Finite Group Databasebeta

← #27140842

Subgroups — #27140842

Order 12416.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 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
1D4×Dic194C211yesmaximal, characteristic, complemented
2#20261082C211yesmaximal, characteristic, complemented
3#20260936C211yesmaximal, characteristic, complemented
4#20260930C211yesmaximal, characteristic
5#20260482C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C22×Dic194—21no—
9C2×#13095500—21no—
10C2×#13095496—21no—
11#15690810C2211yescharacteristic
12#15690809—21no—
13#15690807C2211yescharacteristic
14#15690789—21no—
15#15690787—22no—
16C4×Dic194C2211yescharacteristic
17D4×Dic97—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690732—21no—
20#15690732—22no—
21#15690730—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194—42no—
28C2×Dic194—42no—
29C2×Dic194D411yescharacteristic
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—21no—
33C22×Dic97—22no—
34#13095500—42no—
35#13095496—42no—
36C2×#11715895—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97D411yescharacteristic
39C4⋊Dic97D412yes—
40#13095490—22no—
41#13095490—21no—
42C22⋊Dic97—22no—
43C4×Dic97—21no—
44C2×D4×C97D411yescharacteristic
45C22×C4×C97—21no—
46C2×C8×C97—21no—
47C2×C8×C97—21no—
48C4⋊C4×C97D411yescharacteristic
49C22⋊C4×C97—21no—
50C42×C97C2311yescharacteristic
51C23×C97—21no—
52C22×D97—41no—
53D4×C97—21no—
54D4×C97SD1612yescomplemented
55C2×C4×C97—21no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×Dic97—21no—
62C2×Dic97—21no—
63C2×Dic97—22no—
64C2×Dic97—44no—
65D388—42no—
66D388—41no—
67D388—41no—
68D388—41no—
69D388—41no—
70Dic194—21no—
71Dic194—44no—
72Dic194—41no—
73Dic194—41no—
74Dic194D812yescomplemented
75C8×C97—41no—
76C8×C97—41no—
77#11715895—41no—
78C22×C97—22no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97#12011yescharacteristic
82C2×D97—81no—
83C2×D97—81no—
84C2×D97—81no—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97C2×D811yescharacteristic
88C4×C97C2×SD1611yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C4×C97#13711yescharacteristic
91Dic97—22no—
92Dic97—42no—
93Dic97—41no—
94C2×C97—42no—
95C2×C97#44711yescharacteristic
96C2×C97#44811yescharacteristic
97C2×C97#45411yescharacteristic
98D97—161no—
99#1627—971nomaximal, Sylow
100C97#162711yescharacteristic, complemented, Sylow
101D4×Q8—971no—
102#486—971no—
103#460—971no—
104#459—971no—
105#332—971no—
106Q8⋊C8—971no—
107D4⋊C8D9711yescharacteristic, complemented
108C22×Q8—1941no—
109C2×SD16—1941no—
110C2×D8—1941no—
111C4⋊Q8—971no—
112C4⋊Q8—1941no—
113#127—971no—
114C22⋊Q8—1942no—
115C22⋊Q8—1941no—
116C4×Q8—971no—
117C4×D4—1941no—
118C4×D4C2×D9711yescharacteristic
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#102—1941no—
122#102—1942no—
123#102—1941no—
124C4×C8C2×D9711yescharacteristic
125C2×Q8—971no—
126C2×Q8—3882no—
127C2×Q8—3882no—
128C2×Q8—1942no—
129C2×Q8—1941no—
130C2×D4—1941no—
131C2×D4—1941no—
132C2×D4—1941no—
133C2×D4D38811yescharacteristic
134C22×C4—21no—
135C22×C4—1942no—
136SD16—3882no—
137D8—3882no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1941no—
141C4⋊C4—971no—
142C4⋊C4—972no—
143C4⋊C4—1942no—
144C4⋊C4—1941no—
145C4⋊C4—1941no—
146C4⋊C4D38811yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—1942no—
149C42—1941no—
150C42C22×D9711yescharacteristic
151C23—21no—
152C23—3881no—
153Q8—972no—
154Q8—3884no—
155Q8—3881no—
156Q8—3881no—
157Q8—1941no—
158D4—21no—
159D4—3882no—
160D4—3881no—
161D4—3881no—
162D4—3881no—
163D4—3881no—
164D4#1309550012yescomplemented
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—3884no—
169C2×C4—1941no—
170C2×C4—1942no—
171C2×C4—1941no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×D38811yescharacteristic
175C8—41no—
176C8—41no—
177C8—3881no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—7761no—
182C22—7761no—
183C22—7761no—
184C22#1569078011yescharacteristic
185C4—21no—
186C4—41no—
187C4—3882no—
188C4—3881no—
189C4—1942no—
190C4D8×D9711yescharacteristic
191C4C2×#1309550011yescharacteristic
192C4#1569083711yescharacteristic
193C4#1569084311yescharacteristic
194C2—42no—
195C2—15521no—
196C2#2026087011yescharacteristic
197C2#2026087611yescharacteristic
198C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radical#20260930C2
hypercenterD4⋊C8D97
hypocenterC97#1627
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. D4×Dic194C2semidirect
2. #20261082C2semidirect
3. #20260936C2semidirect
5. #20260482C2semidirect
6. #20260467C2semidirect
7. D4⋊C8×C97C2semidirect
54. D4×C97SD16semidirect
74. Dic194D8semidirect
100. C97#1627semidirect
107. D4⋊C8D97semidirect
164. D4#13095500semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202610822
#202609362
#202609302
#202604822
#202604672
D4⋊C8×C972
#162797