Finite Group Databasebeta

← #27140358

Subgroups — #27140358

Order 12416.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D97C211yesmaximal, characteristic
2#20260790C212yesmaximal, complemented
3#20260475C211yesmaximal, characteristic, complemented
4#20260474C212yesmaximal, complemented
5#329×C97C211yesmaximal, characteristic, complemented
6C2×Q8×D97C411yescharacteristic
7C2×D4×D97C411yescharacteristic
8#15690799C411yescharacteristic
9#15690798C2211yescharacteristic
10#15690797C411yescharacteristic
11#15690796—22no—
12C22⋊C4×D97—22no—
13C42×D97C2211yescharacteristic
14#15690739—22no—
15#15690722—22no—
16#15690721C2212yescomplemented
17#124×C97C2211yescharacteristic
18#97×C97C2212yescomplemented
19C23×D97—21no—
20C2×Dic194—21no—
21C2×Dic194C2×C411yescharacteristic
22Q8×D97—42no—
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25D4×D97—42no—
26C2×C4×D97—21no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D412yes—
29C2×#11715895—22no—
30C22⋊Dic97—22no—
31#13095488—22no—
32#13095488—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D412yes—
35C2×Q8×C97C2×C411yescharacteristic
36C2×D4×C97C2×C411yescharacteristic
37C2×C8×C97—22no—
38C22⋊C4×C97—22no—
39C42×C97C2311yescharacteristic
40C23×C97—21no—
41C22×D97—21no—
42C22×D97—44no—
43C22×D97—42no—
44C22×D97C22⋊C411yescharacteristic
45Q8×C97—41no—
46D4×C97—41no—
47C2×C4×C97—21no—
48C2×C4×C97C22×C411yescharacteristic
49C2×C4×C97C2×D412yes—
50C22⋊D97—42no—
51C2×Dic97—21no—
52C2×Dic97C22⋊C412yes—
53C2×Dic97C22⋊C411yescharacteristic
54D388—41no—
55C4×D97—22no—
56C4×D97—42no—
57C4×D97—42no—
58Dic194—42no—
59Dic194—41no—
60C8×C97—42no—
61#11715895—42no—
62C22×C97—41no—
63C22×C97—42no—
64C22×C97C2×C22⋊C411yescharacteristic
65C2×D97—22no—
66C2×D97—41no—
67C2×D97—42no—
68C2×D97—84no—
69C2×D97#10012yes—
70C4×C97—21no—
71C4×C97—22no—
72C4×C97—41no—
73Dic97—21no—
74Dic97—22no—
75Dic97—41no—
76C2×C97—81no—
77C2×C97C2×#10011yescharacteristic
78C2×C97#42112yes—
79D97—22no—
80D97—81no—
81#1533—971nomaximal, Sylow
82C97#153311yescharacteristic, complemented, Sylow
83C2×#124—971no—
84#432—972no—
85#329—971no—
86#329—972no—
87#329D9711yescharacteristic, complemented
88C22×Q8—971no—
89C22×D4—971no—
90#124—971no—
91#124—971no—
92#124—971no—
93#124—1942no—
94#124C2×D9711yescharacteristic
95C2×C22⋊C4—1942no—
96C2×C42—971no—
97C4⋊C8—1942no—
98C22⋊C8—1942no—
99#97—972no—
100#97C2×D9712yescomplemented
101C24—1941no—
102C2×Q8—971no—
103C2×Q8—3882no—
104C2×Q8—1941no—
105C2×Q8C4×D9711yescharacteristic
106C2×D4—971no—
107C2×D4—3882no—
108C2×D4—1941no—
109C2×D4C4×D9711yescharacteristic
110C22×C4—972no—
111C22×C4—971no—
112C22×C4—1941no—
113C2×C8—22no—
114C2×C8—1942no—
115C22⋊C4—22no—
116C22⋊C4—1942no—
117C22⋊C4—1942no—
118C22⋊C4—1942no—
119C42—971no—
120C42—972no—
121C42C22×D9711yescharacteristic
122C23—21no—
123C23—971no—
124C23—3882no—
125C23—3884no—
126C23—1941no—
127Q8—41no—
128Q8—3881no—
129Q8—3882no—
130D4—41no—
131D4—3881no—
132D4—3882no—
133C2×C4—21no—
134C2×C4—972no—
135C2×C4—971no—
136C2×C4—3882no—
137C2×C4—3882no—
138C2×C4—1942no—
139C2×C4—1941no—
140C2×C4C2×C4×D9711yescharacteristic
141C2×C4D4×D9712yes—
142C8—42no—
143C8—3882no—
144C22—41no—
145C22—42no—
146C22—7764no—
147C22—972no—
148C22—3881no—
149C22—3882no—
150C22—1942no—
151C22C22⋊C4×D9711yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—3881no—
156C4—1941no—
157C4—1942no—
158C2—81no—
159C2—7761no—
160C2—1942no—
161C2#100×D9711yescharacteristic
162C2#2026075012yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC42×C97C23
hypercenter#329D97
hypocenterC97#1533
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260790C2semidirect
3. #20260475C2semidirect
4. #20260474C2semidirect
5. #329×C97C2semidirect
16. #15690721C22semidirect
18. #97×C97C22semidirect
82. C97#1533semidirect
87. #329D97semidirect
100. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#124×D972
#202607902
#202604752
#202604742
#329×C972
#153397