Finite Group Databasebeta

← #27157692

Subgroups — #27157692

Order 12416.

272 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261803C211yesmaximal, characteristic
2#20261746C212yesmaximal
3(C2×#87604)⋊Q8C211yesmaximal, characteristic
4#20261607C212yesmaximal
5#101×D97C211yesmaximal, characteristic
6Q8×#87604—22no—
7Q8×#87604—22no—
8#15690889—22no—
9#15690889C2212yes—
10C2×C4⋊#87604C2211yescharacteristic
11C2×C4×#87604C2211yescharacteristic
12#15690879—22no—
13#15690878—22no—
14(C2×#87604)⋊C4—22no—
15C2×Q8×D97C2211yescharacteristic
16M16×D97C2212yes—
17#15690729C411yescharacteristic, complemented
18#15690728C412yescomplemented
19#101×C97C411yescharacteristic, complemented
20C22×#87604—21no—
21C22×#87604—21no—
22C4⋊#87604—22no—
23C4⋊#87604—22no—
24C4⋊#87604—22no—
25C4⋊#87604D412yes—
26C4⋊#87604D412yes—
27C22⋊#87604—22no—
28C4×#87604—24no—
29C4×#87604—22no—
30C4×#87604—22no—
31C2×Dic194C2×C411yescharacteristic
32C2×Dic194Q812yes—
33Q8×D97—22no—
34Q8×D97—22no—
35C2×C4×D97D412yes—
36C2×C4×D97C2311yescharacteristic
37#13095495C2×C412yes—
38#13095494—22no—
39C8×D97—22no—
40C2×Q8×C97C2×C411yescharacteristic
41M16×C97C2×C412yes—
42C22×D97C2×D411yescharacteristic
43C2×#87604—22no—
44C2×#87604—22no—
45C2×#87604—22no—
46C2×#87604—41no—
47C2×#87604—41no—
48C2×#87604—42no—
49C2×#87604—41no—
50Q8×C97—22no—
51C2×C4×C97C4⋊C412yes—
52C2×C4×C97C22×C411yescharacteristic
53C2×Dic97C4⋊C412yes—
54C2×Dic97C2×Q811yescharacteristic
55C4×D97—22no—
56C4×D97—22no—
57C4×D97C2×D412yes—
58C4×D97#4112yes—
59Dic194—22no—
60Dic194—24no—
61C8×C97—22no—
62#11715895—22no—
63C22×C97C2×C4⋊C411yescharacteristic
64C2×D97—21no—
65C2×D97—21no—
66C2×D97#12011yescharacteristic
67C2×D97#12311yescharacteristic
68#87604—42no—
69#87604—42no—
70#87604—42no—
71#87604—81no—
72C4×C97—22no—
73C4×C97C4×D412yes—
74Dic97—22no—
75Dic97C22⋊Q812yes—
76C2×C97—21no—
77C2×C97#38511yescharacteristic
78D97—21no—
79D97#45411yescharacteristic
80D97#45611yescharacteristic
81#1857—971nomaximal, Sylow
82C97#185711yescharacteristic, complemented, Sylow
83#519—971no—
84#428—972no—
85C2×#101—971no—
86#391—971no—
87#339—972no—
88C22×Q8—971no—
89C2×M16—972no—
90C4×Q8—1942no—
91C4×Q8—1942no—
92#117—972no—
93#117—1942no—
94C2×C4⋊C4—971no—
95C2×C42—971no—
96#107—1942no—
97#106—1942no—
98#101—971no—
99#101—972no—
100#101#8760411yescharacteristic, complemented
101#95—1942no—
102C2×Q8—972no—
103C2×Q8—971no—
104C2×Q8—1942no—
105C2×Q8—1942no—
106C2×Q8C2×#8760411yescharacteristic
107C22×C4—971no—
108C22×C4—972no—
109C22×C4—1941no—
110C22×C4—1941no—
111M16—972no—
112M16—1942no—
113M16C2×#8760412yes—
114C2×C8—1942no—
115C4⋊C4—972no—
116C4⋊C4—972no—
117C4⋊C4—1942no—
118C4⋊C4—1942no—
119C4⋊C4—1942no—
120C22⋊C4—1942no—
121C42—1942no—
122C42—1942no—
123C42—1944no—
124C23—971no—
125Q8—22no—
126Q8—1944no—
127Q8—1942no—
128C2×C4—971no—
129C2×C4—972no—
130C2×C4—972no—
131C2×C4—972no—
132C2×C4—3881no—
133C2×C4—3882no—
134C2×C4—3881no—
135C2×C4—3881no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4—1942no—
139C2×C4—1942no—
140C2×C4—1942no—
141C2×C4C4⋊#8760412yes—
142C2×C4C22×#8760411yescharacteristic
143C8—22no—
144C8—1942no—
145C22—971no—
146C22—971no—
147C22—1941no—
148C22—1941no—
149C22C2×C4⋊#8760411yescharacteristic
150C4—22no—
151C4—7761no—
152C4—972no—
153C4—3882no—
154C4—3882no—
155C4—3882no—
156C4—1942no—
157C4D4×#8760412yes—
158C2—21no—
159C2—971no—
160C2—971no—
161C2—1941no—
162C2#2026168211yescharacteristic
subgroupG/H
centerC2#20261682
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#101×C97C4
socleC2×C97#385
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#101#87604
hypocenterC97#1857
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
17. #15690729C4semidirect
18. #15690728C4semidirect
19. #101×C97C4semidirect
82. C97#1857semidirect
100. #101#87604semidirect

Maximal subgroups

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

subgroupindex
#202618032
#202617462
(C2×#87604)⋊Q82
#202616072
#101×D972
#185797