Finite Group Databasebeta

← #27140676

Subgroups — #27140676

Order 12416.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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#20260885C212yesmaximal
2#20260652C211yesmaximal, characteristic
3#20260472C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#544×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C2×#13095498—22no—
8C2×#13095488—21no—
9C2×#13095488—21no—
10#15690730—22no—
11C22⋊#11715895C2212yes—
12#15690717C2211yescharacteristic
13C4⋊Q8×C97—22no—
14#125×C97—22no—
15C22⋊Q8×C97—21no—
16C22⋊Q8×C97C2212yes—
17#117×C97—21no—
18C2×C4⋊C4×C97C2211yescharacteristic
19C23×D97—21no—
20C2×D388—22no—
21C2×D388—42no—
22C22×Dic97—21no—
23#13095498—44no—
24C2×#11715895—22no—
25#13095488—42no—
26#13095488—42no—
27C2×Q8×C97—21no—
28C2×Q8×C97D412yes—
29C22×C4×C97D411yescharacteristic
30C22×C4×C97D411yescharacteristic
31C22×C4×C97C2311yescharacteristic
32C4⋊C4×C97—21no—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—21no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97—22no—
37C4⋊C4×C97—22no—
38C4⋊C4×C97D412yes—
39C22⋊C4×C97—22no—
40C42×C97—22no—
41C23×C97C2×D411yescharacteristic
42C22×D97—22no—
43C22×D97—41no—
44C22×D97—41no—
45C22×D97—41no—
46C22×D97—41no—
47C22×D97—41no—
48C22×D97—41no—
49Q8×C97—42no—
50Q8×C97—81no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97SD1612yescomplemented
59C2×C4×C97C2×D412yes—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63D388—24no—
64D388—42no—
65#11715895—42no—
66C22×C97—21no—
67C22×C97—21no—
68C22×C97#12011yescharacteristic
69C22×C97C2×SD1611yescharacteristic
70C22×C97#13711yescharacteristic
71C2×D97—42no—
72C2×D97—42no—
73C2×D97—42no—
74C2×D97—81no—
75C2×D97—81no—
76C2×D97—81no—
77C2×D97—81no—
78C4×C97—22no—
79C4×C97—22no—
80C4×C97—42no—
81C4×C97—41no—
82C4×C97—42no—
83Dic97—81no—
84C2×C97—21no—
85C2×C97—21no—
86C2×C97#44811yescharacteristic
87C2×C97#45411yescharacteristic
88C2×C97B(4,𝔽2)11yescharacteristic
89D97—82no—
90#1599—971nomaximal, Sylow
91C97#159911yescharacteristic, complemented, Sylow
92#544D9711yescharacteristic, complemented
93#450—972no—
94#394—971no—
95#328—972no—
96#324—971no—
97C22×D4—971no—
98C2×SD16—1942no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D9712yes—
103#117—21no—
104C2×C4⋊C4C2×D9711yescharacteristic
105C2×C22⋊C4—1941no—
106C2×C22⋊C4—1941no—
107#102—1942no—
108C22⋊C8—972no—
109#95—971no—
110C24—1941no—
111C2×Q8—21no—
112C2×Q8C22⋊D9712yes—
113C2×D4—3882no—
114C2×D4—1942no—
115C22×C4—1941no—
116C22×C4C22⋊D9711yescharacteristic
117C22×C4C22⋊D9711yescharacteristic
118C22×C4C22×D9711yescharacteristic
119SD16—3884no—
120C2×C8—1942no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D9712yes—
128C22⋊C4—22no—
129C22⋊C4—3882no—
130C22⋊C4—3882no—
131C42—22no—
132C23—3881no—
133C23—3881no—
134C23—3881no—
135C23—3881no—
136C23—3881no—
137C23—3881no—
138C23—1942no—
139C23C2×C22⋊D9711yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—3882no—
143D4—1944no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4#1309549812yescomplemented
155C2×C4C2×C22⋊D9712yes—
156C8—3882no—
157C22—21no—
158C22—21no—
159C22—7761no—
160C22—7761no—
161C22—7761no—
162C22—7761no—
163C22—3882no—
164C22—3882no—
165C22—3882no—
166C22#1569077911yescharacteristic
167C22C2×#1309549811yescharacteristic
168C22#1569084111yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—7761no—
175C2—21no—
176C2—21no—
177C2—7762no—
178C2#2026087411yescharacteristic
179C2#2026090811yescharacteristic
180C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#544×C97C2
socleC22×C97#120
prime radicalC22×D388C22
hypercenter#544D97
hypocenterC97#1599
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #20260472C2semidirect
4. #20260461C2semidirect
5. #544×C97C2semidirect
58. C2×C4×C97SD16semidirect
91. C97#1599semidirect
92. #544D97semidirect
154. C2×C4#13095498semidirect

Maximal subgroups

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

subgroupindex
#202608852
#202606522
#202604722
#202604612
#544×C972
#159997