Finite Group Databasebeta

← #27140910

Subgroups — #27140910

Order 12416.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261455C211yesmaximal, characteristic
2#20261086C211yesmaximal, characteristic, complemented
3#20261031C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic
5#20260480C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic
8#15690810C2211yescharacteristic
9#15690810C2211yescharacteristic
10#15690803—22no—
11#15690797—21no—
12#15690789—21no—
13C4×Dic194C2211yescharacteristic
14#15690775—21no—
15#15690769—21no—
16#15690744—21no—
17#15690741—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690735—21no—
20#15690735—22no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194D411yescharacteristic
28C2×#11715895—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97D411yescharacteristic
33C4⋊Dic97D412yes—
34#13095490—22no—
35#13095490—21no—
36C22⋊Dic97—22no—
37C22⋊Dic97—22no—
38C4×Dic97—22no—
39C4×Dic97—21no—
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C2×C8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C22⋊C4×C97—21no—
46C42×C97C2311yescharacteristic
47C23×C97—21no—
48D4×C97—21no—
49D4×C97—21no—
50C2×C4×C97—21no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—22no—
59C2×Dic97—41no—
60Dic194—21no—
61Dic194—21no—
62Dic194—41no—
63Dic194—41no—
64Dic194—42no—
65C8×C97—41no—
66C8×C97—41no—
67#11715895—41no—
68C22×C97—41no—
69C22×C97—41no—
70C22×C97—41no—
71C22×C97#12011yescharacteristic
72C4×C97—21no—
73C4×C97—41no—
74C4×C97#13511yescharacteristic
75C4×C97#13511yescharacteristic
76C4×C97#13711yescharacteristic
77C4×C97#13711yescharacteristic
78Dic97—41no—
79Dic97—42no—
80Dic97—41no—
81Dic97—81no—
82C2×C97—81no—
83C2×C97#45211yescharacteristic
84C2×C97#45211yescharacteristic
85C2×C97#45411yescharacteristic
86#1638—971nomaximal, Sylow
87C97#163811yescharacteristic, complemented, Sylow
88#556—971no—
89#487—971no—
90#477—971no—
91#476—971no—
92#332—971no—
93Q8⋊C8—971no—
94D4⋊C8D9711yescharacteristic
95C4⋊Q8—971no—
96C4⋊Q8—971no—
97#126—1942no—
98#124—1941no—
99C22⋊Q8—1941no—
100C4×Q8—971no—
101C4×Q8—1941no—
102C4×D4C2×D9711yescharacteristic
103#117—1941no—
104#107—1941no—
105#106—1941no—
106C4⋊C8—971no—
107C4⋊C8C2×D9711yescharacteristic
108#103—1942no—
109#103—1941no—
110#102—1941no—
111C4×C8C2×D9711yescharacteristic
112C2×Q8—971no—
113C2×Q8—1941no—
114C2×Q8—1941no—
115C2×D4D38811yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—1941no—
120C4⋊C4—972no—
121C4⋊C4—971no—
122C4⋊C4—1942no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4D38811yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C42—1942no—
132C42—1941no—
133C42C22×D9711yescharacteristic
134C23—21no—
135Q8—3882no—
136Q8—3881no—
137Q8—3881no—
138Q8—1941no—
139Q8—1941no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3881no—
146C2×C4—1941no—
147C2×C4—1942no—
148C2×C4—1941no—
149C2×C4D4×D9711yescharacteristic
150C2×C4D4×D9711yescharacteristic
151C2×C4C2×D38811yescharacteristic
152C8—41no—
153C8—41no—
154C8—3881no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1569078011yescharacteristic
159C4—21no—
160C4—41no—
161C4—7761no—
162C4—3881no—
163C4—3882no—
164C4—3881no—
165C4#1569082911yescharacteristic
166C4#1569083211yescharacteristic
167C4#1569083911yescharacteristic
168C4#1569084311yescharacteristic
169C2—81no—
170C2#2026089811yescharacteristic
171C2#2026089911yescharacteristic
172C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4⋊C8D97
hypocenterC97#1638
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. #20261031C2semidirect
5. #20260480C2semidirect
6. #20260467C2semidirect
87. C97#1638semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202610862
#202610312
#202610252
#202604802
#202604672
D4⋊C8×C972
#163897