Finite Group Databasebeta

← #27157054

Subgroups — #27157054

Order 12416.

234 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261736C211yesmaximal, characteristic, complemented
2C4×#13095521C211yesmaximal, characteristic, complemented
3#20261585C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5#20261270C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×#13095519—21no—
8#15690877—21no—
9C4⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897C2212yes—
12#15690805—21no—
13#15690798C411yescharacteristic
14#15690795—21no—
15#15690790C411yescharacteristic
16#15690785C411yescharacteristic
17#15690775C2211yescharacteristic
18#15690772—21no—
19#15690772C2211yescharacteristic
20#15690768—21no—
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic
23#13095521—24no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×Dic194C2×C411yescharacteristic
28C2×D388—21no—
29C2×D388C2×C411yescharacteristic
30C2×C4×D97—21no—
31C2×C4×D97C2×C411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C4⋊Dic97—21no—
35#13095490—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C4⋊C4×C97C2×C411yescharacteristic
49C42×C97C2×C411yescharacteristic
50C22×D97—21no—
51C22×D97C22⋊C411yescharacteristic
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60C2×Dic97C2×D411yescharacteristic
61D388—22no—
62D388—42no—
63C4×D97—22no—
64C4×D97—22no—
65C4×D97—21no—
66C4×D97—21no—
67C4×D97—41no—
68C4×D97—41no—
69Dic194—22no—
70#11715897—24no—
71#11715897—42no—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—21no—
74C2×D97—21no—
75C2×D97—41no—
76C2×D97—41no—
77C2×D97—41no—
78C2×D97#10412yes—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97#13112yes—
84Dic97—21no—
85Dic97—21no—
86Dic97—41no—
87Dic97AGL(1,ℤ/8)11yescharacteristic
88Dic97#13711yescharacteristic
89C2×C97#40811yescharacteristic
90C2×C97#41711yescharacteristic
91C2×C97C2×#10411yescharacteristic
92D97—22no—
93D97—81no—
94#1483—971nomaximal, Sylow
95C97#148311yescharacteristic, complemented, Sylow
96#529—971no—
97C2×C4⋊C8—971no—
98C4×M16—971no—
99Q8⋊C8—972no—
100D4⋊C8—972no—
101C2×M16—1941no—
102C22×C8—1941no—
103#126—1941no—
104#125#8760411yescharacteristic
105#124—971no—
106#123—1941no—
107C22⋊Q8—971no—
108#121—971no—
109C4×Q8—971no—
110C4×D4—971no—
111C4×D4—1941no—
112#117—1941no—
113C2×C42—971no—
114C4⋊C8—972no—
115C4⋊C8—1941no—
116#97—1941no—
117C4×C8—972no—
118C2×Q8—971no—
119C2×D4—971no—
120C2×D4—1941no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—1941no—
125M16—1944no—
126C2×C8—3882no—
127C2×C8—1942no—
128C2×C8—1942no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4C2×#8760411yescharacteristic
134C4⋊C4C2×#8760411yescharacteristic
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C22⋊C4—1941no—
138C22⋊C4—1941no—
139C22⋊C4—1941no—
140C42—971no—
141C42—971no—
142C42—971no—
143C42—1941no—
144C42C2×#8760411yescharacteristic
145C23—971no—
146C23—1941no—
147Q8—1942no—
148D4—3882no—
149D4—1942no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—971no—
153C2×C4—971no—
154C2×C4—971no—
155C2×C4—3881no—
156C2×C4—3881no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4—1942no—
160C2×C4—1941no—
161C2×C4—1942no—
162C2×C4C22⋊#8760411yescharacteristic
163C2×C4C22⋊#8760411yescharacteristic
164C2×C4C22×#8760411yescharacteristic
165C8—3882no—
166C8—1944no—
167C22—972no—
168C22—3881no—
169C22—3881no—
170C22—3881no—
171C22—1941no—
172C22—1941no—
173C22C2×C22⋊#8760411yescharacteristic
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—971no—
179C4—971no—
180C4—3881no—
181C4—1941no—
182C4—1941no—
183C4#1569089812yes—
184C2—7761no—
185C2—1942no—
186C2#2026171011yescharacteristic
187C2#2026172611yescharacteristic
188C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690785C4
cosocleC4×Dic97C23
hypercenter#125#87604
hypocenterC97#1483
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #20261736C2semidirect
2. C4×#13095521C2semidirect
3. #20261585C2semidirect
4. #20261583C2semidirect
95. C97#1483semidirect

Maximal subgroups

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

subgroupindex
#202617362
C4×#130955212
#202615852
#202615832
#202612702
#148397