124—21no—
11C29×#123—22no—
12C4×Q8×C29C2211yescharacteristic
13C4×Q8×C29C2211yescharacteristic
14C4×D4×C29—21no—
15C4×D4×C29—21no—
16C2×#90785—21no—
17#11745231C2211yescharacteristic
18#11745183—21no—
19#11745180—22no—
20#11745158—21no—
21#11745156—21no—
22C4⋊#6597C2211yescharacteristic
23C4⋊#6597C2211yescharacteristic
24C4×#6597C2211yescharacteristic
25C2×Q8×C29D411yescharacteristic
26C2×Q8×C29D411yescharacteristic
27C2×D4×C29—21no—
28C2×D4×C29—21no—
29C2×D4×C29D412yes—
30C22×C4×C29—22no—
31C2×Dic58—21no—
32C4⋊C4×C29—21no—
33C4⋊C4×C29—21no—
34C4⋊C4×C29D411yescharacteristic
35C4⋊C4×C29D411yescharacteristic
36C22⋊C4×C29—22no—
37C22⋊C4×C29—22no—
38C22⋊C4×C29—22no—
39C42×C29—21no—
40C42×C29—21no—
41C42×C29C2311yescharacteristic
42#90785—42no—
43C4⋊Dic29—21no—
44C4⋊Dic29—21no—
45C2×#6597—21no—
46C2×#6597—21no—
47C2×#6597—21no—
48C23×C29—22no—
49Q8×C29—21no—
50Q8×C29—21no—
51Q8×C29—21no—
52Q8×C29—21no—
53D4×C29—42no—
54D4×C29—41no—
55D4×C29—41no—
56D4×C29—41no—
57D4×C29—41no—
58C2×C4×C29—21no—
59C2×C4×C29—21no—
60C2×C4×C29—21no—
61C2×C4×C29—21no—
62C2×C4×C29—82no—
63C2×C4×C29C2×D411yescharacteristic
64C2×C4×C29C2×D411yescharacteristic
65C2×C4×C29C2×D411yescharacteristic
66C2×Dic29—41no—
67Dic58—41no—
68Dic58—41no—
69#6597—41no—
70#6597—41no—
71#6597—41no—
72#1640—291nomaximal, Sylow
73C22×C29—42no—
74C22×C29—42no—
75C22×C29—42no—
76C22×C29#12011yescharacteristic
77C4×C29—21no—
78C4×C29—41no—
79C4×C29—41no—
80C4×C29—41no—
81C4×C29—41no—
82C4×C29#13511yescharacteristic
83C4×C29#13511yescharacteristic
84C4×C29AGL(1,ℤ/8)11yescharacteristic
85C4×C29#13711yescharacteristic
86Dic29—81no—
87#559D2911yescharacteristic
88#486—291no—
89#475—291no—
90#462—291no—
91#331—291no—
92Q8⋊C8—291no—
93Q8⋊C8—291no—
94C2×C29—82no—
95C2×C29#44911yescharacteristic
96C2×C29#45211yescharacteristic
97C2×C29#45511yescharacteristic
98C2×Q16—581no—
99C4⋊Q8—291no—
100#127C2×D2911yescharacteristic
101#124—21no—
102#124—21no—
103#123—22no—
104C4×Q8C2×D2911yescharacteristic
105C4×Q8C2×D2911yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108#106—581no—
109C4⋊C8—291no—
110C4⋊C8—291no—
111#103—581no—
112#103—581no—
113#102—582no—
114C4×C8—291no—
115C29#164011yescharacteristic, complemented, Sylow
116C2×Q8—581no—
117C2×Q8C22⋊D2911yescharacteristic
118C2×Q8C22⋊D2911yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D2912yes—
122C22×C4—22no—
123Q16—1162no—
124C2×C8—581no—
125C2×C8—581no—
126C2×C8—581no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—581no—
130C4⋊C4—581no—
131C4⋊C4C22⋊D2911yescharacteristic
132C4⋊C4C22⋊D2911yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D2911yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—1161no—
145Q8—1161no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—82no—
156C2×C4—1161no—
157C2×C4C2×C22⋊D2911yescharacteristic
158C2×C4C2×C22⋊D2911yescharacteristic
159C2×C4C2×C22⋊D2911yescharacteristic
160C8—1161no—
161C8—1161no—
162C8—1161no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1174531511yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—2321no—
173C4#1174531111yescharacteristic
174C4#1174531211yescharacteristic
175C4#1174531211yescharacteristic
176C4#1174531311yescharacteristic
177C2—82no—
178C2#1427130811yescharacteristic
179C2#1427132511yescharacteristic
180C2#1427134111yescharacteristic
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#559C2
socleC22×C29#120
prime radicalC29×#127C22
hypercenter#559D29
hypocenterC29#1640
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #14271448C2semidirect
3. #14271373C2semidirect
5. Q8⋊#6597C2semidirect
6. Q8⋊#6597C2semidirect
115. C29#1640semidirect

Maximal subgroups

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

subgroupindex
#142715092
#142714482
#142713732
#142709072
Q8⋊#65972
C29×#5592
#164029