Finite Group Databasebeta

← #27142138

Subgroups — #27142138

Order 12416.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690798C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic, complemented
3#20260741C211yesmaximal, characteristic
4#20260739C211yesmaximal, characteristic
5C2×#15690728C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7#339×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11#15690798—22no—
12#15690798C414yescomplemented
13C4×Dic194—22no—
14#15690775—22no—
15#15690769C2211yescharacteristic
16#15690766—22no—
17C2×#13095488—22no—
18#15690732—22no—
19#15690730—22no—
20#15690728—22no—
21C2×M16×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23C2×C42×C97C2211yescharacteristic
24C23×D97—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C2×Dic194D412yes—
31C2×D388—21no—
32C2×D388—21no—
33C2×D388—21no—
34C2×D388—21no—
35C2×D388—21no—
36C2×D388C2×C412yes—
37C22×Dic97D412yes—
38#13095495—21no—
39#13095495—21no—
40C2×#11715895—21no—
41C4⋊Dic97D412yes—
42#13095490—24no—
43C22⋊Dic97—21no—
44#13095488—28no—
45C4×Dic97—24no—
46C4×Dic97—21no—
47C22×C4×C97—21no—
48C22×C4×C97C2311yescharacteristic
49M16×C97—21no—
50M16×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97D412yes—
53C22⋊C4×C97—21no—
54C42×C97—21no—
55C42×C97—21no—
56C42×C97C2×C412yes—
57C23×C97C2×D411yescharacteristic
58C22×D97—22no—
59C22×D97—44no—
60C22×D97—41no—
61C22×D97—41no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—42no—
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97#4111yescharacteristic
70C2×C4×C97#4111yescharacteristic
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97C22⋊C414yes—
75D388—22no—
76D388—22no—
77D388—41no—
78D388—41no—
79Dic194—22no—
80Dic194—22no—
81Dic194—22no—
82Dic194—22no—
83C8×C97—41no—
84#11715895—41no—
85C22×C97—22no—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97#12011yescharacteristic
88C22×C97#12311yescharacteristic
89C2×D97—44no—
90C2×D97—42no—
91C2×D97—82no—
92C2×D97—82no—
93C4×C97—42no—
94C4×C97—42no—
95C4×C97C4×D411yescharacteristic
96C4×C97C4×D411yescharacteristic
97C4×C97#12111yescharacteristic
98C4×C97#12111yescharacteristic
99Dic97—24no—
100Dic97—42no—
101C2×C97—21no—
102C2×C97—21no—
103C2×C97#38611yescharacteristic
104C2×C97#45412yes—
105D97—82no—
106#1888—971nomaximal, Sylow
107C97#188811yescharacteristic, complemented, Sylow
108C2×#124—971no—
109#519—971no—
110#418—971no—
111#418—971no—
112C2×#101—971no—
113#339—971no—
114#339D9711yescharacteristic, complemented
115C22×Q8—971no—
116C22×D4—971no—
117C2×M16—971no—
118C2×M16C2×D9711yescharacteristic
119#124—974no—
120#124—1942no—
121C4×Q8—1942no—
122C4×Q8—1942no—
123#117—971no—
124#117—1942no—
125#117C2×D9711yescharacteristic
126C2×C22⋊C4—1942no—
127C2×C42C2×D9711yescharacteristic
128#102—1942no—
129#102—1942no—
130#101—1942no—
131C24—1941no—
132C2×Q8—972no—
133C2×Q8—972no—
134C2×Q8—1941no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×Q8—1941no—
138C2×D4—972no—
139C2×D4—1941no—
140C2×D4—1941no—
141C2×D4—1941no—
142C2×D4—1941no—
143C2×D4—1941no—
144C22×C4—21no—
145C22×C4—972no—
146C22×C4C22×D9711yescharacteristic
147M16—21no—
148M16—21no—
149M16—1941no—
150M16—1941no—
151C2×C8—21no—
152C2×C8—1941no—
153C4⋊C4—972no—
154C4⋊C4—1944no—
155C4⋊C4D38812yes—
156C22⋊C4—21no—
157C22⋊C4—1948no—
158C22⋊C4—1941no—
159C42—21no—
160C42—21no—
161C42—1944no—
162C42—1941no—
163C42C4×D9712yes—
164C23—3884no—
165C23—3881no—
166C23—3881no—
167C23—1942no—
168C23D4×D9711yescharacteristic
169Q8—1942no—
170Q8—1942no—
171Q8—1942no—
172Q8—1942no—
173D4—3881no—
174D4—3881no—
175D4—1942no—
176D4—1942no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—974no—
181C2×C4—1942no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4C2×C4×D9711yescharacteristic
185C2×C4D4×D9711yescharacteristic
186C2×C4D4×D9711yescharacteristic
187C2×C4C2×D38811yescharacteristic
188C2×C4#1309551111yescharacteristic
189C2×C4#1309551211yescharacteristic
190C8—41no—
191C8—3881no—
192C22—22no—
193C22—7762no—
194C22—7762no—
195C22—3884no—
196C22—3882no—
197C22C22⋊C4×D9711yescharacteristic
198C22#1569078011yescharacteristic
199C22#1569079511yescharacteristic
200C4—42no—
201C4—42no—
202C4—3882no—
203C4—1944no—
204C4#1569077211yescharacteristic
205C4C4×D38811yescharacteristic
206C4#1569078311yescharacteristic
207C4#1569078511yescharacteristic
208C2—21no—
209C2—21no—
210C2—7762no—
211C2#2026061311yescharacteristic
212C2#2026090912yes—
subgroupG/H
centerC22#15690780
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97#120
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#339D97
hypocenterC97#1888
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
5. C2×#15690728C2semidirect
6. #20260495C2semidirect
7. #339×C97C2semidirect
12. #15690798C4semidirect
107. C97#1888semidirect
114. #339D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907982
#202612122
#202607412
#202607392
C2×#156907282
#202604952
#339×C972
#188897