Finite Group Databasebeta

← #27157142

Subgroups — #27157142

Order 12416.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261706C211yesmaximal, characteristic, complemented
2#20261678C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#121×D97C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×C22⋊#87604—21no—
8#15690867C2212yescomplemented
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yescomplemented
12C2×D4×D97—21no—
13C2×D4×D97C411yescharacteristic, complemented
14C22×C4×D97C2211yescharacteristic
15#15690785C2211yescharacteristic
16#15690784C411yescharacteristic, complemented
17#15690783—21no—
18#15690782—21no—
19#15690781C2211yescharacteristic
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22#121×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095521—42no—
25C2×#11715897—22no—
26C22⋊#87604—42no—
27C23×D97—21no—
28C23×D97D411yescharacteristic
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34D4×D97—42no—
35D4×D97—44no—
36C22×Dic97D411yescharacteristic
37C2×C4×D97—21no—
38C2×C4×D97—42no—
39C2×C4×D97C2×C411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97C2311yescharacteristic
43C4⋊Dic97C2×C411yescharacteristic
44#13095490—21no—
45C22⋊Dic97—21no—
46#13095488—21no—
47#13095488—21no—
48C2×D4×C97—21no—
49C2×D4×C97C2×C411yescharacteristic
50C22×C4×C97C2×C411yescharacteristic
51C4⋊C4×C97C2×C411yescharacteristic
52C22⋊C4×C97—21no—
53C23×C97—21no—
54C23×C97C22⋊C411yescharacteristic
55C22×D97—22no—
56C22×D97—22no—
57C22×D97—21no—
58C22×D97—41no—
59C22×D97—42no—
60C22×D97—42no—
61C22×D97—42no—
62C22×D97C22⋊C411yescharacteristic
63C22×D97C2×D411yescharacteristic
64C2×#87604—42no—
65C2×#87604—42no—
66C2×#87604—42no—
67D4×C97—41no—
68D4×C97—42no—
69C2×C4×C97—21no—
70C2×C4×C97—41no—
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C22×C411yescharacteristic
73C22⋊D97—42no—
74C22⋊D97—42no—
75C22⋊D97—42no—
76C2×Dic97—21no—
77C2×Dic97—41no—
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80D388—41no—
81D388—42no—
82C4×D97—22no—
83C4×D97—22no—
84C4×D97—41no—
85C4×D97—41no—
86C4×D97—42no—
87#11715897—42no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97C2×C22⋊C411yescharacteristic
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—44no—
100C2×D97—41no—
101C2×D97—41no—
102C2×D97—41no—
103C2×D97—84no—
104C2×D97#9912yes—
105C2×D97AGL(1,ℤ/8)12yes—
106#87604—82no—
107C4×C97—21no—
108C4×C97—21no—
109C4×C97—41no—
110Dic97—21no—
111Dic97—21no—
112Dic97—41no—
113C2×C97—41no—
114C2×C97—81no—
115C2×C97C2×#9911yescharacteristic
116C2×C97#41811yescharacteristic
117C2×C97#42111yescharacteristic
118D97—22no—
119D97—22no—
120D97—81no—
121#1510—971nomaximal, Sylow
122C97#151011yescharacteristic, complemented, Sylow
123C2×#121—971no—
124#407—971no—
125#381—971no—
126#327—972no—
127#327—972no—
128C22×D4—971no—
129C22×D4—1941no—
130C23×C4—971no—
131C2×M16—1941no—
132#121—971no—
133#121—971no—
134#121—971no—
135#121—1941no—
136#121—1941no—
137#121#8760411yescharacteristic, complemented
138C2×C4⋊C4—971no—
139C2×C22⋊C4—1941no—
140C2×C22⋊C4—1941no—
141C22⋊C8—972no—
142C22⋊C8—1941no—
143#95—972no—
144#95—1941no—
145C24—971no—
146C24—1941no—
147C2×D4—21no—
148C2×D4—971no—
149C2×D4—3884no—
150C2×D4—3882no—
151C2×D4—1941no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4C2×#8760411yescharacteristic
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—3882no—
162C22×C4—1941no—
163C22×C4—1942no—
164C22×C4C2×#8760411yescharacteristic
165M16—3882no—
166C2×C8—1942no—
167C4⋊C4—971no—
168C4⋊C4—1941no—
169C4⋊C4C2×#8760411yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3882no—
172C22⋊C4—1941no—
173C22⋊C4—1941no—
174C22⋊C4—1941no—
175C23—21no—
176C23—971no—
177C23—971no—
178C23—3881no—
179C23—3882no—
180C23—3882no—
181C23—3882no—
182C23—1942no—
183C23—1942no—
184C23—1941no—
185C23C22⋊#8760411yescharacteristic
186D4—41no—
187D4—42no—
188D4—3881no—
189D4—3882no—
190D4—3882no—
191D4—3882no—
192D4—3882no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—971no—
196C2×C4—971no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—3882no—
200C2×C4—3882no—
201C2×C4—3882no—
202C2×C4—3882no—
203C2×C4—3881no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1941no—
207C2×C4C22⋊#8760411yescharacteristic
208C2×C4C22×#8760411yescharacteristic
209C8—3882no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—7764no—
217C22—972no—
218C22—972no—
219C22—3884no—
220C22—3881no—
221C22—3881no—
222C22—3881no—
223C22—1941no—
224C22—1941no—
225C22—1941no—
226C22—1941no—
227C22C2×C22⋊#8760411yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—7762no—
232C4—3881no—
233C4—1941no—
234C4—1941no—
235C2—41no—
236C2—81no—
237C2—7761no—
238C2—1942no—
239C2—1942no—
240C2D97⋊#9911yescharacteristic
241C2#2026172711yescharacteristic
242C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#121×D97C2
cosocleC2×C4×D97C23
hypercenter#121#87604
hypocenterC97#1510
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261706C2semidirect
2. #20261678C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
8. #15690867C22semidirect
11. (C2×#87604)⋊C4C22semidirect
13. C2×D4×D97C4semidirect
16. #15690784C4semidirect
20. C4⋊C4×D97C4semidirect
22. #121×C97C4semidirect
122. C97#1510semidirect
137. #121#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202616782
#202615872
#202615862
#121×D972
#151097