Finite Group Databasebeta

← #27142259

Subgroups — #27142259

Order 12416.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261502C211yesmaximal, characteristic
2#20260744C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic
4#20260491C212yesmaximal, complemented
5#536×C97C211yesmaximal, characteristic
6#15690817—24no—
7C2×#13095495—22no—
8C2×#13095495C2212yes—
9#15690769C2212yes—
10#15690734—24no—
11#15690725—22no—
12C2×#41×C97C2211yescharacteristic
13C22×Q8×C97C2211yescharacteristic
14C4⋊Q8×C97C412yescomplemented
15#124×C97C412yescomplemented
16#123×C97—22no—
17C22⋊Q8×C97—22no—
18#117×C97C2211yescharacteristic
19#13095495—24no—
20#13095495—22no—
21#13095495—22no—
22C2×#11715895—24no—
23C2×#11715895D414yes—
24C4⋊Dic97—22no—
25C22⋊Dic97—22no—
26C4×Dic97—22no—
27#41×C97—22no—
28#41×C97—22no—
29C2×Q8×C97—21no—
30C2×Q8×C97—22no—
31C2×Q8×C97—22no—
32C2×Q8×C97C2×C412yes—
33C2×Q8×C97C2×C411yescharacteristic
34C2×D4×C97C2×C411yescharacteristic
35C2×D4×C97D412yes—
36C22×C4×C97—21no—
37C22×C4×C97D412yes—
38C22×C4×C97C2311yescharacteristic
39C4⋊C4×C97—24no—
40C4⋊C4×C97—21no—
41C22⋊C4×C97—24no—
42C22⋊C4×C97—21no—
43C42×C97C2×C412yes—
44C23×C97C22⋊C412yes—
45C23×C97C2×D411yescharacteristic
46Q8×C97—22no—
47Q8×C97—24no—
48Q8×C97—41no—
49Q8×C97—41no—
50D4×C97—24no—
51D4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—42no—
57C2×C4×C97C22⋊C412yes—
58C2×C4×C97C22×C411yescharacteristic
59C2×C4×C97C2×D412yes—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97#4112yes—
62C2×Dic97—42no—
63#11715895—24no—
64C22×C97—22no—
65C22×C97—41no—
66C22×C97—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#12112yes—
69C4×C97—22no—
70C4×C97—42no—
71C4×C97—42no—
72C4×C97C4×D412yes—
73C4×C97#12011yescharacteristic
74C4×C97#12311yescharacteristic
75Dic97—82no—
76C2×C97—22no—
77C2×C97—21no—
78C2×C97—42no—
79C2×C97#38611yescharacteristic
80#1909—971nomaximal, Sylow
81C97#190911yescharacteristic, complemented, Sylow
82#568—971no—
83#536D9711yescharacteristic
84#419—972no—
85#410—971no—
86H(3,ℤ/4)—972no—
87C2×#41C2×D9711yescharacteristic
88C22×Q8C2×D9711yescharacteristic
89#131—1944no—
90C2×M16—972no—
91C2×M16—1942no—
92C4⋊Q8Dic9712yescomplemented
93#124Dic9712yescomplemented
94#123—22no—
95C22⋊Q8—22no—
96#117—972no—
97#117C2×D9711yescharacteristic
98#103—1944no—
99#99—1942no—
100#41—22no—
101#41—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic9712yes—
106C2×Q8C2×Dic9711yescharacteristic
107C2×D4C2×Dic9711yescharacteristic
108C2×D4C22⋊D9712yes—
109C22×C4—21no—
110C22×C4C22⋊D9712yes—
111C22×C4C22×D9711yescharacteristic
112M16—1944no—
113M16—1942no—
114M16—1942no—
115C2×C8—974no—
116C2×C8—1944no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—1942no—
120C22⋊C4—24no—
121C22⋊C4—21no—
122C22⋊C4—1942no—
123C42—1942no—
124C42C2×Dic9712yes—
125C23C22⋊Dic9712yes—
126C23C2×C22⋊D9711yescharacteristic
127Q8—22no—
128Q8—24no—
129Q8—41no—
130Q8—41no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—3882no—
139C2×C4C22⋊Dic9712yes—
140C2×C4C22×Dic9711yescharacteristic
141C2×C4D4×D9712yes—
142C2×C4C2×C22⋊D9711yescharacteristic
143C2×C4#1309551012yes—
144C8—1944no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×C22⋊Dic9711yescharacteristic
149C22#1569078212yes—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—7762no—
154C4D4×Dic9712yes—
155C4#1569077811yescharacteristic
156C4#1569079111yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#536×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#536D97
hypocenterC97#1909
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20260744C2semidirect
4. #20260491C2semidirect
14. C4⋊Q8×C97C4semidirect
15. #124×C97C4semidirect
81. C97#1909semidirect
92. C4⋊Q8Dic97semidirect
93. #124Dic97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607442
#202607092
#202604912
#536×C972
#190997