Finite Group Databasebeta

← #27157042

Subgroups — #27157042

Order 12416.

260 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261754C211yesmaximal, characteristic, complemented
2#20261736C211yesmaximal, characteristic, complemented
3D4⋊#11715897C214yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5C2×#13095519—21no—
6#15690877—21no—
7C4⋊#11715897—21no—
8C4⋊#11715897C2212yes—
9#15690867—21no—
10C4×#11715897C2212yes—
11C2×#13095510—21no—
12#15690810C411yescharacteristic
13#15690797—21no—
14#15690781—21no—
15#15690781C412yes—
16D4×Dic97—21no—
17D4×Dic97C2212yes—
18C42×D97C2211yescharacteristic
19#127×C97C411yescharacteristic
20#13095519—42no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23#13095510—42no—
24#13095510—42no—
25C2×Dic194—21no—
26C2×C22⋊D97—22no—
27C22×Dic97—22no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C412yes—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97C2311yescharacteristic
38C2×D4×C97—21no—
39C2×D4×C97—21no—
40C2×D4×C97C2×C412yes—
41C42×C97C2×C411yescharacteristic
42C23×C97—22no—
43C22×D97C22⋊C411yescharacteristic
44D4×C97—24no—
45D4×C97—42no—
46D4×C97—42no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22×C411yescharacteristic
50C22⋊D97—44no—
51C2×Dic97—22no—
52C2×Dic97—42no—
53C2×Dic97—42no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C2×D411yescharacteristic
56C2×Dic97C2×D411yescharacteristic
57C4×D97—22no—
58C4×D97—41no—
59C4×D97D812yes—
60C4×D97SD1612yes—
61Dic194—42no—
62#11715897—42no—
63#11715897—42no—
64C22×C97—42no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—21no—
69C2×D97—21no—
70C2×D97#10212yes—
71C4×C97—21no—
72C4×C97#10212yes—
73C4×C97#13112yes—
74Dic97—21no—
75Dic97—21no—
76Dic97—42no—
77Dic97C2×D811yescharacteristic
78Dic97C2×SD1611yescharacteristic
79C2×C97—82no—
80C2×C97#40811yescharacteristic
81C2×C97C2×#10211yescharacteristic
82C2×C97#42111yescharacteristic
83D97—22no—
84#1480—971nomaximal, Sylow
85C97#148011yescharacteristic, complemented, Sylow
86#532—971no—
87#431—971no—
88C2×C4⋊C8—971no—
89D4⋊C8—974no—
90C2×#41—1941no—
91C22×C8—1941no—
92C4⋊Q8—971no—
93#127#8760411yescharacteristic
94#124—1941no—
95#121—972no—
96#121—1941no—
97C4×D4—972no—
98C4×D4—1941no—
99C2×C42—971no—
100C4⋊C8—972no—
101C4⋊C8—1941no—
102C4⋊C8—1941no—
103C22⋊C8—1941no—
104C4×C8—972no—
105#41—3882no—
106#41—3882no—
107C2×Q8—1941no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—1942no—
111C2×D4C2×#8760412yes—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4—971no—
115C22×C4—1942no—
116C2×C8—3882no—
117C2×C8—1942no—
118C2×C8—1942no—
119C4⋊C4—972no—
120C4⋊C4—1941no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—971no—
124C42—971no—
125C42—971no—
126C42C2×#8760411yescharacteristic
127C23—22no—
128C23—971no—
129Q8—3882no—
130D4—24no—
131D4—42no—
132D4—42no—
133D4—3884no—
134C2×C4—971no—
135C2×C4—972no—
136C2×C4—972no—
137C2×C4—971no—
138C2×C4—971no—
139C2×C4—3881no—
140C2×C4—3882no—
141C2×C4—3882no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22⋊#8760411yescharacteristic
146C2×C4C22×#8760411yescharacteristic
147C8—3882no—
148C8—3882no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—972no—
153C22—1941no—
154C22—1941no—
155C22C2×C22⋊#8760411yescharacteristic
156C4—21no—
157C4—971no—
158C4—971no—
159C4—3882no—
160C4—1941no—
161C4—1941no—
162C4#1569087212yes—
163C4#1569089712yes—
164C2—82no—
165C2—1942no—
166C2#2026170911yescharacteristic
167C2C2×#1569087211yescharacteristic
168C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC4×Dic97C23
hypercenter#127#87604
hypocenterC97#1480
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261754C2semidirect
2. #20261736C2semidirect
3. D4⋊#11715897C2semidirect
85. C97#1480semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202617362
D4⋊#117158972
#202612832
#148097