Finite Group Databasebeta

← #27157189

Subgroups — #27157189

Order 12416.

234 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C4×#13095520C211yesmaximal, characteristic, complemented
2C2×#15690865C211yesmaximal, characteristic, complemented
3#20261592C212yesmaximal, complemented
4#20261591C212yesmaximal, complemented
5#20261271C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8#15690865—21no—
9#15690865C2212yes—
10#15690865C2212yes—
11C4×#11715897—21no—
12#15690805—21no—
13#15690801C2211yescharacteristic
14#15690799C2211yescharacteristic
15#15690794—21no—
16#15690784C411yescharacteristic
17Q8×Dic97C411yescharacteristic
18#15690772C411yescharacteristic
19#15690770—21no—
20#15690768—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23#13095520—24no—
24C2×#11715897—24no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×D388C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C22×Dic97D411yescharacteristic
30C22×Dic97D411yescharacteristic
31C2×C4×D97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C2×Q8×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C22⋊C4×C97—21no—
50C23×C97C22⋊C411yescharacteristic
51C22×D97—21no—
52Q8×C97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—42no—
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—21no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66D388—41no—
67C4×D97—42no—
68#11715897—24no—
69#11715897—24no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97#10412yes—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—41no—
75C2×D97—41no—
76C2×D97—41no—
77C4×C97—21no—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97—41no—
81Dic97—21no—
82Dic97—21no—
83Dic97—21no—
84Dic97—41no—
85Dic97#10412yes—
86C2×C97—22no—
87C2×C97#41311yescharacteristic
88C2×C97C2×#10411yescharacteristic
89C2×C97C2×#10411yescharacteristic
90D97—81no—
91#1524—971nomaximal, Sylow
92C97#152411yescharacteristic, complemented, Sylow
93#529—971no—
94C4×M16—971no—
95C2×#97—971no—
96#330—972no—
97#329—972no—
98C2×M16—1941no—
99C22×C8—1941no—
100#126—1941no—
101#125—971no—
102#124—971no—
103#123—1941no—
104C22⋊Q8#8760411yescharacteristic
105#121—971no—
106C4×Q8—971no—
107C4×D4—971no—
108C4×D4—1941no—
109#117—1941no—
110C2×C42—971no—
111#97—972no—
112#97—972no—
113#97—1941no—
114C4×C8—1941no—
115C2×Q8C2×#8760411yescharacteristic
116C2×D4—971no—
117C2×D4—1941no—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4—1941no—
121C22×C4C2×#8760411yescharacteristic
122M16—1944no—
123C2×C8—1942no—
124C2×C8—1942no—
125C2×C8—1944no—
126C4⋊C4—21no—
127C4⋊C4—971no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4C2×#8760411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—1941no—
134C22⋊C4—1941no—
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—971no—
141C42—1941no—
142C23—1941no—
143C23C22⋊#8760411yescharacteristic
144Q8—41no—
145D4—3881no—
146D4—3882no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—971no—
153C2×C4—971no—
154C2×C4—3882no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4—1942no—
158C2×C4C22⋊#8760411yescharacteristic
159C2×C4C22×#8760411yescharacteristic
160C8—1944no—
161C8—1944no—
162C22—21no—
163C22—21no—
164C22—3881no—
165C22—3881no—
166C22—3881no—
167C22#1569087512yes—
168C22C2×C22⋊#8760411yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—972no—
174C4—3881no—
175C4—1941no—
176C4—1941no—
177C4—1941no—
178C2—22no—
179C2—7761no—
180C2#2026172211yescharacteristic
181C2D97⋊#10411yescharacteristic
182C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690784C4
cosocleC4×Dic97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1524
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×#13095520C2semidirect
2. C2×#15690865C2semidirect
3. #20261592C2semidirect
4. #20261591C2semidirect
92. C97#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#130955202
C2×#156908652
#202615922
#202615912
#202612712
#152497