Finite Group Databasebeta

← #27157873

Subgroups — #27157873

Order 12416.

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

normal subgroups only

This group has 320 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#20261739C211yesmaximal, characteristic, complemented
2C2×#15690875C212yesmaximal, complemented
3C2×#15690874C212yesmaximal, complemented
4#20261671C211yesmaximal, characteristic, complemented
5#20261284C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×C4×#87604C2212yes—
8C4⋊#11715897—22no—
9#15690875—24no—
10#15690874—24no—
11(C2×#87604)⋊C4—22no—
12C2×#13095512C2211yescharacteristic
13C2×#13095510C2211yescharacteristic
14#15690808C411yescharacteristic, complemented
15#15690806C411yescharacteristic, complemented
16#15690798C411yescharacteristic, complemented
17#15690783—22no—
18#15690770—22no—
19C42×D97C2211yescharacteristic
20#124×C97C411yescharacteristic, complemented
21C22×#87604—22no—
22#13095521—22no—
23#13095521—22no—
24C2×#11715897—22no—
25C4×#87604—22no—
26C4×#87604D414yescomplemented
27#13095512—22no—
28#13095512—22no—
29#13095510—22no—
30#13095510—22no—
31C2×Dic194C2×C411yescharacteristic
32C2×D388—21no—
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97—21no—
35C2×C22⋊D97—22no—
36C22×Dic97—21no—
37C2×C4×D97—21no—
38C2×C4×D97D412yes—
39C2×C4×D97C2311yescharacteristic
40#13095490—22no—
41#13095488—22no—
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97D412yes—
44C2×Q8×C97C2×C411yescharacteristic
45C2×D4×C97C2×C411yescharacteristic
46C22⋊C4×C97—22no—
47C42×C97C2×C411yescharacteristic
48C23×C97—21no—
49C22×D97—21no—
50C22×D97C2×D411yescharacteristic
51C2×#87604—24no—
52C2×#87604—44no—
53Q8×C97—22no—
54D4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C412yes—
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—44no—
59C22⋊D97—42no—
60C2×Dic97—21no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97C22⋊C412yes—
64C2×Dic97C2×D411yescharacteristic
65D388—22no—
66D388—42no—
67C4×D97—42no—
68C4×D97—41no—
69C4×D97—41no—
70C4×D97C2×D412yes—
71C4×D97#4112yes—
72Dic194—22no—
73#11715897—42no—
74C22×C97—41no—
75C22×C97—42no—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—22no—
78C2×D97—41no—
79C2×D97—42no—
80C2×D97#12112yes—
81#87604—44no—
82C4×C97—22no—
83C4×C97—41no—
84C4×C97C4×D412yes—
85Dic97—22no—
86Dic97—41no—
87Dic97#12411yescharacteristic
88Dic97#12711yescharacteristic
89C2×C97—81no—
90C2×C97#39011yescharacteristic
91C2×C97#42112yes—
92D97—22no—
93D97—81no—
94#1963—971nomaximal, Sylow
95C97#196311yescharacteristic, complemented, Sylow
96#532—971no—
97#423—971no—
98C2×#104—972no—
99C2×#104—972no—
100#376—971no—
101C2×#41—971no—
102C2×#41—971no—
103C2×M16—972no—
104C4⋊Q8—971no—
105#127—971no—
106#124—971no—
107#124#8760411yescharacteristic, complemented
108#121—1942no—
109C4×D4—1942no—
110C2×C42—971no—
111C2×C42—972no—
112C4⋊C8—1942no—
113#104—1944no—
114#104—1944no—
115#95—1942no—
116#41—1942no—
117#41—1942no—
118#41—1942no—
119#41—1942no—
120C2×Q8—971no—
121C2×Q8C2×#8760411yescharacteristic
122C2×D4—971no—
123C2×D4—1941no—
124C2×D4—1942no—
125C2×D4—1941no—
126C2×D4C2×#8760411yescharacteristic
127C22×C4—971no—
128C22×C4—972no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4—1942no—
132M16—1942no—
133M16—1942no—
134C2×C8—1942no—
135C4⋊C4—1942no—
136C22⋊C4—22no—
137C22⋊C4—1942no—
138C42—972no—
139C42—971no—
140C42—974no—
141C42—1942no—
142C42C2×#8760411yescharacteristic
143C23—21no—
144C23—971no—
145C23—1941no—
146Q8—22no—
147Q8—1942no—
148D4—22no—
149D4—3882no—
150D4—3882no—
151D4—3884no—
152D4—1942no—
153C2×C4—21no—
154C2×C4—971no—
155C2×C4—972no—
156C2×C4—972no—
157C2×C4—972no—
158C2×C4—3882no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—3881no—
163C2×C4—3884no—
164C2×C4—1944no—
165C2×C4—1941no—
166C2×C4C22⋊#8760412yes—
167C2×C4C22×#8760411yescharacteristic
168C8—3882no—
169C22—41no—
170C22—42no—
171C22—972no—
172C22—3881no—
173C22—3882no—
174C22—1942no—
175C22C2×C22⋊#8760411yescharacteristic
176C4—22no—
177C4—41no—
178C4—971no—
179C4—971no—
180C4—3884no—
181C4—3881no—
182C4—1942no—
183C4D4×#8760412yes—
184C2—81no—
185C2—7761no—
186C2—1942no—
187C2#2026168711yescharacteristic
188C2#2026173212yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC2×C4×D97C23
hypercenter#124#87604
hypocenterC97#1963
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261739C2semidirect
2. C2×#15690875C2semidirect
3. C2×#15690874C2semidirect
4. #20261671C2semidirect
14. #15690808C4semidirect
15. #15690806C4semidirect
16. #15690798C4semidirect
20. #124×C97C4semidirect
26. C4×#87604D4semidirect
95. C97#1963semidirect
107. #124#87604semidirect

Maximal subgroups

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

subgroupindex
#202617392
C2×#156908752
C2×#156908742
#202616712
#202612842
#196397