Finite Group Databasebeta

← #27140909

Subgroups — #27140909

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#20261459C211yesmaximal, characteristic
2#20261087C211yesmaximal, characteristic, complemented
3#20261031C211yesmaximal, characteristic, complemented
4#20261026C211yesmaximal, characteristic
5#20260465C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic
7#332×C97C211yesmaximal, characteristic, complemented
8#15690810C2211yescharacteristic
9#15690805—22no—
10#15690799—21no—
11#15690790—21no—
12Q8×Dic97—21no—
13C4×Dic194C2211yescharacteristic
14C4×D388C2211yescharacteristic
15#15690768—21no—
16#15690744—21no—
17#15690742—21no—
18#15690735—21no—
19#15690733—22no—
20#15690732—21no—
21C4×#11715895C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27C2×D388D411yescharacteristic
28C2×C4×D97—21no—
29C2×#11715895—21no—
30C4⋊Dic97—22no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97D411yescharacteristic
34C4⋊Dic97D411yescharacteristic
35#13095490—21no—
36#13095488—22no—
37#13095488—22no—
38#13095488—21no—
39C4×Dic97—22no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×C8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D412yes—
46C42×C97C2311yescharacteristic
47C22×D97—21no—
48Q8×C97—41no—
49Q8×C97—41no—
50C2×C4×C97—22no—
51C2×C4×C97C2×D411yescharacteristic
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×Dic97—21no—
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97—41no—
58D388—21no—
59D388—21no—
60C4×D97—41no—
61C4×D97—41no—
62Dic194—21no—
63Dic194—21no—
64Dic194—42no—
65C8×C97—41no—
66C8×C97—41no—
67#11715895—41no—
68C22×C97#12011yescharacteristic
69C2×D97—41no—
70C2×D97—41no—
71C2×D97—41no—
72C4×C97—21no—
73C4×C97—42no—
74C4×C97#13511yescharacteristic
75C4×C97#13511yescharacteristic
76C4×C97#13711yescharacteristic
77C4×C97#13711yescharacteristic
78Dic97—41no—
79Dic97—41no—
80Dic97—41no—
81Dic97—81no—
82C2×C97#45211yescharacteristic
83C2×C97#45211yescharacteristic
84C2×C97#45411yescharacteristic
85D97—81no—
86#1638—971nomaximal, Sylow
87C97#163811yescharacteristic, complemented, Sylow
88#556—971no—
89#487—971no—
90#477—971no—
91#476—971no—
92#332D9711yescharacteristic, complemented
93Q8⋊C8—971no—
94D4⋊C8—971no—
95C4⋊Q8—971no—
96C4⋊Q8C2×D9711yescharacteristic
97#126—1942no—
98#124—1941no—
99C22⋊Q8—1941no—
100C4×Q8—971no—
101C4×Q8—1941no—
102C4×D4—971no—
103#117—1941no—
104#107—1941no—
105#106—1941no—
106C4⋊C8C2×D9711yescharacteristic
107C4⋊C8C2×D9711yescharacteristic
108#103—1941no—
109#103—1942no—
110#102—1941no—
111C4×C8—971no—
112C2×Q8—21no—
113C2×Q8—971no—
114C2×Q8—1941no—
115C2×D4—971no—
116C22×C4—1941no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—1941no—
120C4⋊C4—21no—
121C4⋊C4—971no—
122C4⋊C4—971no—
123C4⋊C4—1942no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4D38812yes—
128C22⋊C4—1942no—
129C22⋊C4—1942no—
130C22⋊C4—1941no—
131C42—1942no—
132C42—1941no—
133C42C22×D9711yescharacteristic
134C23—1941no—
135Q8—41no—
136Q8—41no—
137Q8—3882no—
138Q8—1941no—
139Q8—1941no—
140D4—1941no—
141D4—1941no—
142C2×C4—22no—
143C2×C4—3881no—
144C2×C4—3881no—
145C2×C4—3881no—
146C2×C4—1941no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4D4×D9711yescharacteristic
150C2×C4D4×D9711yescharacteristic
151C2×C4C2×D38811yescharacteristic
152C8—41no—
153C8—41no—
154C8—3881no—
155C22—3881no—
156C22—3881no—
157C22—3881no—
158C22#1569078011yescharacteristic
159C4—21no—
160C4—42no—
161C4—7761no—
162C4—3881no—
163C4—3881no—
164C4—3881no—
165C4#1569082911yescharacteristic
166C4#1569083111yescharacteristic
167C4#1569083911yescharacteristic
168C4#1569084211yescharacteristic
169C2—7761no—
170C2#2026089811yescharacteristic
171C2#2026090011yescharacteristic
172C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#332D97
hypocenterC97#1638
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20261087C2semidirect
3. #20261031C2semidirect
5. #20260465C2semidirect
7. #332×C97C2semidirect
87. C97#1638semidirect
92. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202614592
#202610872
#202610312
#202610262
#202604652
#202604622
#332×C972
#163897