Finite Group Databasebeta

← #27141945

Subgroups — #27141945

Order 12416.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261210C211yesmaximal, characteristic
2#20260775C212yesmaximal, complemented
3C2×#15690727C211yesmaximal, characteristic
4#20260495C212yesmaximal, complemented
5#390×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7D4×Dic97—22no—
8D4×Dic97—22no—
9#15690769C2212yes—
10C2×C22⋊Dic97—22no—
11#15690743—22no—
12#15690741—22no—
13#15690727C414yescomplemented
14C22×D4×C97C2211yescharacteristic
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C22⋊C4×C97—22no—
17C2×C42×C97C2211yescharacteristic
18C22×Dic97—22no—
19C22×Dic97—22no—
20#13095495—22no—
21#13095495C2×C414yescomplemented
22C2×#11715895—22no—
23C4⋊Dic97D412yes—
24C4⋊Dic97D412yes—
25C22⋊Dic97—24no—
26C22⋊Dic97—22no—
27C4×Dic97—22no—
28C24×C97D412yes—
29C2×D4×C97—22no—
30C2×D4×C97—22no—
31C2×D4×C97C2×C412yes—
32C2×D4×C97Q812yes—
33C22×C4×C97—21no—
34C22×C4×C97—21no—
35C22×C4×C97C2311yescharacteristic
36C4⋊C4×C97—22no—
37C22⋊C4×C97—44no—
38C42×C97—22no—
39C23×C97—22no—
40C23×C97—22no—
41C23×C97—42no—
42C23×C97—42no—
43C23×C97C4⋊C414yes—
44C23×C97C2×D411yescharacteristic
45D4×C97—24no—
46D4×C97—24no—
47C2×C4×C97—22no—
48C2×C4×C97—41no—
49C2×C4×C97—41no—
50C2×C4×C97—42no—
51C2×C4×C97—41no—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D412yes—
54C2×C4×C97C2×Q811yescharacteristic
55C2×C4×C97#4112yes—
56C2×Dic97—22no—
57C2×Dic97—22no—
58C2×Dic97—82no—
59C2×Dic97—82no—
60#11715895—24no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—24no—
64C22×C97—44no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97—82no—
68C22×C97C2×C4⋊C411yescharacteristic
69C22×C97#12011yescharacteristic
70C22×C97#12311yescharacteristic
71C4×C97—42no—
72C4×C97—81no—
73C4×C97C4×D412yes—
74C4×C97C22⋊Q812yes—
75Dic97—42no—
76Dic97—42no—
77C2×C97—21no—
78C2×C97—21no—
79C2×C97—44no—
80C2×C97#38511yescharacteristic
81C2×C97#45311yescharacteristic
82C2×C97#45511yescharacteristic
83#1856—971nomaximal, Sylow
84C97#185611yescharacteristic, complemented, Sylow
85#518—971no—
86#428—972no—
87C2×#100—971no—
88#390D9711yescharacteristic
89#339—972no—
90C22×D4C2×D9711yescharacteristic
91C2×M16—972no—
92C4×D4—1942no—
93C4×D4—1942no—
94#117—972no—
95C2×C4⋊C4C2×D9711yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—1942no—
98C2×C42C2×D9711yescharacteristic
99#107—1942no—
100#106—1942no—
101#100—974no—
102C24C22⋊D9712yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4Dic19412yes—
106C2×D4C4×D9712yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—1942no—
110C22×C4—1942no—
111C22×C4C22×D9711yescharacteristic
112M16—974no—
113M16—1942no—
114C2×C8—1942no—
115C4⋊C4—22no—
116C4⋊C4—972no—
117C4⋊C4—972no—
118C22⋊C4—44no—
119C22⋊C4—1944no—
120C22⋊C4—1942no—
121C42—22no—
122C42—1942no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23#1309549014yes—
128C23C2×C22⋊D9711yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—7762no—
137C2×C4—7762no—
138C2×C4—1942no—
139C2×C4—1942no—
140C2×C4C2×C4×D9711yescharacteristic
141C2×C4D4×D9712yes—
142C2×C4C2×Dic19411yescharacteristic
143C2×C4#1309551012yes—
144C8—1944no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×#1309549011yescharacteristic
153C22#1569077811yescharacteristic
154C22#1569079111yescharacteristic
155C4—42no—
156C4—81no—
157C4—3882no—
158C4—3882no—
159C4#1569077012yes—
160C4#1569078712yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#2026060711yescharacteristic
165C2#2026090211yescharacteristic
166C2#2026091211yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#390×C97C2
socleC22×C97#120
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenter#390D97
hypocenterC97#1856
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20260775C2semidirect
4. #20260495C2semidirect
13. #15690727C4semidirect
21. #13095495C2×C4semidirect
84. C97#1856semidirect

Maximal subgroups

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

subgroupindex
#202612102
#202607752
C2×#156907272
#202604952
#390×C972
#185697