Finite Group Databasebeta

← #27157871

Subgroups — #27157871

Order 12416.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261740C211yesmaximal, characteristic, complemented
2C2×#15690875C212yesmaximal, complemented
3C2×#15690874C212yesmaximal, complemented
4C42×#87604C211yesmaximal, characteristic, complemented
5#20261284C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×C4×#87604—22no—
8C2×C4×#87604C2212yes—
9#15690877—22no—
10#15690875—24no—
11#15690874—24no—
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—28no—
26C4×#87604—22no—
27C4×#87604D414yescomplemented
28#13095512—22no—
29#13095512—22no—
30#13095510—22no—
31#13095510—22no—
32C2×Dic194C2×C411yescharacteristic
33C2×D388—21no—
34C2×D388C2×C411yescharacteristic
35C2×C22⋊D97—21no—
36C2×C22⋊D97—22no—
37C22×Dic97—21no—
38C2×C4×D97—21no—
39C2×C4×D97D412yes—
40C2×C4×D97C2311yescharacteristic
41#13095490—22no—
42#13095488—22no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97D412yes—
45C2×Q8×C97C2×C411yescharacteristic
46C2×D4×C97C2×C411yescharacteristic
47C22⋊C4×C97—22no—
48C42×C97C2×C411yescharacteristic
49C23×C97—21no—
50C22×D97—21no—
51C22×D97C2×D411yescharacteristic
52C2×#87604—28no—
53C2×#87604—24no—
54Q8×C97—22no—
55D4×C97—22no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C412yes—
58C2×C4×C97C22×C411yescharacteristic
59C22⋊D97—42no—
60C22⋊D97—44no—
61C2×Dic97—21no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97C22⋊C412yes—
65C2×Dic97C2×D411yescharacteristic
66D388—22no—
67D388—42no—
68C4×D97—24no—
69C4×D97—41no—
70C4×D97—41no—
71C4×D97C2×D412yes—
72C4×D97#4112yes—
73Dic194—22no—
74#11715897—42no—
75C22×C97—41no—
76C22×C97—42no—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97—22no—
79C2×D97—41no—
80C2×D97—42no—
81C2×D97#12112yes—
82#87604—28no—
83C4×C97—22no—
84C4×C97—41no—
85C4×C97C4×D412yes—
86Dic97—41no—
87Dic97#10414yescomplemented
88Dic97#12411yescharacteristic
89Dic97#12711yescharacteristic
90C2×C97—81no—
91C2×C97#39011yescharacteristic
92C2×C97C2×#10412yes—
93D97—22no—
94D97—81no—
95#1962—971nomaximal, Sylow
96C97#196211yescharacteristic, complemented, Sylow
97#532—971no—
98#423—971no—
99C2×#104—972no—
100C2×#104—972no—
101C43—971no—
102C2×#41—971no—
103C2×#41—971no—
104C2×M16—972no—
105C4⋊Q8—971no—
106#127—971no—
107#124—971no—
108#124#8760411yescharacteristic, complemented
109#121—1942no—
110C4×D4—1942no—
111C2×C42—971no—
112C2×C42—972no—
113C2×C42—1942no—
114C4⋊C8—1942no—
115#104—1944no—
116#104—1944no—
117#41—1942no—
118#41—1942no—
119#41—1942no—
120#41—1942no—
121C2×Q8—971no—
122C2×Q8C2×#8760411yescharacteristic
123C2×D4—971no—
124C2×D4—1941no—
125C2×D4—1942no—
126C2×D4—1941no—
127C2×D4C2×#8760411yescharacteristic
128C22×C4—971no—
129C22×C4—972no—
130C22×C4—1941no—
131C22×C4—1941no—
132C22×C4—1942no—
133M16—1942no—
134M16—1942no—
135C2×C8—1942no—
136C4⋊C4—1942no—
137C22⋊C4—22no—
138C22⋊C4—1942no—
139C42—972no—
140C42—971no—
141C42—974no—
142C42—1942no—
143C42—1948no—
144C42C2×#8760411yescharacteristic
145C23—21no—
146C23—971no—
147C23—1941no—
148Q8—22no—
149Q8—1942no—
150D4—22no—
151D4—3882no—
152D4—3884no—
153D4—3882no—
154D4—1942no—
155C2×C4—21no—
156C2×C4—972no—
157C2×C4—971no—
158C2×C4—972no—
159C2×C4—972no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—3881no—
163C2×C4—3881no—
164C2×C4—1944no—
165C2×C4—1941no—
166C2×C4—1944no—
167C2×C4—1948no—
168C2×C4C22⋊#8760412yes—
169C2×C4C22×#8760411yescharacteristic
170C8—3882no—
171C22—41no—
172C22—42no—
173C22—972no—
174C22—3881no—
175C22—3882no—
176C22—1942no—
177C22C2×C22⋊#8760411yescharacteristic
178C4—22no—
179C4—41no—
180C4—971no—
181C4—971no—
182C4—974no—
183C4—3881no—
184C4—1948no—
185C4D4×#8760412yes—
186C2—81no—
187C2—7761no—
188C2—1942no—
189C2#2026168711yescharacteristic
190C2D97⋊#10412yes—
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#1962
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. #20261740C2semidirect
2. C2×#15690875C2semidirect
3. C2×#15690874C2semidirect
4. C42×#87604C2semidirect
14. #15690808C4semidirect
15. #15690806C4semidirect
16. #15690798C4semidirect
20. #124×C97C4semidirect
27. C4×#87604D4semidirect
87. Dic97#104semidirect
96. C97#1962semidirect
108. #124#87604semidirect

Maximal subgroups

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

subgroupindex
#202617402
C2×#156908752
C2×#156908742
C42×#876042
#202612842
#196297