Finite Group Databasebeta

← #27157096

Subgroups — #27157096

Order 12416.

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

normal subgroups only

This group has 246 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
1#20261757C211yesmaximal, characteristic, complemented
2#20261753C211yesmaximal, characteristic, complemented
3D4⋊#11715897C214yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5#15690877—21no—
6C4⋊#11715897C2212yes—
7#15690867—21no—
8#15690867—21no—
9#15690865—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
20C2×#11715897—22no—
21C2×#11715897—22no—
22#13095510—44no—
23C2×Dic194—21no—
24C2×C22⋊D97—22no—
25C22×Dic97—22no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97C2×C412yes—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33C4×Dic97D411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97C2311yescharacteristic
36C2×D4×C97—21no—
37C2×D4×C97—21no—
38C2×D4×C97C2×C412yes—
39C42×C97C2×C411yescharacteristic
40C23×C97—22no—
41C22×D97C22⋊C411yescharacteristic
42D4×C97—24no—
43D4×C97—42no—
44D4×C97—42no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C22⋊D97—44no—
49C2×Dic97—22no—
50C2×Dic97—44no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54C4×D97—22no—
55C4×D97—22no—
56C4×D97—41no—
57Dic194—42no—
58#11715897—42no—
59#11715897—42no—
60C22×C97—42no—
61C22×C97—42no—
62C22×C97—42no—
63C22×C97C2×C22⋊C411yescharacteristic
64C2×D97—21no—
65C2×D97—21no—
66C2×D97—21no—
67C4×C97—21no—
68C4×C97—21no—
69C4×C97#13112yes—
70Dic97—21no—
71Dic97—21no—
72Dic97—42no—
73Dic97AGL(1,ℤ/8)12yes—
74C2×C97—82no—
75C2×C97#40811yescharacteristic
76C2×C97#41811yescharacteristic
77C2×C97#42111yescharacteristic
78D97—41no—
79#1496—971nomaximal, Sylow
80C97#149611yescharacteristic, complemented, Sylow
81#532—971no—
82#432—971no—
83#431—971no—
84D4⋊C8—974no—
85C2×#41—1941no—
86C4⋊Q8—971no—
87#127#8760411yescharacteristic
88#124—1941no—
89#121—972no—
90#121—1941no—
91C4×D4—972no—
92C4×D4—1941no—
93C2×C42—971no—
94C4⋊C8—972no—
95C4⋊C8—1941no—
96C22⋊C8—1941no—
97C22⋊C8—1941no—
98#97—1941no—
99C4×C8—972no—
100#41—3884no—
101C2×Q8—1941no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—1942no—
105C2×D4C2×#8760412yes—
106C22×C4—971no—
107C22×C4—971no—
108C22×C4—971no—
109C22×C4—1942no—
110C2×C8—1942no—
111C2×C8—1942no—
112C4⋊C4—972no—
113C4⋊C4—1941no—
114C22⋊C4—1942no—
115C22⋊C4—1942no—
116C42—971no—
117C42—971no—
118C42—971no—
119C42C2×#8760411yescharacteristic
120C23—22no—
121C23—971no—
122Q8—3882no—
123D4—24no—
124D4—42no—
125D4—42no—
126D4—3884no—
127C2×C4—971no—
128C2×C4—971no—
129C2×C4—971no—
130C2×C4—3881no—
131C2×C4—3884no—
132C2×C4—1942no—
133C2×C4—1942no—
134C2×C4—1942no—
135C2×C4C22⋊#8760411yescharacteristic
136C2×C4C22⋊#8760411yescharacteristic
137C2×C4C22×#8760411yescharacteristic
138C8—3882no—
139C8—3882no—
140C22—42no—
141C22—42no—
142C22—42no—
143C22—1941no—
144C22—1941no—
145C22—1941no—
146C22C2×C22⋊#8760411yescharacteristic
147C4—21no—
148C4—21no—
149C4—972no—
150C4—3882no—
151C4—1941no—
152C4—1941no—
153C4#1569089712yes—
154C2—82no—
155C2—3881no—
156C2#2026170911yescharacteristic
157C2#2026172711yescharacteristic
158C2#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#1496
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. #20261757C2semidirect
2. #20261753C2semidirect
3. D4⋊#11715897C2semidirect
80. C97#1496semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202617532
D4⋊#117158972
#202612832
#149697