Finite Group Databasebeta

← #28389836

Subgroups — #28389836

Order 14464.

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×D452C211yesmaximal, characteristic
2#20298994C211yesmaximal, characteristic, complemented
3#20298800C211yesmaximal, characteristic
4#20298345C211yesmaximal, characteristic, complemented
5Q8⋊#11744751C211yesmaximal, characteristic, complemented
6#20298325C211yesmaximal, characteristic
7#475×C113C211yesmaximal, characteristic, complemented
8C2×Q8×D113—21no—
9C2×#14268634—21no—
10#16813622—21no—
11#16813622C2211yescharacteristic
12#16813602—22no—
13#16813602—21no—
14C4×D452—21no—
15C4×D452C2211yescharacteristic
16#16813553—22no—
17C4⋊#11744751C2211yescharacteristic
18#16813546—21no—
19C4×#11744751C2211yescharacteristic
20C4⋊Q8×C113C2211yescharacteristic
21C4×Q8×C113C2211yescharacteristic
22#106×C113—21no—
23C4⋊C8×C113C2211yescharacteristic
24#103×C113—21no—
25C2×Dic226—22no—
26C2×Dic226—21no—
27Q8×D113—42no—
28Q8×D113—42no—
29C2×D452D411yescharacteristic
30C2×C4×D113—22no—
31C2×C4×D113—21no—
32#14268634—42no—
33C2×#11744751—21no—
34C2×#11744751—21no—
35C4⋊Dic113—22no—
36C4⋊Dic113—21no—
37C4⋊Dic113D411yescharacteristic
38C4⋊Dic113D412yes—
39#14268624—22no—
40#14268624—21no—
41C2×Q8×C113—21no—
42C2×Q8×C113D411yescharacteristic
43C2×C8×C113—21no—
44C4⋊C4×C113—21no—
45C4⋊C4×C113—21no—
46C4⋊C4×C113—21no—
47C4⋊C4×C113D411yescharacteristic
48C42×C113—21no—
49C42×C113C2311yescharacteristic
50C22×D113—21no—
51Q8×C113—21no—
52Q8×C113—41no—
53Q8×C113—41no—
54Q8×C113SD1612yescomplemented
55C2×C4×C113—21no—
56C2×C4×C113—21no—
57C2×C4×C113—41no—
58C2×C4×C113C2×D411yescharacteristic
59C2×C4×C113C2×D411yescharacteristic
60C2×C4×C113C2×D411yescharacteristic
61C2×Dic113—22no—
62C2×Dic113—21no—
63D452—21no—
64D452SD1612yes—
65C4×D113—44no—
66C4×D113—41no—
67C4×D113—41no—
68Dic226—44no—
69Dic226—41no—
70Dic226—41no—
71C8×C113—41no—
72#11744751—22no—
73#11744751—41no—
74C22×C113#12011yescharacteristic
75C2×D113—22no—
76C2×D113—41no—
77C2×D113—41no—
78C4×C113—22no—
79C4×C113—21no—
80C4×C113—41no—
81C4×C113—81no—
82C4×C113C2×SD1611yescharacteristic
83C4×C113C2×SD1611yescharacteristic
84C4×C113AGL(1,ℤ/8)11yescharacteristic
85C4×C113#13711yescharacteristic
86Dic113—41no—
87Dic113—42no—
88C2×C113#44811yescharacteristic
89C2×C113#45011yescharacteristic
90C2×C113#45611yescharacteristic
91D113—42no—
92#1651—1131nomaximal, Sylow
93C113#165111yescharacteristic, complemented, Sylow
94D4×Q8—1131no—
95#498—1131no—
96#475D11311yescharacteristic, complemented
97#461—1131no—
98#332—1131no—
99Q8⋊C8—1131no—
100D4⋊C8—1131no—
101C22×Q8—2261no—
102C2×SD16—2261no—
103C4⋊Q8—1131no—
104C4⋊Q8—2261no—
105C4⋊Q8C2×D11311yescharacteristic
106C22⋊Q8—2262no—
107C22⋊Q8—2261no—
108C4×Q8C2×D11311yescharacteristic
109C4×D4—1131no—
110C4×D4—2261no—
111#106—21no—
112#106—2262no—
113C4⋊C8—1131no—
114C4⋊C8C2×D11311yescharacteristic
115#103—21no—
116#103—2261no—
117C4×C8—1131no—
118C2×Q8—21no—
119C2×Q8—4522no—
120C2×Q8—4522no—
121C2×Q8—2262no—
122C2×Q8—2261no—
123C2×Q8C22⋊D11311yescharacteristic
124C2×D4—1131no—
125C22×C4—2262no—
126C22×C4—2261no—
127SD16—4522no—
128C2×C8—21no—
129C2×C8—2261no—
130C2×C8—2261no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1131no—
135C4⋊C4—1132no—
136C4⋊C4—2262no—
137C4⋊C4—2261no—
138C4⋊C4C22⋊D11311yescharacteristic
139C22⋊C4—2262no—
140C22⋊C4—2261no—
141C42—21no—
142C42C22×D11311yescharacteristic
143C23—2261no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—4524no—
148Q8—4521no—
149Q8—4521no—
150Q8#1426863412yescomplemented
151D4—1132no—
152D4—2261no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—4524no—
157C2×C4—4521no—
158C2×C4—4521no—
159C2×C4—2262no—
160C2×C4—2261no—
161C2×C4D4×D11311yescharacteristic
162C2×C4D4×D11311yescharacteristic
163C2×C4C2×C22⋊D11311yescharacteristic
164C8—41no—
165C8—4521no—
166C8—2262no—
167C22—4521no—
168C22—4521no—
169C22—2262no—
170C22#1681359011yescharacteristic
171C4—21no—
172C4—22no—
173C4—41no—
174C4—81no—
175C4—4522no—
176C4—4521no—
177C4SD16×D11311yescharacteristic
178C4C2×#1426863411yescharacteristic
179C4#1681364711yescharacteristic
180C4#1681365111yescharacteristic
181C2—4522no—
182C2#2029873511yescharacteristic
183C2#2029874811yescharacteristic
184C2#2029877911yescharacteristic
subgroupG/H
centerC22#16813590
commutator, cosocleC42×C113C23
FrattiniC42C22×D113
Fitting#475×C113C2
socleC22×C113#120
prime radicalC2×D452D4
hypercenter#475D113
hypocenterC113#1651
LCSG ▷ C42×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20298994C2semidirect
4. #20298345C2semidirect
5. Q8⋊#11744751C2semidirect
7. #475×C113C2semidirect
54. Q8×C113SD16semidirect
93. C113#1651semidirect
96. #475D113semidirect
150. Q8#14268634semidirect

Maximal subgroups

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

subgroupindex
Q8×D4522
#202989942
#202988002
#202983452
Q8⋊#117447512
#202983252
#475×C1132
#1651113