Finite Group Databasebeta

← #27157866

Subgroups — #27157866

Order 12416.

344 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#15690873C212yesmaximal, complemented
2C2×#15690872C212yesmaximal
3#20261694C211yesmaximal, characteristic, complemented
4#20261681C211yesmaximal, characteristic, complemented
5#124×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604—22no—
8C2×C4⋊#87604C2212yes—
9#15690873—24no—
10#15690872—24no—
11#15690865—22no—
12C2×Q8×D97C2211yescharacteristic
13C2×D4×D97C2211yescharacteristic
14#15690799C411yescharacteristic, complemented
15#15690798C411yescharacteristic, complemented
16#15690797C411yescharacteristic, complemented
17#15690796—22no—
18C22⋊C4×D97—22no—
19C42×D97C2211yescharacteristic
20#124×C97C411yescharacteristic, complemented
21C22×#87604—22no—
22#13095519—22no—
23#13095519D414yes—
24C2×#11715897—22no—
25C4⋊#87604—22no—
26C4⋊#87604—22no—
27C4⋊#87604—44no—
28C23×D97—21no—
29C2×Dic194—21no—
30C2×Dic194C2×C411yescharacteristic
31Q8×D97—22no—
32Q8×D97—22no—
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97—21no—
35D4×D97—22no—
36D4×D97—22no—
37C2×C4×D97—21no—
38C2×C4×D97D412yes—
39C2×C4×D97C2311yescharacteristic
40C22⋊Dic97—22no—
41#13095488—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×D97—41no—
52C22×D97—41no—
53C22×D97—42no—
54C22×D97—42no—
55C22×D97C2×D411yescharacteristic
56C2×#87604—42no—
57C2×#87604—44no—
58Q8×C97—22no—
59D4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97C22⋊C412yes—
62C2×C4×C97C22×C411yescharacteristic
63C22⋊D97—42no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C412yes—
66C2×Dic97C2×D411yescharacteristic
67D388—22no—
68C4×D97—22no—
69C4×D97—22no—
70C4×D97—41no—
71C4×D97—41no—
72C4×D97C2×D412yes—
73C4×D97#4112yes—
74Dic194—22no—
75Dic194—42no—
76#11715897—24no—
77C22×C97—41no—
78C22×C97—42no—
79C22×C97C2×C22⋊C411yescharacteristic
80C2×D97—41no—
81C2×D97—42no—
82C2×D97—82no—
83C2×D97—81no—
84C2×D97—81no—
85C2×D97#12411yescharacteristic
86C2×D97#12711yescharacteristic
87C2×D97AGL(1,ℤ/8)12yes—
88C2×D97#13712yes—
89#87604—82no—
90C4×C97—22no—
91C4×C97—41no—
92C4×C97C4×D412yes—
93Dic97—22no—
94Dic97—41no—
95Dic97#12112yes—
96C2×C97—81no—
97C2×C97#39011yescharacteristic
98C2×C97#41712yes—
99D97—81no—
100D97#48912yes—
101D97#49611yescharacteristic
102D97#49711yescharacteristic
103#1959—971nomaximal, Sylow
104C97#195911yescharacteristic, complemented, Sylow
105C2×#124—971no—
106C2×#103—972no—
107C2×#102—972no—
108C2×#97—971no—
109#384—971no—
110C22×Q8—971no—
111C22×D4—971no—
112C22×C8—972no—
113#124—971no—
114#124—971no—
115#124—971no—
116#124—1942no—
117#124#8760411yescharacteristic, complemented
118C2×C4⋊C4—972no—
119C2×C4⋊C4—1942no—
120C2×C22⋊C4—1942no—
121C2×C42—971no—
122#103—1944no—
123#102—1944no—
124#97—1942no—
125C24—1941no—
126C2×Q8—971no—
127C2×Q8—1941no—
128C2×Q8—1942no—
129C2×Q8—1942no—
130C2×Q8C2×#8760411yescharacteristic
131C2×D4—971no—
132C2×D4—1942no—
133C2×D4—1942no—
134C2×D4—1941no—
135C2×D4C2×#8760411yescharacteristic
136C22×C4—971no—
137C22×C4—972no—
138C22×C4—1941no—
139C22×C4—1942no—
140C2×C8—974no—
141C2×C8—1942no—
142C2×C8—1942no—
143C4⋊C4—3884no—
144C4⋊C4—1942no—
145C4⋊C4—1942no—
146C22⋊C4—22no—
147C22⋊C4—1942no—
148C22⋊C4—1942no—
149C22⋊C4—1942no—
150C42—972no—
151C42—971no—
152C42C2×#8760411yescharacteristic
153C23—21no—
154C23—971no—
155C23—3881no—
156C23—3881no—
157C23—3882no—
158C23—3882no—
159C23—1941no—
160Q8—22no—
161Q8—3882no—
162Q8—1942no—
163D4—22no—
164D4—3882no—
165D4—1942no—
166C2×C4—21no—
167C2×C4—971no—
168C2×C4—972no—
169C2×C4—972no—
170C2×C4—972no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3884no—
174C2×C4—3882no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1941no—
178C2×C4C22⋊#8760412yes—
179C2×C4C22×#8760411yescharacteristic
180C8—1944no—
181C22—41no—
182C22—42no—
183C22—7761no—
184C22—7762no—
185C22—7761no—
186C22—971no—
187C22—971no—
188C22—972no—
189C22—972no—
190C22—3881no—
191C22—3882no—
192C22C2×C22⋊#8760411yescharacteristic
193C4—22no—
194C4—41no—
195C4—7762no—
196C4—972no—
197C4—3881no—
198C4—1942no—
199C4D4×#8760412yes—
200C2—81no—
201C2—7761no—
202C2—971no—
203C2—972no—
204C2—971no—
205C2#2026168711yescharacteristic
206C2#2026172612yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C22
cosocleC2×C4×D97C23
hypercenter#124#87604
hypocenterC97#1959
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. C2×#15690873C2semidirect
3. #20261694C2semidirect
4. #20261681C2semidirect
14. #15690799C4semidirect
15. #15690798C4semidirect
16. #15690797C4semidirect
20. #124×C97C4semidirect
104. C97#1959semidirect
117. #124#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908732
C2×#156908722
#202616942
#202616812
#124×D972
#195997