Finite Group Databasebeta

← #27157565

Subgroups — #27157565

Order 12416.

612 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 86 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×#15690869C212yesmaximal, complemented
2C2×#15690868C212yesmaximal, complemented
3#20261678C212yesmaximal, complemented
4C22×#13095511C211yesmaximal, characteristic
5C2×C22⋊#87604C2214yes—
6#15690869—24no—
7#15690868—24no—
8(C2×#87604)⋊C4—24no—
9C2×#13095511—24no—
10C2×#13095511—24no—
11C2×#13095511—24no—
12C22×Dic194C411yescharacteristic, complemented
13C22×D388C2211yescharacteristic
14C22×C22⋊D97C411yescharacteristic, complemented
15C22×C22⋊D97C2211yescharacteristic
16C22×C4×D97C411yescharacteristic, complemented
17C22×C4×D97C2211yescharacteristic
18C23×C4×C97C411yescharacteristic, complemented
19C22×#87604—24no—
20C22⋊#87604—24no—
21C22⋊#87604—24no—
22C23×D97C2×C411yescharacteristic
23C23×D97C2311yescharacteristic
24#13095511—232no—
25C2×Dic194—22no—
26C2×Dic194—22no—
27C2×Dic194—22no—
28C2×D388—22no—
29C2×D388—22no—
30C2×D388D414yes—
31C2×C22⋊D97—22no—
32C2×C22⋊D97—22no—
33C2×C22⋊D97—22no—
34C2×C22⋊D97—22no—
35C2×C22⋊D97—22no—
36C2×C22⋊D97D414yes—
37C22×Dic97C2×C411yescharacteristic
38C22×Dic97C2×C411yescharacteristic
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41C2×C4×D97—22no—
42C2×C4×D97—22no—
43C2×C4×D97—22no—
44C2×C4×D97D414yes—
45C24×C97C2×C411yescharacteristic
46C22×C4×C97—22no—
47C22×C4×C97—22no—
48C22×C4×C97—22no—
49C22×C4×C97C2×C411yescharacteristic
50C22×C4×C97C2×C411yescharacteristic
51C23×C97—21no—
52C23×C97—21no—
53C23×C97—42no—
54C23×C97C22⋊C412yes—
55C23×C97C22×C411yescharacteristic
56C22×D97—21no—
57C22×D97—21no—
58C22×D97—42no—
59C22×D97—42no—
60C22×D97C22⋊C412yes—
61C22×D97C2×D412yes—
62C22×D97C2×D412yes—
63C22×D97C2×D412yes—
64C2×#87604—48no—
65C2×#87604—44no—
66C2×C4×C97—28no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97C22⋊C412yes—
72C2×C4×C97C22⋊C412yes—
73C22⋊D97—28no—
74C22⋊D97—28no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97C22⋊C412yes—
80C2×Dic97C22⋊C412yes—
81D388—28no—
82C4×D97—28no—
83C4×D97—28no—
84Dic194—28no—
85C22×C97—22no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97C2×C22⋊C411yescharacteristic
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97C2×C22⋊C411yescharacteristic
93C2×D97—24no—
94C2×D97—22no—
95C2×D97—42no—
96C2×D97—42no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—42no—
100C2×D97#12014yes—
101#87604—84no—
102C4×C97—22no—
103C4×C97—22no—
104Dic97—22no—
105Dic97—22no—
106C2×C97—21no—
107C2×C97—21no—
108C2×C97—42no—
109C2×C97#37911yescharacteristic
110C2×C97#41012yes—
111D97—24no—
112D97—42no—
113#1782—971nomaximal, Sylow
114C97#178211yescharacteristic, complemented, Sylow
115C22×#41—971no—
116C2×#99—972no—
117C2×#99—972no—
118#381—972no—
119C2×#41—1944no—
120C2×#41—1944no—
121C2×#41—1944no—
122C22×Q8—971no—
123C22×D4—971no—
124C22×D4—971no—
125C22×D4—971no—
126C23×C4—971no—
127C23×C4—971no—
128C23×C4#8760411yescharacteristic, complemented
129C2×C22⋊C4—974no—
130#99—1944no—
131#99—1944no—
132#95—1944no—
133C24—971no—
134C24—971no—
135C24C2×#8760411yescharacteristic
136#41—19432no—
137C2×Q8—1942no—
138C2×Q8—1942no—
139C2×Q8—1942no—
140C2×D4—974no—
141C2×D4—974no—
142C2×D4—1942no—
143C2×D4—1942no—
144C2×D4—1942no—
145C2×D4—1942no—
146C2×D4—1942no—
147C2×D4—1942no—
148C2×D4—1942no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—971no—
153C22×C4—971no—
154C22×C4—974no—
155C22×C4—1942no—
156C22×C4—1942no—
157C22×C4—1942no—
158C22×C4—1942no—
159C22×C4—1944no—
160C22×C4—1942no—
161C22×C4C2×#8760411yescharacteristic
162C22×C4C2×#8760411yescharacteristic
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—972no—
169C23—972no—
170C23—972no—
171C23—972no—
172C23—3882no—
173C23—3882no—
174C23—1941no—
175C23—1941no—
176C23C22⋊#8760412yes—
177C23C22×#8760411yescharacteristic
178Q8—1948no—
179D4—1948no—
180D4—1948no—
181D4—1948no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—972no—
188C2×C4—972no—
189C2×C4—3884no—
190C2×C4—3888no—
191C2×C4—1948no—
192C2×C4—1948no—
193C2×C4—1941no—
194C2×C4—1941no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4C22⋊#8760412yes—
198C2×C4C22⋊#8760412yes—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—974no—
205C22—3882no—
206C22—3882no—
207C22—3882no—
208C22—3882no—
209C22—3882no—
210C22—1942no—
211C22—1944no—
212C22C2×C22⋊#8760411yescharacteristic
213C22C2×C22⋊#8760411yescharacteristic
214C22C2×C22⋊#8760411yescharacteristic
215C4—22no—
216C4—22no—
217C4—7764no—
218C4—1942no—
219C4—1942no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—3882no—
224C2—1944no—
225C2#2026167611yescharacteristic
226C2#2026171412yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×#13095511C2
cosocleC23×D97C23
hypercenterC23×C4#87604
hypocenterC97#1782
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690869C2semidirect
2. C2×#15690868C2semidirect
3. #20261678C2semidirect
12. C22×Dic194C4semidirect
14. C22×C22⋊D97C4semidirect
16. C22×C4×D97C4semidirect
18. C23×C4×C97C4semidirect
114. C97#1782semidirect
128. C23×C4#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908692
C2×#156908682
#202616782
C22×#130955112
#178297