Finite Group Databasebeta

← #27157991

Subgroups — #27157991

Order 12416.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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×#15690870C211yesmaximal, characteristic, complemented
2C2×#15690868C211yesmaximal, characteristic, complemented
3#20261637C212yesmaximal, complemented
4#20261635C212yesmaximal, complemented
5#20261294C211yesmaximal, characteristic
6C2×#13095520—21no—
7C2×C22⋊#87604—21no—
8#15690870—21no—
9#15690870C2212yes—
10#15690868—21no—
11#15690868C2212yes—
12C22×C22⋊D97C2211yescharacteristic
13#15690793C411yescharacteristic, complemented
14#15690793C2211yescharacteristic
15#15690791—21no—
16#15690782—21no—
17#15690782—21no—
18#15690779C411yescharacteristic, complemented
19#15690779C2211yescharacteristic
20#15690778—21no—
21C2×C22⋊Dic97C411yescharacteristic, complemented
22C22×D4×C97C411yescharacteristic, complemented
23C22×#87604—41no—
24#13095520—22no—
25#13095520—41no—
26C2×#11715897—41no—
27C22⋊#87604—22no—
28C22⋊#87604—41no—
29C23×D97D411yescharacteristic
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C22⋊D97—41no—
33C2×C22⋊D97—41no—
34C2×C22⋊D97C2×C411yescharacteristic
35C2×C22⋊D97D411yescharacteristic
36C2×C22⋊D97D411yescharacteristic
37C2×C22⋊D97C2311yescharacteristic
38C22×Dic97—21no—
39C22×Dic97D411yescharacteristic
40#13095490—21no—
41#13095490—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45C22⋊Dic97C2×C411yescharacteristic, complemented
46C22⋊Dic97C2×C411yescharacteristic, complemented
47#13095488—21no—
48#13095488—21no—
49C24×C97C2×C411yescharacteristic
50C24×C97C2×C411yescharacteristic, complemented
51C2×D4×C97—21no—
52C2×D4×C97—21no—
53C2×D4×C97—41no—
54C2×D4×C97—41no—
55C22×C4×C97C2×C411yescharacteristic, complemented
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—41no—
60C23×C97—81no—
61C23×C97—81no—
62C23×C97C22⋊C411yescharacteristic
63C23×C97C22⋊C411yescharacteristic
64C23×C97C22×C411yescharacteristic
65C22×D97—21no—
66C22×D97—21no—
67C22×D97—81no—
68C22×D97C22⋊C411yescharacteristic
69C22×D97C2×D411yescharacteristic
70C2×#87604—42no—
71C2×#87604—82no—
72D4×C97—42no—
73D4×C97—41no—
74D4×C97—41no—
75C2×C4×C97—21no—
76C2×C4×C97—41no—
77C22⋊D97—41no—
78C22⋊D97—41no—
79C22⋊D97—41no—
80C22⋊D97—41no—
81C22⋊D97—81no—
82C22⋊D97—81no—
83C2×Dic97—21no—
84C2×Dic97—21no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—81no—
88C2×Dic97C22⋊C411yescharacteristic
89C2×Dic97C2×D411yescharacteristic
90#11715897—42no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—81no—
100C22×C97—81no—
101C22×C97—82no—
102C22×C97#9911yescharacteristic
103C22×C97#9911yescharacteristic
104C22×C97C2×C22⋊C411yescharacteristic
105C2×D97—21no—
106C2×D97—21no—
107C2×D97—42no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—81no—
111#87604—82no—
112C4×C97—41no—
113Dic97—21no—
114Dic97—21no—
115Dic97—41no—
116Dic97—41no—
117C2×C97—21no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—41no—
121C2×C97—41no—
122C2×C97—81no—
123C2×C97C2×#9911yescharacteristic
124D97—41no—
125D97—41no—
126#2121—971nomaximal, Sylow
127C97#212111yescharacteristic, complemented, Sylow
128#534—971no—
129C2×#100—971no—
130C2×#99—971no—
131#352—972no—
132AΣL(1,𝔽16)—972no—
133C22×D4—971no—
134C22×D4#8760411yescharacteristic, complemented
135C2×M16—1941no—
136#123—971no—
137#123—971no—
138#123—1941no—
139#121—1941no—
140#121—1941no—
141#120—971no—
142#120—971no—
143#120—1941no—
144C2×C22⋊C4—971no—
145C2×C22⋊C4—1941no—
146#100—972no—
147#100—1941no—
148#99—972no—
149#99—1941no—
150C24—971no—
151C24C2×#8760411yescharacteristic
152C24C2×#8760411yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—971no—
158C2×D4—971no—
159C2×D4—971no—
160C2×D4—971no—
161C2×D4—3881no—
162C2×D4—3881no—
163C2×D4—1941no—
164C2×D4—1941no—
165C22×C4—971no—
166C22×C4—3881no—
167C22×C4—1941no—
168C22×C4C2×#8760411yescharacteristic, complemented
169M16—3881no—
170M16—1942no—
171C2×C8—3881no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C22⋊C4—971no—
175C22⋊C4—971no—
176C22⋊C4—3881no—
177C22⋊C4—1941no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4—1941no—
181C22⋊C4—1942no—
182C22⋊C4—1941no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—7761no—
190C23—971no—
191C23—971no—
192C23—1941no—
193C23—1941no—
194C23C22⋊#8760411yescharacteristic
195C23C22⋊#8760411yescharacteristic
196C23C22×#8760411yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—7761no—
201D4—7761no—
202D4—3881no—
203D4—3881no—
204D4—3881no—
205D4—3881no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—7762no—
209C2×C4—7761no—
210C2×C4—971no—
211C2×C4—971no—
212C2×C4—3882no—
213C2×C4—1941no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4—1941no—
217C8—3882no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—7761no—
230C22—3881no—
231C22—3881no—
232C22—3882no—
233C22—1941no—
234C22—1941no—
235C22#1569086811yescharacteristic
236C22#1569086811yescharacteristic
237C22C2×C22⋊#8760411yescharacteristic
238C4—41no—
239C4—7762no—
240C4—3881no—
241C4—3881no—
242C4—1941no—
243C4—1941no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—3881no—
251C2—3881no—
252C2C2×#1569086811yescharacteristic
subgroupG/H
centerC2C2×#15690868
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×D4×C97C4
socleC2×C97C2×#99
prime radical#20261294C2
cosocleC2×C22⋊D97C23
hypercenterC22×D4#87604
hypocenterC97#2121
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690870C2semidirect
2. C2×#15690868C2semidirect
3. #20261637C2semidirect
4. #20261635C2semidirect
13. #15690793C4semidirect
18. #15690779C4semidirect
21. C2×C22⋊Dic97C4semidirect
22. C22×D4×C97C4semidirect
45. C22⋊Dic97C2×C4semidirect
46. C22⋊Dic97C2×C4semidirect
50. C24×C97C2×C4semidirect
55. C22×C4×C97C2×C4semidirect
127. C97#2121semidirect
134. C22×D4#87604semidirect
152. C24C2×#87604semidirect
168. C22×C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908702
C2×#156908682
#202616372
#202616352
#202612942
#212197