Finite Group Databasebeta

← #27157259

Subgroups — #27157259

Order 12416.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261758C211yesmaximal, characteristic, complemented
2#20261753C211yesmaximal, characteristic, complemented
3#20261594C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261282C211yesmaximal, characteristic
6#15690877C2212yes—
7#15690877C2212yes—
8#15690867—21no—
9#15690867—21no—
10#15690865—21no—
11C4×#11715897—21no—
12C2×#13095511—21no—
13#15690809C2211yescharacteristic
14#15690806C2211yescharacteristic
15#15690796—21no—
16#15690785C411yescharacteristic
17#15690783—21no—
18#15690781C411yescharacteristic
19C4×D388C411yescharacteristic
20C4×C22⋊D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25#13095511—44no—
26C2×Dic194—21no—
27C2×D388—21no—
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×D4×C97C2×C411yescharacteristic
45C22×C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2×C411yescharacteristic
49C23×C97—21no—
50C22×D97—21no—
51C22×D97C22⋊C411yescharacteristic
52D4×C97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—42no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—42no—
59C22⋊D97—42no—
60C22⋊D97—42no—
61C2×Dic97—21no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65D388—41no—
66D388—42no—
67C4×D97—22no—
68C4×D97—22no—
69C4×D97—41no—
70C4×D97—42no—
71Dic194—42no—
72#11715897—42no—
73#11715897—42no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97—21no—
79C2×D97—21no—
80C2×D97—21no—
81C2×D97—41no—
82C2×D97—41no—
83C2×D97—41no—
84C4×C97—21no—
85C4×C97—21no—
86C4×C97—21no—
87C4×C97—41no—
88Dic97—21no—
89Dic97—41no—
90Dic97#13512yes—
91Dic97AGL(1,ℤ/8)12yes—
92C2×C97—81no—
93C2×C97#41311yescharacteristic
94C2×C97#41611yescharacteristic
95C2×C97#41811yescharacteristic
96D97—41no—
97D97—81no—
98#1553—971nomaximal, Sylow
99C97#155311yescharacteristic, complemented, Sylow
100#532—971no—
101#432—971no—
102#431—971no—
103#332—972no—
104#331—972no—
105C2×#41—1941no—
106C4⋊Q8—971no—
107#127—971no—
108#124—1941no—
109#121—971no—
110#121—971no—
111#121—1941no—
112C4×D4—971no—
113C4×D4—1941no—
114C4×D4#8760411yescharacteristic
115C2×C42—971no—
116C4⋊C8—972no—
117C4⋊C8—972no—
118C22⋊C8—1941no—
119C22⋊C8—1941no—
120#97—1941no—
121C4×C8—1941no—
122#41—3884no—
123C2×Q8—1941no—
124C2×D4—971no—
125C2×D4—1941no—
126C2×D4—1941no—
127C2×D4—1941no—
128C2×D4—1941no—
129C2×D4C2×#8760411yescharacteristic
130C22×C4—21no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—971no—
134C22×C4—1941no—
135C2×C8—1942no—
136C2×C8—1942no—
137C4⋊C4—971no—
138C4⋊C4—1941no—
139C4⋊C4C2×#8760411yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—1941no—
142C22⋊C4—1941no—
143C22⋊C4—1941no—
144C42—971no—
145C42—971no—
146C42—971no—
147C42C2×#8760411yescharacteristic
148C23—21no—
149C23—971no—
150C23—1941no—
151Q8—3882no—
152D4—41no—
153D4—3881no—
154D4—3882no—
155D4—3882no—
156D4—3882no—
157D4—3882no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3882no—
164C2×C4—3881no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1941no—
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—3882no—
172C8—3882no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—3881no—
177C22—3881no—
178C22—3881no—
179C22—1941no—
180C22—1941no—
181C22—1941no—
182C22C2×C22⋊#8760411yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—972no—
188C4—972no—
189C4—3881no—
190C4—1941no—
191C2—81no—
192C2—7761no—
193C2—3881no—
194C2#2026172111yescharacteristic
195C2#2026172511yescharacteristic
196C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC4×Dic97C23
hypercenterC4×D4#87604
hypocenterC97#1553
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261758C2semidirect
2. #20261753C2semidirect
3. #20261594C2semidirect
4. #20261593C2semidirect
99. C97#1553semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202617532
#202615942
#202615932
#202612822
#155397