Finite Group Databasebeta

← #27140693

Subgroups — #27140693

Order 12416.

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

normal subgroups only

This group has 256 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#20260890C212yesmaximal, complemented
2#20260647C211yesmaximal, characteristic
3#20260472C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic
5#543×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095501—22no—
8C2×C22⋊Dic97—21no—
9#15690733—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C22⋊Q8×C97—22no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97—22no—
16C22⋊Q8×C97C2212yes—
17C2×C22⋊C4×C97—21no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C2×Dic194—22no—
20C2×Dic194—42no—
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095501—44no—
24C2×#11715895—22no—
25C22⋊Dic97—42no—
26C24×C97D411yescharacteristic
27C2×Q8×C97—21no—
28C2×Q8×C97D412yes—
29C22×C4×C97—21no—
30C22×C4×C97—21no—
31C22×C4×C97D411yescharacteristic
32C22×C4×C97C2311yescharacteristic
33C4⋊C4×C97—21no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97D412yes—
37C22⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97Q1612yescomplemented
45C23×C97C2×D411yescharacteristic
46Q8×C97—42no—
47Q8×C97—81no—
48C2×C4×C97—22no—
49C2×C4×C97—21no—
50C2×C4×C97—22no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97—81no—
54C2×C4×C97—81no—
55C2×C4×C97C2×D412yes—
56C2×Dic97—22no—
57C2×Dic97—41no—
58C2×Dic97—41no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62Dic194—24no—
63Dic194—42no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97—22no—
68C22×C97—42no—
69C22×C97—41no—
70C22×C97—41no—
71C22×C97—81no—
72C22×C97#12011yescharacteristic
73C22×C97C2×Q1611yescharacteristic
74C22×C97#13711yescharacteristic
75C4×C97—22no—
76C4×C97—41no—
77C4×C97—42no—
78C4×C97—42no—
79Dic97—42no—
80Dic97—81no—
81C2×C97—21no—
82C2×C97—21no—
83C2×C97—42no—
84C2×C97#45111yescharacteristic
85C2×C97#45511yescharacteristic
86C2×C97B(4,𝔽2)11yescharacteristic
87#1602—971nomaximal, Sylow
88C97#160211yescharacteristic, complemented, Sylow
89#543D9711yescharacteristic
90#451—972no—
91#393—971no—
92#328—972no—
93#323—971no—
94C22×Q8—971no—
95C2×Q16—1942no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D9712yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—1941no—
102C2×C22⋊C4C2×D9711yescharacteristic
103#103—1942no—
104C22⋊C8—972no—
105#95—971no—
106#95—1941no—
107C24C22⋊D9711yescharacteristic
108C2×Q8—21no—
109C2×Q8—3882no—
110C2×Q8—1942no—
111C2×Q8C22⋊D9712yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—1941no—
115C22×C4—1941no—
116C22×C4C22⋊D9711yescharacteristic
117C22×C4C22×D9711yescharacteristic
118Q16—3884no—
119C2×C8—1942no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D9712yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—3882no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1309550112yescomplemented
133C23C2×C22⋊D9711yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—3882no—
137Q8—1944no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—3881no—
146C2×C4—3881no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—3881no—
150C2×C4—1942no—
151C2×C4C2×C22⋊D9712yes—
152C8—3882no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1569077911yescharacteristic
161C22C2×#1309550111yescharacteristic
162C22#1569084111yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—7761no—
168C4—3882no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2026089111yescharacteristic
173C2#2026091311yescharacteristic
174C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#543×C97C2
socleC22×C97#120
prime radicalC24×C97D4
hypercenter#543D97
hypocenterC97#1602
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20260890C2semidirect
3. #20260472C2semidirect
44. C23×C97Q16semidirect
88. C97#1602semidirect
132. C23#13095501semidirect

Maximal subgroups

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

subgroupindex
#202608902
#202606472
#202604722
#202604592
#543×C972
#160297