Finite Group Databasebeta

← #27140838

Subgroups — #27140838

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#20261081C211yesmaximal, characteristic, complemented
3#20260932C211yesmaximal, characteristic, complemented
4#20260480C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#459×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9C2×#13095499—21no—
10#15690810C2211yescharacteristic
11#15690809—21no—
12#15690789—21no—
13#15690787—22no—
14C4×Dic194C2211yescharacteristic
15D4×Dic97—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690731—21no—
18#15690731—22no—
19C4×#11715895C2211yescharacteristic
20C2×D8×C97—21no—
21#127×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194—42no—
28C2×Dic194—42no—
29C2×Dic194D411yescharacteristic
30C22×Dic97—22no—
31#13095499—42no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36C4⋊Dic97D412yes—
37#13095490—22no—
38#13095490—21no—
39C22⋊Dic97—22no—
40C4×Dic97—21no—
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97D411yescharacteristic
45C22×C4×C97—21no—
46D8×C97—42no—
47C2×C8×C97—21no—
48C4⋊C4×C97D411yescharacteristic
49C22⋊C4×C97—21no—
50C42×C97C2311yescharacteristic
51C23×C97—21no—
52C23×C97—41no—
53D4×C97—21no—
54D4×C97—42no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97SD1612yescomplemented
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97—22no—
69C2×Dic97—44no—
70Dic194—21no—
71Dic194—44no—
72Dic194—41no—
73Dic194—41no—
74Dic194D812yescomplemented
75C8×C97—41no—
76#11715895—41no—
77#11715895—41no—
78C22×C97—22no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—81no—
82C22×C97—81no—
83C22×C97—81no—
84C22×C97#12011yescharacteristic
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—161no—
96C2×C97#44711yescharacteristic
97C2×C97#44811yescharacteristic
98C2×C97#45411yescharacteristic
99#1627—971nomaximal, Sylow
100C97#162711yescharacteristic, complemented, Sylow
101D4×Q8—971no—
102#486—971no—
103#460—971no—
104#459D9711yescharacteristic
105#332—971no—
106Q8⋊C8—971no—
107D4⋊C8—971no—
108C22×Q8—1941no—
109C2×SD16—1941no—
110C2×D8—21no—
111C4⋊Q8—971no—
112C4⋊Q8—1941no—
113#127C2×D9711yescharacteristic
114C22⋊Q8—1942no—
115C22⋊Q8—1941no—
116C4×Q8—971no—
117C4×D4—1941no—
118C4×D4C2×D9711yescharacteristic
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#102—21no—
122#102—1941no—
123#102—1942no—
124C4×C8—971no—
125C2×Q8—971no—
126C2×Q8—3882no—
127C2×Q8—3882no—
128C2×Q8—1942no—
129C2×Q8—1941no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D9711yescharacteristic
134C22×C4—21no—
135C22×C4—1942no—
136SD16—3882no—
137D8—42no—
138C2×C8—21no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—971no—
142C4⋊C4—972no—
143C4⋊C4—1942no—
144C4⋊C4—1941no—
145C4⋊C4—1941no—
146C4⋊C4C22⋊D9711yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—1942no—
149C42—1941no—
150C42C22×D9711yescharacteristic
151C23—21no—
152C23—41no—
153Q8—972no—
154Q8—3884no—
155Q8—3881no—
156Q8—3881no—
157Q8—1941no—
158D4—21no—
159D4—42no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4#1309549912yescomplemented
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×C22⋊D9711yescharacteristic
175C8—41no—
176C8—3881no—
177C8—3881no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22#1569077811yescharacteristic
185C4—21no—
186C4—41no—
187C4—3882no—
188C4—3881no—
189C4—1942no—
190C4D8×D9711yescharacteristic
191C4C2×#1309549911yescharacteristic
192C4#1569083411yescharacteristic
193C4#1569084011yescharacteristic
194C2—42no—
195C2—161no—
196C2#2026086711yescharacteristic
197C2#2026087311yescharacteristic
198C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#459×C97C2
socleC22×C97#120
hypercenter#459D97
hypocenterC97#1627
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. D4×Dic194C2semidirect
2. #20261081C2semidirect
3. #20260932C2semidirect
4. #20260480C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
59. D4×C97SD16semidirect
74. Dic194D8semidirect
100. C97#1627semidirect
164. D4#13095499semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202610812
#202609322
#202604802
#202604652
D4⋊#117158952
#459×C972
#162797