Finite Group Databasebeta

← #27141020

Subgroups — #27141020

Order 12416.

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#20261131C211yesmaximal, characteristic, complemented
2#20261015C211yesmaximal, characteristic, complemented
3#20260940C211yesmaximal, characteristic, complemented
4#20260481C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#547×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic
10#15690743—21no—
11#15690741—22no—
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690733—21no—
15#15690731—21no—
16C4×#11715895C2211yescharacteristic
17C2×#41×C97—21no—
18C4⋊Q8×C97C2211yescharacteristic
19#123×C97—22no—
20C22⋊Q8×C97—21no—
21#121×C97—21no—
22C4×D4×C97C2211yescharacteristic
23C4×D4×C97C2211yescharacteristic
24C2×C4⋊C4×C97—21no—
25C2×Dic194—21no—
26#13095499—42no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32#41×C97—42no—
33#41×C97—42no—
34C2×Q8×C97—21no—
35C2×D4×C97—22no—
36C2×D4×C97D411yescharacteristic
37C2×D4×C97D411yescharacteristic
38C22×C4×C97—22no—
39C22×C4×C97—21no—
40C22×C4×C97—21no—
41C4⋊C4×C97—22no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97D411yescharacteristic
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D412yes—
46C22⋊C4×C97—22no—
47C22⋊C4×C97—21no—
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C23×C97—21no—
52Q8×C97—41no—
53Q8×C97—41no—
54D4×C97—21no—
55D4×C97—21no—
56D4×C97—21no—
57D4×C97—44no—
58D4×C97SD1612yescomplemented
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—44no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×Dic97—41no—
71Dic194—41no—
72Dic194—41no—
73#11715895—22no—
74#11715895—41no—
75#11715895—41no—
76C22×C97—22no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97#12011yescharacteristic
83C4×C97—21no—
84C4×C97—41no—
85C4×C97—41no—
86C4×C97—42no—
87C4×C97C2×SD1611yescharacteristic
88C4×C97#13511yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91Dic97—81no—
92C2×C97—42no—
93C2×C97—81no—
94C2×C97#44911yescharacteristic
95C2×C97#45011yescharacteristic
96C2×C97#45611yescharacteristic
97#1654—971nomaximal, Sylow
98C97#165411yescharacteristic, complemented, Sylow
99#547D9711yescharacteristic
100#498—971no—
101#474—971no—
102#461—971no—
103#332—971no—
104D4⋊C8—971no—
105D4⋊C8—971no—
106C2×#41—21no—
107C2×SD16—1941no—
108C4⋊Q8—971no—
109C4⋊Q8C2×D9711yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D9711yescharacteristic
114C4×D4C2×D9711yescharacteristic
115C2×C4⋊C4—21no—
116#107—1941no—
117#106—1942no—
118C4⋊C8—971no—
119C4⋊C8—971no—
120#103—1941no—
121#102—1941no—
122C4×C8—971no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—1941no—
127C2×D4—22no—
128C2×D4C22⋊D9711yescharacteristic
129C2×D4C22⋊D9711yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—3882no—
134C2×C8—1941no—
135C2×C8—1941no—
136C2×C8—1941no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—1941no—
140C4⋊C4—1941no—
141C4⋊C4C22⋊D9711yescharacteristic
142C4⋊C4C22⋊D9712yes—
143C4⋊C4C22⋊D9711yescharacteristic
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D9711yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—3881no—
153Q8—3881no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1309549912yescomplemented
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—3881no—
168C2×C4C2×C22⋊D9711yescharacteristic
169C2×C4C2×C22⋊D9711yescharacteristic
170C2×C4C2×C22⋊D9711yescharacteristic
171C8—3881no—
172C8—3881no—
173C8—1942no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1569077911yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—7761no—
186C4C2×#1309549911yescharacteristic
187C4#1569083011yescharacteristic
188C4#1569083511yescharacteristic
189C4#1569083611yescharacteristic
190C2—42no—
191C2—81no—
192C2#2026088011yescharacteristic
193C2#2026088611yescharacteristic
194C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#547×C97C2
socleC22×C97#120
hypercenter#547D97
hypocenterC97#1654
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20261131C2semidirect
2. #20261015C2semidirect
3. #20260940C2semidirect
4. #20260481C2semidirect
5. D4⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
58. D4×C97SD16semidirect
98. C97#1654semidirect
158. D4#13095499semidirect

Maximal subgroups

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

subgroupindex
#202611312
#202610152
#202609402
#202604812
D4⋊#117158952
#547×C972
#165497