Finite Group Databasebeta

← #27157196

Subgroups — #27157196

Order 12416.

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

normal subgroups only

This group has 264 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
2C4×#13095521C211yesmaximal, characteristic, complemented
3#20261591C212yesmaximal, complemented
4#20261590C212yesmaximal, complemented
5#124×D97C211yesmaximal, characteristic
6C2×#13095521—21no—
7#15690877—21no—
8#15690867—21no—
9#15690865C2212yescomplemented
10#15690865C2212yescomplemented
11C4×#11715897—21no—
12C2×Q8×D97C411yescharacteristic
13C2×D4×D97C411yescharacteristic
14#15690799C2211yescharacteristic
15#15690798C411yescharacteristic
16#15690797C2211yescharacteristic
17#15690796—21no—
18#15690796—21no—
19C22⋊C4×D97—21no—
20C22⋊C4×D97—21no—
21C42×D97C2211yescharacteristic
22#124×C97C411yescharacteristic
23#13095521—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C23×D97—21no—
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29Q8×D97—42no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32D4×D97—42no—
33C2×C4×D97—21no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—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×D97—42no—
54C22×D97—42no—
55C22×D97—42no—
56C22×D97C22⋊C411yescharacteristic
57Q8×C97—41no—
58D4×C97—41no—
59C2×C4×C97—21no—
60C2×C4×C97C22⋊C411yescharacteristic
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22×C411yescharacteristic
63C22⋊D97—42no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C2×D411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68D388—41no—
69C4×D97—22no—
70C4×D97—22no—
71C4×D97—41no—
72C4×D97—42no—
73Dic194—42no—
74Dic194—41no—
75#11715897—24no—
76#11715897—42no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—41no—
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—84no—
87C2×D97#10412yes—
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—41no—
92Dic97—21no—
93Dic97—21no—
94Dic97—41no—
95Dic97#10012yes—
96C2×C97—81no—
97C2×C97C2×#10011yescharacteristic
98C2×C97C2×#10411yescharacteristic
99C2×C97#42111yescharacteristic
100D97—22no—
101D97—81no—
102#1527—971nomaximal, Sylow
103C97#152711yescharacteristic, complemented, Sylow
104C2×#124—971no—
105#432—971no—
106C4×M16—971no—
107#329—972no—
108#329—972no—
109C22×Q8—971no—
110C22×D4—971no—
111C2×M16—1941no—
112#124—971no—
113#124—971no—
114#124—971no—
115#124—1941no—
116#124—1941no—
117#124#8760411yescharacteristic
118C2×C22⋊C4—1941no—
119C2×C22⋊C4—1941no—
120C2×C42—971no—
121C4⋊C8—1941no—
122C22⋊C8—1941no—
123#97—972no—
124#97—972no—
125C4×C8—1941no—
126C24—1941no—
127C2×Q8—971no—
128C2×Q8—3882no—
129C2×Q8—1941no—
130C2×Q8C2×#8760411yescharacteristic
131C2×D4—971no—
132C2×D4—3882no—
133C2×D4—1941no—
134C2×D4C2×#8760411yescharacteristic
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139M16—1944no—
140C2×C8—1942no—
141C2×C8—1942no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—1941no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C22⋊C4—1941no—
149C22⋊C4—1941no—
150C42—971no—
151C42—971no—
152C42—971no—
153C42C2×#8760411yescharacteristic
154C23—21no—
155C23—971no—
156C23—3882no—
157C23—3882no—
158C23—3882no—
159C23—1941no—
160Q8—41no—
161Q8—3881no—
162Q8—3882no—
163D4—41no—
164D4—3881no—
165D4—3882no—
166C2×C4—21no—
167C2×C4—971no—
168C2×C4—971no—
169C2×C4—971no—
170C2×C4—3881no—
171C2×C4—3882no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4—1941no—
175C2×C4C22⋊#8760411yescharacteristic
176C2×C4C22⋊#8760411yescharacteristic
177C2×C4C22×#8760411yescharacteristic
178C8—3882no—
179C8—1944no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—7764no—
184C22—972no—
185C22—3881no—
186C22—3881no—
187C22—3881no—
188C22—1941no—
189C22—1941no—
190C22C2×C22⋊#8760411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—41no—
195C4—972no—
196C4—3881no—
197C4—1941no—
198C4—1941no—
199C2—81no—
200C2—7761no—
201C2—1942no—
202C2D97⋊#10011yescharacteristic
203C2D97⋊#10411yescharacteristic
204C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1527
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. C4×#13095521C2semidirect
3. #20261591C2semidirect
4. #20261590C2semidirect
9. #15690865C22semidirect
10. #15690865C22semidirect
103. C97#1527semidirect

Maximal subgroups

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

subgroupindex
#202617582
C4×#130955212
#202615912
#202615902
#124×D972
#152797