Finite Group Databasebeta

← #27157122

Subgroups — #27157122

Order 12416.

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261707C211yesmaximal, characteristic, complemented
2C2×(C2×#87604)⋊C4C211yesmaximal, characteristic, complemented
3#20261589C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#20261242C211yesmaximal, characteristic
6C23×#87604—21no—
7C2×#13095520—21no—
8#15690867—21no—
9#15690867C2212yes—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12C22×C4×D97C2211yescharacteristic
13#15690791—21no—
14#15690786C411yescharacteristic, complemented
15#15690786C2211yescharacteristic
16#15690781C411yescharacteristic, complemented
17#15690781C2211yescharacteristic
18#15690778—21no—
19C4×C22⋊D97—21no—
20C4×C22⋊D97—21no—
21#15690769C411yescharacteristic, complemented
22C2×#41×C97C411yescharacteristic, complemented
23C22×#87604—24no—
24C22×#87604—22no—
25C22×#87604—42no—
26C22×#87604—42no—
27#13095520—42no—
28C2×#11715897—22no—
29C23×D97D411yescharacteristic
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—42no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97C2311yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic, complemented
39C4⋊Dic97C2×C411yescharacteristic, complemented
40#13095490—21no—
41#13095490—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45#13095488—21no—
46#13095488—21no—
47C4×Dic97—21no—
48#41×C97—42no—
49C2×Q8×C97C2×C411yescharacteristic, complemented
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic, complemented
52C22×C4×C97—21no—
53C22×C4×C97C2×C411yescharacteristic
54C23×C97—21no—
55C23×C97C22⋊C411yescharacteristic
56C22×D97—22no—
57C22×D97—22no—
58C22×D97—21no—
59C22×D97—21no—
60C22×D97C22⋊C411yescharacteristic
61C22×D97C2×D411yescharacteristic
62C2×#87604—24no—
63C2×#87604—48no—
64C2×#87604—42no—
65C2×#87604—42no—
66Q8×C97—41no—
67D4×C97—41no—
68D4×C97—42no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—42no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22×C411yescharacteristic
74C22⋊D97—42no—
75C22⋊D97—41no—
76C22⋊D97—41no—
77C2×Dic97—22no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97C22⋊C411yescharacteristic
81C2×Dic97C2×D411yescharacteristic
82C4×D97—41no—
83C4×D97—41no—
84C4×D97—41no—
85C4×D97—41no—
86#11715897—42no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97#9912yes—
93C22×C97C2×C22⋊C411yescharacteristic
94C2×D97—24no—
95C2×D97—24no—
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—21no—
100C2×D97#10412yes—
101C2×D97AGL(1,ℤ/8)11yescharacteristic
102C2×D97#13711yescharacteristic
103#87604—44no—
104C4×C97—21no—
105C4×C97—21no—
106C4×C97—41no—
107Dic97—21no—
108Dic97—21no—
109Dic97—41no—
110Dic97—41no—
111C2×C97—22no—
112C2×C97—81no—
113C2×C97C2×#9911yescharacteristic
114C2×C97#41711yescharacteristic
115C2×C97C2×#10411yescharacteristic
116D97—21no—
117D97—22no—
118D97—21no—
119#1504—971nomaximal, Sylow
120C97#150411yescharacteristic, complemented, Sylow
121#525—971no—
122#407—971no—
123C2×#95—971no—
124#328—972no—
125#327—972no—
126C2×#41#8760411yescharacteristic, complemented
127C23×C4—971no—
128C23×C4—1941no—
129C2×M16—1941no—
130#123—1941no—
131C22⋊Q8—971no—
132C22⋊Q8—971no—
133#121—971no—
134#121—971no—
135#120—1941no—
136C4×D4—1941no—
137C4×D4—1941no—
138#117—971no—
139C22⋊C8—972no—
140C22⋊C8—1941no—
141#95—972no—
142#95—1941no—
143C24—971no—
144#41—42no—
145C2×Q8C2×#8760411yescharacteristic, complemented
146C2×D4—21no—
147C2×D4—1941no—
148C2×D4—1941no—
149C2×D4C2×#8760411yescharacteristic, complemented
150C22×C4—21no—
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—3882no—
157C22×C4—3882no—
158C22×C4—3882no—
159C22×C4—1942no—
160C22×C4—1944no—
161C22×C4C2×#8760411yescharacteristic
162M16—3882no—
163C2×C8—1942no—
164C4⋊C4—971no—
165C4⋊C4—971no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C22⋊C4—1941no—
172C22⋊C4—1941no—
173C42—1941no—
174C23—21no—
175C23—971no—
176C23—971no—
177C23—1941no—
178C23—1942no—
179C23—1941no—
180C23—1942no—
181C23C22⋊#8760411yescharacteristic
182Q8—41no—
183D4—41no—
184D4—42no—
185D4—3881no—
186D4—3881no—
187D4—3882no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—971no—
192C2×C4—971no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4—3882no—
198C2×C4—3888no—
199C2×C4—3882no—
200C2×C4—1944no—
201C2×C4—1941no—
202C2×C4—1941no—
203C2×C4—1942no—
204C2×C4C22⋊#8760411yescharacteristic
205C2×C4C22×#8760411yescharacteristic
206C8—3882no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—971no—
213C22—971no—
214C22—972no—
215C22—1944no—
216C22—1944no—
217C22—1941no—
218C22—1941no—
219C22—1941no—
220C22—1941no—
221C22#1569086812yes—
222C22C2×C22⋊#8760411yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—3884no—
227C4—3881no—
228C4—3881no—
229C4—1941no—
230C4—1941no—
231C2—22no—
232C2—81no—
233C2—1941no—
234C2—1941no—
235C2—1942no—
236C2C2×#1569086811yescharacteristic
237C2#2026172611yescharacteristic
238C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261242C2
cosocleC2×C4×D97C23
hypercenterC2×#41#87604
hypocenterC97#1504
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261707C2semidirect
2. C2×(C2×#87604)⋊C4C2semidirect
3. #20261589C2semidirect
4. #20261586C2semidirect
14. #15690786C4semidirect
16. #15690781C4semidirect
21. #15690769C4semidirect
22. C2×#41×C97C4semidirect
38. C4⋊Dic97C2×C4semidirect
39. C4⋊Dic97C2×C4semidirect
49. C2×Q8×C97C2×C4semidirect
51. C2×D4×C97C2×C4semidirect
120. C97#1504semidirect
126. C2×#41#87604semidirect
145. C2×Q8C2×#87604semidirect
149. C2×D4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
C2×(C2×#87604)⋊C42
#202615892
#202615862
#202612422
#150497