Finite Group Databasebeta

← #27157254

Subgroups — #27157254

Order 12416.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261738C211yesmaximal, characteristic
2C2×#15690877C211yesmaximal, characteristic
3#20261595C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261287C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8#15690877—21no—
9#15690877C2212yes—
10#15690877C2212yes—
11C4⋊#11715897—21no—
12#15690809C2211yescharacteristic
13#15690808C2211yescharacteristic
14#15690800—21no—
15#15690789C411yescharacteristic
16#15690787—21no—
17Q8×Dic97C411yescharacteristic
18#15690775—21no—
19#15690775C411yescharacteristic
20#15690766—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23#13095520—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—42no—
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29C22×Dic97D411yescharacteristic
30C22×Dic97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38C22⋊Dic97—21no—
39C4×Dic97—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
51Q8×C97—41no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97SD1612yes—
64C2×Dic97Q1612yes—
65C2×Dic97C2×D411yescharacteristic
66C2×Dic97C2×D411yescharacteristic
67Dic194—41no—
68Dic194—41no—
69Dic194—41no—
70#11715897—42no—
71#11715897—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97#10312yes—
75C22×C97C2×C22⋊C411yescharacteristic
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—41no—
79C4×C97—41no—
80Dic97—21no—
81Dic97—41no—
82Dic97—41no—
83Dic97#10312yes—
84Dic97C2×SD1611yescharacteristic
85Dic97C2×Q1611yescharacteristic
86Dic97AGL(1,ℤ/8)11yescharacteristic
87Dic97#13711yescharacteristic
88C2×C97—22no—
89C2×C97#41311yescharacteristic
90C2×C97C2×#10311yescharacteristic
91C2×C97#41711yescharacteristic
92#1550—971nomaximal, Sylow
93C97#155011yescharacteristic, complemented, Sylow
94#533—971no—
95#423—971no—
96C2×C4⋊C8—971no—
97#333—972no—
98#332—972no—
99C2×M16—1941no—
100C22×C8—1941no—
101C4⋊Q8—971no—
102C4⋊Q8—971no—
103#125—1941no—
104C22⋊Q8—971no—
105C22⋊Q8—1941no—
106C22⋊Q8#8760411yescharacteristic
107C4×Q8—971no—
108C4×Q8—971no—
109C4×Q8—1941no—
110#117—1941no—
111C2×C42—971no—
112C4⋊C8—972no—
113C4⋊C8—972no—
114C4⋊C8—1941no—
115C4⋊C8—1941no—
116C2×Q8—971no—
117C2×Q8—1941no—
118C2×Q8C2×#8760411yescharacteristic
119C22×C4—971no—
120C22×C4—971no—
121C22×C4C2×#8760411yescharacteristic
122M16—3882no—
123C2×C8—3882no—
124C2×C8—1942no—
125C2×C8—1942no—
126C4⋊C4—21no—
127C4⋊C4—971no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4C2×#8760411yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1941no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—971no—
141C42—1941no—
142C42—1941no—
143C23C22⋊#8760411yescharacteristic
144Q8—41no—
145Q8—3881no—
146Q8—3881no—
147Q8—3881no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—972no—
152C2×C4—972no—
153C2×C4—971no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4—1941no—
160C2×C4—1941no—
161C2×C4C22⋊#8760411yescharacteristic
162C2×C4C22×#8760411yescharacteristic
163C8—3882no—
164C8—3882no—
165C22—21no—
166C22—21no—
167C22#1569087312yes—
168C22C2×C22⋊#8760411yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—971no—
174C4—972no—
175C4—971no—
176C4—971no—
177C4—971no—
178C4—3881no—
179C4—3881no—
180C4—1941no—
181C2—22no—
182C2#2026172211yescharacteristic
183C2C2×#1569087311yescharacteristic
184C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1550
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
3. #20261595C2semidirect
4. #20261594C2semidirect
93. C97#1550semidirect

Maximal subgroups

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

subgroupindex
#202617382
C2×#156908772
#202615952
#202615942
#202612872
#155097