Finite Group Databasebeta

← #27157058

Subgroups — #27157058

Order 12416.

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

normal subgroups only

This group has 218 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#20261751C211yesmaximal, characteristic
2C4×#13095520C211yesmaximal, characteristic
3#20261585C212yesmaximal, complemented
4Q8⋊#11715897C212yesmaximal, complemented
5#20261287C211yesmaximal, characteristic
6C2×#13095520—21no—
7C4⋊#11715897C2212yes—
8C22⋊#11715897—21no—
9#15690865—21no—
10C4×#11715897—21no—
11C4×#11715897C2212yes—
12#15690809C411yescharacteristic
13#15690808C411yescharacteristic
14#15690800—21no—
15#15690789C411yescharacteristic
16#15690787—21no—
17Q8×Dic97C2211yescharacteristic
18#15690775—21no—
19#15690775C2211yescharacteristic
20#15690766—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23#13095520—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C23×C97C22⋊C411yescharacteristic
50Q8×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C2×Dic97—22no—
57C2×Dic97—22no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64Dic194—22no—
65Dic194—42no—
66#11715897—24no—
67#11715897—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97C2×C22⋊C411yescharacteristic
72C4×C97—21no—
73C4×C97—41no—
74C4×C97—41no—
75C4×C97#13112yes—
76Dic97—21no—
77Dic97—21no—
78Dic97—41no—
79Dic97—41no—
80Dic97#10412yes—
81Dic97#13712yes—
82C2×C97—41no—
83C2×C97#40811yescharacteristic
84C2×C97#41911yescharacteristic
85C2×C97C2×#10411yescharacteristic
86#1485—971nomaximal, Sylow
87C97#148511yescharacteristic, complemented, Sylow
88#533—971no—
89#431—971no—
90C4×M16—971no—
91Q8⋊C8—972no—
92Q8⋊C8—972no—
93C2×M16—1941no—
94C4⋊Q8—971no—
95C4⋊Q8—971no—
96#125—1941no—
97C22⋊Q8—971no—
98C22⋊Q8—1941no—
99C22⋊Q8#8760411yescharacteristic
100C4×Q8—971no—
101C4×Q8—971no—
102C4×Q8—1941no—
103#117—1941no—
104C2×C42—971no—
105C4⋊C8—972no—
106C22⋊C8—1941no—
107#97—1941no—
108C4×C8—972no—
109C4×C8—1941no—
110C2×Q8—971no—
111C2×Q8—1941no—
112C2×Q8C2×#8760411yescharacteristic
113C22×C4—971no—
114C22×C4—971no—
115C22×C4C2×#8760411yescharacteristic
116M16—1944no—
117C2×C8—1942no—
118C2×C8—1942no—
119C4⋊C4—21no—
120C4⋊C4—971no—
121C4⋊C4—1941no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4C2×#8760411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—1941no—
130C42—971no—
131C42—971no—
132C42—971no—
133C42—971no—
134C42—1941no—
135C42—1941no—
136C23C22⋊#8760411yescharacteristic
137Q8—22no—
138Q8—3882no—
139Q8—1942no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—1942no—
148C2×C4—1942no—
149C2×C4—1941no—
150C2×C4—1941no—
151C2×C4C22⋊#8760411yescharacteristic
152C2×C4C22×#8760411yescharacteristic
153C8—3882no—
154C8—1944no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22C2×C22⋊#8760411yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—972no—
163C4—972no—
164C4—3881no—
165C4—3881no—
166C4—1941no—
167C4—1941no—
168C4#1569089712yes—
169C2—41no—
170C2#2026170811yescharacteristic
171C2#2026172811yescharacteristic
172C2D97⋊#10411yescharacteristic
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#1485
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261585C2semidirect
4. Q8⋊#11715897C2semidirect
87. C97#1485semidirect

Maximal subgroups

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

subgroupindex
#202617512
C4×#130955202
#202615852
Q8⋊#117158972
#202612872
#148597