Finite Group Databasebeta

← #27157131

Subgroups — #27157131

Order 12416.

346 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 60 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
2C4×C22⋊#87604C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#121×D97C211yesmaximal, characteristic
6C2×#13095520—21no—
7C2×C22⋊#87604—21no—
8C2×C4×#87604—21no—
9#15690867—21no—
10#15690867C2212yescomplemented
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#13095520—42no—
25C2×#11715897—22no—
26C22⋊#87604—24no—
27C4×#87604—42no—
28C23×D97—21no—
29C23×D97D411yescharacteristic
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35D4×D97—42no—
36D4×D97—44no—
37C22×Dic97D411yescharacteristic
38C2×C4×D97—21no—
39C2×C4×D97—42no—
40C2×C4×D97C2×C411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×C4×D97C2311yescharacteristic
44C4⋊Dic97C2×C411yescharacteristic
45#13095490—21no—
46C22⋊Dic97—21no—
47#13095488—21no—
48#13095488—21no—
49C2×D4×C97—21no—
50C2×D4×C97C2×C411yescharacteristic
51C22×C4×C97C2×C411yescharacteristic
52C4⋊C4×C97C2×C411yescharacteristic
53C22⋊C4×C97—21no—
54C23×C97—21no—
55C23×C97C22⋊C411yescharacteristic
56C22×D97—22no—
57C22×D97—22no—
58C22×D97—21no—
59C22×D97—41no—
60C22×D97—42no—
61C22×D97—42no—
62C22×D97—42no—
63C22×D97C22⋊C411yescharacteristic
64C22×D97C2×D411yescharacteristic
65C2×#87604—24no—
66C2×#87604—42no—
67C2×#87604—42no—
68D4×C97—41no—
69D4×C97—42no—
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22×C411yescharacteristic
74C22⋊D97—42no—
75C22⋊D97—42no—
76C22⋊D97—42no—
77C2×Dic97—22no—
78C2×Dic97—21no—
79C2×Dic97C22⋊C411yescharacteristic
80C2×Dic97C2×D411yescharacteristic
81D388—41no—
82D388—42no—
83C4×D97—22no—
84C4×D97—22no—
85C4×D97—41no—
86C4×D97—41no—
87C4×D97—42no—
88#11715897—42no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97C2×C22⋊C411yescharacteristic
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—21no—
100C2×D97—44no—
101C2×D97—41no—
102C2×D97—41no—
103C2×D97—41no—
104C2×D97—84no—
105C2×D97#10412yes—
106C2×D97AGL(1,ℤ/8)12yes—
107#87604—44no—
108C4×C97—21no—
109C4×C97—21no—
110C4×C97—41no—
111Dic97—21no—
112Dic97—41no—
113Dic97#9912yes—
114C2×C97—41no—
115C2×C97—81no—
116C2×C97C2×#9911yescharacteristic
117C2×C97#41811yescharacteristic
118C2×C97C2×#10411yescharacteristic
119D97—22no—
120D97—22no—
121D97—81no—
122#1507—971nomaximal, Sylow
123C97#150711yescharacteristic, complemented, Sylow
124C2×#121—971no—
125#407—971no—
126C4×C22⋊C4—971no—
127#327—972no—
128#327—972no—
129C22×D4—971no—
130C22×D4—1941no—
131C23×C4—971no—
132C2×M16—1941no—
133#121—971no—
134#121—971no—
135#121—971no—
136#121—1941no—
137#121—1941no—
138#121#8760411yescharacteristic, complemented
139C2×C4⋊C4—971no—
140C2×C22⋊C4—1941no—
141C2×C22⋊C4—1941no—
142C2×C42—1941no—
143C22⋊C8—972no—
144C22⋊C8—1941no—
145#95—972no—
146C24—971no—
147C24—1941no—
148C2×D4—21no—
149C2×D4—971no—
150C2×D4—3884no—
151C2×D4—3882no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4C2×#8760411yescharacteristic
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—3882no—
163C22×C4—1941no—
164C22×C4—1942no—
165C22×C4C2×#8760411yescharacteristic
166M16—3882no—
167C2×C8—1942no—
168C4⋊C4—971no—
169C4⋊C4—1941no—
170C4⋊C4C2×#8760411yescharacteristic
171C22⋊C4—21no—
172C22⋊C4—1941no—
173C22⋊C4—1941no—
174C22⋊C4—1944no—
175C22⋊C4—1941no—
176C42—3882no—
177C23—21no—
178C23—971no—
179C23—971no—
180C23—3881no—
181C23—3882no—
182C23—3882no—
183C23—3882no—
184C23—1942no—
185C23—1942no—
186C23—1941no—
187C23C22⋊#8760411yescharacteristic
188D4—41no—
189D4—42no—
190D4—3881no—
191D4—3882no—
192D4—3882no—
193D4—3882no—
194D4—3882no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—971no—
198C2×C4—971no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—3882no—
202C2×C4—3882no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4—1944no—
208C2×C4—1941no—
209C2×C4C22⋊#8760411yescharacteristic
210C2×C4C22×#8760411yescharacteristic
211C8—3882no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—7764no—
219C22—972no—
220C22—972no—
221C22—3884no—
222C22—3881no—
223C22—3881no—
224C22—3881no—
225C22—1941no—
226C22—1941no—
227C22—1941no—
228C22—1941no—
229C22C2×C22⋊#8760411yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—972no—
234C4—3884no—
235C4—3881no—
236C4—1941no—
237C2—41no—
238C2—81no—
239C2—7761no—
240C2—1942no—
241C2—1942no—
242C2D97⋊#9911yescharacteristic
243C2#2026172711yescharacteristic
244C2D97⋊#10411yescharacteristic
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#1507
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. #20261707C2semidirect
2. C4×C22⋊#87604C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
10. #15690867C22semidirect
11. (C2×#87604)⋊C4C22semidirect
13. C2×D4×D97C4semidirect
16. #15690784C4semidirect
20. C4⋊C4×D97C4semidirect
22. #121×C97C4semidirect
123. C97#1507semidirect
138. #121#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
C4×C22⋊#876042
#202615872
#202615862
#121×D972
#150797