Finite Group Databasebeta

← #46810169

Subgroups — #46810169

Order 34432.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#31771001C211yesmaximal, characteristic, complemented
2#31770885C211yesmaximal, characteristic, complemented
3#31770810C211yesmaximal, characteristic, complemented
4#31770351C211yesmaximal, characteristic, complemented
5D4⋊#14524130C211yesmaximal, characteristic, complemented
6D4⋊#14524130C211yesmaximal, characteristic, complemented
7#547×C269C211yesmaximal, characteristic
8C2×#16864390—21no—
9#22575453C2211yescharacteristic
10#22575386—21no—
11#22575384—22no—
12C4⋊#14524130C2211yescharacteristic
13C4⋊#14524130C2211yescharacteristic
14#22575376—21no—
15#22575374—21no—
16C4×#14524130C2211yescharacteristic
17C2×#41×C269—21no—
18C4⋊Q8×C269C2211yescharacteristic
19#123×C269—22no—
20C22⋊Q8×C269—21no—
21#121×C269—21no—
22C4×D4×C269C2211yescharacteristic
23C4×D4×C269C2211yescharacteristic
24C2×C4⋊C4×C269—21no—
25C2×Dic538—21no—
26#16864390—42no—
27C2×#14524130—21no—
28C2×#14524130—21no—
29C2×#14524130—21no—
30C4⋊Dic269—21no—
31C4⋊Dic269—21no—
32#41×C269—42no—
33#41×C269—42no—
34C2×Q8×C269—21no—
35C2×D4×C269—22no—
36C2×D4×C269D411yescharacteristic
37C2×D4×C269D411yescharacteristic
38C22×C4×C269—22no—
39C22×C4×C269—21no—
40C22×C4×C269—21no—
41C4⋊C4×C269—22no—
42C4⋊C4×C269—21no—
43C4⋊C4×C269D411yescharacteristic
44C4⋊C4×C269D412yes—
45C4⋊C4×C269D411yescharacteristic
46C22⋊C4×C269—22no—
47C22⋊C4×C269—21no—
48C22⋊C4×C269—21no—
49C42×C269C2311yescharacteristic
50Dic538—41no—
51Dic538—41no—
52C2×Dic269—41no—
53#14524130—22no—
54#14524130—41no—
55#14524130—41no—
56C23×C269—21no—
57C23×C269—21no—
58Q8×C269—41no—
59Q8×C269—41no—
60D4×C269—21no—
61D4×C269—21no—
62D4×C269—21no—
63D4×C269—44no—
64D4×C269SD1612yescomplemented
65C2×C4×C269—21no—
66C2×C4×C269—21no—
67C2×C4×C269—22no—
68C2×C4×C269—44no—
69C2×C4×C269—41no—
70C2×C4×C269—41no—
71C2×C4×C269—41no—
72C2×C4×C269—41no—
73C2×C4×C269C2×D411yescharacteristic
74C2×C4×C269C2×D411yescharacteristic
75C2×C4×C269C2×D411yescharacteristic
76C22×C269—22no—
77C22×C269—41no—
78C22×C269—41no—
79C22×C269—41no—
80C22×C269—41no—
81C22×C269—41no—
82C22×C269#12011yescharacteristic
83C4×C269—21no—
84C4×C269—41no—
85C4×C269—41no—
86C4×C269—42no—
87C4×C269C2×SD1611yescharacteristic
88C4×C269#13511yescharacteristic
89C4×C269AGL(1,ℤ/8)11yescharacteristic
90C4×C269AGL(1,ℤ/8)11yescharacteristic
91Dic269—81no—
92C2×C269—42no—
93C2×C269—81no—
94C2×C269#44911yescharacteristic
95C2×C269#45011yescharacteristic
96C2×C269#45611yescharacteristic
97C269#165411yescharacteristic, complemented, Sylow
98#1654—2691nomaximal, Sylow
99#547D26911yescharacteristic
100#498—2691no—
101#474—2691no—
102#461—2691no—
103#332—2691no—
104D4⋊C8—2691no—
105D4⋊C8—2691no—
106C2×#41—21no—
107C2×SD16—5381no—
108C4⋊Q8—2691no—
109C4⋊Q8C2×D26911yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D26911yescharacteristic
114C4×D4C2×D26911yescharacteristic
115C2×C4⋊C4—21no—
116#107—5381no—
117#106—5382no—
118C4⋊C8—2691no—
119C4⋊C8—2691no—
120#103—5381no—
121#102—5381no—
122C4×C8—2691no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—5381no—
127C2×D4—22no—
128C2×D4C22⋊D26911yescharacteristic
129C2×D4C22⋊D26911yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—10762no—
134C2×C8—5381no—
135C2×C8—5381no—
136C2×C8—5381no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—5381no—
140C4⋊C4—5381no—
141C4⋊C4C22⋊D26911yescharacteristic
142C4⋊C4C22⋊D26912yes—
143C4⋊C4C22⋊D26911yescharacteristic
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D26911yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—10761no—
153Q8—10761no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1686439012yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—10761no—
168C2×C4C2×C22⋊D26911yescharacteristic
169C2×C4C2×C22⋊D26911yescharacteristic
170C2×C4C2×C22⋊D26911yescharacteristic
171C8—5382no—
172C8—10761no—
173C8—10761no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#2257542211yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—21521no—
186C4C2×#1686439011yescharacteristic
187C4#2257547311yescharacteristic
188C4#2257547811yescharacteristic
189C4#2257547911yescharacteristic
190C2—42no—
191C2—81no—
192C2#3177075011yescharacteristic
193C2#3177075611yescharacteristic
194C2#3177078811yescharacteristic
subgroupG/H
centerC22#22575422
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting, prime radical#547×C269C2
socleC22×C269#120
hypercenter#547D269
hypocenterC269#1654
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #31771001C2semidirect
2. #31770885C2semidirect
3. #31770810C2semidirect
4. #31770351C2semidirect
5. D4⋊#14524130C2semidirect
6. D4⋊#14524130C2semidirect
64. D4×C269SD16semidirect
97. C269#1654semidirect
158. D4#16864390semidirect

Maximal subgroups

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

subgroupindex
#317710012
#317708852
#317708102
#317703512
D4⋊#145241302
#547×C2692
#1654269