Finite Group Databasebeta

← #27140872

Subgroups — #27140872

Order 12416.

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

normal subgroups only

This group has 214 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#20261442C211yesmaximal, characteristic
2#20261115C211yesmaximal, characteristic, complemented
3#20261031C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic
5#20260483C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic
8C2×Dic388—22no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690801—21no—
12#15690789—21no—
13#15690787—22no—
14C4×Dic194C2211yescharacteristic
15D4×Dic97—21no—
16C2×C4⋊Dic97—21no—
17#15690744—21no—
18#15690741—21no—
19C4⋊#11715895C2211yescharacteristic
20#15690735—21no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27C2×Dic194D412yes—
28C22×Dic97—22no—
29Dic388—44no—
30C2×#11715895—21no—
31C4⋊Dic97—22no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37#13095490—22no—
38#13095490—21no—
39C22⋊Dic97—22no—
40C4×Dic97—21no—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97—21no—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49D4×C97—21no—
50D4×C97Q1612yescomplemented
51C2×C4×C97—21no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—22no—
60C2×Dic97—41no—
61C2×Dic97—82no—
62Dic194—21no—
63Dic194—21no—
64Dic194—42no—
65Dic194—42no—
66C8×C97—22no—
67C8×C97—41no—
68#11715895—41no—
69C22×C97—22no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97#12011yescharacteristic
73C4×C97—21no—
74C4×C97—41no—
75C4×C97C2×Q1611yescharacteristic
76C4×C97#13511yescharacteristic
77C4×C97#13711yescharacteristic
78C4×C97#13711yescharacteristic
79Dic97—41no—
80Dic97—42no—
81Dic97—41no—
82Dic97—81no—
83C2×C97—42no—
84C2×C97#45111yescharacteristic
85C2×C97#45211yescharacteristic
86C2×C97#45311yescharacteristic
87#1632—971nomaximal, Sylow
88C97#163211yescharacteristic, complemented, Sylow
89#554—971no—
90#494—971no—
91#477—971no—
92#476—971no—
93#333—971no—
94Q8⋊C8—971no—
95D4⋊C8D9711yescharacteristic
96C2×Q16—1942no—
97C4⋊Q8—971no—
98C4⋊Q8—971no—
99#125—1941no—
100C22⋊Q8—1942no—
101C22⋊Q8—1941no—
102C4×Q8—971no—
103C4×D4—1941no—
104C4×D4C2×D9711yescharacteristic
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—971no—
115C2×Q8—1941no—
116C2×D4D38811yescharacteristic
117C22×C4—21no—
118C22×C4—1942no—
119Q16—3884no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1941no—
123C4⋊C4—971no—
124C4⋊C4—1942no—
125C4⋊C4—1942no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4D38811yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—1942no—
134C42—1941no—
135C42C22×D9711yescharacteristic
136C23—21no—
137Q8—3882no—
138Q8—3882no—
139Q8—1941no—
140Q8—1941no—
141D4—21no—
142D4Dic38812yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—7762no—
147C2×C4—3881no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4D4×D9711yescharacteristic
152C2×C4D4×D9711yescharacteristic
153C2×C4C2×D38811yescharacteristic
154C8—22no—
155C8—41no—
156C8—3881no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1569078011yescharacteristic
161C4—21no—
162C4—41no—
163C4—7761no—
164C4—3881no—
165C4—3881no—
166C4—3882no—
167C4C2×Dic38811yescharacteristic
168C4#1569082911yescharacteristic
169C4#1569083911yescharacteristic
170C4#1569084311yescharacteristic
171C2—42no—
172C2#2026089311yescharacteristic
173C2#2026089811yescharacteristic
174C2#2026090411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4⋊C8D97
hypocenterC97#1632
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261115C2semidirect
3. #20261031C2semidirect
5. #20260483C2semidirect
6. #20260467C2semidirect
50. D4×C97Q16semidirect
88. C97#1632semidirect
142. D4Dic388semidirect

Maximal subgroups

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

subgroupindex
#202614422
#202611152
#202610312
#202610252
#202604832
#202604672
D4⋊C8×C972
#163297