Finite Group Databasebeta

← #27140675

Subgroups — #27140675

Order 12416.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261366C211yesmaximal, characteristic, complemented
2#20260886C212yesmaximal
3#20260471C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#394×C97C211yesmaximal, characteristic
6C2×#13095499—22no—
7#15690809—22no—
8#15690801—22no—
9#15690789—21no—
10#15690789C2212yes—
11#15690769—21no—
12#15690731—22no—
13C22⋊#11715895C2212yes—
14C22×D4×C97C2211yescharacteristic
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C22⋊C4×C97—21no—
17C2×C22⋊C4×C97—21no—
18#95×C97C2211yescharacteristic
19C2×Dic194—21no—
20C2×Dic194D412yes—
21#13095499—44no—
22C2×#11715895—22no—
23C4⋊Dic97—21no—
24C4⋊Dic97—21no—
25C4⋊Dic97—21no—
26C4⋊Dic97D412yes—
27#13095490—22no—
28#13095490—22no—
29C22⋊Dic97—22no—
30C4×Dic97—22no—
31C24×C97—21no—
32C2×D4×C97—22no—
33C2×D4×C97—42no—
34C22×C4×C97—21no—
35C22×C4×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C4⋊C4×C97—22no—
39C22⋊C4×C97—42no—
40C22⋊C4×C97—42no—
41C23×C97—22no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97—41no—
48C23×C97C2×D411yescharacteristic
49D4×C97—24no—
50D4×C97—42no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97SD1612yescomplemented
60C2×C4×C97C2×D412yes—
61C2×Dic97—22no—
62C2×Dic97—22no—
63Dic194—42no—
64Dic194—81no—
65#11715895—42no—
66C22×C97—21no—
67C22×C97—21no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97—42no—
71C22×C97—81no—
72C22×C97—81no—
73C22×C97—81no—
74C22×C97—81no—
75C22×C97#12011yescharacteristic
76C22×C97C2×SD1611yescharacteristic
77C22×C97#13711yescharacteristic
78C4×C97—22no—
79C4×C97—22no—
80C4×C97—41no—
81C4×C97—81no—
82Dic97—42no—
83Dic97—42no—
84C2×C97—21no—
85C2×C97—21no—
86C2×C97—82no—
87C2×C97#44811yescharacteristic
88C2×C97#45411yescharacteristic
89C2×C97B(4,𝔽2)11yescharacteristic
90#1599—971nomaximal, Sylow
91C97#159911yescharacteristic, complemented, Sylow
92#544—971no—
93#450—972no—
94#394D9711yescharacteristic
95#328—972no—
96#324—971no—
97C22×D4C2×D9711yescharacteristic
98C2×SD16—1942no—
99C4⋊Q8—1942no—
100#125—1942no—
101C22⋊Q8—972no—
102C22⋊Q8—1941no—
103#117—1941no—
104C2×C4⋊C4C2×D9711yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—1942no—
108C22⋊C8—972no—
109#95C2×D9711yescharacteristic
110C24—21no—
111C2×Q8—972no—
112C2×Q8—1941no—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4C22⋊D9711yescharacteristic
117C22×C4C22⋊D9711yescharacteristic
118C22×C4C22×D9711yescharacteristic
119SD16—3884no—
120C2×C8—1942no—
121C4⋊C4—22no—
122C4⋊C4—972no—
123C4⋊C4—1942no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1942no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—1942no—
131C42—1942no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×C22⋊D9711yescharacteristic
140Q8—7761no—
141Q8—3882no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4#1309549912yescomplemented
155C2×C4D4×D9712yes—
156C8—3882no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1569077811yescharacteristic
167C22C2×#1309549911yescharacteristic
168C22#1569084011yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—81no—
173C4—3882no—
174C4—3882no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#2026087311yescharacteristic
179C2#2026090711yescharacteristic
180C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#394×C97C2
socleC22×C97#120
prime radicalC22×D4×C97C22
hypercenter#394D97
hypocenterC97#1599
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #20261366C2semidirect
3. #20260471C2semidirect
4. #20260461C2semidirect
59. C2×C4×C97SD16semidirect
91. C97#1599semidirect
154. C2×C4#13095499semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202608862
#202604712
#202604612
#394×C972
#159997