Finite Group Databasebeta

← #27140873

Subgroups — #27140873

Order 12416.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261445C211yesmaximal, characteristic
2#20261115C211yesmaximal, characteristic, complemented
3#20261030C211yesmaximal, characteristic, complemented
4#20261026C211yesmaximal, characteristic
5#20260483C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×Dic388—22no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690802—21no—
12#15690788—22no—
13#15690786—21no—
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16C4⋊C4×D97—21no—
17#15690743—21no—
18#15690742—21no—
19C4⋊#11715895C2211yescharacteristic
20#15690734—21no—
21#15690732—21no—
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194D412yes—
27C2×D388D411yescharacteristic
28C2×C4×D97—21no—
29C2×C4×D97—22no—
30Dic388—44no—
31C2×#11715895—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37#13095490—22no—
38#13095490—22no—
39#13095488—22no—
40#13095488—21no—
41C2×Q8×C97D411yescharacteristic
42C2×C8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—21no—
49Q8×C97—21no—
50Q8×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—22no—
58C2×Dic97—41no—
59D388—21no—
60D388Q1612yescomplemented
61C4×D97—41no—
62C4×D97—41no—
63C4×D97—82no—
64Dic194—42no—
65Dic194—42no—
66C8×C97—22no—
67C8×C97—41no—
68#11715895—41no—
69C22×C97#12011yescharacteristic
70C2×D97—22no—
71C2×D97—41no—
72C2×D97—41no—
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—41no—
76C4×C97C2×Q1611yescharacteristic
77C4×C97#13511yescharacteristic
78C4×C97#13711yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—41no—
81Dic97—42no—
82Dic97—81no—
83C2×C97#45111yescharacteristic
84C2×C97#45211yescharacteristic
85C2×C97#45311yescharacteristic
86D97—42no—
87#1632—971nomaximal, Sylow
88C97#163211yescharacteristic, complemented, Sylow
89#554—971no—
90#494—971no—
91#477—971no—
92#476—971no—
93#333—971no—
94Q8⋊C8D9711yescharacteristic, complemented
95D4⋊C8—971no—
96C2×Q16—1942no—
97C4⋊Q8—971no—
98C4⋊Q8—971no—
99#125—1941no—
100C22⋊Q8—1942no—
101C22⋊Q8—1941no—
102C4×Q8C2×D9711yescharacteristic
103C4×D4—971no—
104C4×D4—1941no—
105C2×C4⋊C4—1941no—
106#107—1941no—
107#106—1941no—
108C4⋊C8—971no—
109C4⋊C8C2×D9711yescharacteristic
110#103—1941no—
111#102—1941no—
112C4×C8C2×D9711yescharacteristic
113C2×Q8—972no—
114C2×Q8—1941no—
115C2×Q8D38811yescharacteristic
116C2×D4—971no—
117C22×C4—1942no—
118C22×C4—1941no—
119Q16—3884no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—971no—
125C4⋊C4—1942no—
126C4⋊C4—1942no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4D38811yescharacteristic
132C22⋊C4—1942no—
133C22⋊C4—1941no—
134C42—21no—
135C42C22×D9711yescharacteristic
136C23—1941no—
137Q8—21no—
138Q8—21no—
139Q8—3882no—
140Q8—3882no—
141D4—972no—
142D4—1941no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—7762no—
146C2×C4—3881no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4D4×D9711yescharacteristic
152C2×C4D4×D9711yescharacteristic
153C2×C4C2×D38811yescharacteristic
154C8—22no—
155C8—41no—
156C8—3881no—
157C22—3881no—
158C22—3881no—
159C22—1942no—
160C22#1569078011yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—7761no—
165C4—3881no—
166C4—3882no—
167C4Q16×D9711yescharacteristic
168C4#1569083211yescharacteristic
169C4#1569083911yescharacteristic
170C4#1569084311yescharacteristic
171C2—3882no—
172C2#2026089211yescharacteristic
173C2#2026089911yescharacteristic
174C2#2026090411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenterQ8⋊C8D97
hypocenterC97#1632
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261115C2semidirect
3. #20261030C2semidirect
5. #20260483C2semidirect
7. Q8⋊C8×C97C2semidirect
60. D388Q16semidirect
88. C97#1632semidirect
94. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202614452
#202611152
#202610302
#202610262
#202604832
#202604642
Q8⋊C8×C972
#163297