Finite Group Databasebeta

← #27152190

Subgroups — #27152190

Order 12416.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#15690839C214yesmaximal, complemented
2#20261283C211yesmaximal, characteristic
3C4⋊Q8×D97C211yesmaximal, characteristic, complemented
4#20261126C214yesmaximal, complemented
5#20261115C211yesmaximal, characteristic, complemented
6#20261114C211yesmaximal, characteristic, complemented
7#20261107C211yesmaximal, characteristic, complemented
8C4×#13095494C211yesmaximal, characteristic, complemented
9#492×C97C211yesmaximal, characteristic, complemented
10C2×#13095510C2212yes—
11C2×Q8×D97C2212yes—
12#15690839—216no—
13C2×Dic388C2214yes—
14C2×#13095501C2214yes—
15C2×#13095499C2214yes—
16C2×#13095494C2212yes—
17#15690810C2211yescharacteristic
18#15690810C2211yescharacteristic
19#15690809—22no—
20#15690808C2212yes—
21#15690797C2212yes—
22#15690781—22no—
23D4×Dic97—22no—
24C4⋊C4×D97—22no—
25C42×D97C2211yescharacteristic
26#15690720C2212yes—
27C4×#11715895C2211yescharacteristic
28C2×SD16×C97C2214yes—
29C4⋊Q8×C97C2211yescharacteristic
30#127×C97C2211yescharacteristic
31C4×C8×C97C2211yescharacteristic
32#13095510—28no—
33C2×Dic194—22no—
34C2×Dic194C2312yes—
35C2×Dic194C2312yes—
36Q8×D97—28no—
37C2×C22⋊D97—22no—
38C22×Dic97—22no—
39C2×C4×D97—22no—
40C2×C4×D97D412yes—
41C2×C4×D97C2311yescharacteristic
42Dic388—28no—
43#13095501—28no—
44#13095499—28no—
45#13095494D418yescomplemented
46C2×#11715895C2312yes—
47C4⋊Dic97—22no—
48C4⋊Dic97—22no—
49#13095490—24no—
50C22⋊Dic97—24no—
51C4×Dic97D412yes—
52C4×Dic97C2311yescharacteristic
53C2×Q8×C97C2312yes—
54C2×D4×C97—22no—
55C2×D4×C97C2312yes—
56SD16×C97—28no—
57C2×C8×C97C2312yes—
58C4⋊C4×C97—22no—
59C42×C97C2311yescharacteristic
60C23×C97—22no—
61C22×D97C2×D411yescharacteristic
62Q8×C97—24no—
63D4×C97—24no—
64D4×C97—44no—
65C2×C4×C97—22no—
66C2×C4×C97C2×D412yes—
67C2×C4×C97C2411yescharacteristic
68C22⋊D97—44no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—44no—
72C2×Dic97C2×D412yes—
73C2×Dic97C2×D411yescharacteristic
74C4×D97—24no—
75C4×D97—44no—
76C4×D97C2×D414yes—
77Dic194—24no—
78Dic194—24no—
79Dic194—44no—
80C8×C97C2×D414yes—
81#11715895C2×D414yes—
82C22×C97—42no—
83C22×C97—44no—
84C22×C97C22×D411yescharacteristic
85C2×D97—22no—
86C2×D97#12712yes—
87C4×C97—42no—
88C4×C97#13714yescomplemented
89C4×C97C22×D412yes—
90Dic97—22no—
91Dic97—42no—
92Dic97—42no—
93Dic97#12712yes—
94C2×C97—82no—
95C2×C97C2×#12711yescharacteristic
96C2×C97C2×#13712yes—
97D97—22no—
98#3152—971nomaximal, Sylow
99C97#315211yescharacteristic, complemented, Sylow
100C2×#137—974no—
101#532—971no—
102C2×C4⋊Q8—971no—
103#497—974no—
104#494—971no—
105#494—971no—
106#492—971no—
107#492D9711yescharacteristic, complemented
108C4×M16—971no—
109C2×#41—972no—
110C22×Q8—972no—
111#137—19416no—
112C2×Q16—974no—
113C2×Q16—974no—
114C2×SD16—974no—
115C2×SD16C2×D9714yes—
116C2×M16—972no—
117C4⋊Q8—972no—
118C4⋊Q8—971no—
119C4⋊Q8—971no—
120C4⋊Q8—1942no—
121C4⋊Q8C2×D9711yescharacteristic
122#127C2×D9711yescharacteristic
123#124—972no—
124#121—1942no—
125C4×D4—1942no—
126C2×C4⋊C4—1942no—
127C2×C42—971no—
128#97—972no—
129C4×C8—971no—
130C4×C8C2×D9711yescharacteristic
131#41—1948no—
132C2×Q8—972no—
133C2×Q8—972no—
134C2×Q8—1942no—
135C2×Q8—1948no—
136C2×Q8C22×D9712yes—
137C2×D4—22no—
138C2×D4—1942no—
139C2×D4C22×D9712yes—
140C22×C4—971no—
141C22×C4—972no—
142C22×C4—1942no—
143C22×C4—1942no—
144Q16—1948no—
145Q16—1948no—
146SD16—28no—
147SD16—1948no—
148M16—978no—
149C2×C8—972no—
150C2×C8C22×D9712yes—
151C4⋊C4—22no—
152C4⋊C4—1944no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C22⋊C4—1944no—
156C42—972no—
157C42—971no—
158C42C22×D9711yescharacteristic
159C23—22no—
160C23—971no—
161Q8—24no—
162Q8—3884no—
163Q8—1944no—
164Q8—1944no—
165D4—24no—
166D4—44no—
167D4—3884no—
168C2×C4—22no—
169C2×C4—971no—
170C2×C4—974no—
171C2×C4—972no—
172C2×C4—3884no—
173C2×C4—3884no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1944no—
177C2×C4D4×D9712yes—
178C2×C4C23×D9711yescharacteristic
179C8—974no—
180C8D4×D9714yes—
181C22—44no—
182C22—42no—
183C22—972no—
184C22—1942no—
185C22C2×D4×D9711yescharacteristic
186C4—42no—
187C4—972no—
188C4—3882no—
189C4—3882no—
190C4—1942no—
191C4#1569083914yescomplemented
192C4C2×D4×D9712yes—
193C2—82no—
194C2—1942no—
195C2#127×D9711yescharacteristic
196C2C2×#1569083912yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#492×C97C2
socleC22×C97C22×D4
prime radical#20261283C2
hypercenter#492D97
hypocenterC97#3152
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#15690839C2semidirect
3. C4⋊Q8×D97C2semidirect
4. #20261126C2semidirect
5. #20261115C2semidirect
6. #20261114C2semidirect
7. #20261107C2semidirect
8. C4×#13095494C2semidirect
9. #492×C97C2semidirect
45. #13095494D4semidirect
88. C4×C97#137semidirect
99. C97#3152semidirect
107. #492D97semidirect
191. C4#15690839semidirect

Maximal subgroups

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

subgroupindex
C2×#156908392
#202612832
C4⋊Q8×D972
#202611262
#202611152
#202611142
#202611072
C4×#130954942
#492×C972
#315297