Finite Group Databasebeta

← #27142966

Subgroups — #27142966

Order 12416.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261316C211yesmaximal, characteristic
2C2×#15690736C212yesmaximal, complemented
3#20260737C212yesmaximal, complemented
4#20260600C211yesmaximal, characteristic, complemented
5#343×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2212yes—
7#15690790—22no—
8#15690785—22no—
9#15690784—22no—
10C2×C4⋊Dic97C2211yescharacteristic
11C2×C4×Dic97C2211yescharacteristic
12#15690736—24no—
13#15690735—22no—
14#15690732—22no—
15#15690717—22no—
16C2×M16×C97C2212yes—
17C2×C4⋊C4×C97C2211yescharacteristic
18#13095511—22no—
19#13095511—22no—
20#13095511—22no—
21#13095511—22no—
22C2×Dic194D412yes—
23C2×D388—22no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C2×C4×D97—22no—
29C4⋊Dic97—22no—
30C4⋊Dic97—22no—
31#13095488—24no—
32C4×Dic97—21no—
33C4×Dic97D412yescomplemented
34C22×C4×C97D412yes—
35C22×C4×C97C2311yescharacteristic
36M16×C97—22no—
37M16×C97—22no—
38C2×C8×C97—22no—
39C4⋊C4×C97—22no—
40C23×C97C2×D411yescharacteristic
41C22×D97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—42no—
44C2×C4×C97C2×D411yescharacteristic
45C2×C4×C97C2×D412yes—
46C2×C4×C97#4112yes—
47C2×C4×C97#4111yescharacteristic
48C22⋊D97—42no—
49C22⋊D97—42no—
50C2×Dic97—22no—
51C2×Dic97—22no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54C2×Dic97—42no—
55C2×Dic97—41no—
56D388—22no—
57D388—22no—
58D388—82no—
59C4×D97—42no—
60C4×D97—42no—
61Dic194—22no—
62Dic194—22no—
63C8×C97—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97#12011yescharacteristic
67C22×C97#12111yescharacteristic
68C22×C97#12411yescharacteristic
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—42no—
72C4×C97—42no—
73C4×C97#12112yes—
74C4×C97#12312yes—
75Dic97—42no—
76Dic97—42no—
77Dic97—81no—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97#39411yescharacteristic
81C2×C97#45411yescharacteristic
82C2×C97#45511yescharacteristic
83D97—82no—
84#2040—971nomaximal, Sylow
85C97#204011yescharacteristic, complemented, Sylow
86#537—971no—
87C2×#104—972no—
88#417—972no—
89#382—971no—
90#343D9711yescharacteristic, complemented
91C2×#41—972no—
92C2×M16C2×D9712yes—
93C22⋊Q8—1942no—
94#121—1942no—
95#121—1942no—
96C2×C4⋊C4—971no—
97C2×C4⋊C4C2×D9711yescharacteristic
98C2×C42—971no—
99#104—1944no—
100#103—1942no—
101#102—1942no—
102#95—1942no—
103#41—1942no—
104#41—1942no—
105#41—1942no—
106#41—1942no—
107C2×Q8—972no—
108C2×D4—972no—
109C2×D4—1942no—
110C2×D4—1942no—
111C22×C4—1941no—
112C22×C4—1941no—
113C22×C4—1942no—
114C22×C4D38812yes—
115C22×C4C22×D9711yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—1942no—
121C4⋊C4—1942no—
122C22⋊C4—1944no—
123C42—972no—
124C42—1941no—
125C23—1942no—
126C23C2×D38811yescharacteristic
127Q8—1942no—
128Q8—1942no—
129D4—7762no—
130D4—3882no—
131D4—3882no—
132D4—1942no—
133D4—1942no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3882no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4—3882no—
141C2×C4—3882no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4D4×D9711yescharacteristic
145C2×C4D4×D9712yes—
146C2×C4#1309551112yes—
147C2×C4#1309551211yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—3882no—
152C22—3882no—
153C22—3882no—
154C22#1569078011yescharacteristic
155C22#1569078511yescharacteristic
156C22#1569079811yescharacteristic
157C4—42no—
158C4—7761no—
159C4—3882no—
160C4—3882no—
161C4#1569078312yes—
162C4#1569079512yes—
163C2—21no—
164C2—21no—
165C2—7762no—
166C2#2026065311yescharacteristic
167C2#2026091011yescharacteristic
168C2#2026091511yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#343×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#343D97
hypocenterC97#2040
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#15690736C2semidirect
3. #20260737C2semidirect
4. #20260600C2semidirect
5. #343×C97C2semidirect
33. C4×Dic97D4semidirect
85. C97#2040semidirect
90. #343D97semidirect

Maximal subgroups

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

subgroupindex
#202613162
C2×#156907362
#202607372
#202606002
#343×C972
#204097