Finite Group Databasebeta

← #27140889

Subgroups — #27140889

Order 12416.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261389C211yesmaximal, characteristic
2#20261086C211yesmaximal, characteristic, complemented
3#20261026C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic, complemented
5#20260480C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690794—22no—
12#15690790—21no—
13#15690784—21no—
14C4×D388C2211yescharacteristic
15C2×C4⋊Dic97—21no—
16#15690742—21no—
17#15690741—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690735—22no—
20#15690732—21no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25#13095510—42no—
26#13095510—42no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×D388D411yescharacteristic
30C2×C22⋊D97—22no—
31C22×Dic97—22no—
32C2×C4×D97—21no—
33C2×#11715895—21no—
34C4⋊Dic97—22no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97D411yescharacteristic
39C4⋊Dic97D412yes—
40#13095488—22no—
41#13095488—21no—
42C2×D4×C97D411yescharacteristic
43C22×C4×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50C22×D97—21no—
51D4×C97—21no—
52D4×C97SD1612yes—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C22⋊D97—44no—
60C2×Dic97—21no—
61C2×Dic97—22no—
62C2×Dic97—44no—
63C2×Dic97—41no—
64D388—21no—
65D388—21no—
66C4×D97—41no—
67C4×D97—41no—
68Dic194—41no—
69Dic194—41no—
70Dic194—42no—
71C8×C97—41no—
72C8×C97—41no—
73#11715895—41no—
74C22×C97—22no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C2×D97—41no—
79C2×D97—41no—
80C2×D97—41no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97C2×SD1611yescharacteristic
84C4×C97#13511yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97#13711yescharacteristic
87Dic97—42no—
88Dic97—41no—
89Dic97—81no—
90C2×C97—42no—
91C2×C97#45011yescharacteristic
92C2×C97#45211yescharacteristic
93C2×C97#45411yescharacteristic
94D97—81no—
95#1635—971nomaximal, Sylow
96C97#163511yescharacteristic, complemented, Sylow
97#547—971no—
98#487—971no—
99#476—971no—
100#476—971no—
101#332—971no—
102D4⋊C8—971no—
103D4⋊C8D9711yescharacteristic, complemented
104C2×#41—1941no—
105C4⋊Q8—971no—
106C4⋊Q8—971no—
107#123—1942no—
108C22⋊Q8—1941no—
109#121—1941no—
110C4×D4—971no—
111C4×D4C2×D9711yescharacteristic
112C2×C4⋊C4—1941no—
113#106—1941no—
114#106—1941no—
115C4⋊C8—971no—
116C4⋊C8C2×D9711yescharacteristic
117#103—1942no—
118#102—1941no—
119#102—1941no—
120C4×C8C2×D9711yescharacteristic
121#41—3882no—
122#41—3882no—
123C2×Q8—1941no—
124C2×Q8—1941no—
125C2×D4—971no—
126C2×D4—1942no—
127C2×D4D38811yescharacteristic
128C22×C4—21no—
129C22×C4—1942no—
130C22×C4—1941no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—1941no—
134C4⋊C4—972no—
135C4⋊C4—971no—
136C4⋊C4—1942no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C4⋊C4—1941no—
140C4⋊C4D38811yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—1942no—
143C22⋊C4—1941no—
144C42C22×D9711yescharacteristic
145C23—21no—
146C23—1941no—
147Q8—3882no—
148Q8—3881no—
149Q8—3881no—
150D4—21no—
151D4—3884no—
152D4—1941no—
153D4—1941no—
154D4#1309550012yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—3881no—
159C2×C4—3884no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1941no—
163C2×C4—1942no—
164C2×C4D4×D9711yescharacteristic
165C2×C4D4×D9711yescharacteristic
166C2×C4C2×D38811yescharacteristic
167C8—41no—
168C8—41no—
169C8—3881no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—3881no—
174C22—3881no—
175C22—3881no—
176C22#1569078011yescharacteristic
177C4—21no—
178C4—41no—
179C4—7761no—
180C4—3881no—
181C4—3882no—
182C4C2×#1309550011yescharacteristic
183C4#1569082911yescharacteristic
184C4#1569083811yescharacteristic
185C4#1569083911yescharacteristic
186C2—42no—
187C2—7761no—
188C2#2026088811yescharacteristic
189C2#2026089811yescharacteristic
190C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radical#20261389C2
hypercenterD4⋊C8D97
hypocenterC97#1635
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
2. #20261086C2semidirect
3. #20261026C2semidirect
4. #20261025C2semidirect
5. #20260480C2semidirect
6. #20260464C2semidirect
7. D4⋊C8×C97C2semidirect
96. C97#1635semidirect
103. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610862
#202610262
#202610252
#202604802
#202604642
D4⋊C8×C972
#163597