Finite Group Databasebeta

← #27157256

Subgroups — #27157256

Order 12416.

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

normal subgroups only

This group has 252 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#20261758C211yesmaximal, characteristic, complemented
2#20261736C211yesmaximal, characteristic, complemented
3#20261595C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261284C211yesmaximal, characteristic
6C2×#13095519—21no—
7#15690877C2212yes—
8#15690877C2212yes—
9C4⋊#11715897—21no—
10#15690867—21no—
11#15690865—21no—
12C2×#13095512C411yescharacteristic
13C2×#13095510C411yescharacteristic
14#15690808C2211yescharacteristic
15#15690806C2211yescharacteristic
16#15690798C411yescharacteristic
17#15690783—21no—
18#15690783—21no—
19#15690770—21no—
20#15690770—21no—
21C42×D97C2211yescharacteristic
22#124×C97C411yescharacteristic
23#13095519—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26#13095512—42no—
27#13095510—42no—
28C2×Dic194C2×C411yescharacteristic
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C22×Dic97—21no—
35C2×C4×D97—21no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39#13095490—21no—
40#13095490—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97C2311yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C2×D4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C42×C97C2×C411yescharacteristic
51C23×C97—21no—
52C22×D97—21no—
53C22×D97C22⋊C411yescharacteristic
54Q8×C97—41no—
55D4×C97—41no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C22⋊D97—42no—
61C22⋊D97—42no—
62C22⋊D97—41no—
63C22⋊D97—41no—
64C2×Dic97—21no—
65C2×Dic97—42no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C2×Dic97C2×D411yescharacteristic
69D388—41no—
70D388—42no—
71C4×D97—22no—
72C4×D97—22no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—42no—
76Dic194—41no—
77#11715897—42no—
78#11715897—42no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97C2×C22⋊C411yescharacteristic
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—41no—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97#10012yes—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—41no—
93Dic97—21no—
94Dic97—41no—
95Dic97#13512yes—
96Dic97AGL(1,ℤ/8)11yescharacteristic
97Dic97#13711yescharacteristic
98C2×C97—81no—
99C2×C97C2×#10011yescharacteristic
100C2×C97#41611yescharacteristic
101C2×C97#41711yescharacteristic
102D97—22no—
103D97—81no—
104#1551—971nomaximal, Sylow
105C97#155111yescharacteristic, complemented, Sylow
106#532—971no—
107#432—971no—
108C2×C4⋊C8—971no—
109#333—972no—
110#331—972no—
111C2×#41—971no—
112C2×#41—971no—
113C22×C8—1941no—
114C4⋊Q8—971no—
115#127—971no—
116#124—971no—
117#124#8760411yescharacteristic
118#121—1941no—
119#121—1941no—
120C4×D4—1941no—
121C4×D4—1941no—
122C2×C42—971no—
123C4⋊C8—972no—
124C4⋊C8—972no—
125C4⋊C8—1941no—
126C22⋊C8—1941no—
127#97—1941no—
128#41—3882no—
129#41—3882no—
130C2×Q8—971no—
131C2×Q8C2×#8760411yescharacteristic
132C2×D4—971no—
133C2×D4—1941no—
134C2×D4—1941no—
135C2×D4—1941no—
136C2×D4—1941no—
137C2×D4C2×#8760411yescharacteristic
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—1941no—
142C22×C4—1941no—
143C2×C8—3882no—
144C2×C8—1942no—
145C2×C8—1942no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C42—971no—
153C42—971no—
154C42—971no—
155C42C2×#8760411yescharacteristic
156C23—21no—
157C23—971no—
158C23—1941no—
159Q8—41no—
160Q8—3881no—
161D4—41no—
162D4—3881no—
163D4—3882no—
164D4—3882no—
165D4—3881no—
166D4—3881no—
167D4—3882no—
168C2×C4—21no—
169C2×C4—971no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—3882no—
173C2×C4—3882no—
174C2×C4—1941no—
175C2×C4—1942no—
176C2×C4—1941no—
177C2×C4—1942no—
178C2×C4—1941no—
179C2×C4C22⋊#8760411yescharacteristic
180C2×C4C22⋊#8760411yescharacteristic
181C2×C4C22×#8760411yescharacteristic
182C8—3882no—
183C8—3882no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—972no—
188C22—3881no—
189C22—3881no—
190C22—3881no—
191C22—1941no—
192C22—1941no—
193C22C2×C22⋊#8760411yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—972no—
199C4—971no—
200C4—971no—
201C4—3881no—
202C4—1941no—
203C2—81no—
204C2—7761no—
205C2—1942no—
206C2D97⋊#10011yescharacteristic
207C2#2026172511yescharacteristic
208C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1551
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261758C2semidirect
2. #20261736C2semidirect
3. #20261595C2semidirect
4. #20261593C2semidirect
105. C97#1551semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202617362
#202615952
#202615932
#202612842
#155197